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Edited by O. M. Belotserkovskii Computational Center Moscow, USSR
M. N. Kogan Moscow Physicotecfmicallnstitute Dolgoprudny, USSR
S. S. Kutateladze and A. K. Rebrov Institute of Thermophysics Novosibirsk, USSR
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library 01 Congress Cataloging in Publication Oata
International Symposium on Rarefied Gas Oynamics (l3th: 1982: Novosibirsk, R.S.F.S.R.) Rarefied gas dynamics.
"Revised versions 01 selected papers lrom the Thirteenth International Symposium on Rarefied Gas Oynamics, held July 1982, in Novosibirsk, USSR"-T.p. verso.
Bibliography: p. Includes index. 1. Rarelied gas dynamics-Congresses. 1. Belotserkovskii, O. M. (Oleg Mikhailovich) II. Title.
QC168.155 1982 533/2 85-3719
Revised versions 01 selected papers lrom the Thirteenth International Symposium on Rarelied Gas Oynamics, held July 1982, in Novosibirsk, USSR
© 1985 Springer Science+ Business Media New York Originally published by P1enum Press, New York in 1985
Softcover reprint of the hardcover 1 st edition 1985
AII rights reserved
No part 01 this book may be reproduced, stored in a retrieval system, or transmitted, in any lorm or by any means, electronic, mechanical, photocopying, microlilming,
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ISBN 978-1-4612-9497-9 ISBN 978-1-4613-2467-6 (eBook) DOI 10.1007/978-1-4613-2467-6
X. COLLISIONAL PROCESSES
ANALYTIC.AL :roRMULAE .R>R CROSS SECTIONS AND
RATE CONSTANTS OF ELEMENTARY PROCESSES IN GASES
G. V. Du.brovskiy, A. V. Bogdanov, Yu. Ji:. Gorbachev, L. F. Vyunenko, V. A. Pavlov, and V. M. Strelchenya
Leningrad State University Leningrad, 198904, USSR
In our review article presented here a quasiclassical kinetic equation has been euggested witn a collision integral expressed through the scattering T-matrix in the quasiclassical approximation. Now we shall give some results on cross sections (CS) and rate constants (RC) calculations carried out within a simplified version of this expression (a generalized eieanal formula, GEF).
I. GEN.li.o'RALIZED EICONAL :roRMULAE FOR CROSS SECTIOUS
The GEF for a differential CS (DCS) can be written as1
<5 <~ ,q) = ~~~~ ~dj exp(iA} ,11) f;n<f rf 2,
~n<f) = ~ (:::)~ e;lCP(~ ~0 ) ( exp(it. S/b) - S'if). (I. I) - ~ . ~ . Here ~= p- p' ~ the relative momentum transfer, q is the quantum numbere change, \I ie a set of N angle variables of the internal motion of collidin~ moleculea, '§ ~s their _telatille posi tion radiusvector in the point of maximal approach (j' = (f ,+)), t. S is the claasical action increment
+00
.6.S(~,\i0) =- s dt V(f+ ;(t)t,\90 +~(t)t, t(t)), -- .. ~ where V is the interaction potential, v(t),v(t) stand for relative velocity and internal motion frequenciea vectora and are to be found from the trajectory problern aolution. (Here we shall use an eieanal approximation).
Thua to calculate CS (I.I) one haa to express both the molecu-
697
lar Rarnil t~ni~ and the ~ntera.ction potential in the action-angle va.riablee I-~ (H ~ H( I)), to choose an adequate model for claesical trajectory, and to do the neceesa.r~ integrale. For dia.t~rnice the rotating Moree oscillator (r10) model may be used for H( I ) with the interaction potential V of a general type
(I.2)
Here r 1 2 are the i~teratsmic distancee in the molecu1es, 2( I 2 are the anelee between R and r 1_2 , P (x) are Legendre polynomiai~. Depending on the transition ubder cSnsideration (RT, RR, RV, VT, VV) different terme in Fq.(I,2) ahould be reta.ined.
For polyatomics the rigid rotor - harmonic oecillator model is the eimpliest one with the Hamiltonia.n H being
H(~) ""Bj(j +I)+ (A- C)l2 + 1i.t)lk(~+I/2), it..(j,l,~).(I.3) k=I
A,B,C in Fq. (I. 3) a.re the molecule rotational oonata.nts (A))-B~O~, j is the rotational quantum number corresponding to the molecule total momentum, 1 ia the"quantum number" aasociated with the momentum projection onto the symmetry axi~ for a symmetric to~ molecu1e, and being calculat!i through the real rotationa.l spectrum 1 ~ (Bj(j+I) - E .k k) ~(A - C) for asymmetric tops. J - +
In many applications the inelaatic tra.naitions influence on the ela.etic acattering can be neg1ected, that reaulta in the following formu1ae for DCS and CS
00
G (q) "" 2~ ~ db b I ~ 12'
Goh• oa., 1- E -= 300 °1<. 6'o·r "Z. 1-;=2. A A 2.- E "' 't50 °\( ~ 2.,-{=lf
30 3-E-=-618°1<. 30 3 - br:: 6 1 - E -=-1 &8 °\( i
20 zo
io iO z. . ' 3 E "K ~ 0 '
0 618 :fG8
a b
Fig. I. 'lhe Ar- N2 rotational excitation es va j' (a) and E (b).
698
with (;" (®) being the elastic ecattering DCS, r. (q, j>(®)) the "inelastlc ecattering profile" determined by the potintial anieotropy.
2. DIA'IDMIC MOLECULES
RT-proceeees. With the interaction potential for Ar - B2 system taken from Ref. ,, lin can be obtained as followa
lfYint2 = ((2j'+I)/(2j+I) (E-E*)/E)I/2 J 212(F/h), (2.1)
qj
where F~~.= -2~J"'~d..-2sh-\2Sivfd. (2E/.}t)1/ 2), q. = j- j','ll~ (j + j'- I~, E • E8 (I- (b/j') ), E*= Er(j') - Er(jJ~ Er(j) = B JtJ +I), E = E .- EJ2, J'C- is Ar - N2 reduoea ma.ss, B is the B mole8ule rltational oonetant, JP ie the maximal appr8ach dist;gce whioh ~ould be2ex:preseed through the impact parameter b from the equation b = 5' (I- W (f )/E ), where W (R) = C exp(-..I.R) is the elaetic interaction poteßtial. !n the furthgr calculations the foll~wing values4or parameters gave been taken: el. = 2.9,6; C = I.853"10 ; ~ = 3•10 ; B = 9. 2 • IO- ( a. u.). ihe ,,resul te of the CS calcula tions are preslnted in Fig. I. Eqs.(I.4) and (2.!) have been ueed to calculate the rotational excitation RC, ite dependence ve quantum number and temperature being given in Fig. 2.
V~processes. The deeactivation RC for a process N2(m + I) + li {0) •N2(m) + N (o) has been calculated as an example of the rotatfonally averaged2VT traneition RC. Under the adiabatic conditions Eq. (!.4) gives the following analytical expression
The conven tional no ta tion is mainely ueed here, and V is the 0
699
e~tio co111eions frequency, n is the mo1ecular number density, ~era!~!ep!~:n~~:fficient of (r-r8 ) containing term in the in-
The results of the calcu1ations of the Re according to Eq. (2.2) (Tab1~ 1) show a good agreement with the trajectory calculations data • ..
VV'-:processes. Let us consider CO(O)+N2(11oo(l)+N2(o) process RC as an example of a vv.• exchange reaction. 'l'w'o types of analytical approximations for this RC have been proposed.
An analytical fit to the numerical calculations of CS (1.4) has been considered. The rotationally averaged es of this process äe a function of energy in log-log ecale has been approximated by a strait line to yield an expression
"J 6 . -5 _1,644( -1) 2 ~ 01•10-o. 14•10 BI cm a0 •
With this es parametrization the Re under consideration has been calcu1ated in an analytical form
~~(T) = k*(T/T0 )'t, (2.3)
where k*=l.5·10-14 cm3/s, ~ =1.144, T0= 350K.
'lhe secong. approximation can be obtained by integration in Eq. ( 1.1) over f in adiaba tic limi t ( 'I/ d.. v>>1) • In this case assuming the interaction potential to be
V(R,r1,r2) a W0 (R) ( l+avvr1r 2),
with avv being a conetant and W (R) being a ~...orse potential, the RC of VV' exchange in CO-N2 sye~em becomes (n is CO molecule vibrational quantum number, m is that of N2 one)
700
ro~(T) = 't2({)1/2(nt- ~)(m+ i)(~ (fdß- ~(2.fD)l/2)l ( 1+ 2ftE*H(T0-T))(stdr (~1/2l/3exp(- ~- ~~(T0-T)), ~ • ~ (1 - 2x (n + 1/2))/2, p1a (<iißd f4 2kfr)1/3, (2.4)
n en e r J
, ... , 1 , T>T0 and f»f, T<T0 , ~j •IVnj -Ymjl'
To • ('i{kd(.r)l/2)-1· ( 2E*)3/2/( 3' ~/3_ ,~/33)3/2.
Table 1. The comparison of the RC of the process N2(1) + N2(o) ~ 2~2(0) (upper line) with trajectory calculations (bottom line)
---------· ---
-------------------------------------------------------------------------TK 1000 2000 3000 4000
In Table 2 the results obtained using Eqs. (2.3) and (2.~ are compared with the other theoretical and experimental data 7.
The cornparison shows that both formulae obtained are in good agreement with both the trajectory caloulations and experirnent. But while Eq. (2.3) has been deduced for one specific process only, bq. (2.4) may be applied to an arbitrary desactivation VV'-reaction of t~e. kind AB(n1 ) + CD(n2) ~ AB(n1+ 1) + CD(n2- 1) in adiabatic condJ.tJ.one.
VV-proceseee. AB an example the following process hae been considered N2(m) + N (n) + N (~1) ~ N2(n-l) using the previous eectione techniques. ~e first approach again gives Eq. (2.3) but with k* and l depending on m and n and still not depending on the
Table 2.
TK
a --b
0
d
e
The RC of the CO~~) +3N2(1) ~CO(l) + N2(o) process multip1ied by 10 orn /e ( a- the reeuits obtained with the help of' Eq. (2.3), b- the eame f'or Eq. (2.4), c- tr~jectory calculations data 5, d- e~perimental data , e - collinear model estimations ).
------100 150 200 250 300 350 500 1000 2000
------------0.35 0.56 0.79 1.02 1.26 1.5 2.45 4.98 n.o
---- -------- 0.87 - 1.2 2-37 ----- ----
0.53 0.64 0.78 0.93 1.1 1.3 2.0 4.0 6.5
0.41 o.59 o.a6 1.2 1.4 1.5 2.0 6.0 ----
0.04 0.15 0.32 0.52 0.73 0.95 -----
701
0 4 6
( a)
8
i
0
1.-~~~ 2 2-~'=lt 3-6~"b 'i -r =s
~tso 618 i6S
(b)
Fig.2. The Ar-N rotational excitation RC vs j'(a) and T (b). 2
-i5 10
10-n
0
]{ 10 cm} $
s 10-1!>
5 0
i 0
n, tn
to-1>
10-i'l
20 1{0 11, 0,2. 1 (a) (b)
Fig. 3. The comparieon of N2(m) + N2(n) ~N2(m+l) + N2(n-l) VV RC calculated us4ng Eq.( 2.4)(denee line) with a trajectory results (dotted line); (a) Re ve n dependence (m=O), (b) RC ve T dependence.
tempe~ture. We suggeet the foll~wing param~trization ~*= Am(n+1)~ eXE(3 'b lm-n-11), ~-~ b + ~ (m+n) - a2(m-n) - c (m-n)_5, ~= 5·5· 10_14, a 231.58•10 , b = 1.55, b = 0.18, c = 9.810 , A = 1.8• 10 cm /s, T = 500K.
0
In the second approach the RC ie again given by Eq. (2.4). The comparieo2 of the calculatione ueing Eq.(2.4) with the trajectory results ie given in Fig. 3.
702
3. POLY A'IDHIC l~LECli'LES
In the caae of rotational excitation of H20 molecule by a charged particle, taking into account only long-range charge (Q) -dipole (d) interaction potential, Eq. (1.4) with the Hamiltonian (1. 3) givea for the RC of RT - tranai tion j .1. ~ jf1f an exprea-aion 1 1
1/2 I K - ac;;t (2jf+ 1) iq j( '/ f f*)1-21 q j I 3(1 + E*/kTel) )(.
- 31/2 2ji+ 1 (l(l-+qj) f(l+q1- ~2qj))2 (3.1)
where g ='SiQd~ 1/2,P *' >. 1 , 2 are parametera depending ~n molecule quantum numbere, V = lvj - ).2 'Y1 1, e1 = (( qj V f p*) JA)( X(2kT)-l )l/3, ..P * ia the maximal approach diatance .when impact paparameter b ~o. Regretfully the absence of experimental data doea not give ua a poaaibility to check up Eq. (3.1).
For the vibrational excitation of H20 molecule by a neutral structure1eea partic1e (V~tranaitions ) the inelaatic interaction potential hae been aaeumed to take a form
where Qk are the normal vibrational coordinatea of H20 molecule. In thia caae for the total es of a tranaition with ail the three modea vibrational quantum numbers change the following expression has been obtained
703
where (5 is the elaetic ecattering es. The integration of es {3.2) with only one qi! 0 correeponding to the minimal Yk ((010)• -.(000} traneition) under adiabatio conditione gige~ for the RC Eq. (2.2). Comparieon with the experimental data ' ]a that Eq. (2.2) gives the correct order of the magnitude of 1 and correotly deecribee its temperature dependance. Unfortu ~ely the more detailed comparieon is not poeeible due to the experimental data abeence.
REFERENCES
1. G.V. Dubrovekiy, A.V. Bogdanov. A general quaeiclaeeical approximation of the ~Operator in action - angle variables, Chem. Phyl!· Letters. 62:89 (1979). 2. G. V. Dubrovekiy, L .• F. Vyunenko. Generalized eikonal method for the rotational vibrational excitation of diatomice, J. Exp. Theor. Phye. BOr 66 (1981) - in Ruseian. 3· C. Nyeland, G.D. Billing. Approximative treatmente of rotational relaxation, Chem. Phys. 40: 103 (1979). 4. G.D. Bi11ing, E.R. Fischer. VV and VT rate coefficiente in N2 by a quantum - claeeical model, Chem. PbySo 43: 395 (1979). 5. G.D. Bi11ing. Vibration - vibration energy transfer in CO co1li-
. . 14 14 15 15 d1ng w1th N2, N N, and N2, Chem. ?hys. 50: 165 (1980). g• D.C. Allen, C.J. Simpson. Vibrational energy exchange between CO and isotopes of N2, Chem. Phye. 45: 203 (1980). 1. X. Chapuieat, G.Bergeron. Anharmonicity effects in the collinear co11ision of two diatornie molecules, ~~em. PhyB· 67: 397 (1979). B. J. Finzi, F.E. Hovia, V.N. Panfilov, P. Heea, c •. :s. Noore. Vibrationa1 relaxation of water vapor, J. Chem. Phys. 67: 4053 (1977). 9. R.T.V. Kung, R.F. Center. High temperature vibrationa1 re1axation of H2o by H20, He, Ar, and N2, J. Chem. Phye. 62: 2187 {1975).
704
RELAXATION OF VELOCITY DISTRIBUTION OF ELECTRONS COOLED (HEATED)
BY ROTATIONAL EXCITATION (DE-EXCITATION) OF N2
Katsuhisa Koura
First Aerodynamics Division National Aerospace Labaratory 1880 Jindaiji-Machi Chofu, Tokyo, Japan
INTRODUCTION
It is well known1 that low-energy electrons in molecules lose or gain the energy through the rotational excitation or deexcitation of molecules. When the electrons are so diluted in molecules that the effect of the electron-electron collision is negligible as compared with that of the electron-molecule collision, the behavior of the electrQn velocity distribution in the electron cooling or heating process is not well understood. It is not obvious that the electrons cooled or heated by the molecular rotational excitation or de-excitation obey the Maxwell velocity distribution, although the Maxwell distribution is often assumed for the theoretical interpretation of the measured results in the cross-modulation experiments 2 ' 3 and the D region of the ionosphere.q In fact, it is shown 5 that the stationary electron velocity distribution, which depends on the initial velocity distribution, in molecules with two energy levels is not the Maxwell distribution and reveals the sawtooth pattern for the initial Maxwell distribution, where the electron-electron and electron-molecule elastic collisions are ignored as compared with the electronmolecule inelastic collisions.
The purpose of this paper is to study the relaxation of the velocity distribution of low-energy electrons cooled or heated by the rotational excitation or de-excitation of nitrogen molecules with much more rotational energy levels than two 5 using the Monte Carlo simulation 6 in the system where the electrons are so diluted in the heat-bath molecules that the effect of the electron-electron collision is negligible as compared with that of the electronmolecule collision. It is investigated whether the electrons
705
maintain the initial Maxwell distribution in the cooling process, which corresponds to the cross-modulation experiment, or in the heating process, which corresponds to the shock-wave heating, and the electron temperature obeys that of Mentzoni and Row 3 for the local Maxwell distribution. It is examined whether the initial o-fUnction velocity distribution, being far from the Maxwell distribution, actually approaches the Maxwell distribution through the rotational excitation and de-excitation only.
RELAXATION MODEL
The time evolution of the electron velocity distribution function f(v) in the spatially homogeneous system, where the electrons with the number density ne=ff(v)dv are diluted in the heat-bath molecules with the number density n(>>ne) and the electron-electron collision is ignored, is described by the Boltzmann equation7
df(v)/at = J J,f[f(v')fJ,(wJ,)gJ/gJ'- f(w)fJ(vJ)]
xgiJJ'(g,Q)~dvJ' (1)
where t is the time, IJJ'(g,n) is the differential cross section for the electron scatterin~ into a solid angle n with the molecular rotational transition J+J' Le+M(J)+e+M(J'), M(J) being the heat-bath molecules M in the Jth rotational level] and the change in the electron velocity (v+v') and the molecular velocity of M(J) (wJ+ vj•), and g=lv-vJI is the relative velocity. fJ(VJ) is the velocity distribution fUnction of M(J) taken as the Maxwell distribution at the heat-bath temperature T
where vJ=IvJI• IDM is the molecular mass, k is the Boltzmann constant, and nJ is the number density of M(J) taken as the Boltzmann distribution at the heat-bath temperature T
(2)
(3)
in which n=~JllJ, Qr=~JgJexp(-EJ/kT), gJ=(2s+l)(s+a)(2J+l) is the degeneracy, s is the nuclear s~in (s=l and a=O and l for odd and even J, respectively, for N2), EJ=BoJ(J+l) is the rotational energy, and Ba is the rotational constant (Bo=2.00 cm-1 for N2). 8
Since the electron mass m is much smaller than mM, it may be. assumed that g~v(=lvl>>vJ), g'=lv'-vj• l~v'(=lv' l>>lvJ• 1), and VJ~vJ•· On this assumption, Eq. (1) is simplified for the isotropic electron velocity distribution P(v)=4nv2f(v)/ne [!P(v)dv=l] as 9
706
3P(v)/at = ~ ~' [nJ 1 P(v' )vaJ'J(v') - n~(v)vaJJ' (v)], (4)
where OJJ'(v)=fiJJ'(v,Q)dn is the total cross section for the rotational transition J~J' and satisfies the detailed balance v2~0JJ'(v)=v'2~,0J'J(v'), and the electron velocities v and v' are related by the energy conservation mv212+EJ=mv'212+EJ'· It is noted that the electron-molecule elastic collision (J'=J) is ignored in Eq. (4).
Multiplying Eq. (4) by the electron energy E=mv212 and integratiny over v yield the usual electron energy-loss (gain) rate equation 2 '
d'Eidt = ~ ~' nJ(EJ-EJ' )vaJJ' (v),
where E=!EP(v)dv is the average electron energy related to the electron temperature Te by 'E=3kTel2 and
vaJJ'(v) = JvaJJ'(v)P(v)dv.
(5)
(6)
When P(v) is the Maxwell distribution at the electron temperature Te
PMD(v) =(ml2nkTe) 3124nv2exp(-mv212kTe), (7)
and OJJ'(v) is given by the Gerjuoy-Stein formula (J'=J±2) 2
aJ J+2 (E) = { [(J+2) (J+l)] I [(2J+3) (2J+l~}a0 [l-(4J+6)B01E]112 , '
aJ J-2(E) = { [J(J-1)] I [(2J-1)(2J+l)] }a0 [1+(4J-2)B01E]112 , (8) '
where cro=8nQ2a§ll5, Q is the electric quadrupole moment in units of ea6, and ao is the Bohr radius, VOJJ'(v) is analytically written as 3
vcrJ,J±2 (v) = (8kTelnm) 112sJbJ~(bJ)exp(+bJ)' (9)
where SJ={ [(J+2)(J +l )] I [( 2J+3 ).( 2J+l) J }ao and { [J(J-1)] I [(2J-1)(2J+l)]}ao and bJ=(2J+3)BolkTe and (2J-l)BolkTe for J'=J+2 and J-2, respectively, and Kl(bJ) is the modified Bessel function of the second kind. Using the fact of the smallness of the rotational constant BQ in Eqs. (5) and (9), Mentzoni and Row 3
>btained the simple rate equation dT~Idt=(T-Te)IT with the electron energy relaxat~on time Tlto=3k(TTe)l/218Bo, where to =[nao(8kTinm)l!2J-l is the rotational collision time.
Since the rotational transition cross section is usually given 2 by OJJ' (v) instead of OJJ' (g), the simplified Eq. (4) is solved by the Monte Carlo simulation, although the Monte Carlo simulation6 ' 7 is applicable to the exact Eq. (1). OJJ'(vt for nitrogen molecules is taken as the Gerjuoy-Stein formula LEq. (8)] with10 ' 11 Q~l.04 ea§ in correspondence to the analysis of Mentzoni
707
and Row. 3 The effect of the electron-molecule elastic collision is evaluated for the elastic cross section of the· hard sphere model IJJ (v,Q)=crJJ/4n with10 qJJ~5 A2 (OJJ/Oo~lO). The initial electron velocity distribution P(O) at t=O is taken as the Maxwell distribution [Eq. (7)] at the initial electron temperature Te(O) or the o-function distribution
where v1 is the/initial electron velocity related to Te(O) by vl=[3kTe(O)/m]1 2•
MONTE CARLO SIMULATION
(10)
The isotropic electron velocity distribution P(v) described by Eq. (4) is obtained by the Monte Carlo simulation6 : P(v) at the time t+ßt is obtained from P(v) at the time t by following a large number of simulated electrons during the time step ßt through the molecular rotational excitation and de-excitation collisions. Since the molecular motion is ignored in Eq. (4), the Monte Carlo procedure is somewhat simplified:
(i) At the time t, an electron velocity v is assigned to each of simulated electrons of the number of N by the probability
p(v) = P(v). (11)
(ii) A time interval ßtc between successive electron-molecule collisions is assigned by the probability
p(ßt ) = v exp(-v ßt ), c c
where V is the total collision frequency of electron-molecule collisions given by
(12)
(13)
SJ(v)=LJ'OJJ'(v) being the total scattering cross section. Since V is evaluated at the time t and taken to be constant during ßt, the time step ßt should be so small that the difference between v at t+ßt and V at t, ßv=v(t+ßt)-v(t), satisfies the condition
lßvl/v(t) << 1. (14)
(iii) A collision pair of the electron with a velocity v and the molecule in a rotational level J is selected by the probability
(15)
(iv) A rotational level J' after collision is assigned by the
708
probabili ty
(16)
(v) The electron velocity v' after collision is obtained from the energy conservation as
(17)
The effect of the electron-molecule elastic collision can be evaluated by including J'=J in the procedures (ii)-(iv) [crJJ(v)~oJ and calculating the electron velocity v'=lv' I after elastic collision from the exact momentum and energy conservations
mv'+~vM = mv+~vM,
v'-vM = l~vMin, (18)
where v' and VM are the electron and molecular velocities after collision, respectively, v and VM are those before collision given by v=vno and VM=VMnQ• no and no being isotropic unit vectors, the molecular velocity vM is assigned by the probability
p(vM) = PMD(vM)' (19)
PMD(vM)=4nv~fJ(vM)/nJ being the heat-bath Maxwell distribution, and n is a unit vector directed to a scattering solid angle Q assigned by the probability
p(Q) = I JJ(v ,Q) /a JJ(v). ( 20)
(vi) The procedures (ii)-(v) are carried out until the accumulated time Eßtc exceeds ßt. The electron velocity distribution P(v) at the time t+ßt is obtained from the velocity distribution of simulated electrons.
It is noted that the velocity distribution P(v) obtained by the Monte Carlo simulation is the average distribution in the velocity region (v-ßv/2, v+ßv/2) with the width of ßv,
-( v+ßv/2 ( P v) = fv-ßv/2 P v)dv/ßv. (21)
Owing to the fact that there exist practical limits in the values of the velocity width ßv(~) and the number N(~) of simulated electrons and the heat-bath molecules have much more rotational energy levels than two, the fine structure of P(v) such as the sawtooth pattern indicated by Peyraud5 for the two-energy-level molecules may not be obtained for P(v). The electron temperature Te is obtained from the average V2=fv2P(v)dv~ENV2fN as Te=m~/3k.
709
RESULTS AND DISGUSSION
Since the electron energy should not exceed the molecular vibrational excitation threshold1- 3 in order that the molecular rotational excitation or de-excitation is the dominating electron energy loss or gain mechanism, the results are obtained for the temperature range lOO~Te(O), T~3000 K for the electron cooling [Te(O)>T] and heating [Te(O)<T] in nitrogen molecules. The time step öt is taken as 0.5~öt/to~l0 so that the variation of the collision frequency lövl/v(t) is within a few percent; the convergence of result is checked by decreasing öt/to. The upper limit of the rotational level Jmax is taken so that (nJmax/n)/ (nJ/n)max~O.OOl, where (nJ/n)max is the maximum value of the rotational distribution [Eq. (3)]; the convergence of result is checked by increasing Jmax• The velocity width and the4number of simulated electrons are taken as öv/vo=0.2 and 103~N~l0 , where v0=(2kTe/m)l/2 is the most probable speed.
Cooling Process
The time evolutions of the electron velocity distribution P :p(v/vo)[=P(v)vo] for the initial Maxwelldistribution [P(O)=MD] in the cooling process are presented in Figs. 1-3. P deviates considerably from the Maxwelldistribution (MD): P>MD at v/vo~l or v/vo~2 and P<MD at l~v/vo~2. The degree of deviation increases with increasing Te(O) (Figs. 1 and 2) or decreasing T (Figs. 1 and 3) and is remarkable at Te(O)/TVlO. [Even for Te(0)=500 K and T =300 K, a little deviation is observed.] P deviates from MD to the largest extent at the middle stage of cooling (see Fig. 5) and approaches MD within the statistical uncertainty at the nearly same time when the electron temperature Te reaches T (Fig. 5).
Here, the effect of the electron-molecule elastic collision is evaluated for the elastic cross section OJJ/cro=lO. P and Te for OJJiao=lO are in good agreement with those for OJJ/cro=O: The effect of the elastic collision is actually negligible.
The relaxation of P for the initial o-function distribution [P(O)=o] in the cooling process i$ presented in Fig. 4 for Te(O) =1000 K and T=300 K. The o-function distribution, being far from the Maxwell distribution, approaches MD within the statistical uncertainty; the Maxwellization time TMD defined to be the time when P approaches MD within 5 % is TMDito"'lOO. It is noted that. the Maxwellization (o~) is much faster than the temperature equilibration (Te~) (Fig. 5).
The relaxation of the electron temperature Te for P(O)=MD in the cooling process is presented in Fig. 5 in comparison with the electron temperature TMR of Mentzoni and Row 3 for the local Maxwell distribution (P=MD), which is obtained by solving Eqs. (5) and (9).
710
The effect of Te(O) on t he deviation of Te from TMR is shown for T=300 K. For Te(0)~500 K, Te is in good agreement with TMR, although P indicates a little deviation from MD. For Te(O)~lOOO K, Te is in agreement with TMR in the first half of the cooling process and becomes higher than TMR in the latter h alf owing to the considerable deviation of P from MD, which is also observed for T=lOO K [Te(O)=lOOO K]. The degree of deviation of Te from TMR
1.0
0 .8
0.6
10..
0.4
0 .2
00
P(O) =MD
I I
I
~ I
I I
vlvo
Te(O) = 1000 °K
T = 300 °K
2
t I t0
- -o-- 50 t:::. 100 0 200
3
Fig. 1. Time evolution of electron velocity distribution for the initial Maxwell distribution (MD ) in the cool ing process.
1.0
0.8
0.6 ICL
0.4
0.2
0 0
P(O) =MD Te(O) =3000 °K
T = 300 °K
t I t
t:::. 10 --o-- 50 -{}.- 200
\l 400
2 3
Fig . 2 . Time evolution of electron v elocity distribution for the i n i t ial Maxwell dist ribution (MD ) in t he cooling process .
711
1.0 P(O)=MD
0.8 p'..() I
I 0.6 I
10.. I 6 I
0.4 ~ I
I I{?J
0.2 I I
I
0 0
0
vlv0
Te(O) = 1000 °K
T = 100 °K
2
. t I t0
~ 10 --o-- 50
0 100
3
Fig. 3. Time evolution of electron velocity distribution for the initial Maxwell distribution (MD) in the cooling process.
2.2
2.0
1.8
1.6
1.4
I Cl.
P!O)=o
v/v0
Te(O) = 1000 °K
T = 300 °K
_l!to
10
30 50
3
Fig . 4. Relaxation of electron velocity distribution f or the initial o.- function distribution in the cooling process .
712
7
6 ~ ...... ~ 5
4
0
eiO)=MO ßO)=Ö P =MD T (°K) 0
/:!,
0 0
0 0
0
00 00
300 100
Oooo Ooooooo
0o~--~s~o~~~~oo~--~~s~o~--z~o-o---z~so-----3oLo--~35o tlt0
Fig. 5. Relaxation of electron temperature in the cooling process in comparison with that of Mentzoni and Row (P=MD).
P(O)=MD
10...
0 0
Te(O) = 300 °K T = I000°K
t/to 6 10
--<:r -1 00 0 200
2 3
Fig. 6. Time evolution of electron velocity distribution for t he initial Maxwell distribution (MD ) in the heating process.
713
increases with increasing Te(O) in correspondence to the degree of deviation of P from MD, which is shown for Te(O)=lOOO and 3000 K. The effect of the heat-bath temperature T on the deviation of Te from TMR is shown for T=300 and 100 K [Te(O)=lOOO K]. The degree of deviation increases with decreasing T in correspondence to the degree of deviation of P from MD. The temperature equilibration time TT defined to be the time when Te approaches T within 10 % is larger than that of Mentzoni and Row TMR to the extent of about 25 %: (TT/to, TMR/to) at T=300 K are (90,90) for Te(0)=500 K, (180, 170) for 1000 K, and (360, 290) for 3000 K; (120, 100) for Te(O) =1000 K and T=lOO K.
The relaxation of Te for P(O)=o in the cooling process is also presented in Fig. 5 for Te(O)=lOOO K and T=300 K. Te is lower than TMR; TT/to~l50 is about 10 % less than TMR/to.
Heating Process
The time evolution of P for P(O)=MD in the heating process is presented in Fig. 6 for Te(0)=300 K and T=lOOO K. P deviates slightly from MD (P<MD at v/vo~l or v/vo~2 and P>MD at l$v/vo$2), to the largest extent at the middle stage of cooling, and approaches MD within the statistical uncertainty at about TT/to (see Fig. 8). The degree of deviation for the lower Te(O)=lOO K or the higher T =3000 K is as small as that for Te(0)=300 K and T=lOOO K.
The relaxation of P for P(O)=o in the heating process is presented in Fig. 7 for Te(0)=300 K and T=lOOO K. The o-function distribution approaches MD within the statistical uncertainty; TMD/to~5o is much less than TT/to (Fig. 8).
The relaxation of Te for P(O)=MD in the heating process is presented in Fig. 8 in comparison with TMR (P=MD). Te is lower than TMR· The degree of deviation of Te from TMR remains nearly the same for the decrease in Te(O) (300+100 K) or the increase in T (1000+3000 K). TT is larger than TMR to the extent of about 25 %: (TT/to, TMR/to) at T=lOOO K are (260, 220) for Te(0)=300 K and (300, 240) for 100 K; (800, 690) for Te(0)=300 K and T=3000 K.
The relaxation of Te for P(O)=o in the heating process is also presented in Fig. 8 for Te(0)=300 K and T=lOOO K. Te is lower than that for P(O)=MD; TT/to~270 is about 20 % larger than TMR/to.
CONCLUDING REMARKS
The relaxations of the velocity distribution and temperature of electrons cooled or heated by the rotational excitation or deexcitation of nitrogen molecules are studied using the Monte Carlo simulation. The initial Maxwell electron velocity distribution is
714
2.0
1.8
1.6
1.4
1. 2
IQ..
1.0
0.8
0.6
0.4
0.2
0
P(O) = 8
v/ v0
Te (0) = 300°K
T = 1000°K
2
t /to -fr- 5 -<r 10 0 30
3
Fig. 7. Relaxation of electron velocity distribution for the initial o-function distribution in the heating process.
0.8
0 0 50 100 150 200
t/ to 250
100 1000
300 350
Fig . 8. Relaxation of electron temperature in the heating process in comparison with that of Mentzoni and Row ( P=MD).
715
perturbed considerably in the cooling process and slightly in the heating process. The electron temperature is higher in the cooling process and lower in the heating process than that of Mentzoni and Row for the local Maxwell distribution. The temperature equilibration time is larger than that of Mentzoni and Row to the extent Of about 25 %. The initial O-function electron Velocity distribution actually approaches the Maxwell distribution through the rotational excitation and de-excitation only. The Maxwellization is much faster thari the temperature equilibration.
HEFERENCES
1. H. S. W. Massey, E. H. S. Burhop, and H. B. Gilbody, "Electronic and Ionic Impact Phenomena," Vol. II, Oxford Uni v. , London (1969).
2. E. Gerjuoy and S. Stein, Rotational excitation by slow electrons, Phys. Rev. 97:1671 (1955).
3. M. H. Mentzoni and R. V. Row, Rotational excitation and electron relaxation in nitrogen, Phys. Rev. 130:2312 (1963).
4. A. Dalgarno and R. J. Moffett, Electron cooling in the D region, Planet. Space Sei. 9:439 (1962).
5. N. Peyraud, Dynamics of a free electron gas interacting with neutral atoms. Two-dimensional model, Phys. Fluids 20:2037 (1977).
6. K. Koura, Nonequilibrium velocity distributions and reaction rates in fast highly exotherrnie reactions, ~· Chem. Phys. 59:691 (1973); Nonequilibrium velocity distribution and reaction rate in hot-atom reactions, ~· Chem. Phys. 65:3883 (1976).
7. K. Koura, Hole burning into molecular velocity distribution due to monochromatic radiation and molecular elastic collision, ~· Chem. Phys. 72:268 (1980).
8. G. Herzberg, "Molecular Spectra and Molecular Structure," Van Nostrand, Princeton (1950).
9. L. S. Frost and A. V. Phelps, Rotational excitation and momentum transfer cross sections for electrons in H2 and N2 from transport coefficients, Phys. Rev. 127:1621 (1962).
10. A. G. Engelhardt, A. V. Phelps, and C. G. Risk, Determination of momentum transfer and inelastic collision cross sections for electrons in nitrogen using transport coefficients, Phys. Rev. 135:Al566 (1964).
11. A. ~Phelps, Rotational and vibrational excitation of molecules by low-energy electrons, Rev. Mod. Phys. 40:399 (1968).
716
EFFEOTS OF THE miTIAL MJLEOULAR STATES
m A HIGH-ENERGY SOATTERING OF .WLEOULAR BEAMS
V .B. Leonas and I.D. Rodionov
Space Reaearch Institute, USSR Acade~ of Seiences Moscow V-485, II?8IO, USSR Keldysh Institute of Applied Mathematics USSR Aciademy of Sciences, Moscow A-4?, !2504?, USSR
INTRODUOTION
The elastic scattering of molecular beams is a convenient tool to determine interaction potentials. Extensive and reliable data on interatomic forces have been obtained by applying Versions of the method based on the use of thermal and high-energy (E - l KeV) beams. The problem of deriving potantials (potential energy sur.faces - PES) becomes more complicated when studying interactions with the involvement of molec~.
The most direct way to study the PES experimentally is to measure the vibrational-rotational transition differential cross-sections. A laser fluorescence technique proves to be promising in gaining such data in experiments with molecular beams2. Since the above technique is still being developed it is of obvious interest to extract information on PES from the data without selection (in particular, of final states). For this one needs dynamic manifestations of the components of the molecular PES in the state averaged scattering cross-sections. In this paper the theoretical prediction and experimental discovery of such an effect in the differential cross-section averaged by final states are discussed for small scattering angles corresponding to interaction energies of l to 20 eV. The effect the vibrational rainbow (VR) - is explained by the nonadiabatic vibrational transitions influencing the scattering dynamics during the collision time.
717
In the case of experiments with selection the intramolecular motion effect upon the transition cross-section are also of interest becauae the absence of such an effect leads to an invariancy of experimental data making them less informative. In this paper it is proposed to treat data on the total rotational transition c~oss-sections, which have been recently published in-', as experimental Observation of intramolecular motion effects in the high-energy scattering.
THE DISTOR!ED WAVE APPROXIMATION
Let us restriet ourselves by the case of collisions of atoms with a vibro-rotator and represent PES in the form
where R is the distance between atom and the molecule • s center of mass, 't is the distance between atoms in_ the mol~cule' r is a cosine of the angle between
R. and t. • The higher terms of the PES expansion . determine the rotational transition cross-sections and information on these is a key to a quantitative description of nonequilibrium processes in gaseous systems.
A specific feature of the small angle ( B < 0.01 rad) high energy scattering (HES) is that the elastic scattering cross-sections exceed those of inelastic scattering. In a theoretical analysis it allows.to use a distorted wave approximation, where the 7J' , i , m - states are chosen as a basis, the quantization axis ~ is parallel to the wave vector R , and the ~lu-
ence of channels with changin~ quantum numbers V'", J , m on the channe 1 v-J m .....,. 'IJ'-J m. is ignored. Sucb a choice is due to the smallness of the collisional depolarization cross-sections (change of nL ) in this basis for the small-angle range. For the quality analysis this approach is more preferable as compared to the commonly used sudden approximation one because it is simple (calculation of one-dimensional integrals)1 Beeides it can explain and predict some new effects •
Within the framework of the developed tbeory tbe effective spherically symmetric potential governing the scattering is given by
718
(2) a.u. ,
where {).} is the vibrational constant, JL< is tbe reduced molecular mass. rnt it is shown that in the absence of final state selection the small-angle elastic scattering is described rather satisfactorily by this spherically symmetric potential. The latter is explicitly dependent on vibrational number ~ • It is stimulated the predic~ion and experimental verification of such a dependence •
Let us omit a general expression for inelastic transition amplitude in the distorted wave approximation, obtained from integral Lippman-Shvinger equations~ and consider the case 1 , j' >'> L • The ma.trix elements of PES are expressed. tlu'ough the Vigner D-functions and the rotational transition amplitude is given by
00 +oo L.
.C -"..."l",(6) =-- i j:t,(K61J-C { 5 [L 2f(V.ih 8+ g•) + ~ t) •fP .f.t
(3)
. .I .&j+.f Only the terms with 1AJI~ '" and (-l) =l make a contribution to the amplitude.
Por the m -changing transitions the term 1o(K9l) in (3) is replaced by 1,~ 1 (K6/) • The replacement justifies the assumption on ~e smallness of the depolarization cross-sections in the chosen basis - and allows to ignore the terms for the sake of our analysis.
A VmRATION.AL RAINBOW
An anomalous behavior of the BES differential cross-sectiQns for the systems He-Nz and He-CO has been observed in7. Tbe anomaly consists m a distinct bumps similar
719
to a rainbow pecularities. Subsequent measurements in other systems showed that such anomalies are typical for atom-molecular and molecular ones6 but are not observed for at omic pairs. After an analysis of 2these experimental data a hypothesis was suggested in that explains an origin of the anomaly of the differential cross-sections by some pecularities in the molecular vibrational transition during the collision time for the specific impact parameter values. For the effective potential in the form of (2), at the vibrational excitation a nonadiabatic transitio~takes place between potential surface v:- and v. whose relative position is determined oy the value and sign of Co (/t,) Note that the main points of the hypothesis were the statement (unlike the common one ) that the values
BoCRJN;JiW , C0 (R,)/&1'~ and '/o(fi.) are comparable for the small approach distances.
To verify this V~ hypothesis systematic investigations of HES have been carried out for the He-N2 system. The results are discussed below.
The measurement§ were performed with an apparatus described earlier inr. As a result of modification in the measuring part where a control computer was replaced by one of next generation (a microcomputer) and measuring units by modules of CAMA.C standard. In the measurements He and N2 were used alternat1vely either as a target-gas or a oeam-gas at the beam energies E(He)=0.6, 0.8, !.2, I.5, 2.4, 3.2 KeV; E(N2);I.2, 2.4, 3.2 KeV. In order to obtain~he beam of molecules a resonance charge-exchange of N was used. According to different vibrational level~opulations should be ex~ected for molecules N2 in the beam and in the target {in the last case only the level 1J" ; 0 is populated).
The measured angular distributions or the scattered partielas flux intensities !(~) are given in Fig. 2I for ~ the detector angle range from 2xi03 to 4xiO rad. In Fig. I the rainbow peak positions 'l:p
(ct =tl.. • E) are shown for the scattering of He in N2 in He veraus the collision Velocity U •
The main Observable facts can be summarized as follows: an angular width of the rainbow peak is rather narrow (an equivalent. width by the impact .P.arameter does not exceed 0.2 A); the peak shifts w~th the change of collision energy (velocity); the peak halfwidth and amplitude change with energy; the latter values are different when partielas change their roles; and finally, in the case He-CO the bimodal shape of peak is observed •
The shift of the peak position on the reduced
720
U, em/s ~ig. I. The rainbow peak positions
for the scattering of He in N and N2 in Be veraus the collision2velocity U•
angle scale for different beam energies is a clear indication that the peak's origin is nonpotential (for the potential scattering there is an invariant picture on the scale I·J.a /Cf: ) • Moreover, this shift eliminates any connection between the observed peculiarities and the passage through crossing points of the ground and excited electron states (an angular position of the peculiarities arising in this case is also invariant).
The different peak positions 'l'p for N2 in the target and in the beam mean that tbe vibrationally excited beam molecules move not along the terms v:(~) and V!UO but, according to 8 , rather along the terms vl· (R,) and v:·(~) ' which seems to be responsible for different ma.gn.itudes of 'tp for the same relative velocity.
INTRAMOLECULAR MCY.riON MANIFESTATIONS m TBE ROTATIONAL TRANSITION CROSS-SECTIONS
Experimental data obtained in3 on the rotational transition total cross-sections for H2 - molecules
721
Table I. The Born limit ,B and the mod.ifications of ~ involving the intramolecular motion in comparison with the experimental data
Transi- G{j ... j +Aj)/6(j'-+j'+Aj) minim tion experi- 2 {
ment ~ "a-c ... ) ./3·(· •• ) IAK···f [a.u]
0-2 I.9 I.67 5· 2.8 O.I5 I-~
2~4 0.72 0.857 0.4~ O.GI 0.25 I-~
~---5 o.~8 0.794 0.24 0.44 0.25 I-~
2-4 I.9 I. I I.66 I.44 0.~5 ~-- 5
+ excited by the Ll impact in the energy range from 50 to 400 eV, showed the following specific features. For A.j = 2 and j , f ~ 5 when , E > IOO e V the cross-section raliio s<i -+~ +AJ)/6(} -.J' +AJ) coincides with the o~e ob~~ined in tfte Born limit ra-tio that depends on d , ct and AJ but does not depend on E. The Born limit is considered when E-+ oo • In this case the wave number is conserved during the collision A KJJ• --. 0 • It is assumed tbat anisotro-pic interactions with Pa_(cose) symmetry is the dominant in the rotational excitation in our case. Then in the Born limit G(j-.J +2?16(j' .... f +2) =tiJ , where ,f. = {(3+1)(3+~)(2.:1' +3)(1:1' +IJ}/{(3 +l)(:l~2)(Z':1+3}(2.3 +1) J. If the energy transfer A.E and AE' are equal for the both transitions then the measured cross-section ratios (6(3-.J.}/6(!_",3) and 6(2.-.o)/6(0..,. 2) in the referred paper) are independent of E up to 50 eV,
722
When IAEI >lAE'I the cross-seetion ratio decrea-ses with the energy lowering for E < IOO eV and coincides with the ~ for E > IOO eV. The deviation of the measured ratio from ~ increases with IIAEI -IAE'II growing.
No approximation among those available explains such a bebavior of the rotational transition cross-sections. These approximations yield the equations coinciding with ~ • This is valid for both the developed distorted-wave approximation and the rotational adia-batic ( or sudden) approximations ro-t& •
Honapplicability of the rotational adiabatic approximation means that the intramolecular motion effect on the transition cross-section cannot be ignored. At E = I.2 eV such an effect was noted in '~ for the rotational transitions G(0..,.2) and G(o-...lf) of the L.t+- Ha. system. It is rather amazing tbat the effect survives in the considered energy range too. It is associated with the slow change of collision time '1'- lt* Y.Ji/(J Ec·t·J ( R.• is the radius of action of intermolecular forces) when the collision energy Ec.g. grows and the frequency of the molecule rotation sharply inereases with J: CcJtot = 6~·Cj +1.) •
If we ign.ore AK-".jm,w'm • Z in {3) then tl\e q~tity 5wrn_. •i''" (9) will de4>e:q4 only on Aj=J'-J , wliich corresponds to ~ when d, J >> L • Invol-veme~t.of.this te~~ results in the difference between
6(J-.J+~)/5(J'-..J +AJ) and .f1 • The qualitative aspec~ of this difference can be explained by that exp ( t AK ••• ~) is an oscillating multiplier whieh leads to the reduction of a ma~tude of thi scattering amplitude by the value ,.., I AK ••• R* 1-when I A K • ••• 4' >'> t a-.· ~ t~ing .into account that
E >> 8 and IAIC.··hrviAJ li!+2J +Ad I we obtain from (3) the following modified equation for the cross-section ratio:
(4) G (j -.j + Aj) = ~ . ( ! +2J I + AJ I ) Jb
<:5Cj'-'> j' +Aj') /. + 3J +A d ~ == !~2
Table I gives the numerical results of the J3 . modification by applying the relation (4). It shows that the behavior of the cross-section at E=53 eV is explained correctly from the qualitative point of view (Pig. 7 in 3 ).
723
With the change in the wave number growing for the numerator and the denominator the qualitatively correct approach to the modification of ~ yields good ag-reement with experiment. Note that for the energy range under consideration R.,.. - 5 a. u. therefore for the ratio G(o ~a)/5(! -"'3) given in the table the condition of applicability of (4) is not satisfied, and good quantitative agreement is just impossible. Here it is essential to note only that the direction of modification is qualitatively true. Violation of the limiting relations ( ~ or modified ,1! ) corresponds to the range of intermediate energies where such Violation may be a source of additional information for recovering the potential energy surface from experimental data.
OONCLUSIONS
The above discussion indicates the influence of initial molecular states and intramolecular motions on the HES specific features in the measurements of state-to-state transitions or measurements without selection.
REFERENCES 1. Leonas V .B., Rodionov I.D. Preprint IKI AN SSSR,
698: 1 (!982) - In Russian. 2. Leonas V.B., Rodionov I.D. Preprint IKI AN SSSR,
70~& I (!982) - In Russian. ~. Itoh Y., Kobsyashi N., Kaneko Y., J.Phys. Atom.
Molec. Phys. !4: 679 {I98I). 4. Varshalovich D.A., UOscalev N.A., Khersonsky V.K.
Quantum theory of angular momentum. Moscow, "Nauka" (I9?5) - In Russian.
5· Kalinin A.P., Leonas V.B., Khromov V.N., JETP Pisma, 26: 65 (I9??) - In Russian.
6. Kalinin A.P., Leonas V.B., DAN SSSR, 26!: 65 (I98I) - In Russian.
?. Zubkov B. V., Kalinin A.P., Leonas V .B., PTE, 4: 20! (I9??) - In Russian.
8. McAfce K.B. Jr. et al., J.Phys. B, I4 L:243 (I98I). 9. Takayan.ag:l. K., Geltman S., Ph;ys. Rev. A, !~8: IOO~
(!965). IO. Goldflam R., Kouri D.J., Greens., J.Chem. Phys.
67: 566I (!97?). II.Bhimamlra I., Chem. Phys. Lett., ?3: 328 (I980). I2. Khare V., J .Chem. Phys., 68: 46~I (I978). 13. Dickinson A.s., Oomput. Phys. Commun., I?: 5! (!979). I4. Bitz D.E., Kouri D.J., Chem. Phys., 4?: I95 (!980).
724
DIFFERENTIAL CROSS SECTIONS FOR ION-PAIR FORMATION
WITH SELECTION OF THE EXIT CHANNEL
M.J.P. Maneira, A.J.F. Praxedes*, and A.M.C. Moutinho**
ABSTRACT
Centro de Fisica Molecular das Universidades de Lisboa (INIC) - Complexo I Av. Rovisco Pais - IST 1000 Lisboa, Portugal
Differential cross sections for ion-pair formation on K + I2 collisions with selection of exit channels were measured at laboratory energy of 30, 40 and 60 eV, These experiments were performed in a crossed molecular beam apparatus, An analyser measures the time of flight difference between K+ ions scattered with an angle e, and the corresponding negative ions (I- or Iz) which follow a much longer trajectory. Coincidence techniques are used to obtain the time correlation, This work supports experimentally the general trends, predicted by trajectory surface hopping (TSH) calculations 1 and measurements of energy loss, i.e., contributions due to the dissociative process occur at angles larger than the non dissociative one.
INTRODUCTION
In inelastic collisions between neutral atoms and molecules many processes are possible depending on collision energy, species and initial states involved. Some of these processes leading to charged particle production are particularly important. As a
*Departamento de Fisica do Institute Superior Tecnico (U.T.L.), 1000 Lisboa, Portugal.
**Departamento de Fisica e Ciencia dos Materiais da Faculdade de Ciencias e Tecnologia (U.N.L.), Quinta da Torre, 2825 Monte da Caparica, Portugal.
725
matter of fact, ion-pair formation, collisional ionization, associative ionization, collision induced ionic dissociation and chemi-ionization with rearrangement, can occur through several exit channels. These channels can play important roles as intermediate processes in the bulk phenomena related to flames, plasmas, catalysis, chemical reactions 2 • 3, etc, Cross sections measurements on these processes had been performed by several groups, contributing to give insight into the collision dynamics and to the implementation of collision models.
Ion-pair formation in collisions of alkali atoms (M) with molecules (XY),
+ ( -M + XY - M + XY) - (1)
has been subjected to extensive experimental and theoretical studies 4 • 5•6 , in the past fifteen years, Totalcross section measurements, with and without mass selection of the negative ions, lead to the determination of adiabatic electron affinity, dissociation energy of the negative parent ion and to estimations of the potential coupling, Measurements of the yield per exit channels were also performed7•8, Ion count rates of M+ for different collision energies E were measured as a function of the scattering angle e yielding single differential cross sections 9• In other experiments the energy loss ~E of the process is additionally measured yielding double differential cross sections 1• 1D. Comparison between experimental and calculated cross sections, assuming several models leads to their improvement. The determination of vertical electron affinities, and potential parameters as vibrational frequency w, and equilibrium distance of negative parent ion, has been also possible 1G.
In this work measurements of differential cross section per exit channel are presented for
(2)
Total cross sections for the same collisions were measured by Baede near threshold 11 and Rubbers at higher energies 8• From that work I- fraction is known, In the same system differential cross section measurements with energy analysis have been reported some eV above the threshold 1 ,Predictions from trajectory surface hopping calculations have been also published for this system from 10-100 ev13 •
726
EXPERIMENTAL SET UP
The coincidence experiment was performed in a crossed beam apparatus. A fast a1ka1i beam is produced by a charge exchange source, energy being adjustab1e between 10 and 100 eV. The secondary beam of iodine is obtained by effusion from a mu1ticapi11ary array at room temperature.
A sma11 fie1d is estab1ished across the co11ision region. The positive ions (K+), which keep a 1arge kinetic energy, are s1ight1y deflected downwards ( rp- 15°) and the negative ions (I2 and I-) are extracted upwards and are discriminated by time of f1ight (TOF), To distinguish between a K+ coming from dissociative and non dissociative co11isions, time corre1ation between pu1ses coming from positive ion and negative ion detector is obtained using a coincidence spectrometer.
In fig. 1 the spectra of TOF differences obtained for K + I2 show two peaks which correspond to the output channe1s K+ + I- and K+ + I-2•
Fig. 1.
500
400
(I)
0 0 "'t - 300 (/)
~ 2 ::> 0 u 200
•• • 100~----~----~----~----~----~
0 5 10 15 20 25
Ll t ( ,us)
Spectrum of time of f1ight differences between K+ and the corresponding negative ions (I- or Iz), Obtained at 40 eV co11ision energy (1ab.) in the forward direction. Origin of time sca1e was arbitrari1y chosen.
727
The positive ion detector which rotates in discrete steps araund the collision center, permits to obtain the count rates for each exit channel. This is performed after the selection of the peak corresponding to that channel.
A detailed description of this experiment will be published elsewhere.
EXPERIMENTAL RESULTS
The count rates were obtained in both sides of the forward direction, so that symmetry test was always performed. Several scans are added in the same run so that any beam intensity, vacuum or electronic fluctuations are averaged. Noise which is subtracted is taken as the average count rate at very large angles. Both sides
15
10
Vl .... c: :l
.ri ... (1)
<:!:> 5 c: Vl
<:!:> -b
0~--~----,-----~--~----,---~ 0 100 200 300
EO(eV x degree)
Fig. 2. Single differential cross section obtained at Elab. = 20 eV with planar detection configuration. Symbols represent data points, curve follows moving average of 3. Horizontal bar is the resolution ßT, at T ~ 150 eV x degree.
728
15
10 •
1/l ... c: :J
..0 .... (II
Q:) 5 c: 1/l
Q:)
c
0~----~--~----~-----r----~--~
0 100 200 300
E 8 ( eV x degree)
Fig. 3. Single differential cross section obtained with coincidence configuration at Elab. = 40 eV. Symbols and curve and horizontal bar like in fig. 2.
of angular distributions are added and multiplied by sin 8 to get polar differential cross sections a (9) sin 8 as a function of T = ES. Finally correction of collision volume variation with detection angle is made. The resolution in the T axis, ßT, controls the quality with which relevant structure is seen in each curve, and it is given by the expression
M = 9ßE + EM (3)
Preliminary results of the differential cross section were obtained with the detector in plane and without coincidences, The results at 20 eV are shown in fig. 2 and reproduce correctly the results obtained by Aten et al. 9 in the same range of energies.
In the conditions of the coincidence experiment and at lab. energy of 40 eV the single differential cross section, that is,
729
15
6
10
Vl .... c: :::J
..ci ~
~
<:r:> 5 c: Vl
<:r:>
0
0+---~----,-----r---~----~--~
0 100 200 300
E &(eV x degree J
Fig. 4. Differential cross sections with selection of exit channels (coincidence configuration) obtained at Elab = 40 eV. (*) and (--) refer to dissociative channel, (~) änd (----) refer to non dissociative one, Horizontal bar like in fig. 2.
without exit channel discrimination, is shown in fig. 3. The differential cross sections with selection of each exit channel are presented in fig. 4. Results were normalized to the maxima to show relevant behaviour. The effects of deflection and convolution are important but the main features are not significantly affected.
Similar results were also obtained at 30 and 60 eV.
DISGUSSION
Experimental results are discussed in terms of the model developped by J. Los and co- workers. This model describes the ion pair formation in atom-molecule collisions assuming that diabatic
730
e1ectronic transitions can occur at the seam of the adiabatic potential surfaces, If a diabatic transition occurs at the first crossing the interaction is cova1ent but at the second crossing an electron jump may occur, leading to the formation of an ion pair. In such trajectories the impulse resulting from the forces acting on the fast alkali atom is small and litt1e deflection e is obtained. If the adiabatic transition occurs at the first crossing the interaction changes to ionic and bigger deflection is expected since Coulombic forces are acting. These two types of trajectories lead to ion pair formation and are the so-called covalent and ionic trajectories. They are responsible for the covalent and ionic rainbow structure in the differential cross section, however t~e ionic rainbow is rather enhanced due to the bond Stretching of the negative molecular ion. This behaviour is predicted whenever the negative ion cannot be considered frozen during the collision time which is the case at the energies of the experimental results here reported (T 11 ~ 0.3 Tvib)• These structural features are clear1y observedc~n fig. 2. The covalent structure appears for T < 50 eV degree. At large angles the main contributions are due to the ionic trajectories and two peaks are observed. The peak at about 120 eV degree is attributed to the bond-stretching and the other at 175 to dissociative trajectories.
The main features are not completely hidden either in the measurements shown in fig. 3 or 4, although the structure is fading out. However in the differential cross sections per exit channel, i.e., for Iz and r-, an increase of the percentage of r- is observed for T > 150 eV. This is particularly evidenced by the normalized curves shown in fig. 4.
The general configuration of the polar cross sections does not differ substantially for the two channels. One may conclude that the energy transfer is almost independent of the impact parameter. Only for the ionic trajectoriessome differences were found at large angles.
These direct measurements support the general trends predicted by Eversl3 using trajectory surface hopping calculations. These calculations together with differential cross sections with energy analysis of the product ions 1 strongly suggest that K+ ions coming out from dissociative collisions would suffer higher energy loss and would be scattered at larger angles than those coming from non-dissociative collision. As a matter of fact the differentiation (at T > 150 eV) between the two normalized differential cross sections appears as predicted by the T.S.H. calculations. Indeed from such calculations the ionic rainbow peak would have an enhanced contribution of dissociative atomic ions r-. From these measurements, the r- fraction is at least 29% against a value of 25% measured at forward direction (fig. 1).
731
ACKNOWLEDGMENT
This work is partly supported by Research Contract 138.79.74/JNICT.
We wish to express our appreciation for the collaboration of J.M.C. Louren~o and P.C.C. Silva in mounting the experimental set up.
REFERENCES
1. J. A. Aten, C. W. A. Evers, A. E. de Vries and J. Los, Energy transfer and differential scattering for ion pair formation in Na, K, Cs + I2 collisions, Chem.Phys. 23:125 (1977).
2. S. Wexler, E. K. Parks, Molecular beam studies of collisional ionization and ion-pair formation, Ann. Rev. Phys. Chem., 30:179 (1979).
3. M. J. P. Maneira, A. M. C. Moutinho, Discrimina~ao de iÖes negatives pelo metodo das coincidencias na forma~ao de pares de iÖes em colisoes entre particulas neutras, Publica~ao
4. A.
5. J.
6. K.
7. A.
8. M.
9. J.
10. K.
Interna do Centrode Fisica Molecular, Lisboa (1980). P. M. Baede, Charge transfer between neutrals at hyperthermal energies, Adv. Chem. Phys., 30:463 (1975). Los, A. W. Kleyn, Ion-pair formation, in: "Alkali Halide Vapeurs", P. Davidovits and D. McFaddei.l;' ed,, Acad. Press Inc., New York (1979). Lacmann, Collisional ionization, in: "Potential Energy Surfaces", K. P. Lawley, ed., J. Wiley, New York (1980). M. C. Moutinho, J. A. Aten and J. Los, Temperature dependence of the total cross section for chemi-ionization in a1kali-ha1ögen co11isions, Physica 52:471 (1971). M. Rubbers, A. W. Kleyn and J. Los, Ion-pair formation in alkali-halogen collisions at high velocities, Chem.Phys. 17:303 {1976). A. Aten, G. E. H. Lanting, J. Los, The energy dependence of differential cross sections for ion-pair formation in Na, K, Cs + I2 collisions, Chem.Phys. 19:241 (1977). Lacmann, M. J. P. Maneira, U. Weigmann, A. M. C. Moutinho, Total and double differential cross sections of ion pair formation in collisions of K-atoms with SnCl4 and CCl4, submitted to J.Chem.Phys ••
11. A. P. M. Baede, Thesis, Amsterdam (1972). 12. M. J. P. Maneira, A. J. F. Praxedes, A. M. C. Moutinho,
Apparatus for measurements of differential cross sections with selection of the exit channel, to be published.
13. C. Evers, Trajectory surface hopping study of ionizing collisions between Na,K and, Cs + I 2 in the energy range of 10-100 eV, Chem.Phys., 21:355 (1977).
732
LOW-TEMPERATURE VISCOSITY CROSS SECTIONS MEASURED IN A SUPERSONIC
ARGON BEAM II
ABSTRACT
P. W. Othmer and E. L. Knuth
Chemical, Nuclear, and Thermal Engineering Department School of Engineering and Applied Science UCLA Los Angeles, CA 90024
In an earlier study, 1 values of the thermal-conductivity (viscosity) cross section for argon deduced from velocity distributions measured near the centerline of a free jet deviated unexpectedly from the temperature dependence predicted for low temperatures. Subsequently, the molecular-beam sampling system and data-reduction procedures used in that study were upgraded and the measurements were repeated. Cross-section values were obtained at temperatures more than an.order of magnitude below the normal condensation temperature of argon. The temperature dependence of the deduced effective hard-sphere diameter is in good agreement with the predicted dependence, but the magnitudes are about 10 percent less then the predicted values.
INTRODUCTION
The feasibility of determining values of thermal-conductivity (or viscosity) cross sections from velocity distributions near the centerline of a free jet measured by molecular-beam techniques has been reported in an earlier publication.l This approach has the advantage that, as a consequence of the low density and the nonequilibrium state in the free jet, cross-section values can be obtained at temperatures more than an order of magnitude below the normal condensation temperature of the gas. However, in the earlier study, deduced values of the cross section for argon
733
deviated unexpectedly from the temperature dependence predicted for low temperatures. In the study reported here, the experimental facilities and data-reduction procedures were upgraded significantly, the earlier measurements were repeated, and improved values of the argon cross section were deduced.
APPARATUS
The experimental system used in this study is depicted schematically in Fig. 1. It consists of a source chamber, which houses the Stagnation chamber (movable parallel to the beam centerline) and provides the volume into which the free jet expands, and a collimating-detection chamber, which houses the beam chopper and the beam detector (movable transverse to the beam centerline). The skimmer, located on the wall separating the two forementioned chambers, functioned as a sampling probe. The axial location of the Stagnation chamber was set and measured using a commercial micrometer head. The stagnation pressure was measured using a diaphragm-type pressure gauge. The transverse location of the beam detector (electron-bombardment exciter, ion deflector, and metastable-excited-particle detector) was measured using a scale mounted inside the collimating-detection chamber. In order to improve the signal/noise ratio, the output signals were averaged using a Fabri-Tek Model 1062 digital signal averager.
The most important refinements in the system relative to that used in the earlier studies are (1) the replacement of the ionization detector with the particle-excitation detector (improving the signal/noise ratio), (2) the replacement of the analog signal averager with the digital signal averager (providing a more stable baseline and improving the time resolution per data channel), (3) the replacement of the diffusion pump for the collimating-detection chamber with a higher-speed pump and the addition of a liquidnitragen baffle (reducing the background pressure and the oil partial pressure in the chamber), and (4) the addition of a means for recording the contents of the signal averager directly on IBM cards (facilitating the handling of extensive data sets).
PROCEDURES
With the detector located on the beam centerline and for a given stagnation pressure, the free jet was sampled at 6 axial locations. Then this procedure was repeated at a total of 6 stagnation pressures. Finally from 3 to 6 measurements were made off the beam centerline for each of the 36 combinations of axial location and stagnation pressure. The data consisted of time-offlight signals which contain information on both (1) the velocity distribution in the flight direction and (2) the local number density.
The data were analyzed using an ellipsoidal distribution
734
SOURCE CHAMBER
VARIABLE
PRESSURE
FIRST DEGREE OF FREEDO~l OF EXPERH!ENTAL APPARATUS (VARIABLE P 0 )
COLLIMATING - DETECTION CHAMBER
t::::---._ .l:::f. = TRIGGER SIGNAL VARI"DLE
(Zero Time Pulse no Source) POSITION
METASTABLE MOLECULAR BEAM
DETECTOR \
L'-=-::..!....:'-'-'---'R""P_11-'-') ~-+ -- ---------- () ---
3-D CONICAL UNHEATED SKiill!ER
SECOND DEGREE OF FREEDOM OF EXPERIMENTAL APPARATUS
(VARIABLE x/D)
L. THIRD DEGREE OF FREEDOH OF EXPERHlENTAL
.APPARATUS
(VARIABLE DETECTOR PosiTION, e)
Fig. 1. Schematic Diagram of Principal Components of Molecular-Beam System Used in Velocity-Distribution Measurements.
... .... z 0 ;::: ... 2.4 a:: z Q ..... u z 2.2 iZ
2.0.
1.8 0 .I .2
(TJ. )(T~)z 3--2 -r.. r.. .
12(l-1)1
Tu
tonh 1~ ../1- T•
Tu
(li )I (T.l. ) +3- -4- +4 r.. r. .
3 A ~ ~ 7 .8 TEMPERATURE RATIO, T,IT,1
.9 1.0
Fig. 2. Relaxation Function Appearing in Equation Used in Deducing Effective Hard-Sphere Diameter.
735
function. In terms of the time of flight, t, from the chopper to the detector and the polar angle, 9, between the detector position and the free-jet centerline, this distribution function may be written
9 1/2
( 2:kr11 ) {2~kr.L)
[ m (Lcd u-)2
m ( Lcd 9) 2] x exp - -- cos 9 - - -- -- sin 2 k r" t 2kT..L t
where nd(t,9) is the number density at the detector, ns is·the number density at the skimmer entrance (the local number density in the free jet), Ag is the area of the skimmer orifice, Lsd is the distance from the ski.mmer to the detector, a is the duration of exposure of a point in the beam by a chopper slit, Lcd is the distance from the chopper to the detector, m is the mass per particle, k is Boltzmann's constant, T"is the temperature characterizing the distribution parallel to the jet centerline, T.J.. is the temperature characterizing the distribution perpendicular to the jet centerline, and ü is the hydrodynamic velocity. The local value of the collision cross section was deduced from the measured variation of this distribution function with axial distance in the free jet, which variation results from particle collisions. More specifically the effective hard-sphere diameter, cr, was calculated usingl
2 1/2 2 2 _ B_ (yM ) (yM11 - 3) (- dTH/dr) - 4 TJ./r
cr - --"-I ~ 2 4ns (T11- T..L) (yM/1- 1)
2 -2 where yM11 = mu /kT11 , r is distance along the free-j et axis measured from the source orifice, and N/I is a function of T~/T6 which varies from 15/8 to 3 as ~/T11 varies from 1 to 0 (cf. Fig. 2). For each stagnation pressure and axial distance, the axial parameters Ü and TH were obtained by fitting the model distribution function with 9 = 0 to the time-of-flight measurement made on the beam centerline. A typical measured time-of-flight signal is shown in Fig. 3; deduced values of Tu are indicated as functions of normalized axial distance in Fig. 4. The transverse parameter ~ was obtained then by fitting the angular dependence of this distribution func.tion to the transverse beam density distribution provided by the off-centerline measurements·. (In an alternative procedure, the three parameters ü, T11 and T.L were obtained simultaneously by fittin~ the model distribution function to timeof-flight measurements made at several different transverse detector locations. No significant improvement over deducing values of ü and Tn only from measurements on the beam centerline was realized.) The value of ü was obtained by dividing the value of nsÜ obtained
736
-...1 w
-...1
~ . :; .
. : : .. :
.: ! ; :
':!!I:::
'I:: !
8 •·I
I··-
I····
.:
,,.;_ -~
~-~
••• 4::
I :.:.
P, ~
: ! ; "I~
• ; ! ~ ~!
~ :: :
1 : :
i: ! ::
: : I ! : l i f
··-
-:-::fi
,:p ;;;
u· \'
~:·
" l•l
tjjj I
T
o
Ull~: ~
~ ;;.'~ l'
i! ~ ~.
:~ !;
~I ~:~i
3
00
l(
7.
. :\
;;lj
.:. ·
' :.tj
t. I
.. 1 'II'
I ·:·
-l~ :~i~
T* T
l ;1;: :
;:' ·:··
I ~
--j
from t.he pressure rise in the ~ollimating-detection chamber with the free jet "on" by the value of u obtained as described above. The value of r is the nozzle-skimmer distance measured using the micrometer.
The procedure used here differs slightly from that used2 previously in that previously the dimensionlese parameter yM0 was obtained by fitting the normalized distribution function with 6 = 0 to the normalized centerline time-of-flight measurements, i.e. by fitting
nd (t,O) ( Lcd \4 [ y.~ (Lcd }2] nd (Lei u, 0) = u t J exp - T 'ü t - 1
to the measured centerline distribution; then the hydrodynamic velocity ü was obtained from the conservation-of-energy equation
Ü [1 + (3/y~1 ) + (2/y~)J = 5 kT0 /m 1 where T0 is the temperature in the stagnation chamber. In
principle, the two procedures are equivalent.
RESULTS
The effective hard-sphere diameters deduced for argon from the velocity-distribution measurements are compared in Fig. 5 with the values reported earlierl and with the values obtained by Guggenheim2 from viscosity measurements. Note that the lower end of the temperature range for which values have been deduced here is more than an order of magnitude below the normal condensation temperature (87.5 K) for argon. A smooth curve can be drawn through both the present results and Guggenheim's results. The present results are considered to supercede the earlier ones.
The measured effective hard-sphere diameters are compared in Fig. 6 with the predictions of Hirschfelder, Curtiss and Bird3 and with the lowest-temperature Guggenheim result.2 The abscissa is the reduced temperature T* = kT/e, where e/k = 119.8 K. It is seen that the observed temperature dependence is in good agreement with the predicted dependence, but that the measured values are about 10 percent less than the predicted values.
It is concluded that the unexpected temperature dependence of the argon cross section deduced in the earlier study1 was an artifact of the first-generation system used in that study, and that the system refinements made here facilitate relatively reliable determinations of thermal conductivity (viscosity) cross sections for temperatures as low as an order of magnitude below the normal condensation temperature of the gas.
738
8
.';i b
ri 6 UJ 1-UJ :::;: ~
ö ...J 4 ~ z 0 j:: u UJ V)
I
tl\ V) 0 ():: u
4
.... __
·---- 0 ·--o-. 0
0
6 8 10 20
X PRESENT MEASUREMENTS 0 PREVIOUS ME~SUREMENTS 0 GUGr.ENHEIM2
40 60 BO 100 200 PARALLEL TEMPERATURE, T11 ( Kl
400 600 800
Fig. 5. Effective Hard-Sphere Diameter ms. Parallel Temperature Deduced from Velocity-Distribution Measurements in Argon Free Jets.
E
" "' 0
0
"' lU flU :!: « c UJ
"' UJ :c 0.
"' ' 0
"' « :c w > fu w .... .... UJ
40
30
t/k • 119 .R K
20 -8
"Hs • 3. 4 x 10 cm
10
8 PREDICTE0 3
6
4
3
2
o.o4 o.o6 o.oa o. 1 0.2 0 . 3 o. 4 O.f 0.8 1.0
REDUCED TEMPERATURE. T*
Fig. 6. Effective Hard-Sphere Diameter ms. Reduced Temperature for Argon.
739
REFERENCES
1. E. L. Knuth and S.S. Fisher, "Low Temperature Viscosity Cross Sections Measured in a Supersonic Argon Beam", J. Chem. Phys. 48:1674-1684 (1968).
2. E. A. Guggenheim, "Elements of the Kinetic Theory of Gases", Pergamon Press, London (1960).
3. J. 0. Hirschfelder, C.F. Curtiss, and R.B. Bird, "Molecular Theory of Gases and Liquids", Jolm Wiley and Sons, New York (1954).
740
EXCITED OXYGEN IODINE KINETIC STUDIES
D. Pigache, D. David, J. Bonnet and G, Fournier
Office National d'Etudes et de Recherehes Aerospatiales 92320 Chatillon, France
ABSTRACT
When iodine is mixed to a low pressure excited oxygen flow,ground state iodine molecules are excited to the A ~'111u and B "1i o+J electronic states or dissociated into two ground state atoms. The excitation of some iodine atoms follows and laser action is possible on the atomic iodine transition at 1315 nm. Some doubts remain about the processes involved especially about the dissociation of molecular iodine. Our study of the iodine infrared spectrum leads to an unambiguous determination of the eight first vibrational levels of the A state, The analysis of the visible spectrum yields a new determination of the vibrational distribution for the B state. Thus we are able to rexamine and complete the previoustly accepted reactions scheme which explain the electronic excitation of molecular iodine. Finally some hypothesis are considered in order to explain the dissociation of molecular iodine.
INTRODUCTION
It has been known for some time1 that an intense yellow chemiluminescence is observed when iodine is added to a low pressure flow of oxygen containing singlet metastable oxygen molecules. The visible part of the spectrum _is due to the molecular iodine transition 6)'llo+v"";('~{ The transition I(2 P•t2.)-J('I·p,ll) at 1315 nm .is also observed. The oxygen and iodine energy levels which are involved in this phenomena are shown on fig. 1. The generally accepted reaction scheme is the following1•2 :
(1) 0:< ('Es) + I,_(x 'L;) ~ 02 (''29) ~ z.xl ( 2 P3fz)
(2) 02.l'2j) + 1-t (X 'Ij) ____,. o,_(3i;) +I,_ (A 3 '11 .v) (3) l2. ( A ''IT,u) + O:z.. ('Lls) ~ l2. (B "rro-+v) ... 02 el9-)
(4) ol ( 't.9) + 1 (2 P312) ~ ol ("Ig-) ... r (2· P .,~
741
0.,_ ( 'r;) is created initially by the energy pooling reaction
<s) ol. cllg) + o2 cb~)---. ol. ( 'rn T o2 e~;) and later by another energy pooling reaction
(6) ol ('ß~) +I l~P·t~ - Od'I;) +1 (l.'i'3t<-)
The possibility of laser action on the atomic iodine transition has also been suggested3, and verified4 o However this reaction scheme became recently questionable when it was discovered that the dissociation of iodine takes place even when the concentration of o2.er.;) is drastically reduced by adding water vapor5o It was also found that the rate of reaction (1) was much smaller than previously thought6 o Consequently reaction (1) is probably not the dominant iodine dissociation mechanism in the conditions of the 0~ v~9)-r atom chemical laser as in that case the gas flow contains water vaporo
The A "'11 1v state of iodine is not accessible to O..('~)o Thus, in order to explain the dissociation of molecular iodine by 02.-( 1 ~9) without involving 02 ('l.'J+), it is necessary to consider another excited state of iodine lying underA~~Iv 0 A second reaction involving 02 ('6s) could then lead to dissociationo This intermediate species could be vibrationally excited iodine, an excited molecular complex or the only electronic state lying und er A ~1-r; l.l , that is 1'\i 10 7
The purpose of this work is to reexamine the dissociation and fluorescence processes which take place in the excited oxygen-iodine systemo The results of a detailed analysis of the visible and infrared spectra are presented and discussedo
Fig. 1 - E/ectronic energy Ievels for oxygen and iodine. For 12 (A 3 1rtul: (1) W.G. Brown 10 , (2) D. DavkP.
742
"A-monitor
Pump
Monochromator t [ __j Observation ce/1 oj;-1& _________ __::::L._. I,
GeiN3 ~~==r 02- ~ u O r detector rig rmg
Synchronaus amplifier
Signaloutput
Micro-wave cavity
Fig. 2- Experimental set-up.
EXPERIMENTAL EQUIPMENT
The experimental set up is shown in Fig, 2. A low pressure oxygen flow is discharged in a microwave cavity coupled to a 2.45 GHz, 200 W generator, This results in partial dissociation and excitation of the oxygen flow. A mercury oxyde ring is formed gownstream the cavity in order to remove most of the oxygen atoms
Only the most metastable species 0 ll'A5) remains in the observation cell. However 0~ ( 1 ~'JT) is also present because of reaction (5). The 0;~.( 'D,) \ concentration is measured by •emission spectroscopy. It
./ is equal to 9 to 10 % of the total oxygen concentration at a pressure of 0,1 Torr.
The observation cell is a pyrex tube, 600 mm in length and 42mm in diameter, Typical pressure and flow velocity in the observation cell are 0.1 Torr and 20 m/s. A roots pump provides a base pressure of 10-3 Torr. Its nominal pumping speed is 120 m3/h. The pressure and the flowrate are measured by means of a capacitive pressure gauge MKS Baratron and a thermal flowmeter. The iodine vapor is introdu-ced on the axis, at the upstream end of the Observation cell. Iodine crystals are placed in a pyrex reservoir which can be heated so that the partial pressure of iodine in the observation cell can be adjusted between 0 and a few tens of mTorr.
The detection is made through a quartz window closing the cell extremity in front of the iodine injector. Accordingly the signal is integrated over the entire cell length. The light is analysed with a HRP Jobin-Yvon monochromator 588 mm in focal length opened at f/7 and fitted with a 1200 groves/mm grating for the visible and a 610 groves/mm for the infrared, The detectors are in the first case a photomultiplier RCA31034A whose. sensitivity extends to wavelength up to 900 nm and in the secend case a liquid nitrogen cooled germa·nium detector (Northcoast E0817). In order to improve the signal/ noise ratio for the infrared, the light flux is modulated at the
743
entrance of the monochromator and the detector signal is synchronously amplified. Finally the optical signal and the corresponding wavelength are recorded by means of a X-Y platter or digitally processed for subsequent computation.
THE A-X EMISSION OF IODINE (1100-1700 nm)
Without iodine, the only emission observed is due to the Od'A, ~~o)-01PI; o:o) transition at 1270 nm. When iodine is added to the excited oxygen flow, the strong atomic line of iodine l (~P 1(2) -:I_('l.P3fJ.) at 1315 nm appears. It is brighter than the o~ ('.1,)- 02PE3) transition because its radiative lifetime is
shorter.
In addition, a very contrasted band system takes place on each side of the atomic iodine line and 02 ( 16j) -Oz (?·~5-J band , which can be a priori attributed to the A-X system of iodine (fig. 3).
A detaile4 analysis of this spectrum has been given elsewhere9, The main result of this analysis is that eight vibrational levels of the A ~-rr, v state of iodine, which lie clearly below those described by W.G. Brown 10 are populated (see Fig. 1)
These levels are found to be identical to the eight first vibrational levels reported by Gerstenkorn and col. 12, Äs the process of vibrational relaxation allows to populate the ground vibrational level of the A state, this level is now surely identified. This also confirms that the minim~m of the A state potential curve is effectively lower than accepted before 11,12, In addition to the
a b
X fwn)
Fig. 3 - Infrared spectrum at p = 35m Torr, T = 300 K. a) 02 (lf19) 4 02 f3 I:,g)
b) I fP112)-+ I (2 P3, 2 ).
744
P02 = 0.04 Torr r1)
'
600 700
(1) 02 (I 'L)-+ 0 2 (3 'Lv= 0)
r2J o2 r' 'LJ -+ o2 r 'i:, v = 1 J
A. (nm)
900
Fig. 4 - Visible spectrum at : p = 0.4 and p = 0.04 Torr, T = 3000 K.
A-X emission, the observed infra-red spectra contains a few bands between 900 and 1100 nm which can be assigned to B-X and a few unidentified bands of minor intensity which cannot be attributed to the A-X transition.
THE B-X EMISSION OF IODINE (500 - 900 nm)
Without iodine and at low oxygen pressure (P 02 < 1 Torr), the visible spectrum is rather poor; above all the emission of Ot ('.l:~t)- o.('~~-) at 760 nm. is observed, then the weak dimolar emission of ~X 02('6:>)+2JC0.t{'~9-}at 635 nml. When iodine is added to the oxygen flow, a significant increase of the 02 ( i~)- 0?. P!';) emission is observed because of reaction (6). In addition, a bright yellow emission takes place all along the cell. This emission is due to the B-X system of iodine; it starts at 500 mn and extends to the near infrared (fig. 4). At the total pressure of 0.4 Torr this spectrum has a single maximum at A = 560 mn, but when the the pressure is decreased to 35 m Torr a second maximum appears near 520 mn. This observation incites us to study the vibrational distribution of the B 3Tio+u state of iodine.
A wavelength greater than 560nm, the spectrum of the B-X system shows several peaks which are due to the numerous v' - v" bands; however these bands are far from being resolved since the width of these bands is larger than the shift between two successives bands. Consequently, in order to determine the vibrational distr·ibution of the upper state, it is rather advisable to reconstruct the experimental spectrum by a computation. The spectroscopic constants for the B and X states of iodine allow to determine the position of each transition v' - v". The Franck-Condon factors, recently computed for 0 ~ v' ~ 49 and 0 ~ v" ~ 55 16, lead to the knowledge of the transition probability from a v' level of B to each v" level of X. Then, assuming the thermodynamic equilibrium (T = 300 K) between all the rotation sub-levels the profile of each band can be calculated. Taking into account the detection sensitivity and the apparatus function of the monochromator, several spectra for p = 0.04 Torr to 0.4 Torr have been analysed.
745
The result is remarkable : at low pressure (40 mTorr), the distribution exhibits two peaks, the first near v' = 18, the second near v' = 40 (fig. 5). At the total pressure of 0.4 Torr, the second peak tends to disappear.
DISGUSSION
Considering the reactional scheme outlined in the introduction, it seems normal to observe the A-X system. However, the transition moment between the A and X states of iodine is certainly weaker than between the B and X states (may be by a factor of 100 if one refers to the bromine case 17. So the spectroscopic study of this system in emission is rather difficult and this may explain why it has never been made before.
According to reactions (1) and (2), the Q.,_('r;)molecules can either dissociate iodine or excite it into the A state. If the oxygen molecules relax to ~2.9- II= o , the energy available is 680 cm-1 above the first dissociation limit of iodine and this should favor dissociation via the ''11tv dissociative state. If the oxygen molecules are left in the first vibrational level instead, the energy. provided by Ol.(i}:r) is then equal to 11564.4 cm-1, that is 875 cm-1below the dissociation energy (so dissociation does not occur) and in good coincidence with the energy of 0 2 (A ~rr.v V :7)that is 11409cml 9, 12.
As no vibrational levels of the Astate with v'> 7 are observed although the detection system which extends from 400 to 1800 nm should permit it, it can be concluded that only the eight lowest vibrational levels of the A state are excited and this is a direct confirmation of reaction (2).
Considering now the B state vibrational distributions, reaction (3) can be rewritten :
(3 ') ! 2. ( /-\ 1 Ti1 0 V-:::7) -r 02 l'/.\~ .,r~c)~l2 (B"'n.::,..v ·r) ~O:~.('I.; .,//:.~)
0 10 20 30 40 v' 0 10 20 30 40 v'
a) p = 0.4 Torr b) p = 0.04 Torr
Fig. 5- Vibrational distribution of / 2 (83 1r du).
746
In this reaction, the internal energy of iodine is increased to 19291 cm-1 if the oxygen relaxes to ~r; v:o and to 17735 cm-1 if it relaxes to the 1st vibrational level.
As the two peaks of the observed vibrational distribution of !t..(ß''{\"0+") correspond to energies equal to 19295 cm-1 for v=40 and 17708 cm-1 for v = 18 (see fig,l) in agreement with the internal energy of the iodine molecules after the reaction 3', there is no doubt that reaction 3' occurs through the two possible channels. This observation is a direct confirmation of the reaction sequence 2 and 3.
Previously, Derwent and Thrush gave successively two different analysis. In the first one, they reported a fairly even distribution2; in the second one, they found a peak near V' = 2018. However, the peak near V'= 40 does not appear, and that is in agreement with their observed spectrum which does not exhibit a second maximum near 520 nm. This diversified behaviour of the B-state vibrational distribution can be explained by the collisional relaxation, the distribution at p = 0,04 Torr giving the real excitation processes of the A and B state of iodine.
CONCLUSION
Our observation and analysis of the A-X and B-X emission bands of molecular iodine confirm with additional informations the reaction scheme originally proposed by Arnoldl, Derwent2 and co-workers as far as the excitation of the A and B states of iodine is concerned,
The main mechanism of iodine dissociation remains unknown and the necessary intermediate species has not been identified, However our observation of the A-X band in emission makes possible an unambiguous determination of the energy level of the A state. As the energy difference between the A(?.'if,v) and the ltr2.v states of iodine has been ca1cu1ated by Mu11iken7 to be about 1000 cm-1 the energy level of 3·T12.v should be about 9900 cm-1, 2000 cm above 02. ( '11,2) making excitation transfer from 0;~ ( 161! impossible, Thus, J.2 {'1l2.;~)would appear as an unlikely intermediate species.
As some emission bands remain unidentified, it is believed that an analysis of the infrared spectra with higher resolution and sensitivity than in the present study and with an extension towards longer wavelengths, could contribute further to the understanding of the excited oxygen ·-iodine system.
REFERENCES
1. S.J. Arnold, N. Finlayson, and E.A. Ogryzlo, Some novel energy
747
pooling processes involving O;r..('ö,) , J. of Chem. Phys., 44 : 2529 (1966).
2. R.G. Derwent, D.R. Kearns, and B.A. Thrush, The excitation of iodina by singlet molecular oxygen, Chem. Phys. lett., 6 : 115 (1970).
3. R.G. Derwent and B.A. Thrush, The radiative lifetime of the metastable iodine atom !.(5 1 PIJ7..) , Chem. Phys. lett. 9 : 591 (1971).
4. W.E. McDermott, N.R. Pchelkin, D.J. Benard, and R.R. Bousek, An electronic transition chemical laser, Appl. Phys. lett., 32 : 469 (1978).
5. R.F. Heidner 111, C.E. Gardner, T.M. El-Sayed, and G.I. Segal, Dissociation of I2 in 02 ('11)-! atom transfer laser, Chem. Phys. lett., 81 : 142 (1981).
6. D.F. Muller, R.H. Young, P.L. Houston, and J.R. Wiesenfeld, Direct observation of 12 collisional dissociation by 02(b'~~J Appl. Phys. lett., 38 : 404 (1981). ·
7. R.S. Mulliken, 1odine revisited, J. of Chem. Phys., 55:288 (1971), 8. S.H. Whitlow and F.D. Findlay, Single and double electronic
transitions in molecular oxygen, Can. J. Chem., 45 : 2087 (1967). 9. D. David, Analysis of the A-X system of iodine from its infrared
emission spectrum, Chem. Phys. lett., accepted for publication. 10. W.G. Brown, An infrared absorption band system of iodine, Phys. ~ 38 : 1187 (1931).
11. R.A. Ashby and C.W. Johnson, The vibrational numbering and improved constants of the A3~1u state of I2, J. of Molec. Spectros., 84 : 81 __ (1980 l .
12. S. Gerstenkorn, P. Luc, and J. Verges, On the ground vibrational level of the A '\'tf11, state of the 1.1 molecule, J. Phys. B
13.
14.
15.
16. 17.
18.
19.
748
At. Mol. Phys., 14 : L193 (1981)• J. Wei and J. Tellinghuisen, Parameterizing diatornie spectra "best" spectroscopic constants for the 1'1. B.:. >C transition, J. of Molec. Spectros., 50 : 317 (1974). s. Gerstenkorn, P, Luc, and J. Sinzelle, Study of the iodine absorption Spectrum by means of Fourier spectroscopy in the region 12600-14000 cm-1, J, Physigue, 41 : 1419 (1980). D.H. Rank and B.S. Rao, Molecular constants of the ground state of !1, J. of Molec. Spectros., 13 : 34 (1964). K.K. Yee, unpublished results. M.A.A. Clyne, J.A. Coxon, anq H.W. Cruse, Electronic energy transfer to dissociated "bromine by 02. I z; I 1A.9 Chem. Phys. lett., 6 : 57 (1970). R.G. Derwent and B.A.Thrush, Excitation of iodine by singlet molecular oxygen, Part 1 : Mechanism of the 17.. chemilumines-cence, J. Chem. Soc. Faraday Trans. 11, 68 : 720 (1972), Part 2 : Kinetics of the excitation of the iodine atoms, Discussion Faraday Soc., 53 : 162 (1972). J. Tellinghuisen, Resolution of the visible infrared absorption spectrum of L~ into three contributing transitions, J. of Chem. Phys., 58 : 2821 (1973).
DEI'ERMINATIOO OF ANTISYMMETRie M)J)E ENERGY OF co2 IN.n:x::TED
IN'ID A SlJPERS(N.[C NI'I'R(X;EN FJ..fJil
V.A. Volkov1 A.P. Zuev1 N.N. Ostroukhovl and B.K. Tkachenko
M:>sca.v Physico-Technica1 Institute Do1goprudny1 141700 1 USSR
INTROIJlCI'IOO
The mixing of co2 into a supersonic flow of vibrated-exci ted nitrogen for increasing efficiency and specific ch~2cteristics of gasdynamic 1asers (GDL) is wide1y used at present 1 • The main characterization of a 1aser media is i ts energy reserve 1 the detennination of which opens a nurober of advantages as canpared wi th the tradi tiona1 detennination of co2 vibrationa1 terrperature. First i t is the energy reserve that characterizes the system potentia1ities; second 1 because of the streng dependence of the energy on the vibrationa1 terrperature and detennination of the energy reserve wi th the he1p of measured temperature meanings may 1ead to significant errors.
In spite of a big nurober of various spectroscopica1 methods for deter.mination of en~~ reserve and of vibrationa1 temperature of the rro1ecular gases 1 for 1ow pressures (P ~ 0. I abn) up to now the sirrp1est method1 based on the thick optica1 1ayer rrode 1 was not used. In the present YJOrk on the base of ca1culation of the N2+ :+C02 mixtures absorption coefficient in the rotationa1-vibrationai banCl of ).. = 4. 3 JAID1 the spectra1 interva1s 1 in which at the conditions c1ose to the ones realizab1e in the GDL resonators the emittance of the gas is c1ose to the unity were detennined. As it fo1-1ows fran the known re1ations for radiation intensity of the abso-1ute1y b1ack body and for the energy of a harmonic osci11ator 1 these quantities depend on the temperature identica11y. It is known too 1
that the radiation in the 4.3 JIID band practically is canp1ete1y caused by vibrationa1-rotationa1 transitions within the ~ 3 rrode of co2 • Consequent1y under the asstlllption of the Boltzrrann distribution of rro1ecules on vibrationa1 1eve1s 1 the intensi ty of eigen radiation of the 1aser acti ve media in the detennined spectra1 interva1s is
749
proportional to the specific energy reserve of the i 3 lOOde. Thus to detennine the energy reserve of ~ 3 lOOde it is sufficiently to measure the radiation intensity in the rrentioned spectral interval and IlEk.e the graduation of the :rreasuring systen.
Following the proposed me~ and at the installation, which is analogous to that described in , the investigation of the process of mixing of co2 injected into supersonic flow of vibrated-excited nitrogen and its mixtures was carried out.
CHOICE OF SPECI'RAL INTERVAL FOR MFASUREMENI'S
As it is known, the average absorption coefficients of the gases are low at the pressures, which are characterized for GJL resonators, this is the consequence of the fact that the half width of rotational lines is Im.lch less than the distance between them. Nevertheless it is possible to find the spectral intervals, in which the lines are arranged sufficiently closely, which mi.ght at certain conditions result in sufficiently higher average absorption coefficient. For that purpose the crnputations of the af!r>rption coefficient in the spectral interval of i =2200 - 2400 cm -were carried out by direct sumnation of the sFate rotational lines contributions. ~ tr~sitions: 00°0 - 00 1, cxPl - 00°2, m 0 2 - 00°3, 01 o - 01 1, 02 o-0221 of the isotoce ~12o16 • 00°0 - ci:P1 00°1 - 00°2 01 1o - 0111
. .:rs.:l 2' o o ' o o ' r r of the 1.s~2c16o1 : 00 o- 00 1, 00 1 - 00 2, 01 o - 01 1 of the isotope c 0 0 were taken into account. T§e lines 'positions and their intensities were crnputed according to • For ccnp:>und 9 states the vibrational terrperatures -were introduced according to • The calculations for rotational m.nnbers were carried out up to j=90. The lines contour was assl.lired to be I.orentzian and dependence 2o lines half width on rotational quantum numbers was taken frcm for ni trogen as broadening carp::ment.
The calculations were carried out for T ih =lSOOK: Ttr =r t =SOOK and for sane pressure ~~s. It was foundv'Ef'ia.t in the s~al interval of i =2280 - 2380 cm 2-r the pressure 0.07 atrn the part of the sections, where K 0. Qi cm , constitutes 80% and in the interval of -l =2300 - 2380 cm - nore than 85%. The interval of 2350 -2380 an_1 is filled by lines_~st tightly and the intervals of 2280-2290 an and 2330 - 2340 cm the least tightly. The calculated absorption coefficients -were used to recei ve the estiroates of the emi ttance of mixtures of N2 + co2• In particular in Fig .1 the dependences of the lowest concentration of CO needed to ensure emittance of the order of o. 9 with respect to the firl~ss of radiating layer are shown for interval of 2280 - 2380 an • This interval was chosen frcm considerations of sufficient receiver sensitivity at srna.ll apertures and possibility of utilization of ~ined filters wi.th the half width of band transmission of 80 - 100 an .
The absorption in the boundary layer is the cause of the main
750
t.m 0 Olt 0.4 f. 1.6
Fig.l. co2 concentration in mi.xtures N + co2 1 required for emittance 0. 9 of radiating layer 1 having ihickness L. Tvili = 1500K; Trot = Ttr= 500K. 1 1 2,3- P=O.OS, 0.07 1 1.0 atm.
error in the rreasuremants of absolute intensi ties. The estirnate of the l:x:mndary layer influence can be made with the help of available caTq?Utatiorlf1of2C02 emittance in the 4.3 l.Jm band under different condi tions ' . The estirnate fran above will be obtained under the assumption that within the boundary layer the wide lines case takes place, which is corresponded to pressures greater than 1 atm. The estirnate fran below - in the strong lines case, when lines broadening is detennined by the Doppler rrechanism. At rcan temperature it corresponds to the pressure less than §F hundredth of one atm. Then for s:pectralinterval of 2280 - 2380 an and for co2 contents of 20% absorption in the boundary layer will be less tftan 2% if P. L within it is less than 0.05 atm-an in the assumption of slow lines and P·L 0.2 atm·an for strong lines. For the terrperature of 600K the above rreanings are equal correspondingly to 0.01 and 0.1 atm·cm. Certainly rnore exact estirnations of P-L rrrust be carried out for real experirrent condi tions.
EXPERIMENTAL INSI'ALLATION
The installation consists of the heat chamber 1 nozzle and rreasuremant system. For heating inert gases mi.xtures (ni trogen) impulsive electric dischargewas used, the mixtures of N2+H2o were obtained by burning of H2 in the air. Wi th the increasing of pressure in the chamber the diaphragm between the chamber and nozzle bursts out and the heated mi.xture through a wedge nozzle Sem width flows out into a vacuum container. The half-angle of the nozzle expansion was equal to 6. 3° 1 critical section height - 1 nm. The roan terrq;Jerature co2 fran a balloon was injected through holes with diarreter of 0.6 nm, drilled perpendicularly to forming surfaces of the nozzle at the 10 nm distance fran the critical section. The distance between neighbouring holes was 5 mm, the holes rows on opposi te nozzle walls were displaced one with respect to the other by a half-step.
The terrperature in the chamber was determined by strain gauge lli-412, which registered the increase of the pressure. Within the
751
l:imi ts of experi.Joontal error ( ~ 5%) i t coincided wi th the terrperature detennined by the spontaneous radiation intensity of the heated mixture in the spectral intervals of 'A = 4. 3 .±. o .1 r m.
In the lateral nozzle walls there were sapphire windows for IRradiation registration. The required spectral interval was selected with a narrow-band filter (interf~rnce-dispersion) with half width of the ti:ansmission band in 100 cm and with the center at 4.25_,..,m. By a condenser the radiationwas focused on a sensitive photocell elerent, the signal fran which was registered by oscillograph with narory S8-13.
'lb 'lmderstand the flow structure which is fonred in the supersonie nozzle at streng transversal injection, the visualization of the flow was nade. Flat, profiled nozzle with angle point of 200 mn width and with injection schere analogous to that of the wedge nozzle was used in these experi.Joonts. The cri tical section height was equal to 1 mn, injection holes dianeter 1 mn too, and the distance between neighbouring holes was equal to 1 cm. The triangular ledges of 0. 3 mn height were placed at the surfaces fonning the nozzle along the whole of its width and the cylindrical rrodel of 6 mn dianeter was placed at the nozzle axis (Fig. 2.1).
·, '~ ~ ~ , . -~--·
Fig. 2. Shadow and schliren - photographies of the nozzle and flow within it. 2.1 - nozzle with ledges and rrodel; 2.2, 2.3 -nain flow without injection; 2.2 - stationary, 2.3 - hot; 2. 4, 2. 5-bilateral injection into the nozzle wi thout nain flow, 2.4- n=60, 2.5- n=200; 2.6, 2.7- one-side and bilateral injection into the stationary flow, n=200; 2.8, 2.9 - injection into the hot flow in the nozzle with the ledges and the rrodel and without them, n=200.
752
The Mach m.mbers of the flow were estimated fran the gearetry of the shocks, arising at strearnlining of the ledges and of the m:::rlel.
Two flow regilres - stationary (cold) and :i.npllsive (hot) were investigated. In the first case the main flow was fonned by atrrospheric air. In the second one in the chamber the nitrogenwas heated by :i.npllsive electrical discharge; the stagnation pararooters in the initial nanent were equal correspondently: T = 2500K, P =8 atm. As a result of the analysis of the shadow and s2hliren - pRotographies it was es~~ished that depending on the rreaning of the pararooter n= P. /Poo , where P. - the pressure at the output of the injectionJhole, Poo - the pressure of the \ID.disturbed main stream, two flow regilres can be realized. At srnall maarrings of n (Fig. 2.4, 2.6) neighbouring jets of the injected gas don 1 t interact between themselves and resulting flow is analogaus to the supersonic fl~ with injection of a single jet (slow injection) . At n > n* ::: (S/r) , where r is a radius of the injection hole, the neighbouring jets care to contact and the flow structure is sharply changed (strong injection - Fig. 2.5, 2. 7). When the jets are injected in the diverging nozzle part, as a rule, there are no Mach disks in the jets (Fig. 2.6) and the flow after mixing remains supersonic independently on consurrption of the gas injected (Fig. 2.6). For construction of the calculation m:::rlel which can be applied in the wide range of the relative consurrption of a gas injected, r ' at the installation with the wedge nozzle the :rreasurerrents of the static pressure, P, and pressure behind a straight shock, P' , were perfonned for regilres wi thout injection, with injection, a&i in the flow, fonned by injected gas only ( r = 1). The installation worked in the stationary regilre, the main flow was fonned by the atrrospheric air, injected gas 1 was air too. The measurerrents were carried out in two different nozzle cross sectians, where calculated meanings of Mach numbers were equal to 2. 9 and 4 .0 correspondently. P 1 was measured with the pipe of the total pressure at the nozzle axi~. To measure P gas was taken through holes row in the lateral nozzle wall. In Fig. 3 the dependence of M on meaning of r is presented. The results of calculation on the instantaneous mixing nodel (IMM) are presented here too. They were performed in the ass\llll)tion that the mixing takes place in the injection section (upper curves) and in the section which is situated at one calibre lower on the flow (lower curves), the section height in the injection place being taken for calibre. It is seen, that experiments and calculations at f < f * := 0.15 are in a good agreerrent. At
0 > 0 * the calculation predicts the closing of the flow. However the experimental M meanings at t > r * practically don I t change u:p to 0 =1 and they approxinlately are equal to the M ( f *). M ( f *) with a good pricise relates to the Mach nurober of su:personic flow, which is accelerated in the given nozzle but that achieves the sonic velocity in the section nearly injection place. The best agreerrent between the experiments and calculations takes place, if at r ..:. r*
753
16
~~----------------------.. -·~·- _ ...... ---------------------1
~~~0~--------------g
-~---~~~~~~~~~-----~
Fig. 3. Mach nurober of the flow after mi.xing in dependence on injected gas constl!Tption in sections with canputed .r-F4.0 (1121 • ) and r-F2.9 (3 141 o ). 1 13 ~ calculation on IMM in the injection section1 214 - the sarre in the section at one calibre lower.
the calculation is perfonred on IMM in the section situated on halfcalibre lower on the flow of the place injectionl and at r > r * it is needed to suppose that M is a constant1 which is equa1 to the Mach nurober of the flow 1 that has in the rrentioned section sonic velocity.
THE RESULTS OF EXPERlMENTS AND ANALYSIS
The experirrents were carried out for three different mixtures in the chamber with different Stagnation pararreters; 1 - teclmically pure nitrogen1 T =2500K1 P =8.4 atm; 2 - 95% N2 + 5% H20I T0 =28CX)KI P =9.4 atm1 3 - §5 % N2 + ~5% H20 1 T =18CX)K1 P =5.5 atm.
0 The measurarents were carr1ed o8t in the Rozzle section spaced by 7 mn fran the co2 injection place and where the static pressure of illldisturbed main flow varied within the l:imi.ts of 0.05 - 0.2 atm. As such pressures due to the small nozzle width the gas emittance1 d- 1 was substantially less than 'lmity. That is why in each experi-
rrent the values of ()( were measured. For that the flow additionally was probed by a globar radiation with brightness temperature of 1400K and with the use of a rrechanical rrodulator. Thus in each experirrent three values were registered: r1 - globar radiation intensity. 12 - the total radiation intensity of the globar and of the flowl 13 - the intensity of flow eigen radiation. It is evident that
"'- = (I1 - 1/.)/1!. Then the value I = 13/ c1- is the radiation intensi ty of tlie b ack body at the teft?perature of ~ 3 rrode of co2 1
754
i.e. I 0 is proportional to the specific energy reserve of antisyrrmatric co2 m:Xle.
For calculation of I the system of vibrisional kinetic equations was used in the gen&ally accepted fo:r:rn and tagether with the rnass 1 rrarentum and energy conservation equations the systern was integrated fran the equilibrium state1~ the charrber. The vibrational relaxation rates were taken fran with the ergeption of vibrational relaxation rate of nitrogen at the water • To calculate the change of gasdynamic pararreters caused by the injection the described m:Xlel was usedl i.e. at r <. r* the conservation equations system of a rnass 1 rrarentum and energy was sol ved in the assumption that mixing takes place in the section placed at halfcalibre lCMer on flow fran the injection sectionl and at -r > r * the rrarentum equation was displaced by condition that Mach nurober of the flow in
0.6
(JII
tJ
I 1 rel. units 0
0
0
I 0
0
2 ()- ---,r ....... ,...,
/ ....... / ....... /
/ / ..
• .3 __ _!._._ _ _r__
~
---· . ·--·--·--. ---~--~---- ...
tl.l ()3 04 r Fig. 4. The dependence of specific co2 energy reserve on its con-
surrption. o 1 <t 1 • - the experirrental meanings for mixtures 11 2 1 3 correspondentl y; 11 2 1 3 - calculations on the rrodel described in the text.
this section is equal to the unity. The experirrental and calculated dependence of I on relative
rrolar consurrption of co2 injected1 t 1 are present&i in Fig.4. It is readily seen1 that tfie increase of the rrolar part of the injected co2 decreases specific energy within the ~ 3 - m:Xle of the co2• Linear experirrental dependence for the case of injection into pure mtrogen has a s:inple explanation1 if to suppose that in the mixing process
755
the relaxation losses are absent and that before reaching place of observation the carplete excitation of vibrational levels of the v3 -m:x3e of CO takes place C7tling to V - V ~ge with N2 • In fact if to I\.1 rofes of N2 with specific energy e_ liJ' to rnix n,... m::>les of camon äioxide, then supiXJSing that the s~tic vihratl:'onal energies of N2, e.N, and co2, E..c are equal to each other, at the cbservation place we have
(I)
where f = n"/(n,... + ~). The s:inple linear dependence (I) f~ly oorrelated wi~ tlie extlerinental results if to suppose that E: N is close to the specific nitrogen energy, detennined on translational t:ent:Jerature in the critical nozzle section. The analysis of obtained experinental results for the case of injection into pure nitrogen shcMs that all the CO injected in the observation place is vihrationally excited and ~e relaxation losses arenot great, i.e. the redistrihution of the vibrational energy, pr:iroarily frosen in nitrogen between N and CO takes plaae. Since the mixing is governed by gasdynamic fa~rs, ~ conclusion about excitation of all the co2 at the observation cross section is also valid for other mixtures.
For the mixtures with water additions the experinentally measured neanings of specific energy of the i 3-m:lde of co2 are substantially less than for a dry mixture. The tneory predicts result as a consequence of great vibrational relaxation rates in H2o presence. For the strong injection there exists a quantitative agreenent between the calculation and the experinental results. A substantional diminishing of relaxational losses for a slCM injection might be explained by the flCM structure· downstream the rnixing place at transversal injection. As it is show.n above at a slCM injection the jets diameters are less than the distances between than and the jets don' t interact between themselves. In such a case the nodel of instantaneous rnixing can be insufficiently oorrect to describe the relaxational losses, especially at small distances fran the injection place.
Thus at experinents with relative m::>lar consurtption of the injected co2 equal to 0 <. o. 4 all the injected co2 enters in vibrational exchange with nitrogen. That testifies to the efficiency of the investigated schare for rnixing the gas injected with the main flCM. The experinental results are in satisfactory agreement with carputations based on the supposed in the work semienpirical rnixing m::>del.
1. Croshko V. N. , Soloukhin R. I. Optimal inversi.on regines at thennal excitation by rnixing in supersonic flCM. US-:ffi Academy of Seiences Reports, 211:829 (1973) - In Russian.
2. Cassady P. , Newton I. , and Rose P. A New rnixing gasdynamic laser.
756
AIM Journal, 16:305 (1978). 3. Legai -Scmnaire N. , Legai F. Vibrational distribution of popula
tions and kinetics of ccrN2 -system in the funda:rrental and hazm.:r nie regines. Canadian Jomnal of Physics, 48:1966 (1970).
4. Bahir L.P. Determination of vibrational levels populations of co2 I!Dlecule in gasdynamic lasers on buming products by infrared spectroscopy mathods. Physics Institute Preprint of BSSR Academy of Sciences, 54:37 (1979) - In Russian.
5. Attal B., Drust S., Baily R., Pelat M., Taran J.P. Teclmiques RAmN d' etudes des ecoulemants et des flamres par laser. Spectra 2CXJO, 7:37 (1979).
6. Achazov 0. V. , LabWa. S.A. , Soloukhin R. I. , Fanin N .A. About diagnostics of carbon dioxide I!Dlecular states on resonant of co2-laser. USSR Academy of Seiences Reports, 249:1353 (1979) - Ii'i Russian.
7. Bel.kov P.V., Val'ko V.V., D'yakov A.S., Ostroukhov N.N., Tkachenko B.K. co2-<DL utilizing mi.xtures containing carbon nonoxide. Jomnal of Quant.Electronics, 7:2385 (1980) - In Russian.
8. M:1!latchey R.A., Benedict w.s., Clough S.A., Bu:rch D.T., calfee R.F., Fox K., Rotlunan L.S., Gaking J .S. At:nospheric absorption lines paraneters carpilation. AFCRL Report (1973) •
9. Bahir L.P., Overchenko Ju.V. The determination of vibrational levels populations of co2 I!Dlecule in gasdynami.c lasers by infrared spectroscopy mathods. Journal of Applied Spectroscopy, 30: 44 (1979) - In Russian.
lO.Kondrat'ev K.Ja., M:>skalenko N.I. Th.eJ:mal radiation of the planets. 1977, Hidrareteoizdat, Leningrad - In Russian.
ll.Handbook of infrared radiation fran canbustion gases. Aero-astrodynamics Labaratory Report. 1972, Washington.
12.Khnelinin B.A., Plastinin Ju.A. Radiative and absorptional properties of HP, co2 , CO and OCl I!Dlecules at the tenperatures of 3CXr 3Q(X)K. In "Phys1cal prablem; of gasdynamic" 1975, M:>scow - In Russian.
13.Avduevsky V.S., ~v K.I., Polyansky N.N. Interaction of supersonie f1CM with transversal jet, injected through circular hole in the plate. USSR Acac1erey of Seiences Bulletin. Fluid and Gas Mechanics, 1:193 (1970) - In Russian.
14.Lavrov A.V., Harchenko S.S. Application of instantaneous mi.xing I!Ddel to ana1ysis of gasdynamic laser. J.of Buming and Explosion Physics, 1:147 (1980) - In Russian.
15.1Dsev S.A. Gasdynamic 1asers, 1977 "Nauka", M:>scow - In Russian. 16.Zuev A.P., Tkachenko B.K. Determination of vibrational relaxation
tine of level (v=1) N at collisionals of water vapor, Higher Education Institutes Bulfetine, ser. Physics, 6:84 (1978) - In Russian.
757
XI. MOLECULAR BEAMS
WHERE ARE WE GOING WITH MOLECULAR BEAMS?
John B. Fenn
Department of Chemical Engineering Yale University, New Haven, CT, USA 06520
INTRODUCTION
A love affair with molecular beams spanning nearly a quarter of a century has made this reporter a true believer in their powers. But believers, however true, arenot necessarily successful seers. Indeed, more things have happened in those 25 years than were dreamt of in his original philosophy. Even so, when the organizers of this conference handed him the crystal ball with an invitation to peer into the future, he found the temptation irresistible. May whatever honor he might enjoy in his country of molecular beams and rarefied gas dynamics be spared, even though he dares to impersonate a prophet.
Some reflection on the meaning of this title leads to a realization that "where are we going" has two somewhat different connotations. One has to do with what will be going on along the way. The other is concerned with where the way is taking us. Some reasonably reliable expectations on future activity can be formulated simply by identifying and extrapolating trends in the recent past. Much of the following discussion will be devoted to such identification and extrapolation. More difficult is the problern of discerning what might be revealed by this future activity. The latter aspect of where we are going is of particular interest to the society which tolerates and supports us. Thus, in the last section discretion will give way to valor in a wish list of possible scientific and technological payoffs.
First a couple of caveats. There is a vast literature on the subjects that we touch upon. With much of it, this reporter is woefully unfamiliar. He is especially ignorant of the work done in his host country. This incompetence coupled with the constraints on
761
space means that the references cannot pretend to be complete but are little more than arbitrarily-chosen entries to that literature. Herewith a sincere apology for the many significant omissions. Then there is the problern of what the term "molecular beams" embraces. Moralists have long maintained that in human affairs the. distinction between means and ends inevitably becomes blurred. Science often succumbs to similar smearing. Originally regarded primarily as sources supersonic free jets in vacuo have become so inextricably involved in the whole molecular beam scene that it is difficult to know, either literally or figuratively, where jets leave off and beams begin. Consequently, this report will consider as interchangeable the naive and the knowledgeable translations of that delightful French term "Jets Moleculaire". By the same token, the convolution of means and ends makes somewhat arbitrary the classification and organization of topics to be covered. Many of the items could equally well legitimately occupy a spot in more than one of the groupings to be discussed.
COOLING COUNTS
One of the most important roles of free jets has been as refrigerators par excellence. Indeed, their ability to produce beams of high intensity turns out to stem nearly as much from the low temperatures achieved during the adiabatic expansion as from the related ratio of convective to thermal velocity that results in ~eh number focussing emphasized by Kantrowitz and Grey. A very large fraction of actually realized gain in useful intensity from any free jet source is due to the narrowness of the velocity distribution associated with the low temperatures reached at high Mach numbers or speed ratios. At low enough temperatures the total intensity becomes useful intensity. As understanding of free jet sources increased it became apparent that free jet cooling had other benefits as well. It could result in very low internal temperatures, especially in rotation, and it could bring about the formation of very fragile molecular aggregates such as van der Waals molecules and clusters. We will survey briefly recent developments in each of these three variations on the jet cooling theme.
Cooling Coordinates
A most fruitful extension of translational cooling has been pursued by Toennies and his colleagues. They showed that with very small nozzles and very high source pressures they could achieve very narrow velocity distributions.l,2 With such beams Buck and his coworkers used time-of-flight (TOF) energy loss analysis to measure state-to-state translation-rotation excitation cross sections in H2 , n2 and HD.3,4 It is tobe noted that the high source Reynolds numbers also served to suppress the distribution of rotational states in the molecules. By recognizing and taking advantage of the unique
762
properties of helium (extremely low critical temperature and large viscosity cross section at low temperatures) the Toennies group succeeded in reaching translational temperatures of the order of 10-3 K where the velocity spread is as low as 0.02 FWHM.5 Campargue has obtained similar results.6 More importantly the Toennies team has successfully used these nearly monoenergetic beams in revealing studies of inelastic scattering from surfaces and gas phase molecules. Brusdeylins et al. with the TOF technique have been able to resolve energy losses as small as those associated with single phonon excitation or queuehing when a helium beam is scattered from an alkali halide crystal surface.7 Similarly, Faubel et al. have resolved changes of individual rotational levels in Nz whose rotational levels are much closer together than those in Hz. 8 Each reader is free to speculate as to future extensions, applications and implications of the extreme translational cooling that free jets can effect. The feeling of this scribe is that extension of the surface scattering studies will prove most rewarding. Surfaces with adsorbed species should be particularly interesting. It has taken a long time but we have come a long way since the classic diffraction experiments of Estermann, Frisch and Stern. Much remains to be learned.
Cooling Clarifies
That low translational temperatures could be reached in supersonie jets was reasonably well understood by the mid sixties. 9,10 That internal degrees of freedom were also at least partially cooled had been recognized even earlier by Hagena and Henkes. 11 By 1967 several investigators had measured the rotational temperature in free jets of nitrogen and found that it was almost as low as the translationel temperature. 12-16 The possible "practical" importance of this internal cooling did not emerge until several years later, after the birth of spectroscopy with tunable lasers. In 1973 Sinha et al. hinted at what was to come by using laser induced fluorescence (LIF) to determine the internal states of dimers in a free jet of sodium vapor. 17 Smalley et al. ushered in a new era of spectroscopy with their landmark demonstration of how the elimination of rotational bandspread by free jet cooling could "clean up" the vibrational spectrum of polyatomic molecules. 18 The consequences have been dramatic. There has been a veritable flood of LIF results at an ever accelerating rate. It is probably safe to say that more free jets are now being used for spectroscopy than to generate beams for scattering experiments. Even a cursory review of this effort would require more space and more competence than is available to this reporter. He will content bimself with pointing out that the art has so developed that it is now possible to study fairly directly and in detail the rates and mechanisms for transfer of vibrational energy from one part of a polyatomic molecule to another. 19 It is virtually certain that free jets will continue to be the darlings of the spectroscopists and that we will all. learn much from this infatuation.
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Cooling Couples
A third possible consequence of vigorous cooling is a change in the state of aggregation. Precipitation of a condensed phase in adiabatically expanding gases and vapors has many important applications and implications in both technology and science. That condensation was a phenomena to be reckoned with in free jets in vacuo was first noticed by Becker and Henkes in their TOF studies on beams from free jets of hydrogen.20 That observation led to an active program on cluster formation and properties in Becker's laboratory at the Kernforschungszentrum in Karlsruhe. A major motivation for this program is the possibility of injecting fuel in thermonuclear reactors in the form of high energy clusters obtained by ionizing the neutral clusters, accelerating them electrostatically and then neutralizing them by charge-exchange in order to achieve as much penetration as possible across the field lines in a magnetically confined plasma reactor. Consequently, emphasis has been on the formation and properties of clusters containing several tens to several hundreds of molecules.
Hagena reviewed what had already become an extensive literature on cluster formation by free jets in 1974.21 Since that time interest has expanded further. Clusters, both neutral and charged have become objects of increasing interest and subjects for research in many fields. Because they represent states of aggregation intermediate between gaseous and condensed phases, studies of their properties could lead to an understanding not only of the phase transformation process but also of the structure and nature of condensed phases. We know much less about them than we do about gases. In particular, our knowledge and understanding of liquids and solutions leave much to be desired. Mass spectrometric studies of cluster ion beams of the kind long done by Kebarle and more recently by Castleman and co-workers show great promise for the study of "solution chemistry in the gas phase".22,23 Gspann's adventures with helium clusters provide new perspectives on one of natures most interesting fluids. The latest episode is in these proceedings.24 Of at least equal interest and importance are the properties of clusters per se. Their behaviour is considered responsible for many practically important phenomena such as heterogeneous catalysis, the photographic process and atmospheric pollution, to name but a few.
Free jet expansions into vacuum are particularly versatile and flexible devices for the preparation and study of clusters. Not only do they provide a means of controlling their size by stopping their growth, they also deliver them into the "shelter" of a vacuum, where they can maintain their fragile identities for stud~ by a wide variety of techniques including electron diffraction2 , mass säectrometry26~ photoionization spectroscopy27, Raman spectroscopy2 , scattering2~, and LIF30. For these reasons they have been the basis for many interesting papers presented over the years at these symposia.
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The present conference continues that tradition. A number of papers are devoted to this subject. The one by Sattler comprises a particularly interesting overview.31 Also of note is the communication by Devienne, the founder of this symposium series. He and his colleagues use high energy neutral beams to produce clusters by sputtering from a variety of substrates.32 With these same high energy beams incident on surfaces he anticipated by several years what has become one of the hottest developments in the mass spectrometry of complex organic molecules, Fast Atom Bombardment (FAB) ion sources.33
In a development somewhat separate from but parallel to the effort devoted to the study of relatively large clusters a lot of attention has been paid to their much smaller precursors. In 1961 Bentley36 and Henkes37 reported quite independently the first mass spectrometric observations on dimers and other small polymers produced in free jet expansions. Leckenby et al.38, Milne and Greene39, Milne et al.40, Golomb et al.41, and Gordon et al.42 extended the Bentley-Henkes work with particular emphasis on understanding the dimer formation process. Dimers formed in free jet expansion are interesting for reasons other than their role in nucleation-condensation. In elastic scattering experiments they may help elucidate the effects of asymmetry upon intermolecular potentials and provide a possible approach to the direct study of ternary encounters among gas phase molecules. They have sparked a growing interest as "reagents" in reactive scattering experiments. Moreover, they sometimes have startling properties that challenge theories of chemical structure. Witness the discovery by Klemperer et al. that benzene dimers have a dipole moment.43 A most interesting gas dynamic phenomenon was found by Sinha et al. in their examination of laser induced fluorescence from dimers in free jets of alkali metal vapor. By looking at the polarization of the emitted light they determined that the dimers had a fairly strong tendency to be orient~d so that the plane of rotation was parallel to the jet axis. 17 Subsequent investi~ators have confirmed and extended these most interesting results.44, 5 Finally we should mention that photoelectron spectroscopic investigation of rare gas dimers formed in free jets has revealed an·unexpectedly high density of electronic states.46
Perhaps the most interesting consequence of the ability of free jets to achieve steady-state nonequilibrium populations of ordinarily nonstable species has been the formation in jets of mixed gases very small clusters containing more than one species and commonly re~erred to as Van der Waals molecules. Microwave resonance spectrometry, laser induced fluorescence and mass spectrometry have identified and obtained information on the structure and energy levels of such unlikely species as KAr, Hei2 , co2c12He and aniline argonide. Levy has reviewed these and related developments and brings us up to date with his paper at this meeting.47 Rice and his colleagues have found remarkably high cross sections for the quenching of vibrational energy in supersonic free jets.48 This phenomenon may be associated with van der Waals complex formation and could be related to the startling
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dimer depletion that Yamashita et al. found in jets of Ar and CO containing small quantities of polyatomic species.49 More recentiy, reactive scattering experiments have been carried out in which ArXe was formed from Ar2 + Xe.SO In light of results like these it would appear that a door might be opening on a whole new branch of chemistry involving the structure, properties and reactions of molecules that are so weakly bound that they can be formed only under the collision conditions uniquely achievable by supersonic free jets. We are sure to have increasingly frequent reports of advances on this front.
INTERRUFTORS INTERRUFTED
In the previous section we raced past some of the avenues of research that have been opened up by the ability of free jets to reach very low temperatures. This ability to cool is, of course, not limited to free jets. But cooling is not sufficient in itself to achieve the conditions that have been so fruitful. It is also necessary to prevent the system from reaching the equilibrium state to which the low temperature relates thermodynamically. The happy circumstance in free jet expansions in vacuo is that the decrease in temperature is accompanied by a decrease in density, the absolute value of which is almost at the whim of the investigator who can, to an extent uncouple the temperature from the density by judicious choice of source stagnation"pressure. In this way he or she can arrange to have collision frequency vanish at particular temperature levels in the jet and thus frustrate the system's drive toward equilibrium. It is this "freezing" (in the non-thermal sense!) of non-equilibrium states that so enhances the free jet cooling capabilities. Rebrov's paper reviews the ways in which free jet analysis has provided valuable information on the rates and mechanisms of kinetic processes involved in transport, energy exchange and reaction.51 Therefore we will refrain from discussion of this interesting area of research.
Just as an ability to interrupt a system's progress toward equilibrium is a potent attribute of free jet expansion, so in turn does the interruption of the free jet flow turn out to offer valueable rewards. But it took a long time for this idea to take hold. In the early days of the post Kantrowitz-Grey era many members of the molecular beam community were hesitant to go the free jet source route because of the apparent need for very large pumping speeds. One way of reducing the pumping speed requirement is to decrease the duty cycle by making the nozzle flow intermittent. Some investigators chose to go with shock tube driven free jet beam systems, not only because of the high source gas temperatures they could provide but also because the low duty cycle greatly reduced the need for large pumps. As it turned out the duty cycle was so low and the cost and complication of the shock tube structure so high that shock tube systems never attracted much of a following. A more convenient means
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of decreasing duty cycle and pumping speed need was nozzle pulsing by means of a solenoid valve introduced by Hagena in 1963 and reported on at the 4th RGD Symposium.52 It could provide pulses of any length down to about 1.5 ms. In spite of the prospective savings in pumping speed investigators pretty much ignored pulsed nozzles for nearly fifteen years. Then in 1978 Gentry and Giese described a valve that was capable of producing jet pulses as short as 8 to 10 microseconds.53 Moreover, they had earlier demonstrated that such valves were very effective in crossed beam scattering experiments that resolved single changes in rotational levels in HD + He and HD + HD collisions.54 Here are some of the advantages offered by such intermittent flow systems:
1. Very short pulses virtually eliminate noise due to background gas from the jet flow because the signal due to molecules from the scattering region is sent by the detector before unscattered molecules bouncing from the chamber walls can interfere.
2. Very much higher instantaneous intensities can be achieved for a given pumping speed because: (a) the time average gas load is small; (b) the pumps need only achieve as low a background pressure as may be required for the short time immediately preceding and during the pulse. If the detector is gated the background pressure can go to any value the pumps can talerate in between pulses. In other words, pumping speed can be replaced by chamber volume which costs much less. (c) Because differential pumping stages are not required the distance from nozzle to scattering region can be short. Thus the attenuation of beam intensity with distance due to divergence can be minimized. It is to be remernbered that in crossed beam scattering experiments the Signal increases with the product of the beam intensities whereas the noise goes with the sum. Thus the signal to noise ratio is greatly enhanced by high intensity.
3. The very high instantaueaus gas flows that pulsing makes possible permit the use of nozzles with much larger dimensions than would be possible in steady flow systems. Not only are clogging problems thus eased but also the investigator can achieve high nozzle Reynolds numbers without resorting to the high source gas densities that could bring about unwanted clustering. In other words cooling can be maximized while clustering is minimized. Moreover, as permitted nozzle flow areas increase, flexibility in design increases because fabrication is simplified. One can now actually contemplate contouring to achieve particular flow characteristics, e.g. long sojourn times to promote clustering or relaxation. Note that the ability to go to high Reynolds numbers also means that boundary layer problems in nozzles with diverging sections can be decreased. Another important possibility emerges when large flow areas can be used. Amirov et al. have used an ingenious valve that provides a pulsed planar flow from a slit source.55 Such planar jets offer many advantages for some kinds of experiments.
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4. Pulsed jets are very compatible with pulsed lasers. When a continuous beam or jet is crossed with a laser that is pulsed, all the gas flow during the laser-off periods is wasted along with the pumping speed required to remove it. Perhaps more important is the fact that only a small fraction of the beam molecules can be excited by laser photons. If the same number ~f molecules are compressed into a region of time and space that coincides with the laser pulse a very large fraction of the beam molecules will be treated by the photons.
It is probably this last advantage that has been most responsible for the rapidly growing popularity of pulsed sources. Up to now only the Gentry-Giese group have used pulsed nozzles in crossed beam scattering experiments, but many have adopted pulsed sources in beam -laser experiments. One reason for this difference is the popularity of photon molecule interaction studies. Another is that the valve systems needed in such experiments can be simpler and eheaper because the pulses don't need to be as short. Noise from background gas is generally a much less serious problem in free jet spectroscopy than in scattering studies. Several valve designs have been described and their operating characteristics have been reported.56 At least three companies are affering commercial versions.57 One of the questions that emerges concerns the terminal state of the gas in a jetpulse or "beamlet". If the valve is open long enough to achieve fully developed flow equivalent to that in a continuous jet, then all we know about steady flow jets is applicable and it becomes possible to make a pretty good estimation of the final state of the gas, e.g. the rotational temperature. If the flow does not have time to become fully developed then one must depend upon empirical observations to deduce the final state of the gas. Relatively few detailed experimental studies have been made of the final state of gas from pulsed nozzles. The most elegant one to date is presented in the paper by Lülf and Andresen at this symposium.58 By means of LIF measurements they have obtained a time-resolved description of the distribution of rotational states of NO seeded in He and Ar at several distances along the jet axis. It seems fairly clear that with helium the flow is fully developed in about 40 microseconds. More time is required for argon. Generally applicable conclusions cannot yet be drawn. One firm conclusion is that the challenging problem of understanding and describing non-steady.free jet and nozzle flows at relatively low Reynolds numbers has become of renewed importance and should be an inviting challenge to theorists. The only publication to date specifically on this problem for free jets is a note by Saenger which attempts to determine the minimum time that a valve must be open to achieve fully developed flow.59 Her model seems to underestimate the dependence of minimum flow time on nozzle diameter.
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PHOTONS ARE FOREVER
In the first experiment on record in which a molecular beam was used as an investigative tool Dunnoyer radiated a beam of sodium atoms with a resonance lamp in an attempt to measure the radiation lifetime.60 His time resolution, as we now know, was about four orders of magnitude too low so the experiment was a failure. Beams and photons did not get together again until about sixty years later. Actually, that statement is not entirely accurate. Resonance spectroscopy at radio and microwave frequencies dominated the beam scene from the thirties to the sixties and still constitutes a major segment of molecular beam research. But the wave packets at these frequencies are so large that we don't ordinarily regard them as photons. Nevertheless, the possibility of controlled stimulated emission that makes lasers work was first revealed in the development of the maser at microwave frequencies. Thus, it is entirely fitting that having given birth to lasers beam experiments in their "old age" should have become so dependent on them. Indeed, today it is a poor lab in the molecular beam community that doesn't have at least one laser. This ubiquity has already been evident in the preceding discussion and is so great that it precludes the possibility of any kind of complete overview. Be it sufficient for our present purpose to suggest that in future molecular beam research laser photons will play increasingly major roles in both the preparation and detection of beam molecules used for scattering experiments. Particularly in detection lasers offer the nearest approach to the "ideal" detector, one that can determine the density, velocity, internal state, phase and orientation of molecules before and after the interaction under investigation. Similarly, molecular beams will continue to provide nearly ideal samples of molecules of all kinds for the study by various kinds of spectroscopies of a wide range of consequences of molecule-photon interactions including excitation of nuclear, electronic, rotational and vibrational energy levels that can lead to fluorescence, intramolecular energy transfer, dissociation, ionization and reaction. We will limit more detailed consideration to a few functions of photons that didn't quite fit in the previous categories that have already been glanced at. They seem to deserve attention because they will or should cut wide swaths in the future.
Surfaces Survive
The main original interest in molecular beam methods in this series of symposia was as a means of investigating the exchange of energy and momentum between molecules and surfaces. The Sputnik-generated interest in drag and heat transfer characteristics of satellites and space vehicles was a strong motivating factor. Until very recently almost all microscopic information on these exchanges stemmed from observations on the speed and direction of reflected molecules. This technique can be very powerful and as we have already noted has been brought to new levels of precision and resolution by Toennies and bis colleagues in their scattering experi-
769
ment with ultracold beams. Another new dimension has been added by the use of photons to probe the internal energy states of the departing molecules. Except for some pioneering experiments with electron beam fluorescence reported by Marsden in the 4th of these symposia there seem to have been no attempts to look at these internal energy states until quite recently.61 Then there was a flurry of reports on the use of LIF to look at the accommodation of rotational energy during surface scattering experiments with molecular beams. 62-64 Cavanagh and King used LIF to look at the rotational state of NO molecules thermally desorbed from platinum.65 In perhaps the most elegant experiments to date Kleyn et al. have determined not only the rotational distribution but also the polarization of the ag~ular momentum vector in NO molecules scattered by a silver surface. Meanwhile, making something of a comeback has been the classical infrared emission spectrometry used by Polanyi so successfully for so many years in bis pioneering studies of energy distribution in reaction products.67 Greatly enhanced by the availability of Fourier Transform Spectrometers, this technique has recently made it possible to look at the distribution of internal energy in important molecules such as CO and co2 that don't lend themselves very conveniently to laser-induced fluorescence approaches. We have been able to analyze the distribution of rotational and vibrational energy in CO and co2 molecules vibrationally excited by single collisions with hot metal surfaces.68 Wehave also been able to analyze the distribution of internal energy in product co2 molecules formed by the catalytic oxidation of CO on a platinum surface.69 In sum, photons in the future will help us form pictures about surfaces as well as on them.
Cold Counts Again
One of the great difficulties with infrared photons in molecular beam experiments is their reluctance to enter and leave molecules. The radiation lifetimes of most vibrationally excited species are in the millisecond range, long enough so that most excited molecules are out of the vacuum system before they radiate. Moreover, molecular beam samples are so thin that the problern of weak absorptivities and emissivities is exacerbated. A similar situation is circumvented at microwave frequencies by taking advantage of the difference in focussing properties of molecules that bang on to the photons. In such a case long radiation lifetime becomes a virtue. Recently, Scoles and bis colleagues have developed an anaiogous technique that may well accomplish for spectrometry at infrared frequencies what Rabi's invention of the molecular beam spectrometer achieved in the microwave region of the spectrum. The key is the use of a liquid helium (cold!) bolometer as a detector. The molecular beam is crossed by a modulated photon beam from a tunable infrared laser. When the laser frequency coincides with an absorption frequency of the beam species the detector, which measures the total energy flux in the beam, senses an increase in that energy due to the absorption of photons. Thus the detector output is an AC signal at the frequency of the laser beam modulation and with an amplitude proportional to the number
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of photons. When the intersection angle of the laser and molecular beams is changed from 90 to 45 degrees the resulting doppler shift makes possible the determination of velocity distribution of the beam molecules. Scoles and company have analyzed free jet expansions of CO, NO, co2 and HF. They have made 2nd order Stark effect measurements with COz, have obtained spectra for dimers and observed predissociation.70 An application to ultrahigh resolution spectroscopy was to be presented at this symposium. We are sure to hear much more about this cool and exciting development.
Beams from Beams
The consideration of photans in general and lasers in particular has thus far been preoccupied with their use in the spectrometry of beam molecules per se or for detection purposes. Implicit in these procedures, though it has not been explicitly enlarged upon, is the use of photans to prepare beam molecules in particular states for scattering purposes, the other half of the now-fashionable "state-to -state" chemistry syndrome. In this last section we turn to a more primitive role for laser photons, the production of beams by vaporization of condensed phases. The energy flux densities for which lasers are famous have been widely applied in technology, from the cutting and welding of heavy metal parts to the most delicate of surgical procedures. In the so-called "laser blow off" phenomenon their ability to vaporize very refractory materiale is also well known. Noteworthy for our purpose is the fact that when a high intensity pulse of laser photans interacts with a surface the resulting blast of vapor has many of the characteristics of supersonic free jets. Covington et al. have reviewed the early studies of this yhenomenon and have documented the analogy with free jet expansion.7 Tehranian and Olander have recently pursued this analogy further.72 More relevant to our considerations is the work of Tang and bis colleagues.73 They generate beam pulses by firing a Q-switched ruby laser at a target comrising a thin metal coating on a glass plate, preferably from behind (through the glass). In this way they have generated beams of such materiale as Ho, B, C, Al and Mo and carried out reactive scattering experiments with some of them. Their particular interest was chemiluminescence resulting from reaction with NzO. In a sense the generation of short bursts of vapor by laser pulses is the ultimate in pulsed valves, the time scale being in terms of nanoseconds rather than microseconds. Although the metbad has been applied to rather refractory materials there seems to be no a priori reason why solid substrates of normally gaseaus and liquid materiale could not be used. One wonders, for example, whether a helium cooled substrate of hydrogen or oxygen might provide a source of H or 0 atoms with all the advantages of pulsing that we outlined earlier. Probably more important is the ability of this technique to generate beams of very refractory materials. It would thus permit the molecular beam study of reactions that could not readily be carried out in any other kind of laboratory gear. The feeling here is that much will be gained by exploiting this application of laser power.
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Finally we will mention a most recent development that combines laser blow off with pulsed free jet expansion of rare gases. Smalley's group at Rice has generated copper clusters ranging in size from 1 to 29 atoms by laser vaporization of a rotating copper target rod within the throat of a pulsed nozzle through which heliumwas flowing.74 Detection was by laser photoionization with time of flight mass analysis. Results gave interesting and valuable information about the dependence of ionization cross sections on photoenergy and cluster size.
The crystal ball shows that photons playing their role in these very exciting productions of unusual molecular beams will get rave notices from future reviewers.
ENDS TO END
Up to now this report has been essentially an exercise in laying out prospective roads through terrain we can see but have not yet traversed. The time has come in these closing paragraphs to speculate on what these roads may be leading to. What will be some of the landmarks of our progress along the way into the future? If we let our fancies run free, what can we imagine the payoffs to be? Of course, it is risky to deduce ends by simply looking at means. Dunnoyer did not anticipate the laser nor Stern the atomic clock. Uranium enrichment was not even a gleam in Becker's eye when he decided to put the Kantrowitz-Grey prescription to an experimental test. Scientists can't know in advance, and most of them never find out, whether there is gold or garbage at the end of their rainbows. But the society that sponsored their fun welcomes the golden eggs of technological payoff. If they stop coming the goose may starve, so let's ponder a few possibilities.
Molecular Beam Epitaxy (MBE)
Already one of the hottest high technologies this technique is being vigorously pursued by some fifty laboratories at last count. At least five companies are in business to make equipment and supplies. MBE permits the growth of single-crystal-like films of very precisely controlled sem-conductor compositions on appropriate substrates. It is really a chemical process method for in situ synthesis of complex compounds in layers whose thickness is measured in Angstroms. A most conservative projection is that free jet sources will supplement and maybe supplant the effusive ovens which are the present workhorses. Advantages to be gained include higher deposition rates, better control of film densities and a wider variety of film composition and characteristics, e.g. by exploiting cluster formation during expansion.
772
Heating Via Cooling
We have already mentioned that the Becker group approach to fuelling a magnetic fusion reactor is to inject high energy hydrogen clusters produced in this sequence of steps: ionization of neutral clusters formed in a nozzle expansion, acceleration of the cluster ions, and neutralization by charge exchange. After Searcy75 successfully produced cluster ions of water in a free jet expansion seeded with ions from a corona discharge, this reporter became convinced that an attractive way to make cluster ions of Hz would be to inject protons into expanding hydrogen. DOE and its predecessors refused to show any interest. It is now a pleasure to report that Beuhler and Friedman have indeed successfully accomplished the production of Hz cluster ions by this method.75 In common we share the near-belief that properly focussed a convergent array of accelerated cluster ions might form the basis of an inertially confined fusion reactor. Thus to make hydrogen bot enough to fuse one might.be well advised to make it cold enough to condense!
Scattering Sketching
The human mind is more adept at processing information in the form of pictures than in the form of words and numbers. The suggestion is that surface scattering of helium might provide the basis for making pictures of a surface that the eye can "see". Davisson and Germer's discovery of diffraction by electrons bad to await the invention of the cathode ray screen before LEED could become the powerful tool it is to-day. The surface scattering results by the Toennies team suggest that all we need is an imaging screen for helium in order to replace LEED by LEAD. We might expect the same kind of resolution improvements that Mueller obtained when he transformed the field-electron microscope into the field-ion microscope. Indeed if someone is clever enough to invent a lens, we might even conjure up the neutral helium analog of the field-ion microscope and obtain real pictures of surfaces without the constraints imposed by high fields and necessary tip characteristics. It is not at all clear how such imaging might be achieved but when one sees on the cover of Laser Focus - (November 1975) - the teeboicolor picture of a free jet painted by a laser and then reflects on the advances that have been made in computer assisted tomography, he begins to believe that HASCAT pictures of surfaces may well be developed.
Molecular Beam Medicine
We are conditioned by experience to regard volatility and, therefore, relatively low molecular weight as the necessary prerequisite for admission to the community of beam-forming molecules. The ability to make beams of very fragile clusters began to undermine that prejudice. It should have been shattered completely by some experiments that Dole and bis colleagues reported nearly fifteen years ago.76 They electrosprayed solutions or dispersions of
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polymer molecules into a bath gas of nitrogen. Neutral solvent molecules evaporated from the charged droplets leaving behind charged solute molecules. The resulting gaseous dispersion of macroions was expanded through a small orifice into a vacuum system to produce a beam of macroions. Polystyrene with molecular weight as high as 400,000 was successfully dispersed in this way. There would seem to be no reason why particles of much higher molecular weight might not be treated in the same way. Thus one can contemplate the possibility of generating beams of biologically important molecules, even viruses and bacteria! The day may come when no self-respecting clinical laboratory will be without a molecular beam machine.
ACKNOWLEDGEMENTS
Support from many sources over the years has given me the chance to learn what is in this report. The National Science Foundation, the Office of Naval Research, the Air Force Office of Scientific Research and the Petroleum Research Fund deserve particular mention. In addition, appreciation is due the Alexander von Humboldt Stiftung and the Max Planck Institut für Strömungsforschung in Göttingen for providing me the time and place to write it.
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Wegener, ed.) p 93 (Dekker, N.Y. 1974) P. Kebarle, Ann. Rev. Phys. Chem. 28, 445 (1977) A. W. Castleman, Jr., in "Kineticsof Ion Moleeule Reactions,
p. 295 (Plenum, N. Y. 1979); P. M. Holland and --, J. Chem. Phys. 76, 4204 (1982)
J. Gspan~ This Symp. -- and H. Vollmar, J. Physique 39,C-6 330 (1978)
J. Farges, J. Cryst. Growth 31,79 (1975); B. G. DeBoer and G. D. Stein, Surf. Sei. lo6, 84 Tl98l)
0. F. Hagena and W. Obert, J. Chem. Phys. 56, 1793 (1972) A. Hermann, E. Schumacher and L. Wöste, Helv. Chim. Acta 61, 453
(1978); see also Ref. 46 I. F. Silvera, F. Tommasini and R. J. Wijngaarden, 10th RGD (J.
L. Potter, ed.) v ~. 1295 (1976) and unpublished work. A. Van Deursen, A. van Lumig and J. Reuss, Int. J. Mass Spect.
Ion Phys.l8, 129 (1975); H. Vehmeyer, R. Feltgen, P. Chakrahorti, M. Dukov, F. Torelle and H. Pauly, Chem.Phys.Lett. 42, 597 (1976) --
D. H. Levy, Adv. Chem. Phys.~, 323 (1981) K. Sattler, This symp. F. M. Devienne and M. Teisseire, This symp. D. J. Surman, J. C. Vickerson, J. Chem. Soc.Chem.Comm.324 (1981) M. Barker, R. S. Bordol, R. D. Sedgwick and A.N. Tyler ibid. 325 P. G. Bentley, Nature 190, 432 (1961) W. Henkes, z. Naturforsch. l6a, 842 (1961) R. E. Leckenqy, E. J. Robbi~and P. Trevalion, Proc. Roy. Soc.
A286, 409 (1964) T.~Milne, A. E. Vandegrift, F.T. Greene, J. Chem. Phys. ~.
1552 (1970); ibid. 47, 4095 (1967) and 2J, 2221 (1972) D. Golomb, R. E. Good, R. F. Brown, J. Chem. Phys.52,1545 (1970) A. J. Gordon, y. T. Lee and D. R. Herschbach, J. Chem. Phys. ~.
2393 (1971) J. M. Calo, J. Chem. Phys. 62, 4904 (1974) K. C. Janda, J. C. Hemminge~ J. S. Winn, S. E. Novick, S. J.
Harris and W. Klemperer, J. Chem. Phys. 63, 1419 (1979) M. P. Sinha, A. Schulty, R. N. Zare, J. Chem. Phys.58, 549 (1973) K. Bergmann, U. Hefter and P. Hering, Chem.Phys. 32-,-329 (1978);
ibid. 20, 391 (1976) --P. M. Dehmer, J. Chem. Phys. ~. 1263,3433 (1982); ]2, 5625 (1981) D. H. Levy, This symp. J. Tusa, M. Sulkes and S. A. Rice, J. Chem. Phys. 70, 3136 (1979);
1.?_, 5733 (1980)
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57.
58. 59. 60. 61. 62.
64.
65. 66.
68. 69
70.
71.
72. 73.
74.
75. 76.
776
M. Yamashita, T. Sano, S. Kotake, J:Fenn, J. Chem. Phys. 75, 5355 (1981)
D. R. Worsnop, S. Bülow and D. R. Hersehbaeh, in press A. Rebrov, This symp. 0. F. Hagena, Z. f. Angewandte Phys. 16, 183 (1963); 17, 542 (1964) W. R. Gentry, C. F. Giese, Rev. Sei. Inst. 49, 595 (1978) W. R. Gentry, C. F. Giese, J. Chem. Phys. 67: 5389 (1977) A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett. 83, l (1981) J. B. Cross, J. J. Valentini, Rev. Sei. Inst. 53, 38 Tl982);
T. E. Adams, B. H. Roekney, R. J. Morrison and E. R. Grant, ibid. 52, 1469 (1981); C. E. Otis and P.M. Johnson, ibid. 51, 1128 (1980); A. Auerbaeh and R. McDiarmid, ibid. 51, 1273 Tl980); F. M. Behlen and S. A. Riee, J. Chem. Phys. 75: 5673 (1981); J. Tusa, M. Sulkes, S. A. Riee, C. Jouvet, ibid. ~. 3573 (1982)
Beam Dynamies, 623 E. 57th St. Minneapolis, MN 55417; Laser Teehnies, 6007 Osuna Rd. N.E.,Albu~uer~ue, NM 87109; Quanta Ray, 1250 Charleston Rd., Mt. View, Cal. 94043
H. W. Lülf and P. Andresen, This Symp. K. 1. Saenger, J. Chem. Phys 75, 2467 (1981) 1. Dunoyer, Compt. Rend. 157,:1068 (l913);Le Radium 10,400(1913) D. J. Marsden, 4th RGD (J-:Ii. DeLeeuw, ed) v 2, 565 (Äeadem.l966) G. M. MeClelland, G. D. Kubrak, H. G. Rennagel and R. N. Zare,
Phys. Rev. Lett. 46;831-(1981) D. Ettinger, K. Honma, M. Keil and J. C. Polanyi, Chem. Phys.
Lett. 87, 413 (1981); J. W. Hepburn, F. J. Northrup, G. Ogram, J. C.Polanyi and J. M. Williamson, Chem. Phys. Lett. 85, 127(1982)
F. Frankel, J. Hager, W. Kruger, H. Walther, C. T. Campbell, G. Ertl, H. Kuipers and K. Segner, Phys.Rev. Lett. 46, 152 (1981)
R. R. Cavanagh and D. S. King, Phys.Rev. Lett. 47:-1829 (1981) A. W. Kl~n, A. C. Luntz and D. J. Auerbaeh, Phys. Rev. Lett. ~,1169 (1981); Surf. Sei. 177, 33 (1982); J. Chem. Phys. 76, l, (1982)
D. H. Maylotte, J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 21. 1547 (1972); 1. T. Cowley, D. S. Horne and J. C. Polanyi, Chem. Phys. Lett. 12, 144 (1971)
D. A. Mantell, S. B.JiYali, G. 1. Haller and J. B. Fenn, in press D. A. Mantell, S. B. Ryali, B. Halpern, G. 1. Haller and J. B.
Fenn, Chem. P~s. Lett. 81, 185 (1981) T. E. Gough, R. E. Miller and G. Seoles, Appl.Phys.Lett. 30,
338 (1977) and private eommunieation. --M. A. Covington, G. N. Liu and K. A. Lineoln, AIAA J 15, 1174
(1977); Int. J. Mass. Speet. and Ion Phys 16, 191 (1975) F.Tehranian and D. R. Olander, private eommunieation S. P. Tang, N. G. Utterbaek, J. F. Friehnieht, J. Chem. Phys.64,
3833 (1976); Phys. Fluid. 19, 900 (1976) --D. E. Powers, S. G. Hansen, Ml. E. Geusie, D. 1. Miehalopoulos
and R. E. Smalley, J. Chem. Phys. in press. R. Beuhler and 1. Friedman, Phys.Rev.Lett. 48, 1097 (1982) M. Dole, 1.1. Maek, R. 1. Hines, R. C. Mobley, 1. D. Ferguson, M. B. Aliee, J. Chem. Phys.49, 2240(1968); 52, 4977 (1970)
CESIUM VAPOR JETTARGET PRODUCED WITH A SUPERSONIC NOZZLE
ABSTRACT
A. Athanasiou and O.F. Hagena
Institut für Kernverfahrenstechnik, Kernforschungszentrum Karlsruhe, Postfach 3640, 7500 Karlsruhe Federal Republic of Germany
The paper describes experimental results and operating experiences with an apparatus to produce a jet of Cs as a charge-exchange target for the formation of negative hydrogen ions. The nozzle throat was a slit 2x60 mm2 , and the nominal exit Mach nurober M = 5 was obtained with a SO mm long wedge-like supersonic section. The nozzle Knudsen nurober based on source density and nozzle width was in the range of 0.001-0.005. Source conditions were 3 mbar < p0 < 15 mbar and 600 K < T0 < 720 K. The jettarget was obtained by passing the central 60 % of the nozzle flow field through a "cryogenic" (T ~ 310 K) skimmer. The beam profile measured 15 cm downstream of the nozzle throat ~ielded a target thickness (line density) of up to 3·1015 atomslern . The width was 4.5 cm (FWHM) corresponding to a virtual source close to the nozzle throat. There was no evidence for flow distortions due to the skimmer or for excessive boundary layer growth in the nozzle. Compared to the sonic slit nozzle used in other alkali jettarget systems the supersonic section reduces the circulating cesium by a factor of 3 for the same line density and flow divergence.
Cs loss through the target aperture openings is monitored with a surface ionization detector located 41 cm away from the jet axis. Measurements of the spatial distribution of the Cs loss and its time-dependence when turning the Cs jet on and off show that the observed Cs is mostly coming from Cs scattered and/or evaporated from surfaces seen by the Cs jet. The intensity depends on temperature and surface condition of the inner walls of the jet-target assembly, but changes little with jet intensity. The lowest flux measured with the detector in the central positionwas 4·10 12 atoms
- 2 - 1 d . h 1 2 1019 • h . cm s , compare w1t , · 1n t e Jettarget.
777
INTRODUCTION
Supersonic jets have been employed for some time as gas targets in the study of atomic collision processes. 1 Compared with gas cells a jet target obtained from a slit nozzle allows to produce well localized interaction zones for incident beams having large cross sectional areas and still have a minimum of target gas losses in the direction of the beam. Interest in such gas targets has been revived by the need for alcali metal vapor targets to produce multiampere beams of negative hydrogen ions from low energy positive hydrogen ions 2 or from energetic hydrogen clusters 3 • A survey of previous work can be found in a ~aper by Bacal et al. 4 andin the proceedings of two conferences 5 ' •
The present paper reports the performance characteristics of a cesium jettarget which is part of the neutral beam line being built in Karlsruhe. It will convert accelerated hydrogen clusters into negative ions, which are subsequently accelerated and finally neutralized to give a high-energy beam (> 100 keV) of neutral hydrogen atoms 7 . The design of the cesium jettarget intended to achieve the following goals 7 :
- Supersonic nozzle - cooled skimmer system to increase the fraction 8 of the total nozzle flux Jn which is used as the jettarget flux Jt, 8 = Jt/Jn, from low values 8 < 10 % reported for sonic nozzles4 to at least 50 %.
- Proper control of jet conditions by independent variation of source pressure p0 (vapor pressure corresponding to boiler temperature Tb) and temperature T0 > Tb.
- Minimization of cesium losses out of the target region by low temperatures of the Cs-emitting surfaces including the skimmer and by installing a shut-off valve in the vapor line to produce the jettarget only when needed.
- Low operating temperatures T0 < 700 K and pressures p0 < 50 mbar to have comparatively short start-up times and to avoid electromagnetic pumps for the return of the cesium condensate by placing the boiler somewhat below the cesium condensing surfaces.
EXPERIMENTAL APPARATUS
Figure 1 gives an overall view of the experiment to convert a hydrogen cluster beam into negative hydrogen ions by interaction with a cesium vapor target. The jettarget with nozzle, skimmer and cesium collector is situated within a big vacuum chamber (1m 3 ),
while the cesium supply system with boiler, shut-off valves and
778
Vacuumchamber [}TI Cs Jet Target . Extracl1on
H·- Cluster ~ Beam
~c: HC"f O.mp
1 -ts Ur, (100kV ,lA l
Fig. 1. Schematic of experiment to produce H -ions from accelerated hydrogen clusters by charge exchange in a Cs jettarget.
condensate return line is outside. The entire unit is mounted on a vacuum flange 0 . 35 m~. The two side walls of the vacuum chamber are lined with LHe cryopanels with a combined pumping speed of 100 m3 /s for hydrogen 7 . For the experiments of this paper the ion extraction system was not installed and the chamber was not connected to the cluster injector. This facilitated the installation of diagnostic probes to investigate the properties of the jettarget.
Cesium cycle
Figure 2 shows in more detail the components of the cesium system together with a sectional view of the nozzle, skimmer, and target region. Cesium vapor is produced by heating the liquid cesium in the boiler, with thermocouples outside and inside the boiler controlling the temperature and thus the vapor pressure. On its way to the nozzle chamber (1) the vapor passes an all-metal valve good for up to 725 K and the vapor line which are kept ~ SO K hotter than the boiler. The superheated vapor expands out of the nominal M = 5 supersonic nozzle (2), and condenses on the cooled surfaces of skimmer (4) and collector (8). The liquid condensate returns by gravity back into the boiler.
To produce the jettarget the valve in the vapor line is opened. A steady state is established in less than 100 ms, and the shut-off time constant when closing the valve is about 300 ms. There was no detectable leakage of the valve, and superheating the vapor line avoided the eventual accumulation of cesium between valve and nozzle.
779
es supply
vapor line T = 700 K
vacuum chamber
es boiler T8 = 600 K
nozzle 700 K
sk.immer 30DK
® CD
LlrHI-- ® +------1ffif-tl---®
4cm ____.
® ® 0 0
+t------tt--- ® ®
Fig. 2. Cesium jettarget system and details of the nozzle-skimmercollector assembly.
The pulsed mode of operation possible with the present system decreases the cesium lasses and improves the overall safety characteristics of the system.
The flow rate of the cesium in the condensate return line which is equal to the nozzle flux can be measured by closing the valve to the boiler and observing the rising cesium level as indicated by the pressure gage Po·
Nozzle - Skimmer System
As shown in Fig. 2 the nozzle (2) is a straight-walled converging-divergi ng supersonic nozzle. The relevant nozzle dimensions are .:
Throat are ln x bn = 60 X 2 mm2
Exit area le X be 80 X 20 mm2
Length diverging section zn so mm
The nozzle end plate has a separate heater and a c opper cover plate f or temperature uni formity. Two r adiation shie lds (5) be tween nozzle
780
and skimmer (4) 1ower the radiative heat transfer to the skimmer from about 100 W at T0 = 700 K to typica11y 1ess than 20 W.
The skimmer (4) is made from 1 mm copper p1ate brazed to the register of coo1ing tubes (7). The skimmer opening is 62 x 23 mm2 . Skimmer temperature is monitored by a thermocoup1e mounted in the center of the short side of the skimmer opening. The cesium condensate flows through the annular space between skimmer and outer wall into the co11ector.
The jettarget is formed by the central core of the nozzle f1ow fie1d which passes through the skimmer opening. This mass flux Jt condenses in the cooled cup-like co11ector (8) from where it f1ows together with the cesium from the skimmer back into the condensate return line.
The c1uster beam to be transformed into negative ions enters the target region through an aperture with 5 cm diameter. The exit diameter is enlarged to 7 cm to allow for some widening of the beam due to the interactions with the cesium jettarget. Both aperture openings can be c1osed by a rotating shutter (9).
In order to achieve the design value of the target thickness of ntbt = 2·1015 Cs atoms cm- 2 , assuming 50% of the total nozz1e f1ux Jn cou1d be used for the jettarget flux Jt' the operating conditions of the system are as fo11ows:
Source temperature To 700 K
Source pressure Po 5.93 mbar
Nozzle flux Jn 1.11·1021 Cs atoms s-1
0.246 g Cs s-1
Skimmer Knudsen nurober Kn 0.028 s
Diagnostics
Characterization of the jettarget required three types of measurements:
1. Nozz1e and target mass f1ux, J and J . The coo1ing circuits of skimmer and collector were ~sed ast~a1orimeters. The heat deposited by the condensing cesium vapor was measured through the corresponding increase of coo1ant temperature. With the nozz1e at T = 700 K and condensation at 320 K, the enthalpy re1eased by0 the condensing cesium is obtained from thermodynamic data 9 as ~i = 619 J/g. This gives for the design point with 0.123 g Cs/s condensing on skimmer and collector a power of 76 W which is readily measured by observing the temperature rise oT of the coo1ant after passing through
781
the collector and by knowing the coolant mass flow and specific heat. An independent method to measure J is through observation of the accumulation of liquid in tRe condensate return line.
2. Jettarget density profile n(x) in the target zone. Detailed jet profile data were taken with a small temperature probe in the form of a 8 mm dia. cup collecting the incident cesium. Again the condensing cesium gives its enthalpy to the copper mass of the probe, and the rise of probe temperature is recorded as function of time, with the probe temperature lowered to at least 340 K between subsequent measurements. Turning on the cesium jet produces first a strictly linear increase of probe temperature which is indicative of negligible losses due to conduction, radiation and reevaporation. Thus the slope dT/dt is directly proportional to the incident intensity. The density n at the probe position is obtained by dividing the intensity by the jet velocity which can be approximated by the maximum value possible in a free jet. The final result is:
n = 2 . 0 . lOI~ dT/K (700 K)I/2 _3 dt/s T atoms cm
0
The point-wise determination of the density profile is somewhat time consuming, but has the advantage of simplicity and accuracy.
3. Cesium losses from the target zone. Cesium atoms can escape out of the target system by evaporation from cesium covered surfaces and by scattering of cesium incident on surfaces which can be seen through the aperture openings. The expected intensities are many orders of magnitude less than the intensities inside the jet. They can be measured using the surface ioniziation detector introduced Taylor 10 , which consists of a hot tungsten filament converting all incident cesium atoms into ions which are then measured as electric current. The detector uses a tungsten filament lxl3 mm2 which is on the axis of a liquid nitrogen cooled tube with 35 mm dia. It has a 2 mm dia. entrance opening for the cesium. The detector is 410 mm away from the jet axis and can be moved up and down parallel to the cesium jet and rotated around its own axis. This arrangement gives a high directional sensitivity of the detector and allows to distinguish between cesium signals coming from various parts of the apparatus.
RESULTS AND DISCUSSION
An example of the type of data obtained with the present apparatus is given in Figure 3. It shows as function of time the varia-
782
tion of the temperatures of boiler, skimmer and collector and the temperature change OT of the skimmer and collector coolant. At time zero the cesium flow is turned on. Operation of the jet causes the temperature of the boiling cesium to decrease by about 3 K, while all other temperatures increase due to the heat load of the condensing cesium. A new steady state is reached after some twenty minutes, with the different time constants being due to the different thermal inertia. The increase of the temperature rise OT caused by the jet combined with the coolant mass flow is used to determine the cesium flux to the skimmer, J 5 , and to the collector, Jt. The result for the experiment of Fig. 3 is J = 39 . 3 mg/s, J = 78 . 7 mg/s, and Jn = J 5 +Jt = 118 mg/s. Thus tge fraction of tfie total flux utilized for the jettarget is e = Jt/Jn = 67 % which exceeds the anticipated value of SO %. Similar experiments under different source conditions yielded values for e between 62 and 70 % for the nozzle throat - skimmer distance equal to 7.4 cm. For 8.4 cm the values forewerein the range 57 - 63 %. Without the supersonic section the streamlines of a free jet 7 would give e = 18 . 4 % and 16.2 %, respectively . This increase of the gas efficiency by more than a factor of 3 has of course the consequence of corresponding savings in the necessary cesium flux and in the heating and cooling loads of the cesium cycle. No attempt was made so far to compare the measured e with calculations, which must take into account both the boundary layers within the nozzle and the additional lateral expansion downstream of the nozzle exit.
4 ---2
602 -1--r--r----t 600
--+-~~~~Hr4-~~~~
.. ......... 1-.... -! ; ... ~--r .... ·; 'T"
! ~mer_\ 6T ,.,...,_.
iC. ~ _.\ •1"'-~ ........... .. T ~ Collector -r .\
V 6T
310
306
-5 0 5 10 15 20 25 tIm
Fig. 3. Variation of boiler, skimmer , and collector temperature T and of the temperature change OT of the skimmer and collector coolant during operation of the jettarget .
783
Data for the jettarget profile measured with the temperature probe are given in Fig. 4a for three different source conditions. The width of the profile is 4.7 cm which is about equa1 to the width of a source f1ow coming from the nozz1e throat (4.5 cm for z = 15, 4.2 cm for z = 16 cm). Target thickness corresponding to these profiles ranges between 3.3 and 1.4·1015 Cs atoms cm- 2 • Higher va1ues are possib1e by increasing the source pressure p .
0
Measurements of the cesium 1osses with the Langmuir-Tay1or detector showed a tendency to give 1ower va1ues when the experiment was repeated after some time . This can be due to changes in the surface conditions affecting the ref1ection probabi1ity of incident atoms. Fig.4b gives a recording of the detector signa1. At t = 0, the jet is on, but the shutter is c1osed. Opening the shutter at t = 2.5 minutes gives a factor of ten increase, the f1ux at the detector is about 3.6·1012 atoms cm- 2 s- 1 • This intensity corresponds to a cesium flux at the target aperture opening of 3.6·1014 atoms cm- 2 s- 1 which is equa1 to the f1ux of saturated Cs vapor at 300 K. C1osing the shutter gives again a smal1er signa1, and when the jet is turned off at t = 12.5 minutes the intensity of the cesium at the detector is reduced by another factor of ten. This demonstrates the advantage of pulsing the jet and having a shutter in front of the beam apertures. As far as the intensities are concerned which are found with the shutter c1osed and the jet on, these are caused by the fair1y wide gaps between shutter and target chamber walls. It is p1anned to modify the shutter-co1lector assemb1y so that the gaps are minimized and the surfaces close to the apertures are better cooled. Even without
\•701*
12 3 . 6 ) . Po·Umlllr
.fi z ·15an 111 . • !!.
10
lfl2 . s.i mlllr 4 I
~ 1o-'A: !: 8 j31 _10,z atoms
" 15an
5,7 mlllr 11~
-12 -a -4 o 4 a 12
1 (an)
I ;;; ' cm2-s - 6 ;;;
c: c::n ;;;
~ 4
2
0 5
Fig. 4. a) Jettarget profile measured with a temperature probe a1ong the center1ine of the c1uster beam for different source pressures p0 and distances z between nozz1e throat and cluster beam.
784
b) Time-recording of the Langmuir-Tay1or detector signa1 le1·
these modifications the cesium losses measured 325 mm away from the aperture with the jet on are many orders of magnitude less than the intensity in the center of the jettarget. For the conditions of Fig. 4b they differ by a factor of 3·10- 7 • Fora sodium jettarget the reduction measured 80 cm away was only 4·10-~. 11
REFERENCES
1. B. A. D'yachkov, "~rod~ction of high-energy neutral atoms by conversion of H1 , H2 , and H3 ions in a supersonic lithiumvapor jet", Sov1.et Physics-technical Physics 13:1036 (1969).
2. A. S. Schlachter, P.J. Bjorkholm, D.H. Loyd, L.W. Anderson, W. Häberly, "Charge-Exchange Collisions between Hydrogen Ions and Cesium Vapor in the Energy Range 0.5-20 keV", Phys. Rev. 177:184 (1969).
3. E. W. Becker, H.D. Falter, O.F. Hagena, P.R.W. Henkes, R. Klingelhöfer, H.O. Moser, W. Obert, I. Poth, "Production of Negative Hydrogen Ions from Accelerated Cluster Ions", Nucl. Fusion 17:617 (1977).
4. M. Bacal, H.J. Doucet, G. Labaune, H. Lamain, C. Jacquot, S. Verney, "Cesium supersonic jet for D- production by double electron capture", Rev. Sei. Instr. 53:159 (1982).
5. Proc. Symposium on the Production and Neutralization of Negative Hydrogen Ionsand Beams, Brookhaven 1977. BNL 50727 (1978).
6. Proc. 2nd Symposium on the Production and Neutralization of Neg. Hydrogen Ions and Beams, Brookhaven 1980, BNL 51304 (1981).
7. 0. F. Hagena, P.R.W. Henkes, R. Klingelhöfer, B. Krevet, H.O. Moser, "Negative ion production by charge exchange of hydrogen clusters with a cesium vapor target - status report", in: Ref. 6., p. 263.
8. Ullmanns Encyclopädie der Technischen Chemie, Urban und Schwarzenberg Verlag, München, Band 9 (1975).
9. N.B. Vargaftik, "Tables on the thermophysical Properties of liquids and gases", Halsted Press, J. Wiley and Sons, New York (1975).
10. J.B. Taylor, Z. f. Physik 57:242 (1929). 11. P. Poulsen and E.B. Hooper, Jr., Proc. 8th Symp. Eng. Problems
of Fusion Research, San Franzisco, 1979, IEE Pub.No. 79 CH 1441-4-NPS, Vol. II, p. 676.
785
BASIC FEATURES OF THE GENERATION AND DIAGNOSTICS OF ATOH.IC HYDROGEN BEAMS IN THE GROUND AND METABTABLE
228.f/2-8TATES TO DETERMINE THE FUNDAMENTAL PHY8ICAL CONSTANTS
E.K.Izrailov
D.I.Mendeleyev Research and Manufacturing Complex Leningrad, U.8.S.R.
It is well known that an increased accuracy and confidence in the determination of such fundamental ph3sical constants (FPC) of atomic physics as the Rydberg constant (Roo), the ration of the electron mass to proton mass (mefm~) and the fine structure constant (~) pla, a significan~ part in the solution of major problems of theoretical physics and quantum metrology /1,2/.
Although the quantum theory enables one to calculate in principle m~ physical characteristics and a number of fine effects for elementary particles, it can yield8 the desired' accuracy (forot rv 1 x 10-7 , me/mp"' 1 x 10-and R00 rv 1 x 10-9) only for such elementary atomic systems as hydrogen and hydrogen-like atoms. T.herefore we used the method and modified "beam" equipment which make it possible2to study the appropria~e beams of atoms in the ground (1 S# ) and metastable (2 Su ) states at the required metroi6gy level. We shall use2this method combined with the double photoabsorption method (across the 18-+ 28 transition) /3/ to improve R00 and me/mp or together with the two-quantum spectroscopy method to determine o( (from fre9.uency measurements for 28- 3P and 28- 38 transitions). It seems to us that this approach will vield resonances whose relative widths are within 10-r· to 10-9 and thus it will enable us to determine the FPC with an accuracy permissible.
The metastable 22s#2 st&te of the hydrogen atoms (MHA) whose transitions are easy to realize and calculate is known to have a long-life-time (1:sN0,1 s) in the absence of perturbation and feature a fairly high energy lewel which is significant for quantum metrology and,among other things, for refinement of the FPC.
787
T.here is a number of disinctlve features inherent in the generation and diagnostics of the required beams to be taken into consideration for refining the "beams" e~uipment. 1. In the transition of partielas from the 1 S v2 state to the 2 2 S •h state the excitation probability is low (10-r+ 10-B) indepently of the excitation method, which gives rise to weak fluxes. 2. T.he formation of these fluxes is unfailingly accompanied by losses due to scattering, recoll effects, deviations in the geometr, of the eguipment which result in distortion of the function f(v) in the MHA velocity distribution. Decrease in the life-time ~s due to the electric scattering fields set up by charges also causes a loss in the number of the MHA flux whereas high stability of the MHA flux and its adequate magnitude (.no less than 1 x 10S MHA/s) over the region monitored are the determining factors for the experiment. 3. Low (~ 1%) efficiency of the detectors available MHA /4/, which makes it necessary to improve the monitoring system for atomic hydrogen beams at thermal velocities. 4. The background (especially the ultraviolet radiation) which, as a rule, reduces the signal/
noise ratio to the values below unity. It is obvious from the above that we must strive for the S/N ratios over unity and for the conditions leading to a minimum distortion of the function f(v) in the process of the MHA beam formation. In such a situation analysis of the residual gas for H20 and N2 molecules is important. Quenching of the latter is possible owing to a large -~ cross-section of2their collisions with the MHA (1 x 10 and 1 x 10-13 cm , respectively /5/). 5. T.he ability of atomic ~drogen to intensive recombination. 6. T.he need for setting up the conditions which promote optimization of the time of interaction between the MHA beam and its uniform perturbing field.
However, when the atomic energy drops below 0,3 eV, the degree of dissociation falls (.it is desirable that X?>80%) and the atomic beam intensity in the 12 8-th· state decreases ( i t :i.s desirable that 1JS > 1 x 10i"' at/ (cm2 •ßJ over the exc1tation region). Efficient correlation of these factors governs the optimization referred to.
It is evident from the above that the processes ocurring throughout the experiment amount basically to dissociation of H2 molecules into atoms, 2shaping and diagnostics of a beam of atoms in the 1 S-t;2 state_
its excitation and shaping of the MHA flux in the 2fsy2 state followed by monitoring.
T.he principal units of the equipment developed with regard to the .problems raised and the distinctive features specified above are shown schematically in Fig.1.
788
For the normal operation the atomic beam sources in the 12 So~;2 and 22 8.(;2 states tagether wi th diagnostics s,stems were placed in three independently evacuated dissociator, excitation and detector vacuum chambers.
7 8
Fig.l. Arrangement of principal units of the atomic beam apparatus . 1 . Tungsten oven .*2. Collimating slits. 3. Hisatomic flux stabilizer. 4. Kzs(MHA) exciter. 5. Excitat1on current stabilizer. 6. Wuencher. 7. Limiting slits. 8 Channe! detector. 9. Pulse amplifier. 10. Scaler. 11. Automatie printer. 12. Mass spectrometer. 13. Ion monitaring system.
Thermal dissociation was used to obtain a beam of H-atoms in the 1 2 S-1/2 state at an energy on the order of 0,3 eV. For this purpose an effusion source having the desired degree o1· dissociation (Xrv9()!0) and a su1·ficiently low UV-radiation level Coelow rv 100 pulses/ s) was developed and studied. The principle part of the source was a tungsten oven (W-oven) movable in vacuum. T.he source was operable at hydrogen pressures PH in W-oven placed in dissociator chamber ranging from 13 to 260 Pa and at the oven temperatures from 2200 to 2900 K.
T.he degree of dissociation X for hydrogen in a beam was estimated by different methods.
Firstly, in a purely qualitative way using an atomic hydrogen beam indicator based on a reaction which involves conversion of molybdenum oxide (a change of the indicator colour from yellow into blue). Secondly, through calculations based on a measured pressure of hydrogen in the tungsten oven using a transducer calibrated for hydrogen (with an error of rv3%) and on its true temperature Tu • At Tu = 2865 K and pH = 40 Pa the
789
degree of dissociation X was 96%. T.hirdl,, this quantity was estimated from the ratios f= f , determined using a time of flight mass spectrometer ~d considering the difference in the ionization cross section between hydrogen atoms 6" and molecules b2 • Here I1 and l 2 are the signals for the mass peak heights equal to 1 and 2 with the tungsten oven turned on and off.
The beam was sampled on its axis at about 20 cm from the W-oven and a formula given in /6/ was used to deter-
mine X: X= 1/( f+/2'· 2 · 64/o2 )
As a result, it was found that at Tu = 2865 K and PH = 40 Pa, X = 90%. T.his degree of dissociation was attained owing to a special design (a fairly large and uniform dissociation zones in a narrow tungsten tube), the manufacturing method employed (erosion machining of the tube in kerosene) and proper conditions for the speedy onset of thermodynamic equilibrium state in the dissociator. The oven temperature TH was found accurate to rv 5% from the radiance temperature measurements and the estimation of tungsten emissivity.
When determining the beam intensi ty I 15 , use was made of a specially designed ionizer complete with a measuring tube, which enabled us to find the difference A P between residual pressure and beam pressure in the innizer to a higher accuracy. The value of I1s measured in the detector chamberrv 32 cm from the W-oven at PH = 40 Pa and TH = 2360 K as determined was 1 x 10 1'~ at/ ( cm2 • S) wi th an accuracy of 25%. This beam produced a spot /7/ on the molybdenum indicator well defined for visual observation after 6 minutes exposure. Beam traces of H2 atoms for different parameters of the thermal source are shown in Fig.2.
Fig. 2. Beam traces of H18 atoms for different parameters of the thermal source.
790
T.he excitation chamber accomodated a source of the WIA beam in the 2 2 S.f/2 state in which atomic excitation occured during collisions on an electron beam. It should be emphasized here that in this case use was made :-Of'_ a very fine electron beam, a fairl' weak individual magnetic field which focused the electron flux and did not affect for all practical purposes time~s well as the stabilization of anode current Ia (~ Ia/I 0 N 1%) and automatic monitoring of the basic excitation parameters brought to a digital printer.
When stud,ing the conditions for excitation and shaping of a stable MHA beam, flux stabilization of the thermal source atomic beam based on atomic recombination over the platinum surface was also emplo,ed for the first time. The experiments which involved the stabilizer showed a decrease in the beam fluctuation to 2.0% per shift.
The MHA flux was controlled by a special system operating in the particle counting mode.
For this purpose a channel electron multiplier was used. The monitoring efficiency estimated indirectly was 8%.
The above-mentioned beam stabilization made it possible to improve substantiall' the working conditions of the MHA source, choose its optimum duty and obtain its stable characteristics.
Fig.3. shows the principal characteristic of the MHA source, that is, the signal N produced by MHA at the monitoring system output as agfunction of current Ia. A linear increase in Ng over the range of change of I 4 = = 40 - 370 )JA indicates that secondary processes were minimized. A imi ting current of 380 .f A predetermined the choice of Ia = 300 jA for the exci tation operating current.
X/03 Ni,~/s 20
16 oo
12
8
J,
0 (00 200 300 400
Fig. 3. Principal characteristic of the MHA source, that is, the signal Ng produced by HK~ at the monitoring system output as a function of current Ia.
791
In the course of the study the mass spectrometer was employed for continuous analysis of the dissociation products and the composition of the residual gas and of the beam after its collimation by slits of the thermal source was determined.
A substantial amount of hydrogen present as H20, H? and hydrocarbons evolved from vapours which trace the1r origin to the oil used as lubrificant for diffusion pumps evacuating the dissociation chamber was detected over the excitation region.
With pressure oil for pumps substituted by a special ether and their heaters and nozzles modified, no more oil film formed either over the excitation region or during the flight of MHA to the detector.
Special vacuum-pumping ass 1blies provided a vacuum in the dissociator chamber ( N 10-s Pa) at a high rate of pumping ( over 2 x 10 3 l/ s for hydrogen) and a fairly high "oil-free" vacuum (N 10-6 Pa) in the excitation and detector chambers.
This degree of vacuum was attained through the joint use of a getter ion and a magnetic discharge pumps to obtain a higher rate of the hydrogen beam pumping.
For minimizing the distortion of the function f(v) and the difficulties related to the formation photons and the recoil effect, we had to maintain the excitation electron energy at a level exceeding the threshold of the 2 2 S 92 metastable state for hydrogen by no more than 1 eV. Golding of the exciter electrodes, screening over the MHA excitation region using suitable apertures, the choice of optimum excitation conditions (Ia = 300 }' A; Ua = 13 eV) as well as the use of "oil-freen high speed pumping facilities made it possible to obtain over the range of monitoring a flux of 2 x 105 MHA/s having the desired stability at a relative solid angle of monitoring 12./4CJr = 5 • 10-3 and the S/N=5. At the close of the experiments comprehensive studies of the "beam" equipment were carried out so as to find the causes of the MHA losses, estimate them and test the performance of both sources of the hydrogen atomic beam. T.he possibility of the plotting by experiment an excitation curve (for the MHA in the 22S~2 state, Fig.4) whose shape is similar to the theoretical one served as the performance criterion. In so doing, true values of the electron energies were found by the delayed potential method which made it possible to determine by experiment the proper correction, as distinct from the theoretical estimates commonly used /4/.
792
Fig. 4.
4.0
3.0
2.0
1.0
0
~·· ,,u •,,
I u11l litt
I
t
I
,, I
d' 10 11 12 13
Electron Energy, eV
Excitation curves of the 22s112 state electron impact near the theshold .
in H by
It should be noted that in the experiments related to double photoexcitation of ~ffiA in a beam the need to increase the hydrogen beam density in the 12S1/2 state is most significant due to a low excitation probability and difficulties met with in setting up an adequate power exciting field. This can be achieved through deep freezing of the beam ( to T ~ 4, 2 K) /8/ . The development of such a beam source of "cold" hydrogen atoms will make it possible, apart from stepping up the double photo-absorption process in the 1S - 2S transition, to attain an estimated 25-fold decrease of the main factor specific to the double photoabsorption line broadening in the transverse beam excitation (the flight factor).
REFERENCES
1. V.O.Arutyunov, The essentiale of refining the system of electrical standards, Izmeritelnaya technika, 10:50 (1974) - in Russian. "
2. E.K.Izrailov, The problern of the fine structure constant determination from fine structure experiments on hydrogen, in: "Sixth Internat. Conference on Atomic Physics, "Abstracts, Riga (1978).
793
3. E.V.Baklanov and V,P,Chebotayev, On carrying out the experiments for precision frequenc, measurements on the 1s~2s transition, Optika i spectroskopiya, 38:384 (19?5) - in Russian. -
4. W.E.Lamb, Jr. and R.C.Retherford, Fine structure of hydrogen, Phys. Rey. 81:222 (1951,.
5. W.L.Fite et al. Lifetime of the 2 S~h state of atomic hydrogen, PhYS, Rev., 116:363 (1959).
6. W.L.Fite and R.T.Brackmann, Collisions of electrons with hydrogen atoms. I. Ionization, Phys, Rev., 112:1141 (1958).
?. E.K.Izrailov, A beam source of hydrogen atoms, ~ bory i tecbnika experimenta, 3:28 (19?4) - in RUSS!an.
8. S.B.Crampton et a1. 1 Hyperfine resonance of gaseous atomic hydrogen at ~,2 K, Phys. Rev. Lett., 42:1039 (19?9).
794
OPTICAL PUMPING OF METASTAHLE NEON ATOMS IN A WEAK MAGNETIC FIELD
J.P.C. Kroon, H.C.W. Beijerinck and N.F. Verster
Eindhoven University of Technology Eindhoven, The Netherlands
Optical pumping is a well known method for achieving beam modulation of selected states in a crossed beam experiment, e.g. for a time of flight analysis with a pseudo random correlation method. For these applications optical pumping will have to be known in all detail. Special care is needed when a polarized laserbeam is used, since ßm selection rules have to be taken into account.
We have investigated the optical pumping of a fast metastable neon beam which is crossed at right angles by a dye laser beam. The Ne* beam is produced by a hollow cathode arc (I0 = 1012-1o14s-1sr-1 mm-2). The optically pumped Ne* beam is analysed downstream of the pumping region using timeofflight techniques. 1
Neon has two metastable states denoted ls3 and lss in Paschen notation with total angular momentum J = 0 and J = 2, respectively. Pumping of 1ss state atoms and using non polarized light it was possible to determine the beam composition. We found a population ratio
Ne 1 /(Ne 1 +Ne 1 ) = 0.13 ± 0.02 SJ SS SJ
in good agreement with line intensity measurements in a plasma. 2 Using linearly polarized laserlight and choosing the 1ss to 2p2 (J' = 1) transition we expect that the mJ = ±2 states (40%} are not pumped. Experimentally we found an attenuation depending on the polarization of the laserbeam, ranging from results in agreement with ~ selection rules to results indicating that all 1ss state atoms are pumped. These measurements can be explained by a weak magnetic field (fig. 1).
795
Fig. 1. Magnetic field perpendicular to laserpolarization.
we chose the z-axis along the laser polarization and we consider a metastable atom in a mJ = 2 state entering the pumping region. Without magnetic field this atom is not pumped. However, a weak magnetic field perpendicular to the z-axis will cause a Larmor precession of the J vector araund the B vector, resulting in an admixture of other mJ states and thus in a finite chance of pumping depending an the rotation angle 6t. When the magnetic field is parallel to the z-axis Larmor precession will leave the mJ state undisturbed. The distribution over the mJ states after rotating can be calculate~ using quantum mechanical rotation matrices. As an example we give the relative intensities of the mJ states, having a perpendicular magnetic field and with initial condition mJ = 2 at 6t = 0.
~ -0 1 c ~ -c
~ > -~ G 5 ~
I
Fig. 2. Relative intensities of mJ states as a function of the Larmor precession angle 6t with initial conditions mJ = 2 at 6t = 0. Nurobers are indicating different mJ states.
796
The Larmor precession frequency V~ is equal to the Zeeman Splitting V~= 2.1 MHzlgauss of the pumped metastable level. For atoms with a velocity V = 5000 mls and a pumping region of 2 mm we find that a magnetic field B = 1.2 gauss will already cause a full rotation of the J vector in the pumping region, resulting in the pumping of all 1s5 atoms assuming sufficient laserpower. To apply well known magnetic fields we have mounted three pairs of Helmholtz coils araund the pumping region.
The description of these phenomena is similar to the description of the Hanle effect. In that case a magnetic field causes Larmor precession of the excited state, resulting in the depolarization of the fluorescence light.
We have tried to measure polarization effects using a polarized laserbeam in optical pumping of the 1s5(J = 2} to 2p6(J' = 2} transition. For a J = 2 to J' = 2 transition the m = 0 to m' = 0 transition is forbidden. Therefore in a zero magnetic field 20% of the initial 1s5 atoms should not be pumped. A magnetic field of 1 gauss perpendicular to the laserpolarization would be sufficient to destroy all polarization effects by Larmor precession, resulting in all 1s5 state atoms tobe pumped. However, a remaining inhomogeneaus magnetic field of 0.5 gauss araund the pumping region made it impossible to measure these effects. It will not be too difficult to remove this remaining field and we will look at this in more detail in the near future.
Fig. 3. The percentage of non pumped 1s5 state atoms as a function of the perpendicular magnetic field B~ for different parallel magnetic fie ld B 11 • 1} B11 1.9 gauss. 2} B11 3 .1 gauss. 3} Bll = 4.3 gauss. 4} B11 = 5.6 gauss. 5} B11 = 10.5 gauss.
797
We measured polarization effects by applying a magnetic field parallel to the laserpolarization and looking at the metastable beam attenuation with time of flight techniques. Given the already mentioned distribution over the two metastable states we could measure the amount of non pumped 1ss atoms. Fig. 3 shows the percentage of non pumped 1ss atoms with a velocity of 2000 mls as a function of the perpendicular magnetic field B~ for different values of the magnetic field B 11 • The laserpower was 1. 8 mW and the laserbeam had a diameter (e-2 intensity) of d = 3 mm.
As can be seen the percentage of non pumped 1ss atoms exceeds the predicted 20% when having a few gauss parallel to the laserpolarization. This is due to spontaneaus decay from the excited level to the non pumped m = 0 state which enriches the population of this state. For the transition used 56% of the atoms in the excited state decay to their initial 1ss state. A simple analysis shows that repeated excitation will finally deplete the m # 0 states and will enrich the m = 0 state by a factor 1.85, i.e. 37% of the total 1ss state population before pumping is now in the m = 0 state. This percentage is also shown in fig. 3 by a dotted line. The small difference between the curve with B# = 5.6 gauss and Bll = 10.5 gauss at El = 0 gauss indicates that Zeeman splitting does not play an important role. We conclude that optical pumping having 5 gauss parallel to the laserpolarization results in an almost perfect m = 0 state atomic beam.
References
1. J.P.C. Kroon, H.C.W. Beijerinck and N.F. Verster, TOF analysis of an optically pumped metastable neon beam, J. Chem. Phys. 74: 6528 (1981).
2. B. van der Sijde, Configuration temperatures in a hollow cathode argon arc and transition probabilities of the argon II spectrum, J. Quant. Spec. Radiance Transfer 12: 703 (1972).
798
C02-LASER EXCITATION OF A MOLECULAR BEAM
MONITORED BY SPONTANEOUS RAMAN EFFECT
ABSTRACT
G. Luijks, S. Stolte and J. Reuss
Fysisch Laboratorium, Katholieke Universiteit Toernooiveld, 6525 ED Nijmegen. The Netherlands
The vibrational state populations and rotational temperatures of CF3Br and SF6 have been determined at variable distance from the nozzle XR/D for expanding molecular beams using the spontaneaus Raman effect. Moreover, also at variable XE/D the molecular beam has been excited with a tightly focussed c.w. line tunable co2-laser. The phenomenon . of collision assisted absorption and up-pumping of SF6 molecules has been investigated. With XE/D = 0.5, Raman spectra at XR/D = 4 were observed which correspond to a thermally vibrational heating of the beam. An average vibrational energy equivalent to 6 COz-laser photons could be pumped into the SF6 beam molecules. Excitation with the COz-laser at larger XE/D produced Raman spectra with a preferred v3-mode excitation of the SF6 molecules. Using a predissociation technique, Raman spectra of SF6 clusters have been observed.
I. INTRODUCTION
Encouraged by the sensitivity obtained with the spontaneaus Raman effect as a tool to measure rotational ayd vibrational state distributions in expanding molecular beams 1 , we extended this method to a larger group of molecules. An example obtained with an improved apparatus discussed briefly in Section II, is displayed in Section III. B~ing intrigued about the possibility of using our Raman spetrometer as a detector to investigate the state population transfer, induced by a powerful excitator, we decided to cross our beam with a C.W. C02-laser. Recent experiments of Bernstein, Flynn and co-workers2,3) showed that a considerable amount of COz-laser energy can be pumped into an expanding SF6 beam.
799
The results were interpreted via a collision-assisted absorption and up-pumping mechani5m. Their sensitive bolometer determined the avarage amount of energy pumped into a SF6 molecule. Now, with our Raman technique we are able to obtain direct information about the energy disposal among the different vibrational modes and its distribution over the excited molecules. A few examples of C.W. COz-laser up-pumping of a SF6 molecular beam analyzed with the Raman effect will be shown in Section IV. By choosing the conditions, a thermal as well as a " -mode enhanced distribution was observed for the SF6 molecules. In Sec~ion V we apply the method of co2-laser induced pred1ssociation modulation of SF6-clusters4) to determine their Raman spectrum. A spectrum, mainly attibuted to dimers, was observed in the frequency region associated with the van der Waals stretch vibration.
II. APPARATUS
Our earlier set-up1) has been modified . Changes and major improvements will be given more extensively elsewhereS). Herewe suffice with a brief outline only.
double monochromator •
M1 . M3 .M4 : Iaser- m1rrors
M5 Raman - rofloctor
w1 ant•-rrilect10n wmdow
w2 • w3 Zn S. - wondows
B Brewstc:r - angle w mdow
L1 Ions f: 80mm
Lz Zn S. - Ions f : lJOmm
~ . T0 st.agna11on cond•t•ons
Fig. I. The configuration of the apparatus
800
The molecular beam expanding free from a nozzle (D = 0.25 mm) is crossed by a focussed Ar+-laser, operating in a folded intracavity. configuration as is illustrated in fig. I. Intracavity laser powers up to 500 Watts can be achieved. The scattered Raman light is analyzed by a double monochromator and a photomultiplier system, yielding a typical spectral resolution of 2 cm-I We apply the direct relationship that exists between the Raman line intensities and the state populations to monitor state distributions in the molecular beam, as a function of XR/D, where the distance XR between nozzle and Ar+-beamwaist can be varied easily. A C.W. co2-laser can also cross the molecular beam and is focussed to a beamwaist of 0.4 mm ~ at a variable distance XE from the nozzle by a lens, finely adjustable by a three-axis micrometer system.
III. RELAXATION MEASUREMENTS
The v2-vibrational mode of CF3Br is strongly Raman active. In fig. 2. the v2-vibrational and rotational temperatures of a seeded CF3Br beam (IO % in Ar) are compared with results obtained for pure beams of CF3Br6) and SF65). The rotational temperatures, Trot• are derived from the width (f.w.h.m.) observed for the unresolved O,P-R,S branches. The vibrational temperatures, Tvib• are obtained from a comparison of the Q-branche intensities of the anti-Stokes signal, lAS• and the Stokes signal, Is, as following:
ln IIAS (v- vvib)4I = kTvib (I) Is v + v 'b · hv 'b . V1 V1
Here, v stands for the Ar+-laser frequency and Vvib for the excitation frequency of the vibrational mode considered. The vibrational frequencies of the modes considered in fig. 2. are very close: for SF6 VI = 774.5 cm-I and for CF3Br v2 = 762.0 cm-I. As expected the rotational temperature of the seeded CF3Br beam is very low compared to all pure beam rotational temperatures, which have roughly all the same values. The situation is more complex for Tvib· At a constant stagnation pressure, P0 = I.2 atm. the seeded CF3Br beam has a considerable lower v2-temperature than the pure CF3Br beam. Raising P0 of the pure CF3Br beam from I.2 to 6 atm. lowers Tvib at XR/D ~ 4 to the same values as for the seeded CF3Br beam (P0 = I.2 atm.), whereas Trot is increased~ due to the enhanced V-R relaxation rate. The vibrational temperature of a pure SF6 beam with P0 = 6 atm. remains the highest at larger XR/D.
801
T/To s~ x,+ Po;6atm T(K)
Cf3Br • .o Po;6atm • .• ~ ; 1.2atm
0.8
~. 10'l. C§ Br in Ar •.t ~; 1.2atm 240
-. • ... _. ..
}Tv;b 6. ...
\ 0 • 0.6 """ .... - -+- 180 \, - ..I
' '.
0.4 .. 120 ..... ~::.::::, _
' \ 0.2 60
Trot .... .
2 4 6 8 10 12 14 x/0
Fig. 2. Relaxation measurements in a molecular beam. The vibrational temperatures, Tvib of the VJ-mode of SF6 and the v2-mode of CF3Br are shown together with the rotational temperatures, Trot• They are plotted as function of the distance of Raman prohing from the noz:z:le,XR. The curves are drawn through the experimental points for reasons of clarity of display.
IV. VIBRATIONAL HEATING OF SF6 BY A co2-LASER
The co2-laser, tuned to pump the v3-vibrational mode of SF6 (v3 = 948 cm-1) is aligned in such a way that the beam-waist completely overlaps the molecular beam, as close to the nozzle as possible, i.e. in the collision regime. Since the v3-vibration itself is not Raman active, detection is achieved through the Raman active VJ-hot bands, i.e. the red shifted v1-resonances due to the anharmonic shift of the higher vibrational levels of SF6• The position of the detecting Ar+laser lies at XR/D = 2, which is about 1/2 mm downstream. This alignment introduces a delay time in the order of microseconds between co2-laser excitation and Ar+-laser Raman detection, during which hundreds of relaxing collisions can take place. It appears from our measurements, shown in fig. 3., that during this delay time the v3-excitation of SF6 is completely redistributed over all the SF6-vibrational modes in a thermal way, i.e. only one parameter is left to characterize the distribution: the vibrational temperature.
802
> ,.... in z ..., ,.... z
775 770
0 -5
RAMAN SHIFT (cm ·• ) 765 760 755
5'6 ( \11) THERMALIZED STOKES SPECTRUM
• 010 = 2 ~= 2 atm T0 :JOOK
C02-Ias~r : 10 P(22) , 22 Watt, ••/0:0.5
................. co2 -1a~r in. Tvib = 950 K
--- COrlas~r out. Tvib = 240 K
0 00000090 0000~oo••"''"''''''' ''' '' ' ' '"'''''' '•oou•--.. ._,,, , __ .. _ '••••• • -,, . ...,_",,,,,,.,,,, ,
-10 -15 -20 ANHARI«lNICITY SHIFT (cm·•)
Fig. 3. Raman spectrum with and without C02-laser excitation at a short distance from the nozzle. The vibrational temperatures are obtained from a computer simulation5).
A computer simulation, employing the anharmonicity constants of Aboumajd et al.7) confirmed this thermal concept by reproducing the experimental Raman spectra perfectly5). The resulting vibrational temperatures for C02-laser out and co2-1aser in (vlaser = 942 cm-1, 22 Watt, XE/D = 0.5) are 240 K and 950 K respectively (fig. 3.), and correspond with an average absorption of 6 laser-photons per molecule. In order to calculate the C02-laser-excited spectrum of fig. 3. with sufficient accuracy, it turned out to be necessary to include already 1.5 x 106 vibrationally excited levels of the SF6 molecule. Of course the spectrum of fig. 3. is an example of a large row of thermalized spectra at XE/D = 0.5, taken at various C02-laser lines and powers. The case of fig. 3. was selected because it corresponds to the maximal amount of vibrational heating obtained so far with our equipment. Increasing the value of XE/D has two effects. The Raman spectra loose their thermal character and the average amount of C02-laser photons absorbed decreases. An extreme case is shown in fig. 4. where the Raman and C02-1aser were chosen to overlap at a large distance from the nozzle opening, XE/D = XR/D = 10. Thermal redistribution is found to be practically absent. The Raman spectrum of fig. 4. shows that
803
approximately 10 % of all SF6 molecules are disposed into the v3 level by the co2-laser.
775
0
\ ....
RAMAN SHIFT(cm·' ) 770 765
5f6 (V1) STOKES SPECTRUM
..,0=10 Pa=2atm T0 : 300K
co2 - 1as~r : 10P(16) , 20Watt . x. /0 : 10
co2-1as~r in
COrlascr oul
···-............ x13 = -3.0 cm·l
······ ................... 0 •
-5 -10 ANHARMONICITY SHIFT (cm·')
Fig. 4. Raman spectrum with and without COz-laser excitation at a large distance from the nozzle.
V. SF6-CLUSTERS
When a molecular beam containing SFfi-clusters is irradiated by a co2-laser, tuned to the appropriate frequency, these clusters will absorb a photon, leading to predissociation of the van der Waals bond and the removal of the remaining fragments out of the beam due to recoil effects. Very recently COz-laser predissociation spectra of SFfi-clusters have been observed and analyzed4J. New information about cluster bonds and s tructure can be obtained using this laser predissociation technique to modulate the cluster contributions in the Raman spectrum in the neighbourhood of the Rayleigh transition8). Seeding the SF6-beam with 50 % Ar conditions have been chosen to assure dimer contributions mainly~,9), Moreover, the COz-laser was tuned to the P(8) transition (955 cm-1), where among the small clusters only the (SFQ)z dimer exhibits streng absorption. The resulting co2-laser modulat1on spectrum of fig. 5., recorded at XE/D = XR/D = 2, is observed as an attenuation of the unmodulated Raman spectrum, also shown in fig. 5.
804
-2
Fig. 5.
-3 -4 -5 -S RAMAN SHIFT (CM-"J
( SFs }2 ANTI·STOKES spcctrum
X/0=2. 0=0.25 mm
To=300K, Pc,=3atm 50% SF6 in Ar
C~-lascr MODULATION : = in
-- :out
Raman spectrum with and without co2-laser modulation; X~/D = XR/D = 2. The abscissa correspond to the anti-Stokes shtft of the Raman spectrometer position with respect to the maximal signal at the Rayleigh line. The Raman signal observed in absence of co2-laser modulation is represented by the thin curve. The attenuation signal of the thick curve, produced by the C02-laser modulation is very weak and has been multiplied by a factor of 500. Its zero level has been shifted upwards; resolved structures in this attenuation spectrum are indicated with arrows.
Although there isn't even a qualitative spectral analysis available yet, we associate the structures indicated by the arrows in fig. 5. with the van der Waals stretch vibration of (SF6) 2. Further work is in progress.
REFERENCES
I. G. Luijks, S. Stolte and J. Reuss Chem. Phys., 62, 217, (1981)
2. D.R. Coulter, F.R. Grabiner, L.M. Casson, G.W. Flynn and R.B. Bernstein, J. Chem. Phys., 73, 281, (1980)
805
3. M.I. Lester, D.R. Coulter, L.M. Casson, G.W. Flynn and R.B. Bernstein, J. Phys. Chem., 85, 751, (1981)
4. J. Geraedts, S. Stolte and J. Reuss, Z. Phys. A, 304, 167, (I 982)
5. G. Luijks, J. Timmerman , S. Stolte and J. Reuss, in preparation
6. G. Luijks, S. Stolte and J. Reuss, to be published
7. A. Aboumajd, H. Berger and R. Saint-Loup, J. Mol. Spectr., 78, 486, (1979)
8. H.P. Godfried, Thesis, Amsterdam (1982)
9. J. Geraedts, private communication (1982)
806
TIME-OF-FLIGHT AND ELECTRON BEAM FLUORESCENCE
DIAGNOSTICS: OPTIMAL EXPERIMENTAL DESIGNS
ABSTRACT
N.G.Preobrazhensky_and A.I.Sedelnikov
Institute of Pure and Applied Mechanics Academy of Sciences, Siberian Branch Novosibirsk 630090, USSR
A computational study has been conducted to define the optimum of measurements for the time-of-flight and electron beam fluorescence experiments in rarefied gas stream. Tichonov's regularization and Fisher's matrix techniques are used. New iteration procedure convenient in optimal experimental design is proposed.
INTRODUCTION
In the process of measurements carried out in rarefied gas dynamics an experimentalist usually records a rather complicated response of the detector to the properties of the stream under ~tudy. Resteration of stream characteristics by this response is very often an ill-posed inverse problern of mathematical physics. The success of its solving depends on the amount of useful information which the experimental data contain. Randern noise and systematic distortions caused by the instruments are the main factors decreasing this nontrivial information.
We wish to analise the features of the two well-known types of diagnostic experiments in rarefied gas: time-of-flight and electron beam fluorescence techniques. Here unknown quantities are either velocity distribution function (DF) of molecules in gas stream or some gasdynamic parameters for a priori postulated form of DF. So we meet functional or parameterized inverse problem.
807
BASIC EQUATIONS
If one neglects the distortions along the length of ionization gauge it is possible to write the following integral equat i on for the time-of-flight signal u (o~ tJ1:
( 1)
where
f(v) - velocity DF, K(o~ t) is the instrumental function of the molecular beam chopper having the form of isosceles trapezium with the width of 2o, L is the distance between chopper .and the gauge, Te is a characteristic time of system reaction, 0 1 is a constant.
The dependence of K and u on the half-width o arises because of the following reasons. The systematic distortions of the data due to rotating disk chopper with angular velocity w1 are correlated with o, that is proportional to 1/w1. Hence, the increase of o causes the increase of systematical distortion. In fact with the help of (1) on~ considers the set of integral equations for several regimes of measurements with various values of parameter o.
Besides the increase of systematic distortions due to o the growth of the signal Level u takes place. This growth may be seen in Fig. 1: herein relative units the dependence of it maximum value umon 6 i s obtained by calculations (curve 1 refers to time-of-flight
I . 6
0 . 8
0 ö
Fig. 1. The dependence of maximum u signal intensity on nondimensional half- width 6 of insturnentat function.
808
signal). At Fig. the condition
6=o/y, y=lt2-t1 l/23 t, and t2 have to satisfy
Function z(t) is determined by relation (2) at f(v) being the form exp[-(v-VJ] 2 • For calculations one can assume v=1 and use the variable v in nondimensional units. A monotone rise of u with o causes an increase of signal/noise ratio. In other words it results in decrease of the relative noise Level. This way a rise of o causes simultaneously two effects: an increase of systematic and decrease of random noise distortions.
As result among a variety of regimes of measurements there is a most informative one. To find it is a very impotant for precise measurements and therefore a problern of the optimal experimental design arises. The solution of this problern is carried out by the methods of mathematical simulation.
A similar situation takes place when the DF is measured by electron beam fluorescence technique2,3. The relation between the f(v) and observed fluorescence spectrum u of the gas excited by the beam is
where K is the instrumental function of spectroscope, c is the velocity of Light, vo is the emitted frequency due to electronic excitation.
(3)
In our theoretical study a Fabry-Perot interferometer, spectrograph and monochromator are considered. The curve 2,3 and 4 in Fig.1 are: a) Airy function4 (idealized Fabry-Perot etalon); b) K(v)=c 13
vE[-ö,ö]; K(v)=03 vE[-ö,ö] (spectrograph); c) K(v)=c,(2ö-Jv]) 3
Jv J ~2ö, K(v)=03 lv J > 2o (monochromator) have been taken; c 1 is a gaugeable constant.
The velocity DF has been taken in exponential form exp(-v2 ) 3
with half-width ö in frequency measurement being equal to Vln2v 0 /c. Theinstrumental function half-width ö has been equal for Airy function to <1-r)~v/(2nr) 3 r is the reflection coefficient of the mirrors, ~v is the interval of dispersion.
FUNCTIONAL INVERSE PROBLEM
Hence the problern of DF restoration reduces to solution of a linear integral equation of the 1-st kind, (1) or (3). To overcome
809
the instability due to random errors of measurements the well-known method of Tichonov's regularization is used5.
As follows from the papers6,7 the mentional above interdependente of a signal Level and a half-width ö gives a minimum error to solution at some value öm.
It is shown in6 that the minimum error of restered velocity DF practically correspond to the m1n1mum error of its gasdynamic momenta if the noise absolute Level doesn't depend on the signal u Level.
In Fig. 2 and 3 the measure of the error in restered solution
(4)
is plotted against 5; ß and o2 are described by
_ oo ~(wJze(w) ß(t, ö)-=L ]K( ö, w) l 2+~(w) exp(iwt)dw, (5)
(6)
where Ze and K are Fourier transformations of exact solution Ze and K, S is the spectral density of experimental noise, a and M(w) are regularization parameter and Tichonov's stabilizator, respectively.
p,% a)
10
s I I
1/ /2 / /
~ / / "'- .......... /
, __
I I
I
6 ö 0 0.3 0.6
p,% b)
0 0.5 1.0
Fig. 2. The plots of measure of restoration error egainst parameter ö: time-of-.flight <a> and electron beam fluorescence technique (b).
810
Fig. 2a refers to time-of-flight experiment. Exact initial functions f(v) are
f(v)=exp-(v-V) 2~ V=1
(curve 1) and
(curve 2>; instrumental function K(6,t) is taken in the form of isosceles trapezium with the ratio of upper to Lower bases equal to 0.95. The noise Level of the function u corresponds to condition that for 6=0.5 its error is equal to 5% of Um· Random noise of signal u has been with dispersion cr~. Value au is assumed independent of the function u Level (solid Lines) or proportional to ~(dashed Lines>.
The similar dependencies for electron beam fluorescence experiment are given in Fig. 2b. Here the initial functions are
f(v)=exp(-v 2 )
(curve 1) and
(curve 2>; function K has been Airy function with various halfwidth 6.
As follows from the previous investigation? the choice of regularization parameter 6 effects on the Location of minimum of p(6) - curve only slightly. Criterium B, used in?, has been taken in calculation of curve in Fig. 2.
The curves in Fig. 2 has illustrated the availability of optimum values 6=6m giving the most precise restoration of velocity DF. However, strictly speaking one can't calculate the curves in these figures because of unknown function Ze(w) <see formula (5)). Therefore some approximation is needed. As a first step the Forier transform of regularized solution Za(w) (determined in accordance with5) instead of ze(w) has been tested in (5). The measure of error p obtained with such substitution for the time-of-flight problern is given in Fig. 3a (dashed- dotted Line>. If to compare this result with the curve p(6) determined by use ze(w) in (5) (dashed Line> it is easy to see the great shift of the point of minimum between two curves.
Further, the iterative procedure for the calculation of optimum has been proposed. The idea of iterative process concludes
811
p,%
\ b)
4 1.4
3 1.0
2 tS 0.6 L---~--~--~--~----L---~~
k
·0.4 0.8 1.2 1.6 2 3 4
Fig. 3. The results of iterative procedure use for the search of optimum conditions at iteration number k=1,2,3 and 4 (a). The dependence of minimum position <Sm of iteration number k (b).
in determination of function p(k)(<S) at k-step by use of Forier transform for solution za determined for <Sm at (k-1) - step:
aM(w) [za(w) I - (k-1)1 (k) 00 tS-tS •
ß (t,o)==~ IK(<S,w) l2 +aM(w) exp(-z..wt)dw,
(7)
(8)
where <S~k-l) is the point of minimum for the curve p(k-l)(<S). The first iteration (k=1) corresponds to the mentioned above function p(<S). Four iterations (k=1,2,3 and 4 with respect to the numbering of curves) are drawn in Fig. 3a, minimum ~f curve p(<S) being the initial approximation. The approach of tS~ ) to the optimum value of <Sm <dashed line) with increasing of k is demonstrated in the Fig. 3b.
One obtains similar results when the iteration procedure is used in the optimal experimental desing of electron beam fluorescence technique.
812
1.0
0.8
0.6
0.2 0.6 1.0
Fig. 4. The dependence of rms error an (1) and OT (2) on parameter 8 (electron beam fluorescence technique).
PARAMETERIZED INVERSE PROBLEM
When considering the problern of rarefied gas diagnostics in parameterized formulation, velocity DF is assumed to be Maxwellian and the sought parameters are usually density, meanmass velcoity and translational temperature. Fisher's matrix G is a convenient measure of the amount of useful information in this case. According to Rao-Cramer inequality8 the elements of inverse matrix G-~re the lower Limits for the elements of covariance matrix which gives the statistical estimation of sought parameters.
We investigated the influence of the instrumental function half-width ö on the lower Limits of the dispersions ai where i is the number of sought parameter. The interdependence of ö and the noise relative Level of experimental curve u has been taken into account. The effect of ö influence ori determination of the density n and translational temperature T by electron beam fluorescence technique may be seen in Fig. 4.
813
As ordinates the Lower Limits of the error an (curves 1) and aT <curves 2) are plotted. Experimental noise a~solute Level au was 1% of u~<maximum of signal u) at the case when o=o/y=0.6(ris halfwidth of Maxwellian velocity DF). Instrumental function was accepted in Airy form (idealized interferometer Fabry-Perot). The solid Lines in Fig. 4 correspond to the independence of au from signal Level u~ the dashed Lines- to the case when au is proportional to (um)V-2.
CONCLUSION
It has been the purpose of this paper to demonstrate the impotance of correct choice of diagnostic regimes for time-of-flight and electron beam fluorescence experiments in rarefied gas. The possibilities of this choice are correlated with the existence of most informative measurements for two considered types of diagnostic techniques. Modern methods of regularization and optimal experimental design may be quite effective to define these regimes.
RE FE RENCES
1. W.S.Young, Distortion of Time-of-Flight Signal, Rev. Sei. Instrum., 44:715 (1973). - -
2. E.P.Muntz, Molecular Velocity Distribution Function Measurements in a Flowing Gas, Phys. Fluids, 11:64 <1968).
3. A.A.Bochkarev, P.A.Rappaport and N.I.Timoshenko, The Measurements of Translational Temperature in Low Density Jets, Journal of Applied Mechanics and Technical Physics, 1:30 (1973)- In Russian.
4. V.I.Malyshev, "Introduction to the Experimental Spectroscopy", Nauka, Moscow (1979) - In Russian.
5. A.N.Tichonov and V.Ya.Arsenin, "The Methods of Solution of Noncorrect Problems", Nauka, Moscow (1979) - In Russian.
6. Yu.E.Voskoboinikov, A.E.Zarvin, A.I.Sedelnikov and Ya.Ya.Tomsons, On Decrease of Error for Molecular Distribution Function Determination. In: "Diagnostics of Rarefied Gas Flows", IThPh SO AN SSSR, Novosibirsk (1979) - In Russian.
7. N.G.Preobrazhensky and A.I.Sedelnikov, The Optimization of Spectroscopic Measurements on the Basis of Regularization Methods, Journal of Applied Spectroscopy, 35:592 (1981).
8. Y.Bard,"Nonlinear Parameter Estimation", Acad. Press, N.Y., San Francisco, London (1974).
814
MOLECULAR BEAM TIME-OF-FLIGHT MEASUREMENTS IN A NEARLY FREEJET
EXPANSION OF HIGH TEMPERATURE GAS PRODUCED BY A SHOCK TUBE
ABSTRACT
Norio Takahashi and Koji Teshima
Department of Aeronautical Engineering Kyoto University, Kyoto 606, Japan
Translational relaxation in a nearly freejet expansion from a very high temperature source has been studied using a shock heated gas as a source. It has been shown that Toennies and Winkelmann's theoretical relation can almost be applied to the expansion flow from the high temperature source up to 10,000 K for argon.
INTRODUCTION
Translational nonequilibrium in a freejet expansion has been studied extensively by many authors. 1- 4 Anderson and Fenn1 have reduced a semi-empirical expression of the terminal Mach nurober with assuming a sudden freezing model. Miller and Andres2 have made the theoretical analysis using an ellipsoidal velocity distribution function and the moment method of the Boltzmann equation. Brusdeylins and Meyer3 have made a precise time-of-flight experiment for a wide variety of molecules and have given empirical relations of the terminal speed ratio of monatomic, diatomic and polyatomic molecules separately. Toennies and Winkelmann4 have calculated the terminal speed ratio of monatomic gas with the same method as Miller and Andres', and succeeded to explain the experimentally obtained large speed ratios of helium5,6 from high pressure sources, by taking account of the"quantum mechanical effect on the collision cross section. For the source condition where the classical cross section is effective, they have given the folowing relation;
s"c:o = 94 .A. 0 •53 ,
815
where S00 is the terminal speed ratio and given by Soo=(muoo2f2kToo)l/2, m the mass, u00 the terminal stream velocity, T00 the terminal t~anslational tempe~a~ure41~ the Boltzmann constant and A=PodEl/3~ r 0-4/3 (Torr.cm(mev)li3A2K- ). In the scaling parameter A, Po is the source pressure, To the source temperature, d the orifice diameter, and E and Rm are the potential depth and its location for the Lennard-Jones (12,6) potential, respectively. This relation agrees with the experimental results in a wide range of A. However, most of the experiments have been made for the source with room temperature or lower. The present purpose is to study whether the terminal speed ratio from a high temperature source follows this relation or not.
We have used a shock tube as the source, since it can easily raise gas at very high temperature. Although ·the beam from the shock heated source has a short duration (1-2 msec), it has essentially a very high intensity so that the TOF analysis has been successfully made in a single shot, and then the translational temperature, the stream velocity and thus the speed ratio can be obtained for each shot. In the preliminary experiments with small diameter orifices (100 and 200 ~m), the shock heated gas has been extracted from a very narrow region immersed in a thermal boundary layer developed from the room temperature end wall, and thus the effective source temperature has been limited up to 2,000 K.7 Therefore, we have developed a !arge orifice apparatus to get higher effective source temperatures. Although a short conical nozzle has been used instead of a pure sonic orifice, we have assumed the flow as a nearly freejet, because the nozzle expansion angle is !arge and the ratio of its length to its throat diameterissmall (2.5).
APPARATUS
Experimental apparatus consisted of a molecular beam time-offlight apparatus and a shock tube. Two different geometries of the orifice sections were used as shown in Fig.l(a) and (b): one had a small orifice diameter of 100 or 200 ~m, which was attached on the end wall surface of the shock tube having 89 mm inner diameter and 3 m length, the other had a short length conical nozzle with !arge diameter throat of 3.2 mm, 60° divergent angle and 8 mm divergent section. This was attached to the end wall of the shock tube having 128 mm inner diamter and 4.7 m length. The use of the short-length nozzle in place of the sonic orifice was due to the geometrical restriction of the expansion section as shown in Fig.l(b). In the small orifice apparatus, the expansion chamber was evacuated by a 700 1/s mechanical booster pump, and was kept at a pressure less than 5X1Q-3 Torr. A skimmer with 0.67 mm inlet diameter was used. The skimmer-collimator and the flight chambers were evacuated by 600 1/s oil diffusion pumps, the detection chamber bl a 250 1/s oil diffusion pump and were kept at pressures of 10- , 6xlo-7 and
816
S~immcr-
~ C~mbtr
Shutter
Orific. S~immtr
(a) (b)
Fig.l. Shock tubeend sections and expansion sections:(a) with small diameter orifice of d=lOO or 200 ~m:(b) with shortlength and large-throat nozzle of d=3.2 mm.
-7 3Xl0 Torr, respectively. A quadrupole mass filter and a secondary electron multiplier were used as a beam detector. The time-of-flight system had a slit function of a half width ~tFWHM=38 ~s, a flight length Lf=67 cm and a detector's ionizer width 61=7 mm. Although the gas in the driven section of the shock tube flowed into the expansion chamber continuously before firing the shock tube, the decrease in the setting initial pressure was negligibly small.
On the other hand, in the large orifice apparatus the expansion chamber was 0.92 m3 in volume and was evacuated to a pressure of lo-4 Torr by a 600 1/s diffusion pump. A skimmer with 3 mm inlet diameter was used and was located at 275 mm downstream from the nozzle throat. The flight and the detection chambers were evacuated by 3000 1/s and 250 1/s diffusion pumps to pressures of lo-5 and lo-6 Torr, respectively. A Bayard-Alpert type ionization gaugewas used as the beam detector. The time-of-flight system had ~tFWHM=26 ~s, Lf=l35 cm and 01=15 mm. A piece of 4 ~m polyester film was placed at the throat section in order to prevent the gas in the shock tube from flowing into the expansion chamber before the shock tube was fired, and furthermore a sliding shutter was mounted at the exit of the
817
nozzle and was shut off immediately after the shock heated gas was expanded in order to protect the vacuum system from a sudden increase in the background pressure due to the !arge influx from the shock tube source. The shutter worked about 30 msec after the shock tube firing and the pressure rize in each chamber was less than 0.1 Torr in the expansion chamber, 3xlo-4 Torr in the flight chamber and 10-5 Torr in the detection chamber, respectively.
The experiments were made in a source temperature range of To =800-10,000 K at the source pressure Po=730-2,300 Torr for argon and neon (only in the small orifice apparatus) as the test gas. The detected TOF signals were stored into a transient memory recorder at a sampling rate of 2 ~s/word x 1024 words with photocell signals. These data were transfered into a micro-computer and were fitted to a single Maxwellian velocity distribution function convoluted a chopper slit function on it to determine the parallel translational temperatures and the stream velocities, and thus the speed ratios.
RESULTS AND DISGUSSIONS
Typical TOF spectra are shown in Fig.2(a) for the source condition of Po=l,040 Torrand To=9,150 K andin Fig.2(b) for Po=l,780 Torrand To=4,000 K, with the large orifice apparatus. In Fig.2(a), the intensity of each TOF spectrum is almost constant during its duration time although a gradual increase in the base line of the signal is observed because of the increase of the background molecules in the detection chamber. On the other hand, in Fig.2(b) the
(a)
200 ~s
Po=l,040 Torr and To=9,150 K. (b) T'i't-130.0 K, u00=3.0 x 105 cm/s, SU:l3.24 and Toeff=9,040 K.
200 ~s
Po=l,780 Torrand To=4,000 K. 1)'1=38.0 K, uro=2.02 x 105 cm/s, SW:l6.1 and Toeff=3,910 K.
Fig . 2. Typical TOF spectra and the corresponding photocell signals obtained for argon with the large orifice apparatus.
818
intensity rapidly decreases. This attenuation of the beam intensity began to be observed from the source density corresponding to Po= 860 Torr and To=6,500 K, and became strenger with increasing the source density.
The effective source temperature was calculated from the terminal flow velocity and the terminal temperature determined from the TOF spectrum assuming the energy conservation during the expansion. The comparison of the source temperature between the measured value and the theoretical one which is calculated by the measured incident shock Mach number is shown in Fig.3. The measured effective source temperature agrees very well with the theoretical value for the large orifice case, while for the small orifice case it tends to saturate with increasing the theoretical source temperature; 2,000 K for 200 ~. 1,500 K for 100 ~ with argon, 1,600 K for 200 ~m and 1,000 K for 100 ~ with neon. These significant decrease in the effective source temperature is due to the thermal boundary layer effects, and the detail discussions are given in ref.7.
The obtained speed ratios are plotted against the scaling parameter A in Fig.4 as well as the theoretical relation given by Teennies and Winkelmann.4 In the calculation of A the measured effective source temperature has been used for the small orifice case. The measured speed ratios are much smaller than the theoretical prediction and seem to have a different dependency on A for the
o 3-2mm- Ar
• 200wn- Ar t. .. -Ne
x 1001J.m- Ar ,. .. -~
5 To ttwory(K)
Fig.3. Comparison between the measured effective source temperature and the theoretical source temperature which was calculated by the measured incident shock Mach number.
819
8• Vl
X e• t>.a...i: -x•x •
• • x"x ··-~ • X ·• -.t# • o J .2mm -Ar
• 200/lm-Ar 1> 200/lm-Nt!
x 100 11m-Ar
•100/lm-Nt!
Fig.4. Gorrelation between the measured speeg ratio and a scaling parameter of Toennies and Winkelmann.
small orifice case. This is caused by the limitation of the pumping capacity of the present vacuum system. We have made steady beam experiments with room temperature gas using the same arrangement and have found that the speed ratio of argon agrees with Toennies and Winkelmann's relation up to P0d=2 Torr•cm with 30 ~m orifice, and above this value the speed ratio becomes smaller than their relation. Therefore, we can consider the flow rate at Pod=2 Torr.cm with d=30 ~m is the maximum flow rate in this apparatus, for which we can obtain the theoretically predicted speed ratio. In the shock tube operation, the test gas flows continuously into the expansion chamber at flow rates of 1/3-7 times of this critical value before firing the shock tube, so that the speed ratio has become much smaller than the theoretical prediction due to the insufficiency of the pumping capacity. For the large orifice case, the measured speed ratio is close to the theoretical relation up to A=0.05, but its increase becomes slower than the theoretical relation in the range of A=O.OS-0,08 and above A=0.08 it becomes saturated. The value of A=0.08, corresponding to the source condition of Po=860 Torr and To=6,200 K, consists with the critical point that the intensity of the beam began to attenuate prominently during its duration time as shown in Fig.2(b). Thus, above A=0.08, the flow rate becomes solarge that the expansion chamber is rapidly pressurized to yield an appreciable attenuation, and also that the source density becomes high enough to make the skimmer interaction and the background scattering severe. On the other hand, at ·N:, 0.08, the flow rate becomes small not to pressurize the expansion chamber and furthermore the source density
820
becomes low enough to keep the skimmer Knudsen number high. Thus, these skimming effects become weaker with decreasing A. ForA< 0.05, the measured speed ratio can be considered to be free from these skimming effects, but are still slightly lower than the theoretical relation. This difference in the speed ratio for A<0.05 may be caused by the collision effect contributed by the repulsive potential term, which becomes significant to the collision at the high temperature or the high collision energy. Toennies and Winkelmann have included only the attractive potential term, which dominates the collision process at the lower temperature, in the evaluation of the collision cross section. In the source temperature range of To= 7,000-10,000 K, the contribution of this repulsive term may be significant to the cooling process during the expansion, and this contribution may yield a lower speed ratio than their prediction.
Although the experiments are still in progress, larger values of speed ratios which follow the theoretical relation could be expected for larger values of A {>0.05), if we choose a proper arrangement of the experimental set up to perform an ideal skimming.
REFERENCES
1. J. B. Anderson and J. B. Fenn, Velocity Distributions in Molecular Beams from Nozzle Sources, Phys. Fluids,~. 780 (1965).
2. D. R. Miller and R. P. Andres, Translational Relaxation in Low Density Supersonic Jets, Proc. 6th Int. Symp. R.G.D., Vol.I, L. Trilling and H. V. Wachman, ed., Academic Press, New York, 1385 (1969).
3. G. Brusdeylins and H. D. Meyer, Speed Ratio and Change of Internal Energy in Nozzle Beams of Polyatomic Gases, Proc. 11th Int. Symp. R.G.D., Vol.II, R. Campargue, ed., Commissariat a l'Energie Atomique, Paris, 919 (1979).
4. J. P. Toennies and K. Winkelmann, Theoretical Studies of Highly Expanded Free Jets: Influence of Quantum Effects and a Realistic Intermolecular Potential, J. Chem. Phys., 66, 3965 (1977).
5. R. Campargue, A. Lebehot and J. c. Lemonnier, Nozzle Beam Speed Ratios Above 300 Skimmed in a Zone of Silence of He Freejets, Proc. 10th Int. Symp. R.G.D., Part II, J. L. Potter, ed., AIAA, New York, 1033 {1977).
6. G. Brusdeylins, H. D. Meyer, J. P. Toennies and K. Winkelmann, Production of Helium Nozzle Beams with Very High Speed Ratios, Proc. 10th Int. Symp. R.G.D., Part II, J. L. Potter, ed., AIAA, New York, 1047 (1977).
7. K. Teshima, N. Takahashi and M. Deguchi, An Application of the Molecular Beam Time-of-Flight Technique to Measurements of Thermal Boundary Layer Effects on Mass Sampling from a Shock Tube, Proc. 13th Int. Symp. Shock Tube & Waves, C. E. Treanor and J. G. Hall, ed., State University of New York Press, Albany, New York, 116 (1981).
821
XII. ELECTRON BEAM DIAGNOSTICS
ELECTRON-BEAM DIAGNOSTICS OF HIGH TEMPERATURE RAREFIED
GAS FLOWS
N.G.Gorchakova, L.I.Kuznetsov, and V.N.Yarygin
Institute of Thermophysics Siberian Branch of the USSR Academy of Seiences Novosibirsk, 630090, USSR
INTRODUCTION
Today the electron beam is one of the main methods for investigation of rarefied gas flows allowing us to determine a series of local parameters, such as density, partial concentrations, temperatures of internal degrees of freedom, etc. The main idea of the method is to determine gas parameters based on the radiation spectrum excited by a fast electron beam. The above-mentioned spectrum covers the range from X-ray region to visible one. The relation between the intensities excited by electron beam spectra and local gas parameters is provided either theoretically or by a calibration curve.
The elctron-beam technique is characterized by a good spatial localization in combination with small disturbances in the region under consideration. Ultra-violet and visible beam-induced radiation is most generally employed for diagnostics1-5. However, application of these techniques for the investigation of multicomponent gas flows is connected with difficulties. One of them is the overlap of individual spectrum components in an optic region, which is impossible to be accurately taken into account. The contribution of "alient" spectrum is the function of not only density, but also temperature of internal degrees of freedom of gas molecules.
The other difficulty arises when the temperature of gas under study increases up to several thousand degrees. In this case there appears the underground gas radiation in an optic region which masks the electron-beam-induced radiation. The use of X-ray spectrum region6-8 occured to be the most appropriate for overcoming the above difficulties and allowed the capabilities of electron-beam
825
diagnostics of rarefied gas and plasma to be increased.
The aim of the present paper is to analyse the possibilities of electron-beam diagnostics for probing gas flows in order to develop the measurement method for high temperature rarefied flows of gases (including plasma) and their mixtures.
EXPERIMENTAL ARRANGEMENT
The experiments were performed in a low density chamber9,10. For heating of gases ohrnie or arc heaters were used. The system of measuring intensity in optic and braking X-ray radiation excited by an electron beam was completed with the characteristic X-ray measurement system. The characteristic X-ray radiation spectra were studied by the dispersionless method. The proportional counter with a higher sensitivity in the range of soft X-ray radiation was used as a detector. The signal was amplified by the pre-amplifier selected by the differential discriminator and then passed to the recording potentiometer.
ELECTRON-BEAM TECHNIQUES FOR MEASURING RAREFIED GAS DENSITY
The methods of measuring gas density by the radiation intensity of the optical spectrum range have been developed rather well1-5. In Ref. 5 the generalized tables of spectral transitions are presented which are suitable for the measurement of local gas parameters in C02, CO, H2, H20, N2, He, Ar.
The idea of the density measurement by the braking X-ray radiationwas suggested in Refs. 6,7 and developed in Refs. 8,10. The integraT-intensity of X-ray rad1ation of gaseous target is directly proportional to the number density of gas, and the number of y-quanta recorded by the X-ray detector is expressed as follows:
(1 )
Here c is the coeffifient depending on the interaction cross-section and detector aperture, V is the acceleration voltage of the electron beam, va is the threshold of recording by the detector of y quanta in the region of a soft X-ray radiation, Z is the atomic number, n is the atom concentration, i is the electron beam current.
The characteristic X-ray radiation intensity obtained with the use of Tompson model for determination of the ionization crosssection of internal atom shells8,11 is expressed by
I .=Bin( V- V.)/ (V, 2 - V .)~N J J J
(2)
where Vj is the potential of j-shell ionization, B is the parameter
826
depending on a statistical weight of Level, the probability of transition and frequency of y-quantum <B=idem for some definite kind of atoms and Line of spectra).
It follows from (1) and (2) that the intensitied (or N) of braking and characteristic radiations are directly proportional with the concentration of taget atoms.
The important peculiarity of the electron-beam technique is the possibility to make measurements at any temperature of the object to be studied, since the inner shells do not influence the measurement data. The gas flow velocity does not also influence the results, since the time of y-quanta radiation ranges from 10-8 to 10-13 s.
The investigations of braking X-ray radiation of some gases10 confirm the theoretical assumptions and allow to construct the universal dependence of radiation intensity on determining parameters (Fig. 1).
The investigation of characteristic radiation intensity dependence in Ar over the range i=0.0?-20 ma, v-=3-25 kV, n=5·1020-4.2·1021 m-3 confirm a Linear dependence of the characteristic radiation intensity vs electron beam current and concentration8,10. Presented in Fig. 2 is an example illustrating thecharacteristic X-ray radiation spectrum for Ar. The radiation intensity peak corresponds to Ar K~ß established by comparing spectra of various elements.
The estimates show that the radiation absorption between the detector and electron beam, as well as the effect of fluorescent radiation and disturbances created by the electron beam in the gas do not influence significantly accuracy and Locally of measurements in the X-ray region. The upper Limit of the method with respect to atom concentration is determined mainly by electron beam scattering in the medium. The Lower Limit is determined by the apparatus
I/iZ2
imp./ma.s
10- 1 /
1020
~ ~
y
'15
J' _dt!:
~
-
• N2 t::. 02 o Ar
Fig. 1. Universal calibration curve.
827
Fig. 2.
I,imp./s 2000
1000
0
Ti Fe Cu
~ lL 10 50 100 discrimination, V
Characteristic spectrum of argon.
sensitivity, detector aperture and beam current. Thus for Ar nmax is about 3·1021 m-3 with v-=20 kV and 0.1 m beam Length, and nmin is about 3·1018 m-3 with i=1 ma, a spatial resolution about 10-9 m3 ~ a relative solid angle of the detector of 10-5 , and the signal integration time Less or, approximately, equal 10 s.
It should be noted that the measurement technique for a Local gas density developed in this work by a characteristic radiation increases the possibilities of electron-beam diagnostics, since it allows to measure partial densities in multicomponent gas mixtures and to make investigations in the vicinity of walls. Besides, it was established9 by special measurements that the use of X-ray region Leads to a better spatial Localization of measurements. A typical beam size determined by the radiation intensity in X-ray region is much Less than that measured in an optic region under the same conditions of beam formation. The matter is that the secondary electrons do not effect the radiation excitation in the X-ray spectrum region, since their energy Level does not exceed tens electron-Volt. These data were obtained by using electric ganges11.
PARTIAL CONCENTRATION COMPONENT MEASUREMENTS IN GAS MIXTURES AT HIGH TEMPERATURES
The basic idea of this methods is to use and simultaneously record the beam-induced radiation in gas in different spectrum regions such as X-ray and optical ones.
For gas mixtures (especially at high temperatures) the difficulties arise when using bands recommended for measuring density. Let us consider this problern using the mixture of C02 and N2 heated up to 300 to 1200 K. The analysis12 of N2 and C02 spectra in the
0
optic region (A=2850-5000 A) induced by the electron beam shows that their spectra are widely overlapped. Only in the C02 spectrum (Fig. 3>
828
Fig. 3.
To=300 K n=1,22·1015 cm-3
x/d;20 i =2,36 ma
To=1220 K n=0,3·1015 cm-3
x/d;20 i =4,6 ma
Effect of To on the spectral intensity radiation of C02 •
the bands (B2~-x~ng transition of CO~ ion) over_the wave Length range about 2890 A are actually free of overlapp1ng by N2 bands, and therefore can be used for an independent measurement of C02 concentration in the mixture containing N2 • However, when recording the integral value of its intensity it should be noted that with increasing the Stagnation temperature To (Fig. 3), and, respectively, T~ and Tr at the measurement point, there occurs the intensity redistribution in the CO~ spectrum, and short-wave' subbands intensities of transition increase. This takes place due to excitation of higher vibrational Levels of this electron transition. But in this case the value of integral band-system intensity of B2E~-X2 ng transition helds.
In the N2 spectrum over the above-mentioned wave range there are no bands completely free of C02 spectrum overlapping, sufficiently .intensive and suitable for density measurements. The contribution of C02 into (0-0) and <0-1> first negative system N~ bands usually used for N2 density measurements, is strongly dependent on not only C02 concentration, but also the Levels of stagnation temperature To (Fig. 4), and Tv and Tr respectively. Due to this fact, taking into account the contribution of an "alien" spectrum with the use of an optic region becomes almost impossible.
The above difficulties were overcomed by additional measurements in the X-ray region. The radiation intensity (the braking radiation is used) is proportional to a total component concentration nE: IE=IN2 +Ico2 • The value of I co2 is determined by nco2 known from optic measurements taking into account the calibration curve of the X-ray radiation intensity vs C02 concentration, Ix-rayl i =f(nco2 ).
The difference IE-Ico2 =IN2 determines the contribution of N2
component <I N2 ) and, thus, its concentration nN2 <through the caLibration curve) Ix-ray!i=f(nN2 )).
829
Fig. 4.
+<'I z V)
z u...
+<'I z
.--I
.--.._,
8
6
4
Tr-var t-"'foo:;:::=-r----r-=--+---1 x/ d* = 1 0
300 900 To, K
Temperature effect on integral intensities ratio of N2
and co2 bands.
A similar approach can be also applied for other mixtures, with the excitation spectra overlapping in the optic region.
The density measurement becomes more complicated for gas mixtures heated up to essentially higher temperatures at which there occurs the gas molecule ionization and appears natural plasma fluorescence. The possibilities of electron beam diagnostics at the presence of natural plasma fluorescence were analysed over an optic spectrum range. The spectrum of natural and electron-beam induced radiation of argon plasma over the range 3500 to 6000 Ä was studied. The results show that over the temperature range 2000 to 10000 K in the natural plasma radiation there are all the Lines of single~ ionized argon, Ari and the lines Arii are actually absent. In the meantime, in the electron-beam induced spectrum, tagether with Arl, there is a sufficient number of Arll Lines. Therefore, the lines Arii can be used, in principle, for diagnostics of flows having a natural radiation13. However, a more detailed analysis of spectra induced by the electron beam in plasma shows that with increasing temperature the line intensities become comparable with the Level of continuous plasma spectrum. For example, with increasing gas temperature in the source To of the arc heater from 2000 to 7000 K, the ratio of line intensity Arll at 4609 A to the recombination continuum Level (Fig. 5) decreases from 46 to 1.2, all other things being equal. In this connection it is difficult to use the calibration curve obtained at room temperature for intensity in an optic region vs density. To measure partial concentration in mixtures under the above conditions, it is necessary to record the characteristic radiation of components.
830
Fig. 5.
CONCLUSION
3500 2000 K
Effect of stagnation gas temperature on the Level of recombination continuum in Ar jet.
In the given paper the gas temperature effect on the electronbeam induced radiation spectra in the optic region has been analysed. It has been shown that the application of this region is very Limited, especially for gas mixture diagnostics. Although these peculiarities can be tak~n into account sometimes for density measurements <e.g., corresponding choice and increase of the wave Length region to be registered), radical solution of this problern is the use and simultaneaus recording of radiation in X-ray and optic spectrum regions. The use of the X-ray spectrum region admits, on one hand, to have an independent Local density measurement based on stagnation and characteristic radiation in both pure gases and their mixtures, and, on the other hand, to avoid difficulties connected with the temperature effect. The Latter is of particular importance for investigation of nonequilibrium flows.
The method suggested in our work essentially increases the capabilities of electron-beam diagnostics of high-temperature gases and their mixtures.
REFERENCES
1. E.P.Muntz, D.J.Mardsen, Electron Excitation Applied to the Experimental Investigation of Rarefied Gas Flows, in: "Rarefied Gas Dynamics", Acad. Press, N-Y-L, 2:495 (1963). -
2. S.L.Petrie, Flow Field Analysis in a Low Density Arc-Heated Wind Tunnel, in: "Proc. of the Heat Transf. & Fluid Mech. Inst.", Stanford Univ:-press, Stanford, California <1965).
3. D.J.Sebacher, An Electron Beam Study of Vibrational and Rotational Relaxing Flows of Nitrogen and Air, in: "Proc. of the Heat Transf. & Fluid Mech. Inst.", Stanforduniv. Press, Stanford, California <1966).
4. A.A.Bochkarev, V.A.Kosinov, V.G.Prikhod'ko and A.K.Rebrov, Structure of a Supersonic Jet of an Argon-Helium Mixture in a Vacuum, J. Appl. Mech. Tech. Phys., 11:857 <1970).
831
5. A.A.Bochkarev, V.A.Kosinov, A.K.Rebrov, R.G.Sharafutdinov, Measurement of Gas Flow Parameters Using an Electron Beam, in: "Experimental Methods in Rarefied Gas Dynamics", Inst. of Thermophysics, Novosibirsk (1974) - In Russian.
6. C.A.Ziegler, L.L.Bird, K.H.Olson, J.A.Hull and J.A.Morreal, The Technique for Determining Density Distribution in Low Pressure High Temperature Gases, RSI, 35:450 (1964).
7. A.M.Trohan, Measurement of Gas Flow Parameters by Beam of Fast Electrons, PMTF, 3:81 (1964) - In Russian.
8. L.I.Kuznetsov, A.K.Rebrov, V.N.Yarygin, Diagnostics of Ionized Gas by Electron Beam in X-Ray Spectrum Range, in: "11th Intern. Conf. on Phenomena in Ionized Gases", Pra[!ue (1973).
9. N.G.ZharKova, V.V.Prokkoev, A.K.Rebrov, P.A.Skovorodko, V.N.Yarygin, The Effects of Nonequilibrium Condensation and Vibrational Relaxation in Supersonic Expansion of Carbon Dioxide, in: "Rarefied Gas Dynamics", CEA, R.Compargue, ed., Paris (1979).
10. IN:G.Zharkova, L.I.Kuznetsov, A.K.Rebrov, V.N.Yarygin, Density Measurement of Rarefied Gas and Plasma by Electron Beam, TVT, 1:17 (1976) - In Russian. -
11. L.I.Kuznetsov, Yu.S.Kusner, J.-C.Lengrand, S.A.Palopezhentsev, G.I.Sukhinin, Investigation of Diagnostic Electron Beam Plasma, in: "Diagnostic of Rarefied Gas Flows", Inst. of Thermophysics, Novosibirsk (1979) - In Russian.
12. N.G.Gorchakova, V.N.Yarygin, Measurement of Partial Densities by the Electron Beam Technique in the N2 -C02 Mixture at High Temperatures, in: "Molecular Gas Dynamics". Proc. of the 6th ALL-Union Conference, Inst. of Thermophysics, Novosibirsk, 3:121 (1979) - In Russian.
13. R.B.Fraser, F.Robbin and S.Talbot, Flow Properties of a Partially Ionized Free Jet Expansion, Phys. Fluids, 14:2317 (1971).
832
EXCITATION MODELS USED IN THE ELECTRON BEAM FLUORESCENCE TECHNIQUE
J-C. Lengrand*
Laboratoire d'Aerothermique du C.N.R.S. 4 ter, route des Gardes, F 92190 Meudon (France)
ABSTRACT
The model proposed by Coe et al. for electron beam excitation of nitrogen is discussed and simplified in a way that facilitates its further application to a larger range of experimental conditions, without affecting its validity in the range where it has proved to be successful. Application of the model to relatively high temperatures and to situations involving secondary processes is made possible.
INTRODUCTION
The main difficulty encountered when deriving rotational temperature from spectroscopic data in electron beam excited nitrogen is to use a correct model for the excitation of Nz molecules by electrons, i.e. a model that gives the correct transition probabilities between the rotational quantum levels of the involved species. Since Muntz 1 proposed the first excitation model based on dipole selection rules, a number of other workers suggested extensions of Muntz's model in order to interpret observed discrepancies between measured and expected temperatures. One of these extensions consists in accounting for possible quadrupole interactions between electrons and excited molecules, giving thus an interpretation of the apparent overpopulation of the highest rotational levels, which is frequently observed.
* Research Scientist.
833
The first "quadrupole model" was proposed by Smith 2 and its practical use was improved by Lengrand and Cloupeau 3 • Another one has been proposed by Coe et al. ~. Smith assumes that quadrupole interactions occur only during excitation of N2 molecules by lowenergy secondary electrons, whereas Coe considers that, due to the just ejected electron, quadrupole interactions may occur even during excitation by fast primary electrons.
All these models lead to the construction of an excitation matrix E, that relates the vector N~ (population of rotational level K) to the vector IK' (iutensity of h.ne K' in the R-branch of a given spectral band of the firstnegative system). In order to determine the coefficients of the excitation matrix E, Coe et al. use a set of 10 constants Pi interpreted as the probability of i quadrupole interactions occuring in addition to the initial dipole interaction. The values of these constants are determined experimentally and are found tobe only weakly dependent on experimental conditions. So they can be considered as universal.
Nevertheless, Coe's model has several shortcomings :
a) Since only 10 Pi have been determined and used for the calculation of the matrix E, the model cannot be applied to spectra involving more than 10 lines. Determining additional P. would probably require adjustment of the presently known values. 1
b) The densit~ dependence observed at relatively high density by several workers ' 5 ' 6 as well as the influence of the distance from beam axis on intensity distribution 6 ' 7 is not consistent with the model. It is probably due to secondary processes (quenching, reexcitation of ions, excitation by secondary electrons, etc.). Finding the variation law of ten (or more) values of Pi with respect to density or other experimental conditions, as suggested in Ref. 4 would be a formidable task with probably some degree of indetermination.
The purpose of the present paper is to reduce all fitting parameters in Coe's model to a single one, which eliminates objection a) and greatly simplifies the task of an eventual further extension of the model to cope with the difficulties cited as b).
CONTENTS
Since Pi is the probability of exactly i quadrupole interactions to occur during the excitation process, it follows that the probability of, at least, one additional quadrupole interaction to occur, after i-1 have already occurred, is
(I)
834
(X)
w
U1
O 0
100
20
0
30
0
40
0
K'C
Kttl
)
-1
_..... ~
._'::f
. \..
Jo
F -2
-3
.......
..........
MU
NT
Z'
MO
DE
L
---·-
SM
ITH
'S
MO
DE
L C
L:0
.25
)
----
P.
fro
m
Re
f.4
CC
OE
'S M
OD
EU
I
+ H
= c:
:.te
=0.
31
T
PR
ES
EN
W
OR
K
o H=
cste
=0.
4
... , .....
......
+' '
--..... --........
........
........
.... ·-
Fig
. 1
-C
alcu
late
d li
ne i
nte
nsi
ty d
istr
ibu
tio
n u
sin
g d
iffe
ren
t ex
cit
ati
on
mod
els
We calculated the values of Hi from Coe's values of Pi• As it can be seen from the results in Table I, when increasing i, Hi first increases from 0.3 to 0.59, then decreases to 0.45. This variation is at least partially due to truncations. Limiting the possible number of quadrupole interactions to 9 in Coe's work leads to an overestimation of Pi, progressively increasing with i. Furthermore, when evaluating Hi using Eq.(I), the truncation of series in j at 9 induces a strong underestimation of H8 and H9 • This suggests that in the physical process of successive quadrupole interactions, the probability of an additional one may be a constant H. This hypothesis leads to :
(2)
We recalculated the excitation matrix E defined by Coe using the values of p. (i < 9) from Ref. 4, and then, using Eq. (2) for different value~ of H, with no truncation on the number of quadrupole interactions. The calculated matrix had dimensions 30 x 30, which allowed considering a large number of rotational levels and spectral lines. For given rotational temperatures, we calculated the resulting spectra, using the different matrices E. Some of the results are plotted in fig. I as Log10 (IKr/K'II) vs K1 (K 1+I). As usual, lines with even values of K' were doubled to account for nuclear spin. As can be seen in fig. 1, the spectra obtained using Coe's values of Pi fall between the spectra obtained with the hypotheses H = 0.3 and H = 0.4, the agreement with H = 0.3 being extremely good for the first quaritum levels {up to IKr/K'II ~ Io-2 ). So, replacing Coe's excitation matrix by a matrix calculated with the hypothesis H = cte (say 0.3) would give nearly identical results in the range considered in Ref.4 and allow reducing spectroscopic data obtained at higher temperature, usually with a larger number of lines. Eq. (2) leads to a linear Log Pt vs i plot whereas Coe's Log Pi vs i curve is non-linear. The fact that both sets of Pi lead to nearly identical spectra suggests that the curvature in Coe's curve is due mainly to truncations.
At this point, it is interesting to point out similarities and differenca;between both "quadrupole interaction" models 2 ''~. In the present formulation, H has the same physical meaning as in Smith's model, where it is actually considered as constant, in the sense that it does not vary during successive quadrupole interactions. Smith assumes a variation of H against the energy E of the incident electron. Such a variation does not exist in Coe's model '~ as all excitations are supposed to be due to the fast monoenergetic primary electrons, but it should be introduced in the eventual extension mentioned above (measurements at higher density and/or outside the beam axis). A rigorous treatment should include averaging of every excita-
836
tion matrix coefficient for all possible electron energies (as was donein Ref.3 for Smith's model) and for all the viewed volume. But due to the difficulty of getting reliable information on the spatial and energetic distribution function of electrons, and on the function H(e), an overall correction is probably preferable, for instance by calculating the excitation matrix as previously indicated for a given value of H, empirically fitted to calibration measurements.
Another difference between Smith's and Coe's models lies in the sign of the ~K (variation of rotational quantum number) induced by the successive quadrupole interactions. Smith assumes that all successive ~ occurring during a given excitation have the same sign, whereas Coe assumes that successive interactions may produce arbitrarily positive and negative ~K. It follows that large values of the final ~K (between the initial and final states of the excitation) are much less probable in Coe's model than in Smith's one, and the former tends to be closer to Muntz's dipole model, as illustrated in fig. I. This general trend is found for every choice of the fitting parameters at least when a large number of lines is considered.
A discussion of the validity of the different available excitation models is complicated by the fact that several physical processes (relaxation, condensation) may have the same effect on intensity distribution as a change in excitation model. Such a discussion has been left deliberately beyond the scope of the present paper.
CONCLUSION
The simplified model presented in the present paper is consistent with the original Coe's model in the range where Coe's model has proved to be successful.
It may be applied to spectra involving a large nurober of lines.
It may be extended to situations requiring to account for secondary processes.
Table 1. Probability of a i-th quadrupole interaction to occur after i-1 have already occurred
i 2 3 4 5 6 7 8 9
H. 0.30 1
0.33 0.44 0.49 0.55 0.59 0.59 0.55 0.45
837
REFERENCES
I. E.P. Muntz, Static Temperatures in a Flowing Gas, Phys. of Fluids, 5, I, pp. 80-90 (1962)
2. R.B. Smith, N2 First Negative Band Broadening due to Electron Beam Excitation, in "Rarefied Gas Dynamics", 6th Symposium, !• pp. 1749-1760, Acad. Press, N.Y. (1969)
3. J-C. Lengrand, M. Cloupeau, Application of Smith's Model to Rotational Temperature Measurements in Nitrogen, AIAA J., !I• 6, pp. 812-815 (1974)
4. D. Coe, F. Robben, L. Talbot and R. Cattolica, Rotational Temperatures in Non-Equilibrium Free-jet Expansion of Nitrogen, Phys. of Fluids, 23, 4, pp. 706-714 (1980)
5. H. Ashkenas, Rotational Temperature Measurements in Electron Beam Excited Nitrogen, Phys. of Fluids, 10, 12, pp. 2509-2520, (1967) -
6. B.L. Maguire, Density Effects on Rotational Temperature Measurements in Nitrogen using Electron Beam Excitation Technique, in "Rarefied Gas Dynamics", 6th Symposium, 2, pp. 1761-1782, Acad. Press, N.Y. (1969) -
7. A.K. Rebrov, N.V. Karelov, G.I. Sukhinin, R.G. Sharafutdinov and
838
J-C. Lengrand, Electron Beam Diagnostics in Nitrogen : Secondary Processes, in "Rarefied Gas Dynamics", 12th Symposium, 74, Progress in~stronautics and Aeronautics, pp. 931-945,-S. S. Fisher ed. , AIAA, N. Y. (1981)
ELECTRON - BEAM DIAGNOSTICS IN NITROGEN
MULTIQUANTUM ROTATIONAL TRANSITIONS
R.G.Sharafutdinov, G.I.Sukhinin, A.E.Belikov, N.V.Karelov, and A.E.Zarvin
Institute of Thermophysics, Siberian Branch of the USSR Academy of Sciences. Novosibirsk State University Novosibirsk, 630090, USSR
INTRODUCTION
The method of electron-beam fluorescence technique (EBFT) is widely employed in molecular gases for determining rotational Level populations and rotational temperature. The most important quantities necessary for a successful application of EBFT are the probabilities of rotational transitions upon excitation of molecules by an electron impact from the ground state.
This paper presents the measurements of rotational transition probabilities when exciting molecules N2(X 4Eg,v=O,k) by fast electrons with energy E~IO keV to Ni(B2 Et,v'=O,k7 state. A particular attention is paid to creating Boltzmann equilibrium distribution of N2 molecules with the known temperature both in a pure nitrogen jet and argon jet, where the molecules of N2 are only a small fraction. Todetermine the probabilities of rotational transitions, a simple model is suggested which is based on an adiabatic approximation.
EXPERIMENTAL INVESTIGATIONS
The experiments were performed in the Low density wind tunnel equiped by cryogenic pumps and electron-beam diagnostics. The experimental arrangement is described in detail by Borzenko et all. Let us consider the experimental data on intensity of rotational Lines of the R-branch of the (0-0) F.N.S. N~ band upon excitation by the electron beam obtained at different values of pod* at some point x/d* along the axis of pure nitrogen jet <po is the stagnation pres-
839
sure and d* is the diameter of the sonic nozzle). A typical result is shown in Fig. 1. In free expanding gas there can appear the disturbance of equilibrium distribution of rotational l evels2. With increasing the collision number in the jet, i.e. with increasing pod*, the deviations from the equilibrium conditions decrease. In principle, Boltzmann distribution of rotational Levels of molecules
to' 10 2
0 OOQ
0
- 1
0 C'
0
- 2 0 0
0
0
- 3
I . le _ k_
-4 r 1·k
Fig. 1. Rotational line intensity distribution in spectrum in wide region of pod* . To=295 K; Ti8 =6.1 K; x/d*=38; n/n0 =6.2·10- 5 •
Equilibrium isentropic flow calculation (y=1.4) by the multiquantum model (~k=±1,3,5, •.• ) - solid lines, by the Muntz model C~k=±1) - dashed line (only for k'=3>.
can be established with temperature equal to isentropic one. However, at rather high values of pod* in supercooled flow there can be developed the condensation process leading to condensation energy release into the flow. Therefore, beginning with a certain value of pod*, the high rotational Level populations increase and, consequently, the same occurs with relative intensities of rotational lines Ik '~ that
840
can be observed at the right-hand branc~es of the curves in Fig. 1. The gas states at pod* ranging within the minimum of distributions of intensities Ik' under the conditions illustrated in Fig. 1 are close to the Boltzmann equilibrium distribution of rotational Levels with isentropic temperature Ti8 =T(x/d*J. The comparison of measured intensities with the calculated ones for T(x/d*) by the Muntz dipole modeL3 (dashed Lines in Fig. 1) shows that gas does not achieve the equilibrium state, or alternatively the dipole model is not correct. The Latter conclusion was made by Coe et al~, but the possibilities of nonequilibrium conditions in gas were not analyzed.
In order to estimate uncertainties of gas state the authors investigate Ar-jets with a small impurity of nitrogen mol~ules (Less than 5%) and a big nozzle diameter <d*=15 mm). In this case it is hoped that in the gas mixture the conditions close to equilibrium are sustained, with an isentropic temperature of one-atomic gas. The measurements in Ar-N2 were made over the wide range of pod* (10~pod*~1000). In Ar-N2,opposite of pure N2 (Fig. 1), the relative intensities at small x/d* are independent of pod*. It means that molecules of N2 are in equilibrium with Ar. The equilibrium in Ar-N2 and non-coincidence of the measured intensities with the calculated ones by the dipole model confirm the conclusion made by Coe et al~ about the presence of multiquantum rotational transitions upon excitation of N2 to the state N~(B2~3 v'=03 k') by an electron impact.
PROBABILITIES OF ROTATIONAL EXCITATIONS
The intensities of rotational Lines Ik' according to Rebrov et al? are equal
k' r/! k' X _X Ik,= 2k'+I c kr 2k'+I c{fPk'k(E)Nk+D(ng3r)f<Pk'k(e)>Nk}3 (1)
where ~,, ~ are the populations of rotational Levels in N~(B2E~3 v'=03k'J and N2(X1E3V=03k) respectively, Pk'k(E)- the probabilities of rotational transitions for primary electrons; <Pk'k>- the probabilities of rotational transitions for secondary electrons. D(ng3P) is the rate of nitrogen excitation to the state ~(B2E~3 V'=O) by secondary electrons related to the rate of excitation to this state by primary electrons, c is the constant.
The experimental data on intensities of rotational lines Ik' with the help of <1> at the known populations N{ can be used for determining the probabilities of rotational transitions Pk'k(E) if a sufficient number of experiments with different distributions of Nf have been performed. However, such a way of determining Pk'k(E) is a typical ill-posed problem. It is more expedient to find transition probabilities using any theoretical model parametrizing the
841
transition probability dependence on quantum numbers k and k'. Within the framewerk of the adiabatic approximation for ~-~ electron. transitions we have
(2)
and, respectivel}·, for <Pk 'k>. The value ( k' k l ) i s the (3-j) m1 m2m3
symbol.
In the case of nitrogen ionization only the terms with l=1 ,3,5, ••• enter the sum (2), i.e. the rotational transitions by ~k=±1,±3,±5, ••• are possible. This follows from the properties of (3-j) symbols equal when m;=m2=m3=0:
(Y1 Y2 Y3) 2= (y1+Y2-Y3)!(y1-y2+y3)!(-y1+Y2+y3)! X
0 0 0 (y1+y2+y3+!)!
X [ ! ]2 (p-y1)!(p-y2)!(p-y3)!
(3)
where 2p=y1+y2+y3. Pk'k(E) and <Pk'k> satisfy the normalization condition:
k+l k' k l 2 ~ Pk'k(EJ=~z (EJ ~ (2k'+lJ(o o oJ ='[.Pz (EJ=l. k, l 0 k '= lk-l J l 0
lf in (2) we restriet ourselves to one term only with l=1, i.e. P1o=1, the transition probabilities are reduced to Muntz model:
k' k 1)2-{ (k'+l)/(2k'+l) for Pk'k(E)=( 2k'+l)(o o o - k'/(2k'-l) for
k=k '+f k=k ,_,.
Thus, the transition probability matrix can be expressed by vector Pz 0 (E) (l=l,3,5, ..• ) that simplifies the procedure of finding it from the experimental data. For further simplification of the model the transition probability Pz 0 (E) is expressed using a fitting parameter a determined from the comparison with the experimental data
l l (2l ) l-1 Pz (E)=(2l+l)a I ~ (2l+l)a = +I a (i-a2) 2 (4)
o (3+a 2 ) • l=l,3,5 ...
When a=O, only one dipole term P1o=1 is remained in (4).
The parameter a=0.28~g:g~ was determined from the experimental data in Ar-N2 jet shown in Fig. 2a, when the population Nf corresponds to isentropic temperature T. The calculated intensities for this
842
0.1 x/d• 10 10 x/d* 100 0 0
-0.5 -0.5
-1 _,
5
-1.5 -1.5
-2 -2
a) b) -2.5 - 2 . 5 \
lk, I k, \ Lg(~) Lg (I , · k')\o
' -3 -3
200 150 10C 50 20 10 T, K 100 i'O 50 30 20 10 5 T, K
Fig. 2. The verification of the multiquantum transitions model for equilibrium conditions in supersonic flow.
o - experiment -- Muntz model ---- multiquantum model
a) mixture 0.95 Ar+O.OSN2 To=293 K pod*=140 torr-mm
b) pure N2 To=291 K p od*=493 torr·mm
value of the parameter a are shown in Fig. 2a in dotted Line (solid Lines - Muntz model).
For the experiments in pure nitrogen the most close to the equilibrium at each x/ d* are the conditions corresponding to minimum intensities vs pod* . The calculation results for rotational Line intensities are given in Fig. 2b, under the assumption that the populations are equal to Boltzmann ones with an isentropic temperature for a diatornie gas. The obtained agreement i s satisfactory. In Table 1 there are the values of Pk 'k which are compared with the probabilities for the dipole transitions and those determined by Coe et al ~ . It should be noted that the elements of Coe matrix and those of the model under consideration are in good egreement for !'J. k=lk '-kl < S.
DISCUSSION
Kine t ic considera t ion of populat ions of rota t ional Levels ~
843
Table 1. The probabilities of the rotational transitions by the excitation of molecules into B2 E-state. Upper values are the present results. Middle values are the results of Coe et al,. Lower values are the results of Muntz3.
0 1 2 3
.8277 • 1514 0 0 .8346 0 .1350
1 0
.2759 .6167 1 .2668 0 .6201 0
.3333 .6667
.3700 .5427 2 0 .3706 0 .5384
.4000 .6000
.0216 .3876 3 .0193 0 .3879 0
0 .4286
.0317 .3984 4 0 .0310 0 .3975
0 .4444
.0017 .0352 5 .0021 0 .0344 0
0 0
.0027 .0377 6 0 .0037 0 .0365
0 0
.00014 .0029 7 .0003 0 .0042 0
0 0
4
0
.0950
.0923 0
0·
.5122
.5108
.5714
0
.4061
.4051
.4545
0
.0374
.0382 0
5
.0187
.0233 0
0
.0775
.0747 0
0
.4963 .• 4948 .5555
0
.4117
.4105
.4615
0
6 7
.0020 0 .0051
0
.0111
.0159 0 0
.0088 0 .0124
0
.0699
.0677 0 0
.0657 0 .0635
0
.4865
.4850 0
.5455
.4799 0 .4783
.5365
.4159
.4147 0
.4667
inexpanding flow2 shows that under certain conditions the rotational Level distribÜtion may be non-Boltzmann one. All the experimental data are to be analyzed from this point of view. Fig. 3, for example, represents the experimental distribution of intensiti~s for one of the regimes the isentropic temperature for which is T=22.5 K. It is shown in Fig. 3 that populations N~ are non-Boltzmann. In the meantime under the given conditions Coe et al~ considers the population distribution to be Boltzmann one with a rotational temperature T~39.5 K~ The calculation of intensities Ik' corresponding to this temperature is shown in Fig. 3.by dashed lines. It is evident that for lower lines there is a satisfactory agreement with the experimental data, but on upper lines a certain difference is observed.
844
0 0
- f
-1,5
-2
-2,5
- .3
25 50 75 !00 125 150
0 0 - Nk
k(k+1)
D. - Ik '} experiment l ~l~--~~----~-- -0,5
~------1- 1
nonboltz.
Fig. 3. The comparison of Line intensities Ik ' in the spectrum of FNS and Level populations Nk in X1 E(v=O) state with results of Coe et al~. To=293 K; pod*=13.5 torr·mm; x/d*=B; Ti 8 =22.5 K
The same result is illustrated in Fig. 4, in the form of spectrograms obtained by the authors _ and by Coe et al~ under the same conditions. The intensities of Lower lines Ik' are in good agreeme.nt that attests to the reproductibility of data. However, in this paper4 the number of registered lines is insufficient. Probably, this allows4 to find the distribution of rotational Levels of Nf in Boltzmann form that results in the high Line intensity decrease, as compared with the experimental data.
In conclusion it should be noted that in this work the probabiLities of rotational transitions Pk'k (E} were determined for excitation of molecules N2 (X1 E;~ v=O~ k) to the state N1 (B2E~~ v '=O~ k ') by high energy electrons with E~10 keV. The above adiabatic approximation makes it possible to concretize Pk 'k(E) dependence on quantum numbers k and k' (see expression 2) and reduce determination of them to finding the vector Pz0 (E). For simplicity its dependence
845
0 50 100 150 200 250 0
~ I / k'(k,+1)
~ ~ 0 - x/d .. = l 4: Tis= 40 K
""' ~ ~ • •
-0 - Y./d = 8: Tis= 22 !" • ~ I~ l1 - x /d,..=J2: T. = 7 K
n • ~s -I • I ·2
I ~ I ~ • I •
I , i • • k ... • le -- ... • I 1 ·k ...
·3
4
Fig. 4. The comparison of Line intensity distributions obtained by Coe et al~ (Light signs) and in present work <dark signs).
on l is prescribed in the form of (4), but Later on it can be determined by more precise experiments or theoretically. The Pz0 (E ) dependence on electron energy E must appear near the excitation threshold of the state N~ (B2~3 V 13 k 1 ) only, where the multiquantum transition probability increases.
REFERENCES
1. B.N.Borzenko, N.V.Karelov, A.K.Rebrov, R.G.Sharafutdinov, The experimental study of rotational Level population of molecules in free jet of nitrogen, J.Appl.Mech.Tech.Phys., 17: (1976).
-2. R.G.Sharafutdinov and P.A.Skovorodko, Rotat1onal Levels population kinetics in nitrogen free jet, in: Progress in Astronautics and Aeronautics, S.S.Fisher, ed., AIAA, N.Y., 74:754 (1981).
3. E.P.Muntz, Static temperature measurements in a flowing gas, Phys. Fluids, 5:80 (1962).
4. D.Coe, F.Robben, L.Talbot and R.Cattolica, Rotational temperatures in nonequilibrium free jet expansion of nitrogen, Phys. Fluids, 23:706 (1980).
5. A.K.Rebrov, N.V.Karelov, G.I.Sukhinin, R.G.Sharafutdinov and J.-C.Lengrand, Electron beam diagnostics in nitrogen: secondary processes, in: Progress in Astronautics and Aeronautics, S.S.Fisher, ed., AIAA, N.Y., 74:931 (1981).
846
XIII. FREE JETS, NONEQUILIBRIUM EXPANSIONS
FREE JET AS AN OBJECT OF NONEQUILIBRIUM PROCESSES
INVESTIGATION
A.K. Rebrov
Institute of Thermophysics Siberian Branch of the USSR Academy of Seiences Novosibirsk, 630090, USSR
A current definition of non-equilibrium phenomena includes the notions of the processes in which the degree of the nonequilibrium (the relaxation process depth) effects significantly on gas-dynamical characteristics of flow or its radiation. In the general case gas is considered as a multi-temperature system. The degree of the nonequilibrium is determined by the temperature difference of sub-systems, and the gas system capability to approach equilibrium is defined from the ratio of relaxation time to a characteristic gasdynamical one.
In a free jet of gas expanding into a low-density medium a local relaxation time can vary by orders. To characterize a relaxation capability of gas in a jet it is appropriate to use the criterion
P=z (ljt)dt/ds ~ (1)
where l is the molecule mean free path, t some parameter determining gas state, s the coordinate along the current line, z the number of collisions for attaining equilibrium in any degree of freedom or
l dt between degrees of freedom. The value t da , in its physical sence,
is a relative variation of parameter t along the length e, in the other words, Knudsen local number with respect to its inner scale,
The role of the inner scale 1/t ~; is established for a near con
tinuum by an exact solution of Boltzmann equation for small gradients of parameters1. By the value of this scale it is possible to determine the deviations from nonlinearity of laws of molecular mass transfer, momentum and energy. The comparison of the above scale with the mean
849
free path characterizes the medium discontinuity, and for this purpose it has already been used in literature. The factor z generalizes this criterion for different relaxation processes. For a uniform translationally relaxating gas the value P of the order of 0.01 corresponds to the origin boundary of the nonequilibrium. Bird2 has found for axisymmetrical jet p>0,04.
The use of the above criterion allows to characterize an axisymmetrical free jet axpanding into the low density medium as a very universal object for the nonequilibrium process investigation. In the jet it is possible to organize a wide variaty if not any nonequilibrium processes which can take place in gas flows. Let us consider the capabilities of jet in the region, where the flow is hypersonic <M> 3) and close to the radial one.
Variation of some gas parameter obeys the law fo =B/(s/do)m. Here to is the stagnation parameter, B some constant defined by flow formation, do is the characteristic geometrical size of a source, m > 0 means the decrease in the parameter value during expansion. Assuming that a relative density variation takes form n/no=B(s/do) 2 ,
and the elastic collision cross-section and the number of relaxing collisions are the functions of temperature (a(T) and z(T)>, based on (1), we have for spherical expansion
P=I/B(To/Podo)(s/do)[z(T)/a(T)] (2)
where Po, To are the stagnation parameters in the source, real or effective. This follows from (2) that due to free unlimited expansion, it is possible in principle to obtain any reasonable degree of nonequilibrium at any parameters which are available in a jet expansion. This simple idea has already manifested itself in a great number of works concerned with nonequilibirum processes in a free axisymmetrical j et.
It is worthy to note that for a cylindrical source over the . f h . f l P To z (T) . h . l" b . reg1on o ypersomc ow ~ Podo a(T), 1.e. t e nonequ1 1 rlUm
along the streamline is variated in accordance with temperature variations of the cross-sections of elastic and inelastic interactions. Due to this fact, for example the transition to translational freezing can be absent.
A typical structure of an axisymmetrical jet is shown in Fig. 1 a-c. Not going into the well-known details of its description3-7 it should be noted that the region of a one-dimensional flow in a free jet, vacant from background gas is of particular interest when investigating nonequilibrium processes; it can be a jet core, Mach disk and weakly gradient flow behind Mach disk.
850
Fig. 1
' I
J~ -
a
b
c
(a) 1 - the inviscid expansion zone; 2 - the zone of viscosity or rarefaction effects; 3,4,5 - Mach disk, barul and reflected shock waves, respectively; 6- inviscid section of shock Layer; 7 - inviscid flow behind Mach disk; 8- inviscid flow behind the reflected shock wave; 9- the mi .. ing zone along the jet boundary; 10- the mixing zone behind the triple point. (b) 1 - inviscid expansion zone, 2 - zone of merged shock Layer, mixing zone and shock wave. (c) the case of free expansion into vacuum.
The nonequilibrium processes can be investigated in channels, nozzles, piston systems and bombs provided by different means of explosive initiation. The use of free jets has advantages over the above-mentioned and other methods of performing experiments on physical gasdynamics and kinetics. They are:
851
1) the possibility to create the simplestonedimensional radial stationary flow lending itself to a rigorous theoretical description in some cases;
2> the self-similarity of jet structure and parameter distribution at the given determining criteria;
3) the possibility to isolate the jet section with a onedimensional flow, which is used or investigated, from the wall effect and background gas penetration;
4) the simplicity of creating a stationary flow with extremely low temperatures Cup to hundred 4.h fractions of a degree);
5) the simplicity of controlling relaxation rate over the temperature range under investigation and changing the parameter profiles along the jet in general;
6) the simplicity of realizing the jet flow itself at the presence of necessary pumping means.
Now theseadvantageswill be considered in detail.
1. The flow radiality at a free supersonic expansionwas considered long ago, for example, by M.D.Ladyzhenskii in 1962 8. In 1966 Ashkenas and Sherman3 suggested the dependences used up to date for calculation of axial jet parameters at an isentropic expansion. The flow character in the vicinity of the nozzle, in the region of transition to a radial flow, was refined also during recent years9. The radiality of the flow far from the nozzle exit is proved theoretically and experimentally.
In our papers10,11 the possibility to create a purely radial source <spherical or cylindrical) was demonstrated. However, such a source has some disadvantages, (i) first, the use of it requires creating a relatively high capacity pumping system, (ii) second, it is a problern to get high stagnation parameters. It is difficult to create a radial source with the stagnation temperature of thousands Kelvin degrees or Stagnation pressure at least of tens of atmospheres.
The simplicity of a radial flow as a onedimensional one stimulated the theoretical efforts.
The equilibrium nonviscous flow is rigorously described by analytical dependences. Within the framework of Navier-Stokes equations the asymptotical solutions have been obtained12 and the general solutions have been done with the use of the finite-difference method13-15. Using the kinetic models allows the nonequilibrium effects to be described. The key results were obtained in the works
852
by Hamel and Willis16, Edwards and Cheng17, where the translational nonequilibrium as an anysotropy of molecule velocity distribution function was firstly analysed using the momentum method and introducing the ellipsoidal distribution function. Later on the radial expansion calculations have been carried out by Monte-Carlo method2,18.
Now the possibilities of numerical methods allow to consider practically any relaxation process for which a physical model can be developed. It should be noted that for a twodimensional flow the numerical method in natural coordinate system is very effective. This method, now being developed at the Institute of Thermophysics19, can be traced back to more earlier works2D, where the processes in the jet core are not analysed.
2. The free jet structure, in general in the sense of a relative location of its typical surfaces and parameter distribution is likely to be self-establishing and rather weakly depends on structural details of the source itself at the given Mach number at the nozzle throat. In the works on jet scalingS-7,21 it was established that the Reynolds numbers by nozzle throat parameter, Stagnation and background pressure ratio, adiabatic exponent, Mach number at the nozzle exit are the determining criteria of the free supersonic jet scaling. The value ReL=Re*/VPo/Poo behind sonic nozzles being the criterion determining the viscosity or rarefaction effects. It is shown experimentally in6 that when ReL=const and Po/Poo>>l the jet structure is self-similar, i.e. the characteristic surface location in coordinates x/dovPo/Poo and y/doVPo/Poo is invariable for different conditions in a source. A relative density distribution in the jet field also occurs tobe self-similar. An analogaus approach has been developed also for the jets behind supersonic nozzles7,22.
The use of viscous and inviscid gas jet structure self-similarity simplifies the gasdynamical analysis of jet expansion.
3. The near axial jet section usually exploited for measurements can be disturbed by a direct penetration of background gas or its effect through the location of specific zones. The molecule penetration into the jet core was studied in23-25. The results of these investigations allow to estimate the molecule penetration into the jet core dependent on the flow regime. In our works7,24 the availabi~ lity conditions for undisturbed flow core are determined due to the above Reynolds criterion ReL characterizing viscous effects in the jet. Specifically, for ReL>100 the uniform gas jet in its general structure can be regarded as an object of continuous flow, in which the dimensions of undisturbed flow can be determined based on the above-cited references.
Fig. 2a from24 illustrates, in self-similar Coordinates, the location of typical interfaces between N2 jet and surrounding gas •.
853
REt 100 80 60 40
10 8 6
;?· j ..
D.2
Fig. 2
~ ~ ,., , --L ~ )i V
_7 I V ./ A l,.Y"
/2 ~JV ,/ 0 ,
(a) The Location of characteristic boundaries along the jet axis:1 - the density minimum Location; 2- the diffusion front Location; 3 - the viscosity effect boundary. (b) Flow regimes in a jet from a sonic nozzle. Boundaries of: 1 - the turbulent flow in the mixing zone (ReL~104 ); 2 - the Laminar flow (ReL~103 ); 3 - the origin of the transition to a rarefied regime (ReL~102 ); 4 - the origin of scattering regime ReL~S-10; 5 - the existence of a hypersonic flow in a jet core (Re*~230 ); 6 - the effusion flow; 7- the destruction of the self-similarity at Low pressure ratios.
It also shows the Location of the diffusion front being identified experimentaL Ly.
It is very difficult to take into account the effect of background gas on the relaxation process in the jet core because of uncertainty of energetic state of penetrating molecules. Therefore in the experimental investigations of relaxation processes it is expedient to get rid of this effect by increasing the gas discharge through a nozzle, or alternatively, decreasing background pressure.
If the background gas pressure is negligible, i.e. molecules of background gas do not effect sufficiently on the core flow, the jet formation is determined approximatively by the source conditions
I To only, namely, by the value Kn*~ -R - p d , if there is not an e* o o
essential effect of a spontaneaus condensation. Such characteristics as maximum Mach number, viscosity effect boundaries in the nozzle
854
and the condensation effect boundarie> are the very important characteristics of the expansion.
The problern of the Limited value of Mach number was successfully clarified by theoretical and experime1tal studies in 1965-1968 16,17, 26-29,41. There are some references providing recommendations on calculation of the Limited value of M3ch number, especially for monatomic gases, and temperatures in 3nisotropic repre~entation. A systematic investigation of the tra1slational nonequilibrium due to an electron beam method performed JY Cattolica and others30 and the calculations using the Monte-CarLJ method made by Chatvani18 put the problern of specification of i1teraction potentials at Low temperatures corresponding to a deep expansion.
As it follows from the results obtained in the Institute of Thermophysics31, for N2 jet at room stagnation temperature, the variation of Podo effects on the translational temperature distribution, when Podo <10 Torr mm, that is caused by viscous effects and rotational relaxation. When Podo>10, the boundary Layer effect in the nozzle is negligible. It is close to the boundary, when a hypersonie flow in the jet core can be formed. When Podo>20, the delay of the rotational relaxation is not pronounced in the translational temperature distribution till the bo~ndary of the velocity distribution function anisotropy.
Fig. 2b, firstly given in? shows specific regimes of a diatornie gas free jet behind a sonic nozzle with the background pressure P00 •
This honogramm allows molar and molecular effects in a het to be characterized easily.
4. One of the advantages of a free jet as an experimental object is the simplicity of relaxaticn velocity control. It is performed, for example, at the given Stegnation temperature by varying the stagnation pressure that is equi\alent to a corresponding mass flux variation. In general, the formction of the requested parameter field in the jet has no difficulties in principle, as it is readily carried out by the variation of the ~tagnation parameters and changing of the nozzles with differert sizes. The Limitation of these procedures for the selected gas source scheme can be the condensate appearance in the flow at the elevatEd pressure and the appearance of the rarefaction effects and early freezing of the relaxation energy exchange at the Lowered pressLre. ALLthese processes can be controlled by combination of stagnat'on parameters and nozzle sizes.
5. The possibility to obtain flc·ws with high Mach numbers and correspondingly Low temperatures shoL1ld be especially emphasized. This has played a very important rolt· in the development of molecular beam systems with jet sources. It is worthy to mention the wellknown results of Camparguers group or1 obtaining relatively dense
855
flows with extremely low temperatures. The question is achieving Mach numbers above 300 in He flow that corresponds to temperature in He below 10-2 K. The works of this group have stimulated to a considerable extent the development of a low-temperature spectroscopy of molecular gases based on a maximum simplification of a rotational structure of vibrational bands by gasdynamical cooling, i.e., on vibrational nonequilibrium. This trend has been discussed in detail in the review by Campargue (1978)33
6. One of the most advantages of the jet is also a trivial simplicity of realization of a flow itself, especially if the problem of pre-heating or pre-cooling has been solved. In the most cases the expenses connected with creating the proper source are negligible as compared with the cost of the experiment as a whole.
The peculiarities of performing the experiments with a jet in vacuum does not impose restrictions on the use of diagnostics means. Among them are such typically gasdynamte ones as Pitot tubes, different probes, shlieren and electric charge vizualization methods, as well as comparatively new continuously and quickly developing means - molecular beam, electron beam and Laser beam diagnostic method. It is the development of new diagnostics methods that has allowed to obtain many important fundamental results.
The freezing of a translational relaxation was investigated by several groups26-28 almost simultaneously. With the use of the electron-beam technique Muntz29 performed a direct measurement of the anisotropic velocity distribution function. The works belanging to this time have determined to a considerable extent the direction of further investigations of relaxation processes in jets.
The measurements of nonequilibrium velocity distribution function by Cattolica et al30, molecular-beam registration of Mach numbers above 100 in the jet34 stimulated experimental and theoretical investigations of guantum effects in He jets at low temperatures. The work by Miller, Toenni~, Winkelman33 was the first special research in this trend.
But the earliest study of translational gas relaxation has taken place in the works by Becker's group in Karlsruhe, where beginning from the fiveteenths they studied a gasdynamte separation of uranium isotopes36. These works being · continued till now37, together with the works of other groups38,39 formed the whole trend in rarefied gasdynamics.
The investigations of translational jet relaxation covered the path from the experimental study of acceleration of havy gas in light one36,40 Much number free zing establishment41 to the detailed analysis of the molecular velocity distribution function in binary and ternary mixtures42. rn32 the highest molecule energies (tens of
856
electrovolts) were reached for heavy ;1dmixture gas seeded and accelerated in a Light one.
A rotational nonequilibrium as a deviation from Boltzmann distribution in nitrogenwas firstly dis:overed by Marrone43 by the electron-beam measurements. A systema;ic investigation of this phenomenon was carried out in our works4t,45. Presented at this Symposiumare the recent results46. The no1equilibrium rotational Level populationwas also discovered by oth~r methods47.
One of the earliest experimental studies of a vibrational re·Laxation in a free jet was a vibratio1al freezing of n-butan48, based on determining vibrational temperatur~ from the molecule fragmentation in a mass-spectrometric system. )uch a molecular-beam diagnostics was used fruitfully by many rese3rchers to determine the molecule energy state during expansion in a jet before a skimmer which forms a molecular beam. Based on these experimental results, the model of a sudden freezing of Mach number as w~LL as rotational and vibrational energies were constructed49,50. DJring many years the molecular beam diagnostics served as a main one in the study of the condensation products in a jet and for the co1struction of nucleation models. The key results in this field were obtained in measurements of dimer concentration in a free jet of noble ~ases made by Milne and Green51 and of cluster sizes made by Hagena a1d Obert52. The models of dimer formation53 and nonequilibrium condensation54, the profan efficiency of a classic nucleation theory for nonequilibrium conditions55 are now the consequences of experimentally demonstrated capabilities with a free jet. Recently Lewis and Williams56 have started their investigations of the condensation zone structure in a jet, using the Laser beam scattering. Their results make it possible to use the capabilities of numerical models of nucleation more completely, as they give the nucleation zone structure along the stream Line.
Not Later than 10 years ago there appeared a new trend in a Lowtemperaturespectroscopy in adiabatically cooled flows57. The use of a free jet created premises for turning this rather narrow field into the research frontiers of physics. The caQabilities of these methods were demonstarated by Warton and Levy58. Nowadays there appeared some works dealing with intermolecular relaxation processes in complex molecules59.
The informati~n on an energetic state of jet particles reflects the results of a collective process averaging the effects of elementary processes. Nevertheless, the data on the molecular velocity and internal distribution functions are successfully used for determining either the intermolecular ~otential energy function parameters in elastic and inelastic collisions30 or the energy exchange velocity constants over internal degrees of freedom31,60. For this purpose the kinetic equations descriting relaxation processes during a radial expansion with the given intermolecular potenrial model or
857
the given relaxation velocity constant structure have been solved. The model or structure parameters are determined from the comparison of the solution of the system of equations and experimentally obtained change of parameters.
The possibility itself to obtain a detailed information on an energetic state of molecules stimulated theoretical investigations of relaxation processes at a radial expansion.
A physical content of qualitatively new nonequilibrium effects which are possible in uniform and especially nonuniform molecular gases ·or gas with condensed microparticles Cclusters) is far from being exhaustive, especially after the appearance of new effective diagnostic methods. An example illustrating the above-said is the establishment of the enhancement of a rotational nonequilibrium on upper Levels in the condensing nitrogen flow61; the use of a bolometric technique and IR-spectroscopy made it possible to understand a strong cluster effect on a vibrational relaxation in C02 flow and to measure binding energy in clusters62.
The Laser-bolometric methods63 called the optothermal infrared spectroscopy, Lased-induced fluorescence64 and coherent active spectroscopy of Raman scattering65 made it possible to measure relaxation characteristics of gas in a jet almost without energetic disturbances in a flow.
The stream of qualitatively new results will be Likely tobe continued. However, the situation with the investigations of nonequiLibrium processes in jets is such that a significant part of works transfers just now from the region of the effect discovering to that of the careful study of properties, such as thermophysical, quantummechanical, etc. So, there arises the necessity in high accuracy measurements and data systematization. On the other hand, even nowadays the variety of measuring means is so great, that in a goodequiped Laboratory practically any phenomenon in low density gases can be studied with a full information on energetic balance of the molecular processes. Thus, the requirements to rigorously definite flow formation, preciseness of the parameter measurements and detaiLing of a theoretical description of energy exchange processes continuously grow.
Now Let us consider a specific example. For a jet behind a sonic nozzle at known expansion Laws it is possible to extinguish the region undisturbed by background gas. However, the parameter distribution scattering can be conditioned by the different formation of a near-nozzle jet section for different nozzles due to their structural peculiarities.
The effect of geometrical parameters of an axisymmetrical sonic
858
nozzle on jet characteristics has beer studied in66. The discharge coefficient appeared to change with t~e entry angle of a nozzle and its throat edge curvature from 0.975 to 0.812, it affects the jet parameter distribution at the given stagnation parameters. In67,68 also a significant effect of the nozzle entry conditions on the processes in the jet, in particular at small distances from a nozzle exit are shown. The error in determining the mass flux, as well as Kundsen and Reynolds numbers at relatively great distances from a nozzle can be insignificant. At small distances it is simply inadmissible. At present it is possible to confine oneselves to minimum set of technologically simple nozzles for a beneficial use in physical investigations to eliminate the consideration of extra geometrical parameters. Preferable is the nozzle with a sharp edge in the nigligible thin wall. The experimental use if adequate nozzles by different investigators would simplify the comparison and analysis of the results.
In the experiments on re(axation processes in jets the question often arises about the degree of initial nonequilibrium of expanding gas or the location of a boundary of equilibrium flow. This is of particular importance for "inert" degrees of freedom, such as vibrational and electronic ones, as well as for chemical reactions. The diagnostics possibilities in such a zone are limited, since it is usually situated in the high gas density region. Our experience in the use of especially cryogentic pumping methods shows that the development of technical means for creating great consumption low density flows can provide a new wide research front. A principal problern is achieving equilibrium in the initial expansion region. The question is the use of free jets for investigating nonequilibrium processes in ionized gas, for a wide class of chemical processes including plasmochemical ones.
It was established by experiments with jets6,7,69 that at great pressure drops (Po/Poo>>I) the flow region behind Mach disk in the regime of a continuous flow (ReL>IOO) can have a significantly weak gradient of parameters under the conditions of low density and high temperature when necessary. This flow region can be used as a hightemperature thermostat analogously to the region behind a shock wave in a shock tube. However, there is an undoubted advantage. In the given case it is a stationary flow with the prominent starting zone in the form of a shock wave, in front of which it is possible to introduce a component in the form of admixture, whose behaviour is to be investigated in a shock wave.
The organization of such experiments is possible in principle, but for realization of them in practice it is necessary to provide systematic investigations of the zone behind Mach disk which is subjected to strong effects of shear flows accelerating the transition to turbulent flow. In the work by Avduevsky and others5 it has been found that this transition takes place when ReL> 1000 (fig. 2b).
859
ALL the above considerations mainly concern jets behind sonic nozzles. Obtaining the given gasdynamic field in the case of jets from supersonic nozzles is a complicated problem, because of a variety of additional parameters, such as Mach number at the nozzle exit, nozzle opening angle, boundary Layer thickness on the nozzle walls, badly predictable shock disturbances in the stream core. Technological difficulties of manufacturing a good supersonic nozzle and its Limited possibilities of use due to the change of an effective nozzle contour with changing the stagnation parameters and corresponding changing of its boundary Layer should be taken into account.
As for jets behind soni~ nozzles, if their form is unified for research purposes, there arises the possibility to creat an atlas of such flows for a reasonable number scale Re*, Po/P003 cplcv.
One of the peculiarities of plane nozzles or source with a cylindrical symmetry is a weak change of Local Kundsennumber during expansion. This imposes some restrictions upon the possibility to control relaxation processes. Another disadvantage of the source with plane jets is the necessity in increased consumption of a vacuum system, as compared with the case of axisymmetrical jets.
However, there are some undoubted advantages, such as the possibility to use optic absorption diagnostics methods70, which attract attention to these flows, especially in connection with the gasdynamic Laser development.
The investigation of energy exchange in jets is of particular interest not only because of statement of physical problems on the dynamics of relaxation processes. Technological application of expansion of a uniform gas and heterogeneous media into vacuum are variable. They are such as jet vacuum pumps, gas fans, gaseous accelerators of heavy molecules and clusters, gasdynamical sources of various vehicles, jet technological devices, gasdynamical Lasers, etc. The improvement of these techniques will require the development of analytical and numerical methods for jets with a minimum Limitation of physical content.
The author uses this occasion to thank Prof. J.B.Fenn for equainting with his recent review71 at this Symposium time and to adress readers also to this review reflecting to a certain extent the topic of my paper.
REFERENCES
1. J.O.Hirshfelder, Ch.F.Curtiss, R.B.Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc., New York (1954).
2. G.A.Bird, Molecular Gasdynamics, Oxford <1976).
860
3. H.Z.Ashkenas, F.S.Sherman, The Structure and Utilization of Supersonie Free Jets in Low Density Wind Tunnels, in: "Rarefied Gas Dynamics", 4th Intern. Symp., Acad. Press, v.2, p. 84 (1966).
4. V.S.Avdujevsky, A.V.Ivanov, I.M.Karpman and oth., The Viscous Effect on the Flow in the Initial Part of a Highly Underexpanded Jet, Doklady Akademij Nauk SSSR, 197:46 (1971> - In Russian.
5. E.P.Muntz, B.B.Hamel, B.L.Maguire, Exhaust Plume Rarefaction, AIAA Paper, N 69-657 (1969).
6. V.V.Volchkov, A.V.Ivanov, N.I.Kisljakov and oth., The Low Density Jet behind the Sonic Nozzle at High Pressure Ratio, PMTF, 2:64 (1973) - In Russian.
7. A.K.Rebrov, On the Gasdynamic Structure of the High Pressure Ratio Low Density Jet, in: "Problemi teplofisiki i fisicheskoy hydrodynamiki. Novosibirsk, pp. 262-276 (1974) -In Russian.
8. M.D.Ladizhensky, Analisis of the Hypersonic Flow and Koshi Problem Solution, Prikladnaja matematika i mechanika, 26:289 (1962) - In Russian.
9. A.N.Kraiko, V.V.Shelomovsky, On the Free Expansion of the Twodimensional Ideal Gas Jet, Prikladnaja matematika i mechanika, 44:271 (1980) - In Russian.
10. A.A.Bochkarjov, A.K.Rebrov, S.F.Chekmarjov, On the Hypersonic Spherical Gas Expansion with the Stationary Shock Wave, PMTF, 5:62 (1969) - In Russian.
11. A.A.Bochkaryov, A.K.Rebrov, S.F.Chekmarev, S.F.Sharafutdinov, Experimental Investigation of Spherical and Cylindrical Rarefied Gas Expansion with a Stationary Shock Wave, The 7th RGD Symposium, Book of Abstracts, Italy, v. 2, pp. 401-406 (1970>.
12. M.D.Ladizhensky, On the Viscous Gas Discharge into Vacuum, Prikladnaja matematika i mechanika, 26:642 <1962) - In Russian.-----
13. V.M.Gusev, A.V.Zhbakova, The Flow of a Viscous Heat-Conducting Compressible Fluids into a Constant Pressure Medium, in: "Rarefied Gas Dynamics", 6th Intern. Symp., Acad. Press, v-:1, pp. 847-862 (1969).
14. A.K.Rebrov, S.F.Chekmarjov, The Spherical Expansion of the Viscous Heatconducting Gas into Flooded Space, PMTF, 3:122 (1971) - In Russian.
15. N.C.Freeman, Expansion into Vacuum, AIAA J., v.5,9:1696 (1967>. 16. B.B.Hamel, D.R.Willis, Kinetic Theory of Source Flow Expansion
with Application to the Free Jet, The Physics of Fluids, v. 9, 5:829 (1966).
17. R.H.Edwards, H.K.Cheng, Distribution Function and Temperatures in a Monatomic Gas under Steady Expansion into a Vacuum, in: "Rarefied Gas Dynamics", 5th Intern. Symp., Acad. Press, "V:"1, pp. 819-836 (1967).
18. A.U.Chatwani, M.Fiebig, Source Expansion of Monatomic Gas Mixtures, .i!:!.: "Rarefied Gas Dynamics", 12th Intern. Symp., part II, pp. 785-801 (1981).
19. P.A.Scovorodko, On Method of Computation of the Nozzle Viscous Flow, .i!:!.: "Dynamika razrezhennogo gasa", Trudy vsesoyuznoy konferenzii. Chast 2, Novosibirsk, pp. 143-148 (1980) - In Russian.
861
20. F.Boynton, A.Thomson, Numerical Computation of Steady, Supersonic Two-Dimensional Gas Flow in Natural Coordinates, J. of Comput. Phys., 3:379 (1969).
21. V.N.Gusev, V.V.Michailov, On the Scaling of the Flow with Expanding Jets, Uchenie zapiski ZAGI, 1:22 <1970)- In Russian.
22. J.C.Lengrand, "Calculs de Jets Sous-Detendus Issus de Tuyeres Supersoniques", Rapport 75-4, Laberateire d'Aerotermique, Meudon, France (1975).
23. J.W.Brook, B.B.Hamel, E.P.Muntz, Theoretical and Experimental Study of Background Gas Penetration into Under-expanded Free Jets, The Physics of Fluids, v.18, 5:517 <1975).
24. N.I.Kisljakov, A.K.Rebrov, R.G.Sharafutdinov, The Diffusion Processes in the Low Density Free Jet Mixtion Zone, PMTF, 1:121 <1973) - In Russian.
25. R.Campargue, Aerodynamic Separation Effect of Gas and Isotope Mixtures Induced by Invasion of the Free Jet Shock Wave Structure, Journal of Chem. Phys., v.52, 4:1795 (1970).
26. K.Bier, O.F.Hagena, Optimum Conditions for Generating Supersonic Molecular Beams, in: "Rarefied Gas Dynamics", 4th Intern. Symp., v. II, Acad. Pres~ p. 260 (1966).
27. J.F.Scott, J.A.Phipps, Translational Freezing in Freely Expanding Jets, in: "Rarefied Gas Dynamics", 5th Intern. Symp., v. Il, Acad. Press, pp. 260-275 <1967).
28. N.Abuaf, J.B.Anderson, R.P.Andres, J.B.Fenn, D.R.Miller, Studies of Low Density Supersonic Jets, in: "Rarefied Gas Dynamics", 5th Intern. Symp., v.2, Acad. Press,jpp. 1317-1336 (1967).
29. E.P.Muntz, Measurements of Anisotropie Velocity Distribution Functions in Rapid Radial Expansions, in: "Rarefied Gas Dynamics", 5th Intern. Symp., Acad. Press, v.II, pp.1257-1286 <1967).
30. R.Cattolica, F.Robben, L.Talbot, D.R.Willis, Translational Nonequilibrium in Free Jet Expansions, The Physics of Fluids, v.17, 10:1793 (1974).
31. R.G.Sharafutdinov, P.A.Skovorodko, Rotational Level Population Kinetics in Nitrogen Freejets, in: "Rarefied Gas Dynamics", 12th Intern. Symp., part II, pp. 754-771 (1981).
32. R.Campargue, A.Lebehot, J.C.Lemonnier, Nozzle Beam Speed Ratios above 300 Skimmed in a Zone of Silence of He Free-jets, in: "Rarefied Gas Dynami es", 10th Intern.Symp.,part II,p.103IT1977).
33. R.Campargue, Flow Cooling as Applied to Laser Induced Separation, in: "Von Karman Institute Lecture Series on Aerodynamic SeparatTon of Gasesand Isotopes" (1978).
34. R.Campargue, A.Lebehot, High Intensity Supersonic Molecular Beams with Extremely Narrow Energy Spreads in the 0-37 eV Range,~: "Rarefied Gas Dynamics", 9th Intern. Symp., v. 2, FRG, p.c.II.1 (1974).
35. D.R.Miller, J.P.Toennis, K.Winkelman, Quantum Effects in Highly Expanded Helium Nozzle Beams, in: "Rarefied Gas Dynamics", 9th Intern. Symp., v.2, c.9, RFG (1974).
36. E.W.Becker, R.Schütte, Entmischung der Uranisotope mit der Trenndüse, Z. Naturforschung, ser. a, 11:679 (1956).
862
E.W.Becker, W.Henkes, Geschwindigkeitsanalyse von Laval-Strahlen, Z. Phys., 146:320 (1956).
37. A.U.Chatwani, M.Fiebig, N.K.Mitra, W.Ehrfeld, Nonequilibrium Effects and Their Modeling in Separation Nozzles, in: "Rarefied Gas Dynamics", 12th Intern.Symp., part 1, pp.517-540(1981).
38. R.Campargue, J.B.Anderson, J.B.Fenn, B.B.Hamel, E.P.Muntz, J.R.White, Sur les Methades Aerodynamiques de Separation des Gaz et Isotopes, Entropie, 67:11 (1976).
39. A.V.Bulgakov, Yu.S.Kusner, V.G.Prikhodko and A.K.Rebrov, Separation of Gas Mixture Components in Interacting Flows, ..i!l= "Rarefied Gas Dynami es", 12th Intern.Symp.,part I,pp.607-616 (1981).
40. Ju.N.Beljaev, V.B.Leonas, Formation of the Intensive Molecular Beams, Vestnik MGU, seria fizika, astronomia, 18:34 (1963) - In Russian.
41. J.B.Anderson and J.B.Fenn, Velocity Distributions in Molecular Beams from Nozzle Sources, The Physics of Fluids, v.8, 1:780 (1965).
42. R.J.Cattolica, J.B.Anderson, R.J.Callagher, L.Talbot, Velocity Slip and Translational Nonequilibrium of Ternary Gas Mixtures in Free Jet Expansions, Sandia Lab. energy report 78-8216, pp. 31-76 (1978).
43. P.V.Marrone, Temperature and Density Measurements in Free Jets and Shock Waves, The Physics of Fluids, v.10, 3:521 <1967).
44. B.N.Borzenko, N.V.Karelov, A.K.Rebrov, R.G.Sharafutdinov, Experimental Investigation of the Moleeule Rotational Level Population in Nitrogen Free Jet, PMTF, 5:20 <1976) - In Russian.
45. N.V.Karelov, R.G.Sharafutdinov, A.E.Zarvin, Rotational Relaxation in Nitrogen Freejets in the Transition Regime, in: "Rarefied Gas Dynamics", 12th Intern. Symp., part 2, pp. 742-753 <1981).
46. A.E.Belikov, N.V.Karelov, R.G.Sharafutdinov, A.E.Zarvin, Rotational Relaxation in High-Temperature Free Jets of Nitrogen, in: "Rarefied Gas Dynamics", 13th Intern. Symp., Book of AbstractS, pp. 486-488 (1982).
47. S.G.Kukolich, D.E.Oates, J.H.S.Wang, Rotational Energy Distribution in a Nozzle Beam, J. Chem. Phys., v.61, 11:4686 (1974).
48. T.A.Milne, J.E.Beachey, F.T.Greene, Study of Relaxation in Free Jets Using Temperature Dependence of n-Butane Mass Spectra, J. Chem. Phys., v.56, 6:3007 (1972). -
49. C.G.M.Quah, J.B.Fenn, D.R.Miller, Internal Energy Relaxation Rates from Observations on Free Jets, in: "Rarefied Gas Dynamics", 11th Intern. Symp., v. II, Paris, pp. 885-898 <1979).
50. P.K.Sharma, W.S.Young, W.E.Rodgers, E.Knuth, Freezing of Vibrational Degrees of Freedom in Free-Jet Flows with Application to Jets Containing CO, J. Chem. Phys., v. 62, 2:341 (1975).
51. T.A.Milne, F.T.Greene, Mass Spectrometric Observations of Argon Clusters in Nozzle Beams. I. General Behavior and Equilibrium Dimer Concentrations, J. Chem. Phys., v.47, 10:4095 (1967).
52. O.F.Hagena, W.-J.Obert, Cluster Formation in Expanding Supersonic Jets, J. Chem. Phys., v. 56, 5:1793 (1972).
53. E.L.Knuth, Dimer-Formation Rate Coefficients from Measurements of Terminal Dimer Concentrations in Free-Jet Expansions, l·
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Chem. Phys., v.66, 8:3515 (1977>. 54. O.F.Hagena, Cluster Beams from Nozzle Sources, in: "Molecular
Beams and Low Density Gasdynamics", GasdynamicsSeries, v. 4, P.P.Wegener, pp. 93-181 (1974).
55. A.K.Rebrov, Clusters and Condensation in the Expanding Flow, in: "Moleculjarnaja gasodinamika", Novosibirsk, p. 58-69 (1980) -In Russian.
56. J.W.L.Lewis, W.D.Williams, Profile of an Anisentropie Nitrogen Nozzle Expansion, Physics of Fluids, v.19, 7:959 (1976).
57. D.I.Katajev, A.A.Maltsev, Spectroscopy of Vapors of Low-Volatitily Compounds Super-Cooled in Supersonic Stream, Sov. Phys. JETP, 37, 772 (1973); Zh. Exp. Teor. Fiz., 64:1527 (1973) - In Russian.
58. L.Wharton, D.Levy, Jet Supercooling and Molecular Jet Spectroscopy, in: "Rarefied Gas Dynamics", 11th Intern. Symp., v. II, Paris, pp. 1009-1028 (1979).
59. P.S.H.Fitch, C.A.Haynam, D.H.Levy, Intramolecular Vibrational Relaxation in Jet-Cooled Phthalocyanine, J~ Chem. Phys., v.74, 12:6612 (1981).
60. A.N.Vargin, N.A.Ganina, N.V.Karelov and oth., Rotational Relaxation of Molecular Nitrogen in a Free Jet, PMTF, 3:73 (1979) -In Russian.
61. N.V.Karelov, A.K.Rebrov, R.G.Sharafutdinov, Population of Rotational Levels of Nitrogen Molecules at Nonequilibrium Condensation in a Free Jet, in: "Rarefied Gas Dynamics", 11th Intern. Symp., Paris, v. II,jpp. 1131-1140 (1979).
62. A.A.Vostrikov, S.G.Mironov, A.K.Rebrov, B.E.Semjachkin, Investigation of C02 Molecules Vibrational Relaxation in a Geterogeneous Gas-Cluster Media, Kvantovaja electronika, v.8, 6:1356 (1981) -In Russian.
63. T.E.Gough, R.E.Miller, G.Scoles, Infrared Laser Spectroscopy of Molecular Beams, Appl. Phys. Lett., 30:338 (1977).
64. H.W.Lülf, P.Andresen, Rotational Relaxation of NO in Seeded, Pulsed Nozzle Beams, in: "Rarefied Gas Dynamics", 13th Intern. Symp., Book of Abstracts, Novosibirsk, pp. 476-478 (1982).
65. P.Huber-Wälchli, J.W.Nibler, CARS Spectroscopy of Molecules in Supersonic Free Jets, J. Chem. Phys., v.76, 1:273 (1982>.
66. A.L.Addy, Effects of Axisymmetric Sonic Nozzle Geometry on Mach Disk Characteristics, AIAA J., v. 19, 1:121 (1981>.
67. F.Aerts, H.Hulsman, Population Evolution of Internal States of Na2 in a Free Jet Expansion, in: "Rarefied Gas Dynamics", 11th Intern. Symp., Paris, v.II, p~ 925-934 <1978).
68. D.A.Melnik, U.G.Pirumov, A.A.Sergienko, Jet Propulsion Nozzles, .i!l: "Aeromechanika i gasovaja dynamika",Moskwa, Nauka (1976) -In Russian.
69. L.I.Kuznetsov, A.K.Rebrov, V.N.Yarigin, High Temperature Low Density Argon Jets behind the Sonic Nozzle,PMTF,3:82(1975)~In Russian.
70. A.Amirov, U.Even, J.Jortner, Absorption Spectroscopy of Ultracold Large Molecules in Planar Supersonic Expansions,Chem.Phys.Lett., v.83, 1:1 (1981).
71. J.B.Fenn, Collision Kinetics in Gas Dynamics, in: "Applied Atomic Collision Physics", Acad. Press, v. 5, pp. 349-378 (1982).
864
STATE DEPENDENT ANGULAR DISTRIBUTIONS OF Na2 MOLECULES IN A Na/Na2
FREE JET EXPANSION
ABSTRACT
F. Aerts, H. Hulsman and P. Willems
Physics Department, University of Antwerp (U.I.A.) Universiteitsplein 1 B-2610 Wilrijk, Belgium
The spatial distribution of Sodium dimers (Na2) is studied experimentally in the first part of a free jet expansion of Sodium vapour, using laser induced fluorescence. Angular intensity profiles have been measured for several (v,J) - states of the molecules. Their features turn out to be very different, depending on the values of v,J and the stagnation pressure Po· These results allow to get some insight in the various relaxation- and freezing-processes taking place in the expansion.
1 • INTRODUCTION
In a free jet expansion of gas from a nozzle into vacuum, a substantial cooling occurs of the translational degrees of freedom in the gas1. Fora polyatomic gas a simultaneous cooling of the internal degrees of freedom takes place2 • The relaxation of the various degrees of freedom in the gas is determined by the elastic and inelastic collisions which occur in the course of the expansion. Due, however, to the fact that the molecules undergo but a limited number of collisions before the expansion reaches the free molecular flow regime, the relaxation processes do not continue indefinitely and the situation becomes "frozen"3 in the downstream region of the expa~sion. The difference in efficiency of the Vdrious collisions to cool the different modes of motion in the gas (translational, rotational and vibrational motion) gives rise to strong nonequilibrium situations.
The final "frozen" results of these relaxation phenomena have
865
6 r-------------------------------------------------~
2
0
-6 - I. - 2 0 z(mm)
2 6
Fig. 1. Schematic representation of the part of the expansion that is studied. The small dots indicate the positions at which fluorescence intensities were measured, respectively at y/D = 1.3, 2.6, 3.9, 5.2, and 10.3 (D = O.Smm). The open circles show the positions at which the intensity profiles of the three indicated (v,J)-states attain their halfmaximum value. So the (horizontal) distance between the circles represents the FWHM. The (inner) dashed line indicates the FWHM positions for a supersonic expansion, in the absence of relaxation. The full circle indicates the size of the pin-hole, which scans the intensity profiles in the experiment.
been studied extensively2- 5 , and model calculations6-B have been performed in order to relate such results to the relaxation phenomena upstream in the expansion.
Recently, we have presented a new application of the experimental technique of laser-induced fluorescence to directly study the internal state distribution of dimers in the upstream part o f a Sodium free jet expansion9-11. In this previous work, results have been presented concerning fluorescence intensity measurements alpng the axis of the expansion only. Hereby the internal state distribution was found to be of a Boltzmann type only within the first nozzle diameter (D) away from the nozzle, further downstream (1 to 10 D) the deviations were shown to become very large. In this work we will present results of fluorescence intensity
866
measurements concerning the off-axis regions of the expansion: at several distances along the axis of the expansion measurements were performed across the jet, along a direction perpendicular to the axis, and with an angular spread of 45°, (the actual cross-scans that were taken are indicated in Fig. 1).
The transverse intensity profiles of the jet that are obtained in this way turn out to be strongly dependent on the Vibrationrotation, (v,J)-state of the dimers which are excited, and this dependence increases drastically with increasing stagnation pressure Po·
2. METHOD
The basic method and the experimental set-up have been described in previous papers9-11, most recently and completely in ref. 11. We will, therefore, limit ourselves here .to some basic features, and to some details which are of special interest to this type of "transverse" measurements.
The jet is produced by means of an expansion of Sodium vapour through a convergent nozzle into a vacuum chamber (~ 1o-6Torr) • The exit diameter (D) of the nozzle is O.Smm, and the range of stagnation pressures is: 0.5 Torr.mm ~ p0o ~ 6.0 Torr.mm; T0 ~ 750K. The region of the jet just outside the nozzle, from 0 to ~ 200 downstream, is illuminated with an expanded light beam (diameter 20mm) produced by a single mode Ar+ laser. When the laser frequency coincides with one of the molecular transitions, dimers in the appropriate (v,J)-state are excited and emit fluorescence photons. Because of the Doppler selectivity of the excitation mechanism, only dimers with a small velocity component (~ 10m/sec) along the direction of the light beam are excited. In the expansion such dimers are confined to a thin plane, which originates from the nozzle exit and is oriented perpendicular to the laser light beam. This fluorescing plane is observed through a window in the vacuum tank, and along a direction that is (all but) parallel to that of the exciting-laser light beam. The measured intensity of the fluorescence originating from a certain position in this plane is (roughly) proportional to the density of the excitable particles at that position. The formal description of this proportionality and its evolution in the course of the expansion are given elsewhere12. A few aspects, however, are especially relevant for the present type of transverse measurements.
As explained in ref. 11, the dependence of the axial flow Velocity of the dimers on their internal state, which was found by Bergmann et.a1. 5 (downstream in a molecular beam), could slightly distort the experimental sensitivity with which we detect Na2
867
molecules in various (v,J) - states. As we will not try here to compare our results for different states on the basis of absolute numbers of dimers, the actual sensitivity factors are not needed, so that this velocity slip effect does not pose a problern for the comparison of different states.
There is, however, a flow velocity discrepancy that could distort the individual transverse measurements for each state: the acceleration (and relaxation) of the particles in the expansion is considerably lower in the off-axis regions of the jet than on the axis. Hence the flow velocity, which reaches its terminal value after a few D, will be lower on off-axis streamlines, so that dimers traveling along those streamlines have a slightly higher chance of being detected than the ones traveling on the axis. This effect would artificially enhance the wings of a transverse intensity profile.
An other effect that needs tobe considered is optical pumping. When a particle in a certain (v.,J.)-state is excited it will in general fall back to a differenE giound-state level (vk,Jk) so that it cannot be detected a second time. Consequently, in the downstream part of the expansion the number of excitable (and observed) particles is slightly reduced (artificially) as a result of the excitations in the former part of the jet. Although the overall effect of optical pumping is minimised as much as possible by working with light intensities as low as feasible, a slight distortion of the transverse intensity profiles cannot be avoided for several reasons: the longer path along an off-axis streamline to reach the axial displacement value y/D where the measurement takes place, the lower flow velocity and lower number of (inelastic) collisions experienced by the molecules along such a streamline. The net result is a moderate lowering of the wings relative to the centre of the profile.
So the two aforementioned distortions work in opposite directions. The net result of the velocity discrepancy can be estimated from calculations of the type shown in ref. 12, and the magnitude of the optical pumping effect can be estimated from measurements taken with different light intensities. They both turn out to be of the same order of magnitude, namely a few percent at 45°, and are therefore expected to cancel each other for the major part. Although one might still have to take these effects into account in a (future) quantitative analysis of these data, we will disregard them (from now-on) because in this stage of the analysis we are only interested in revealing the qualitative features of the results, (which are so striking that a slight distortion is quite irrelevant) .
868
;' 1,
~~'% 0 0 0
0 0 0 0 .5. o.a 0 0 0
1: 0.6 0 0 0 0
000 (0,28) cP 0 0
~ .. 00 0 0
8 Torr 12 Torr i 0.2
o.
Fig. 2a. Pressure dependence of transverse intensity profiles for five different vibration-rotation-states, indicated by their quantum numbers (v,J). For all states pressures are as indicated for (0,28). All scans taken at an axial displacement of 5.20. Y is the coordinate along the axis of the expansion, z indicates the distance away from the axis. So z/y=1 corresponds to 45°.
3. RESULTS
Transverse intensity measurements have been performed for four stagnation pressures: Po = 1,3,8 and 12 Torr. Formost pressures, scans were taken at four distances from the nozzle: y/0 = 1.3, 2.6, 5.2 and 10.3, (for p =8 Torr also at y/0=3.9). These scans consist of measurements at 3~ different positions across the jet, as is indicated by the small dots in Fig. 1. At each position fluorescence intensities were recorded for (usually) five (v,J)-states: (0,28), (1,37), (3,13), (3,43) and (5,55), which have internal energies (vibration + rotation) of, respectively: 205, 452, 572, 833 and 1309cm-1 13 . Close to the nozzle, at y/0 = 1.3, intensities were also measured for the states (6,41) and (2,98), which have energies of respectively 1285 and 1820cm-1. In this way we accumulated a total of about 75 scans.
869
An overview of the most relevant results is given in Fig. 2. Fig. 2a illustrates the changes occuring due to increases in stagnation pressure p0 , at a constant y/D value, and Fig. 2b shows the changes in the course of the expansion for a particular Po·
In this paper we do not want to compare the absolute intensities of the various scans, but we want to discuss the qualitative changes of the shape of the curves. So we only consider normalised intensity profiles. To compare curves taken at different y/D values or at different p0 , one particular profile is chosen as a standard and is norm~lised to 1 at z/y = 0. The other profiles are then scaled to this normalised one in a straightforward, but somewhat unusual way: for all curves the intensity values of the wings (at ~ 45° off-axis) are made to coincide with that of the normalised standard profile, (e.g. in this way all curves for a given (v,J)-state in Fig. 2 have been scaled to the corresponding (normalised) curve in the first collumn i.e. the one for the lowest p 0 or at the smallest y/D). The choice of this kind of scaling is argumented as follows.
The observed changes of the transverse intensity profiles reflect the relaxation of the internal energy of the molecules in the course of the expansion. (In previous work concerning the axis of the expansion we found that under these conditions dimerisation ceases beyend about 1o,1 1 butthat relaxation only freezes out for distances y ~ 10D) .10 In this respect it is important to realise that on an axial flowline the relaxation is expected to be much strenger than on a far off-axis one, due to the much higher densities in the centre of the expansion, yielding a higher collision rate, and also due to the somewhat lower temperatures, inducing a strenger deviation from thermal equilibrium. Therefore, the effect of relaxation on the occupancies of the various (v,J)-states is quite drastic in the central region of the expansion, while its wings remain relatively unaffected. For this reason we take the wings of the corresponding fluorescence intensity profile as the reference for the scaling procedure. We do not claim this procedure to be unique, it is just a simple way to assure that the figures emphasize mostly the changes in the centre of the profile, as this is indeed the region where most relaxation occurs.
Concerning Fig. 2 now, we would like to point out a few features, which we will later discuss, (some on the hand of more detailed figures) .
1° When the total (accumulated) relaxation is small, lowest p 0 and/or closest y/D value, the profiles are very similar for all (v,J)-states.
2° At high p0o, most relaxation occurs in the first part of the expansion, as could be expected, but for some states, (1,37) and (3,43), it clearly continues down to y ~ 100, while for others the relaxation seems to freeze out sooner, cfr. (3,13) and (5,55).
870
;, oooo 0 0 ou 0 cPoY ""'oo ..!!.. Q8 0 0 0 0 0 0
0 0 0 0 00 0 0 °o
~ 06 0 0 0 0 0 (0 .28) 0 0 0 0 0 0
'.p>,/'0
Y /0= 10.3 \b~ 3 O• 0 Y/0=1 3 0 0 0 0 0
~ 0.2 foo oc fi'0 Y/0=2.6 °' "1JP0 Y/0 =5.2 -, ._
';t oOOO 8Qb{137) :!. 00 0 0 ?;: ._. 0 0 ii\ 0 0 z 0. 4 0 0
~ 02 .,o 00
0
:j 1 0000 :!. o.a
0 0
~~~ ~ 0.1 0 0
rJf' 'b' cP 'b"~(J) (3,13) "' 0 0 z ._,
0 w 0 ~ 02 .,o 00
._ :) 1. oooo
~bd6::}4J) .:! oa 0 0 1:: 0.6 0 0
@ 0< 00 0
::.' Po 000 ~ 0.2
0.
::> 1. 0000
00oo, BBg{555) ~().. 0 0
~ 0.1 0 0
"' 000 z 0 <
~ 0.2
0. _,_ -o• ._ O> 1 -• -o ~ o o ~ 1 -1. -o..s o o 5 1 -1 -o ~ o o 5 ,
Z/Y Z/Y Z/Y Z/Y
Fig. 2b. Evolution of the intensity profiles in the course of the expansion, for the same five states as in Fig. 2a. For all states axial distances are as indicated for (0,28). All scans taken with a stagnation pressure of 12 Torr. The axial displacement values y/D correspond with those indicated in Fig. 1.
3° Increases in Po cause very drastic changes in the ratio of
the central intensity over that of the wings for several states : (1,37), (3,43) and (5,55); while for others, (0,28) and (3,13), it
remains almost unaffected. 4° These data do not indicate a simple Straightforward rela
tionship between the degree of relaxation for the various states and their internal energies, vibrational rotational or total internal energy: relaxation is stronger for (3,43) than for (5,55), and is weakest for (3,13), not for (0,28) or (1,37), which have lower vibrational and total energies.
5° For (3,43) the relaxation is so strong that on the axis of
the expansion this level is depopulated so much that the centerline
intensity becomes substantially lower that the intensity at about
40° where the overall particle density is much lower.
871
:::)
~
?: "' z ... ,_ ?;
0 (1.37) D (0.28)
-2. -1.5 -1. -0.5
...... :::) ~1.
~ 0 (0.28) ~0.8
~ ;:;
06
0.4
0 . -1. -0 5
• (3.13) 0 (3.43)
Torr
0. 0 .5 1.5 Z/Y
0 . 0.5 Z/Y
0·
2. -2 . Y/D=1.3
C· 1.
0.8
0 .6
04
02
1.
-1.5 -1
0 (3,43) 0 (6,41)
• (5,55) 0 (2,98)
-0.5 0 . 0.5 1. Z/Y
Fig. 3. Fluorescence intensity profiles measured at 1.30, for the lowest and the highest stagnation pressures: 1 and 12 Torr. Fig. a,b,d all give results which extend to 63° off-axis; Fig. c is limited to 45°. The lines in Fig. c show the theoretical intensity predictions for effusive and supersonie flow. Symbols for Fig. b as in Fig. a.
4. DISCUSSION
In this section we will discuss some processes, which can explain the features of the data cited above.
1° In Fig. 3 the data with the lowest relaxation are summarised. Fig. 3a demonstrates that a low pressure, close to the nozzle all curves are very similar. Fig. 3c shows that there is only a 25% difference between the energetically lowest, (0,28), and the highest state, (2,98). Fig. 3c also offers a comparison between the experimental distributions and the theoretical predictions for effusive and supersonic expansions;a cosine- and cosine3-distribution, (in refs. 14,15 the latter representation is shown to be a good approximation) . The fact that the measurements t'ke place at a constant y value, instead of at a constant radius ( y 2+z2 ), makes that these theoretical distributions have to be multiplied by an additional cosine, (assuming point source flow15) . This yields the cos2 and
872
cos4 distributions, which are illustrated. Striking is the agreement between the (0,28)-profile and the supersonic distribution. For this expansion (and temperature) regime the effect of relaxation on the profile of (0,28) is expected to be minimal, (due, partly, to a minimal deviation from thermal equilibrium) . So the agreement indicates that even at this low p D value, the expansion is as good as supersonic. (Relaxation woul~ slightly enhance the peaking of the (0,28)-profile, so, it should actually be marginally higher than the cos4 .) In addition, the fact that the distribution for (2,98), which of all states ~s flattened most by relaxation in Fig. 3a, stays far above the cos clearly confirms that the expansion is nowhere near the effusive limit.
2° The apparent insensitivity of the (3,13)-state for relaxation effects suggest an early onset of freezing, but it is more likely the result of compensatory action of vibrational and rotational relaxation. This can be explained as follows. The occupancy of a (v,J) level may change by collisions in which the vibrational quantum number v changes and by collisions in which only the rotational number J changes. One expects the ßv collisions to have a (much) smaller cross-section than the ßJ ones, (a ßv collision must accomplish an energy transfer of (at least) about 160cm-1, while the rotational constant B =0.15cm-1 13. So that the rotational energy exchange, ßE ~ 2BJ, is much smaller than the vibrational one). Consequently, the vibrational relaxation is only important in the first part of the expansion and "freezes" much earlier than the rotational relaxation. Fora state like (3,13) this may lead to a compensating action of vibrational and rotational relaxation. In the course of the expansion the translational temperature falls from 600K to about 200K. The corresponding evolution of the equilibrium Boltzmann distribution over the internal states is such that v = 3 should monotonically lose occupants, but that J = 13 should first increase its occupation, go through a maximum and then start to loose occupants. The net effect in the expansion is the remarkably constant occupation of (3,13), see Fig. 2b (a detailed Figureis not given as it looks exactly like Fig. 4b). So first ßv and ßJ collisions tend to compensate each others effect, then ßv stops and the occupation of the J = 13 level passes through its equilibrium value so that ßJ collisions become rather ineffective, leaving the profile virtually unchanged throughout the expansion.
The same mechanism can also explain the behaviour of the (0,28) distribution at high p0n. In the first part of the jet the v = o causes substantial gains via strong vibrational relaxation, which clearly outnumber the lasses due to rotational relaxation. Later in the expansion the vibrational relaxation freezes and the rotational one becomes dominant. The net result is first a strong peaking and then again a flattening of the (0,28)-profile, this is shown in Fig. Sa.
873
:) ~1.
C· 1.
!;, 0 1 Torr ~ 8 Torr !Q 0.8 0 3 Torr .... .... ~
0.6
0.~
0.2 (1,37) (3, 13)
0. -1 . - 0.5 0. 0.5 1.
0. -1 -0.5 0 0.5 1.
Z/Y Y/0=5.2 Z/Y ~
:) ~1.
C· 1.
1: 0 1 Torr ~ 8 Torr !Q 0.8 .., ~
0.6
0 .4
0.2 0.2 (5,55)
0. -1. -0.5 0 0.5
0 . -1 -0.5 0 0.5 1.
Z/Y Z/Y
Fig. 4. Pressure dependence of transverse intensity profiles for four different vibration-rotation-states, indicated by their quantum numbers (v,J) . All scans taken at an axial displacement of 5.2D. All scans extend to 45° off-axis. Symbols for Fig . b as in Fig. a, for Fig. d as in Fig. c.
3° Using similar arguments, the effects can be e xplained which are found as a function of p0 (at a fixed distance y/D from the nozzle), s.a. the (0,28) results at 5.2D, (shown in Fig. 2a; the detailed figure (not shown) is exactly analogaus to Fig. Sa). For 1 Torr, freezing of the occupation did occur early in the e xpansion when the distribution of (0,28) was hardly changed by relaxation yet. At 3 Torr, the freezing occured later in the expansion so that the relaxation bad already caused an noticable peaking of the profile. At 8 Torr, the same effect is even strenger, but at 12 Torr the ~J relaxation is so strong and continues so lang that the number of (0,28) molecules is already diminishing again for y ~ 5.2D, which in turn flattens the transverse distribution. An even more complex combination of these effects must be at the basis of the complicated (and somewhat fluctuating) behaviour of the (3,13)-profiles found in Fig. 4b.
4°When the vibration and the rotation energy of a state are both larger than kT, this compensation of gains and lasses does
874
...., b· ::. 0· 1. ~1,
~ 0 1.30 6 5.2 D
~ 08 0.6 0 2.6 D 0 10.3 D ... !z
l -06 0.6
0. ~ 04
0 10.3 D 0.2 0.2
(0,28) (1,37)
0. -1 -0 5 0 0.5 1
0. -1. -0.5 0. 0.5 1.
Z/Y 12 Torr Z/Y
:;; ~1.
C- 1.
~ 0 1.3 D 6 5.2 D ~ 0.8 0 2.6 D 0 10.3 D ... ... ;!;
0.6 (3,43)
0 .4
0 .2 (5,55)
0 -1. -0.5 0 . 0.5 1.
o_ -1. -0.5 0. 0.5 1.
Z/Y Z/Y
Fig. 5. Evolution of the intensity profiles as function of y/D, for four different (v,J)-states. All scans taken with a Stagnation pressure of 12 Torr. Symbols indicating the y/D values are for Fig. d as in Fig. c.
not occur; any lowering of the temperature in the expansion causes a depletion of the population both by vibrational and rotational relaxation. Now, comparing the high (5,55) and (3,43)-states in Figs. 4 and 5, it is clear that the relaxation is much less effective for (5,55) than for (3,43), although the deviation from the equilibrium will be higher for (5,55). This corresponds to lower transition probabilities for higher quantum numbers. Fig. 5 also shows that the relaxation aontinues considerably langer for (3,43) and (1,37) than for (5,55). This indicates that an explanation should in the first place be sought by the ßJ relaxation mechanism. And indeed this behaviour can be understood if one realises that the relaxation is governed by two competitive mechanisms. Firstly, the driving force for relaxation is the deviation from thermal equilibrium, which, for decreasing T, increases strongly for the higher E t• On the other hand, the ability of inelastic collisions to induc~0relaxation decreases under a decreasing T and an increasing Erot• The behaviour of the (5,55)-profiles now indicate that for
875
states with J ~ 50 the inefficiency of the collisions dominates in the latter part of the expansion. This view is confirmed by the data in Fig. 3d: comparing (5,55) with (6,41) we find that these states have both high internal energies, which are close to each other: respectively 1309cm-1 and 1258cm-1 , and yet (5,55) clearly relaxes less than the (lower) (6,41). That this lower relaxation should be attributed to the higher J value is further indicated by the behaviour of the (2,98)-state, which has the highest total energy of all states considered (182ocm- 1), and yet it is flattened less in Fig. 3d than both (5,55) and (6,41); this can hardly (solely) be attributed to its low v value, considering the features of states like (1,37) and (3,43) that relax easily (cfr. also Fig. 4 and 5) •
5° The strongest relaxation effects are found for the (3,43)state (Fig. 4c and Sc). At 10.30 and 12Torr, the centerline intensity is about 50% ofthat at 40°, while in the absence of relaxation the centre is expected tobe 4 times higher than the wings. This indicates that the effects of relaxation are about an order of magnitude stronger along the axis of the expansion than in its wings.
Concluding we can state that the, at first sight, overwhelmingly complex relaxation phenomena in this type of free jet can still be qualitatively understood, starting from a few Straightforward mechanisms determining the relaxation. This gives good reason to hope that in a future quantitative analysis, the information furnished by these results can be translated into effective state dependent cross-sections.
ACKNOWLEDGEMENTS
We gratefully acknowledge financial support from the Belgian science supporting agency F.K.F.O.; one of us, P.W., whishes to thank I.W.O.N.L. for a scholarship. We are also indebted to E. De Langhe and J.P. Huysmans for expert technical assistance. Plots of Fig. 2-5 made with HPLOT, courtesy of the CERN-library.
REFERENCES
1. R. Cattolica, F. Robben, L. Talbot and D.R. Willis, Phys. Fluids 17: 1793 (1974).
2. M.P. Sinha, A. Schultz and R.N. Zare, J. Chem. Phys. 58: 549 (1973).
3. J.B. Anderson and J.B. Fenn, Phys. Fluids 8: 780 (1965). 4. R.J. Gordon, Y.T. Lee and D.R. Herschbach, J. Chem. Phys. 54:
2393 (1971). 5. K. Bergmann, U. Hefter and P. Hering, Chem. Phys. 32: 329 (1978). 6. R.J. Gallagher and J.B. Fenn, J. Chem. Phys. 60: 3487 (1974).
876
7. C.G.M. Quah, Chem. Phys. Letters 63: 141 (1979). 8. H. Lang, in: "Rarefied Gas Dynamics", R. Campargue, ed., C.E.A.,
Paris (1979), p. 823. 9. F. Aerts and H. Hulsman, in: "Rarefied Gas Dynamics", R.
Campargue, ed., C.E.A., Paris (1979), p. 925. 10. F. Aerts and H. Hulsman, Chem. Phys. Letters 72: 237 (1980). 11. F. Aerts, H. Hulsman and P. Willems, Chem. Phys. 68: 233 (1982). 12. P. Willems, H. Hulsman and F. Aerts, Chem. Phys. (1982) in press. 13. P. Kusch and M.M. Hessel, J. Chem. Phys. 68: 2591 (1978). 14. H. Ashkenas and F.S. Sherman, in: "Rarefied Gas Dynamics",
J.H. de Leeuw, ed., Academic Press, New York (1965). p. 84. 15. P. Willems, H. Hulsman and F. Aerts, RGD .(1982).
877
MOLECULAR BEAM TIME-OF-FLIGHT MEASUREMENTS AND MOMENT METHOD CALCULATIONS OF TRANSLATIONAL RELAXATION IN HIGHLY HEATED FREE JETS OF MONATOMIC GAS MIXTURES
A. Chesneau and R. Campargue
Laboratoire des Jets Moleculaires Departement de Physico-Chimie, CEN Saclay, France
INTRODUCTION
The translational relaxation, in free jets of pure or mixed monatomic gases, has been studied extensively from theory and experiment during the last twenty years. Theoretical studies on this subject include moment method calculations performed initially for pure gasesl-3 and then for gas mixtures,4-9 and recent Monte Carlo calculations also for pure gasesiO and gas mixtures.9,11,12 Experimental studies on the same topic have been done by determining parallel translational temperatures, T~ , mainly from time-of-flight (TOF) distributions (measured in molecular beams skimmed from free jets of pure gases3,13-15 and gas mixtures 6,7,16-17) and also from Doppler broadenings (observed in the electron beam induced fluorescence in gas mixture free jets9). In spite of such a considerable amount and wide variety of investigations on this subject,it should be emphasized, as indicated previously by Andersonl8 and largely discussed in our recent paper,17 that much more work must be done for resolving many discrepancies between existing results on gas mixture free j ets. Furthermore, mos t of the previous experimental data have been obtained only at room temperature and, consequently, the variation of the nozzle stagnation temperature T0 in a largerange, as made in the present work, contributes to describing more completely the relaxation phenomena in gas mixture free jets. Finally these investigations are of great interest for understanding and improving the seeded molecular beam technique which has important applications in scattering experiments, spectroscopy, isotope separation, etc. The present results have been obtained by the moment method and the time-of-flight technique.
879
THEORETICAL : THE MOMENT METROD
In the first four moments of the Boltzmann equation, appear the conservation laws for number of particles (number density ni), mo-
+ ( . + . ) mentum IDiUi where mi = mass of part1cle and Ui = flow veloc1ty , stress tensor Pilk and heat flux Qi, as a set of 13 equations for each i species .• This system is not closed, due to the heat flux term arising in the stress tensor evolution equation and, consequently,cannot be solved without simplifications. In the experimental conditions of the present work (source Reynolds number ~ 104), the usual assumption1,4,5,7-9 that hypersonic flow is established before significant non-equilibrium effects occur, is particularly realistic and acceptable for simplifying the calculations. With this approximation (Mach number ~ > 10), the heat flux terms, all proportional to 1/fYt , become negligible and the system is reduced to a closed set of ten equations (one for continuity, three for momentum, six for stress components). Furthermore, the flow on the centerline of an axisymmetric free jet can be modelled, as in most of the previous moment method calculations, by a spherically symmetric source flow.l9 This model reduces the problern to a set of four equations giving a complete description of the flow : one for continuity, one for momentum (along a streamline), and two for the stress components (corresponding to T 11 and T .J. ) •
After such simplifications the main difficulty remains in the evaluation of the non-zero collision terms and the definition of a velocity distribution. Two alternatives are possible. In the first way, an ellipsoidal velocity distribution can be used2,3,6 and the integrals are then expanded to the first order in (T 11 -T~ )/T~ assumed to be small. In the second way followed here, the collision terms are formulated for Maxwell molecules 1,4,5,7,8 (kR-4 potenti~l). This potential is not realistic but it greatly simplifies the formulation of the collision terms and avoids the choice of a velocity distribution. Finally, a more appropriate potential is often introduced in the calculation of the collision integra~. Such a procedure may appear surprising, but Kolodner20 showed that the additional terms,appearing when a velocity distribution function is used,are at least one order of magnitude lower than the basic terms which are found -wi.th the maxwellian approximation. At this point, the expansion is described by the following equations derived from Kolodner20 with the above approximation :
d 2 ar (r piui) = 0 r = radial distance from the nozzle throat (I)
dU. d 2 piui ar1 + ar (nik Tl/i) + r nik (ToCT.J.i)
880
- A .. p.p.v .. (2) 1J 1 J 1J
3 ClkT. Clu. --1 + kT --1 +2
u. n.k T...1.. 2. 2 ni ui ar ni // i ar 1 1 r
A •• 3 ~ p.p. [k(T.-T.) +
m.+m. 1 J J 1 1 J
m.v?. J 1J
3
Clk (T 11 • -T .J. • ) Clu. u. ni ui 1 Clr 1 + 2 ni k (T II i Clr1 - T . .1- i r1 )
A·. 2 2 __2:1_ p.p. [k(TI/ .-T.L .) - k (T 11 .-T..l. .) + m.v .. ]
m.+m. 1 J J J 1 1 J 1J 1 J
[ k(TI/ ij-T..I..ij) + ~ 2 - 3 B .. p.p. v ..
1J 1 J m. m.+m. 1J 1 1 J
m.+m. 2 1 J ( - 3 B .. --2- p. k T111.-T..l. 1.)
1J 2m. 1
where p. 1
1
n.m. is the mass density of the i species, 1 1
(3)
(4)
v .. 1J
u.-u. characterizes the velocity slip effect, 1 J
T. 1
T •. 1J
A .. 1J
B .. 1J
T 11 • + 2 T J.. • 1 1
3
T. T. ( 2. + .....J.. ) ].1 wi th
m. m. 1 J
16 n~ ~ (T .. ) =- 1J 1J
3 m.+m. 1 J
16 n~~ (T .. ) 1J 1J
-15 m.+m. 1 J
m.m. ].1 =2...L
m.+m. 1 J
r.kk 11" . . 1 b 1. d . . 1 . ~~ij are co 1s1on 1ntegra s etween an J part1c es, as def1ned by Hirschfelder et a1.21
Collision integrals have been calculated for Maxwell molecules by Cooper and Bienkowski, 5 also for other repulsive power law potentials by Miller and Andres6, and Willis,8 and for rigid spheres by Anderson7 • To our knowledge, it seems that collision integrals for attractive potentials have never been used in the problern of binary mixture expansions. In this work they have been calculated according to Kibara et al.22 for an inverse power attractive potential :
881
V(R)
V(R)
- C R-6 for R > 0 6
()() for R 0
which is well suited to long distance interaction problems. The values of C6 have been taken from Dalgarno23.
With the additional assumption of complete expansion with negligible velocity slip effect (as it is the case in our experimentsl7 where : ui = Uj = u), the equations (3) and (4) ruling the temperatures become
3 ar. r.J... __ 1+2--1=
2 ar r
ar.J. . ___ 1 +
ar T .Li
2 --= r
A •• m. p. - 3 _2l__2:_ __,1. (T. -T.)
m.+m. u 1 J 1 J
A .• m. p. 2 2.:1..2:. __,1.
m.+m. u 1 J
( Tl. . -T .l. . ) J 1
p. Pi + B . • __,1. (T I.'/ •• -T ..L •• ) + BI I (T 11 • -T ..L • ) 1J u 1J 1J u 1 1
This set of equations has been solved rtumerically by means of a fourth-order Runge-Kutta integration scheme,with the initial condition for small value of R :
T 11 • =T.l.. 1 1 T '' • J
TJ,.. = T. • J 1sentrop1c
The solutions have been obtained for several heated binary mixtures (He-Ar and He-Xe with heavy gas mole fraction from 0.3 to 10 %) also investigated by the TOF technique. The scaled temperature t' and scaled distance r' have been derived from WillisS :
where
882
t I T =r-
r'
0
[ P r*
..!.._E F 0
r* kT 0
6 c 1/3]12/11 ( 6 ReHe)
kT where F 0.63
0
6 c 1/3] -g/ll ( 6 ReHe)
kT E 1 • 7 55
0
r"' = 0. 7 D * = radius of the apparent source sphere ( m = I)
c -6 6HeHe = R potential coefficient for helium-helium particle interaction
EXPERIMENTAL : THE MOLECULAR BEAM TIME-OF-FLIGHT TECHNIQUE
The experimental studies of translational relaxation in monatomic gas mixture free jets, have been performed on molecular beams skimmed from zones of silence of these free jets, according to the principle developed at Saclay.l4,15, The apparatus used, with its sophisticated TOF system, has been described recently in details. 17 The difficulties encountered in previous experiments of this type seem to be avoided in our laboratory thanks to :
(I) very large values of P0 o• up to 500 Torr-em (with nozzle Stagnationpressure P0 = 10-30 bars and nozzle diameter o* = 0.19 and 0.23 mm) yielding high beam intensities, very narrow velocity distributions (i. e. speed ratio S 11 "' 120 for Xe in He) and nearly negligible velocity slip (~ I % from experiment17),
(2) heating of the nozzle up to 3000 K (from II 00 to 2500 K in the present work) primarily for the purpose of avoiding condensation, also for studying the relaxation in a wide temperature range and finally for achieving kinetic energies in the eV range, up to about 35 eV,
(3) the use of an extremely high resolution TOF system, with flight path as long as 4.156 m and very high sensitivity mass-selective detector, which allows velocity distributions to be measured for velocity spreads as low as 0.5 % (Ref. 15) or concentrations as low as 60 p.p.m.17
RESULTS AND DISCUSSION
The theoretical and experimental data reported in this paper have been obtained for highly heated free jets of I to 10% Ar in He (Fig. I) and 0.3 to I % Xe in He (Fig. 2).
The variations of scaled temperature t', with scaled distance r', calculated by the moment method described above and starting from isentropic values close to the nozzle, deviate from these ideal values, with increasing r' in the jet. Thus, the main following characteristics are found theoretically:
(i) T// > T..1.. "' T. t . , which is consistent with all the known previous results;sen rop~c
(ii) T 11 h > T 11 z. ht' which is at least in qualitative agreement with ~?~~ious mo~~nt method5,7-9 and Monte Carlo9,11,12 calculations, but not with the moment method data of Willis and Hamel4 and Nanbu,24 the latter being obtained, without the hypersonie approximation, for a quasi-one-dimensional flow in an axisymmetric hyperbolic nozzle, instead of a free jet,
(iii) T n h decreases with increasing heavy gas mole frac-tion, as predict~~v~efore by Anderson18 and WillisS ; also Tu z. ht decreases at the same time, similarly to T 11 heavy· ~g
883
t'
Gas mlxture
1%Ar lnHe 5%Ar ln He
r'
•
• Ar
T" o He
100
Fig. 1. Variation of scaled temperatures with scaled distance for heated free jets of He-Ar mixtures.
t' 10
Gas mlxture
T,
Moment method Experiment T"
1 XeH
Xe
--- --------•
He
10 r•
Fig . 2. Variation of scaled temperatures with scaled distance for heated free jets of He-Xe mixtures.
884
The experimental values of T 11 , for light and heavy species, appear to be nearly the same as those calculated at the distances r' corresponding to the skimmer entrance (Figs. I and 2) . In other words, the expansion seems to be stopped by the skimming process and, consequently, the measured T 11 seem to be slightly higher than the terminal values in the heated free jets. Also, with the same assumption for r', our moment method curves represent conveniently the room temperature beam TOF data of Anderson7 (Fig. 2) but neither those of Miller and Andres6, nor the T 11 for Ar found lower than for He in the Doppler broadening measurements of Cattolica et al.,9 at a well defined distance r' in this case (Fig. I). It is worth recalling that, in the generation of our room temperature He beams, the expansion goes on dowstream of the skimmer, due mainly to quantum effects3 leading to extremely low temperaturei4,I5 (Tu ~ 6 x Io-3 K). Such effects seem to have a minor influence in heated free jets as in the room temperature free jets produced at low P0 D* values in other laboratories.
REFERENCES
I. B. B. Hamel and D.R. Willis, Kinetic theory of source flow expansion with application to the free jet, Phys. Fluids, 9 : 829 (I966).
2. E. L. Knuth and S. S. Fisher, Low-temperature viscosity cross sections measured in a supersonic argon beam, J. Chem. Phys., 48 : I674 (I968).
3. J. P. Toennies and K. Winkelmann, Theoretical studies of highly expanded free jets : influence of quantum effects and a realistic intermolecular potential, J. Chem. Phys., 66 : 3965 (1977).
4. D. R. Willis and B. B. Hamel, Non-equilibrium effects in spherical expansions of polyatomic gases and gas mixtures, in "Rarefied Gas Dynamics", C. L. Brundin ed., Academic Press, New York, London (I967).
5. A. L. Cooper and C. K. Bienkowski, An asymptotic theory for steady source expansion of a binary gas mixture, in "Rarefied Gas Dynamics", C. L. Brundin ed., Academic Press,New York, London (1967).
6. D. R. Miller and R. P. Andres, Translational relaxation in low density supersonic jets, in "Rarefied Gas Dynamics", L. Trilling and H. Y. Wachman-eds., Academic Press, New York ( I969) •
7. J. B. Anderson, Intermediate energy molecular beams from free jets of mixed gases, Entropie, 18 : 33 (I967).
8. D. R. Willis, Theoretical analysis of the velocity slip effect in free jet expansions of binary and ternary gas mixtures, Sandia Labs. Energy Report, SAND 78-8216 (I978).
9. R. J. Cattolica, R. J. Gallagher, J.B. Anderson, and L. Talbot, Aerodynamic separation of gases by velocity slip in free jet expansions, AIAA Journal, I7 : 344 (I979).
885
10. A. U. Chatwani, Monte Carlo simulation of nozzle beam expansion, in "Rarefied Gas Dynamics", J. L. Potter ed., AIAA, New York (1977).
II. P. Raghuraman, P. Davidovits, and J. B. Anderson, Isotope separation by the velocity-slip process, in "Rarefied Gas Dynamics", J. L. Potter ed., AIAA, New York ( 1977).
12. A. U. Chatwani and M. Fiebig, Source expansion of monatomic gas mixtures, in "Rarefied Gas Dynamics", S. S. Fisher ed., AIAA, New York (1981).
13. J. B. Anderson and J. B. Fenn, Velocitydistributions in molecular beams from nozzle sources, Phys. Fluids, 8 : 780 (1965).
14. R. Campargue and A. Lebehot, High intensity supersonic molecular beams with extremely narrow energy spreads in the 0-37 eV range, in "Rarefied Gas Dynamics", M. Becker and M. Fiebig eds., DFVLR Press, Porz-Wahn, West Germany (1974).
15. R. Campargue, A. Lebehot arid J. C. Lemonnier, Nozzle beam speed ratio above 300 skinnned in a zone of silence cf He freejets, in "Rarefied Gas Dynamics", J. L. Potter ed., AIAA, New York (1977).
16. N. Abuaf, J. B. Anderson, R. P. Andres, J. B. Fenn, and D. R. Miller, Studies of low density supersonic jets, in "Rarefied Gas Dynamics", C. L. Brundin ed., Academic Press-:-New York, London (196 7) •
17. R. Campargue, A. Lebehot, J.C. Lemonnier, and D. Marette, Measured, very narrow velocity distributions for heated, Xe and Ar-seeded nozzle-type molecular beams of He and Hz skinnned from free-jet zones of silence ; Xe energies up to 30 eV, in "Rarefied Gas Dynamics", S. S. Fisher ed., AIAA, New York; (1981).
18. J. B. Anderson, Molecular beams from nozzle sources, in '~1olecular Beams and Low Density Gas Dynamics', Gas Dynamics, P. P. Wegeuer ed., Marcel Dekker Inc., New York (1974).
19. H. Ashkenas and F. S. Sherman, The structure and utilization of supersonic free jets in low density wind tunnels, in "Rarefied Gas Dynamics", J. H. de Leeuw ed., Academic Press, New York (1966).
20. I. I. Kolodner, Moment description of gas mixtures. I, Report NY0-7980, Institute of Mathematical Sciences, New York University (1957).
21. J. 0. Hirschfelder, C. G. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids", John Wiley and Sons, New York (1954).
22. T. Kihara, M. H~ Taylor, and J. 0. Hirschfelder, Transport properties for gases assuming inverse power intermolecular potentials, Phys. Fluids, 3 : 715 (1960).
23. A. Dalgarno, I. H. Morrison, and R. M. Pengelly, Leng range interactions between atoms and molecules, Int. J. Quant. Chem. , 1 : 1 6 1 (I 9 6 7) •
24. K. Nanbu, Velocity slip and temperature difference of gas mixtures in quasi-aue-dimensional nozzle flows, Phys. Fluids, 22 : 998 (1979).
886
ROVIBRATIONAL STATE POPULATION DISTRIBUTIONS OF CO (v ~ 4, J ~ 10)
IN HIGHLY HEATED SUPERSONIC FREE JETS OF CO-Nz MIXTURES
M.A. Gaveau, J. Rousseau, A. Lebehot, and R. Campargue
Laboratoire des Jets Moleculaires Departement de Physico-Chimie, CEN Saclay, France
J.P. Martin
Groupe de Recherehes Thermiques du CNRS Ecole Centrale, Chatenay-Malabry, France
INTRODUCTION
The rovibrational relaxation in free jets is a topic of fundamental importance in molecular spectroscopy, laser isotope Separation, molecular beam applications and also of interest for gas dynamic and chemical lasers. The complexity of this problern results from heat capacity lags in free jet expansions of polyatomic gases, due both to continuous decrease of the collision frequency and to the very different collision numbers required for relaxation of the various modes of motion and, consequently, the successive freezings of the vibrational, rotational and translational temperatures : Tvib >> Trot > Ttrans• This appears only as a very simplified description because one single temperature cannot characterize nonBoltzmann distributions,as observed in a number of laboratories.
Since 1962 and mainly during the last few years, the spatial distributions of rovibrational state populations have been measured in free jets for a variety of gases (Nz, COz, CO, NO, Hz, Dz, CH4, CzHz, SF6, Na2 , etc.) with different spectroscopic techniques.: electron beam induced fluorescence,1,2 one-photon laser induced fluorescence,3 spontaneaus fluorescence,4 spontaneaus Raman scattering,5 coherent anti-Stokes Raman spectroscopy (CARS),6 multiphoton induced fluorescence or iönization (MPI)7. Also many diagnostics have been made on molecular beams skimmed from free jets, in particular by using the time-of-flight, bolometric, MPI, or electron beam8 tech-
887
niques (see Ref. therein). These previous data have been the subject of considerable work and controversy. Finally, even the existence of the non-Boltzmann problern in free jets is still discussed, due to the difficulties in finding an appropriate model of the excitation-emission process, for relating the measured intensity distribution to the initial population distribution. Thus, the appearence and the importance of nonequilibrium effects, in the published wide variety of electron beam fluorescence measurements, depend on the approaches used for interpretation with different ~J selection rules2,8 (Muntz, Coe et al., Rebrov et al., DeKoven et al.). Additional complications may result from collisions (quenching), selfabsorption, secondary electrons, etc.
Contrary to most of the previous studies, this work deals with highly heated free jets produced with the stagnation conditions used in a gas dynamic laser (T0 ~ 2000 K, P0 = 10-30 bars). Spontaneous fluorescence measurements, interpreted by a convenient emission selfabsorption model, are presented here for 20 % CO in N2 and in another paper9 for 20 % CO in Ar.
EXPERIMENTAL
The heated free jets are generated by means of a heating furnaceiO for bringiL6 a high pressure circulating gas stream (~50 bars) to a very high temperature (~ 3000 K). Heating is obtained by the Joule effect through small tubular ducts driving the flow and discharging the hot gases just upstream of a nozzle throat. Such a device makes it possible to achieve a very high stagnation enthalpy in a thermodynamic equilibrium state.ll
In the present work, free jets of 20 % CO in N2 are produced through a sonic nozzle (D* = 0.22 mm in diameter) operated at stagnation pressure Po ~ 30 bars and temperatures T0 close to 1800 K. Thus, with the background pressure, Pt ~ 0.15 Torr, maintained in the nozzle chamber by means of mechanical Roots pumps, the Mach disk is located at about 58 mm from the nozzle throat and the supersonic flow inside the shock barrel is similar to that expanding into a perfect vacuum (zone of silence), at least in the first stages of the jet, now investigated (X/o• ~ 36).
Fig. I. Scheme of the set up
888
The rovibrational relaxation of CO, mixed and expanded with N2 in such heated free jets, is studied by measuring infrared fundamental (~v = I) spontaneaus fluorescence emission of the CO jet molecules (4.6 to 5 ~m). The radiation is collected by an optical system (Fig. 1), mechanically chopped at 1000Hz and spectrally resolved with a scanning grating monochromator. The spectral bandwidth is ~cr = 0.5 cm-1 and the spatial resolution corresponds to ~X = 0.2 mm on the jet axis. The radiation is detected by a liquid nitrogen cooled InSb photovoltaic detector. Then the signal is enhanced through a lock-in amplifier and finally displayed on a (X,T) chart recorder. The nozzle can be moved in order to analyze the 1R emission at different distances XID • from the throat.
RESULTS AND DISCUSSION
Experimental Data
Fluorescence spectra have been recorded for the (v,v-1) systems going from (1,0} to (4,3) at locations Xln• = 18 and 36. The intensity r~:,~ per volume unit of the (v',J') + (v,J) transition can be expresseA as :
, _ 64~4 v',v 4 Exp [-Bv,J'(J'+I)IkTR] I 2 r;,·,;- - 3- (v3 , 3 ) s3 , J ~~------.------ v' IR I 3c ' ' Qv o
R where c = velocity of light,
' vv ,v J' ,J
frequency of the transition,
population number density of v' vibrational level,
transition matrix element for rotation,
rotational constant for the v' vibrational level,
partition function for rotation in the v' level,
dipole transition matrix element for v I + 0.
v' v v' v 4 In the plots of Log [r3 ,•J I (v3 ,•3 ) SJ' Jl versus J'(J'+I), a
Boltzmann rotational distribui1on appeärs as ~ straight line, the slope of which is - Bv' I kTR, and the deviations from rotational equilibrium are indicated by the deviations from this straight line. Nevertheless, it is possible to obtain deviations which are not due to nonequilibrium effects, but result for instance from self-absorption phenomena, nonhomogeneity in temperature and density, etc. The departures from linearity, observed in the present results, seem to be due mainly to emission and self-absorption phenomena occurring in the free j et as well as in the background gas. The spontaneaus fluorescence of the background gas has been measured by recording the
889
spectrum emitted only from the gas surrounding the jet. In this case, the vibrational temperature was TXG ~ 1000 K from the intensity ratio of the (1,0) and (2,1) bands, and the rotational temperature deduced from the slope of the curves was T§G ~ 300 K.
In Fig. 2 are shown, in the defined logarithmic form, the relative intensities of the v + 1, v P(J) lines, measured at x;n• = 18, for the stagnation conditions described above. Three main features can be pointed out in this figure :
(i) the data concerning the (1,0) and (2,1) bands for . > > J'(J'+I) ~ 100, i.e. J' ~ 10, appear tobe due to the fluorescence of the background gas, with negligible contribution from the jet, as observed in the analysis of the pure background emission aiso with T~G ~ 300 K, T~G ~ 1000 K.
(ii) the data concerning the (3,2) and (4,3) bands seem to characterize a pure free jet emission (i.e. without any self-absorption produced by the jet, or the background gas) as indicated by the linearity of the plots showing a Boltzmann rotational equilibrium (at least for the low J numbers) with a rotational temperature, T~J ~ 70 K, given by the slope of the straight lines, and a twolevel (4,3) vibrational temperature, T~J (4,3) ~ 1600 K, deduced from the ratio Ij!.T J I Ij!.T .J
(iii) the data'concerning the(1,0) and (2,1) bands, for J'~ 10, result from the superimposition of the emission and the self-absorption occurring in the free jet and the background gas.
An excellent confirmation of these results is shown in Fig. 3 for x;n• = 36, where are found the following temperatures TfJ 45 K, T~J = 1500 K, T~G = 380 K, and TXG = 1000 K.
Model of the Emission Self-Absorption Process
The entire spectrum recorded for CO can be calculated by taking into account both emission and absorption in the free jet and the background gas. The intensity of the radiation is integrated over the successive media encountered by an optical path , using absorption and emission constants deduced from Einstein coefficients for each transition.12 Thus, a good agreement with the experiment data (Figs. 2 and 3) has been found by assuming Boltzmann vibrational and rotational distributions (with Trot << Tvib) in the free jet and the background gas. Nevertheless, it cannot be asserted that this model is the only way for interpretation.
The calculation is performed by using the temperatures deduced from the experimental data (Figs. 2 and 3). The vibrational temperature T~J is taken as equal T~J (4,3). The stagnation temperature T0 , not measurable in the heated nozzle, is determined from the measured values of T~J on the basis of the following flow model : CO is in a rotational and vibrational equilibrium upstream of the nozzle throat (i.e. Tvib = Trot = Ttrans• with specific heat ratio
890
100
I \
Experiment
Theory
100
' -.., 200
J'(J'+1)
• 1-0 P(J) • 2- 1 P(J) A 3- 2 P(J) lll-3 P(J)
R T 90:300 K
T~0 = 1000 K
300 400
-
Fig. 2. Measured and calculated line intensities vs J'(J'+I) for N2-co free jet (P = 30 bars, T ~ 1800 K, X/n• = 18).
0 0
1oor------.------~------~------~-~
> .., ;.; '> 10
Ol 0
...J
I~ ~-~~ \
I A ',
Experiment • 1- 0 P(J) • 2- 1 P(J) A 3 -2 P(J) l 4- 3 P(J)
Theory
' ' . '' 1 lf I & ',
\ \ . ...... .... • ' . ' +, \
l I \ \
100
• .
200
J'(J'+1)
•
300 400
Fig. 3. Same caption as for Fig. I for X/D 4 = 36.
891
y = 9/7)and only the vibrational energy is frozen downstream of the throat (y = 7/5). Thus, from the gas dynamic relations coupled with the Mach number distribution ·~ (X/n•,y) of Askhenas and Sherman,13 it is found T0 ~ 1800 K and T•~ 1580 K (at the nozzle throat) by using either TfJ ~ 70 K measured at X/D~ = 18, or T~J ~ 45 K at X/n·• = 36. Also, with above model, it is possible to deduce from T~J at X/D* = 18, the value of this temperature at any X/D"' > 18, e.g. T~J = 45 Kat X/n• = 36, exactly the value measured. Finally, it can be seen that r• ~ 1580 K ~ T~J and this consistent with the vibrational freezing of the model at the nozzle throat.
The line intensities computed for X/n• = 18 and 36, by using this relaxation model, are reported as dashed lines in Figs. 2and 3, respectively. The theory is in good agreement with the experim.ent and the present description appears tobe "self-consistent".
Discussion
The present results, interpreted on the basis of a Boltzmann equilibrium in each internal mode, could appear in disagreement with those obtained for CO in supersonic nozzles,14 where a vibrational temperature cannot be defined because the high vibrational levels are overpopulated compared to a Boltzmann distribution15. On the contrary, it seems that T~J is meaningful along the free jet with a constant value nearly equal to the translational temperature at the nozzle throat. This is due to the fact that the free jet expansion is so rapid that the V-Vexchanges betweenexcited CO molecules are not possible, the number of collisions necessary for vibrational relaxation {•v a few thousandsl6) being very much higher than the few collisions per f!sec undergone by the jet molecules. A similar conclusion could be given about CO-N2 V-V exchanges.
Also departures from rotational equilibrium are not observed in the present work, as in previous measurements of fluorescence (spontaneous in COz,4 or induced by electron beam in Nz 1,8) but apparent deviations could be due to the model used2,8 (overpopulation) or the skimmer8 (excess in Trat>· The Raman scattering data have shown only Boltzmann distributions for Nz, COz, CH4, SF6,5 but not for Hz and Dz,5 due to their relatively large rotational collision numbers. In any case, the departures are found to appear m.d increase with increasing J and/or the rarefaction (low collision frequency at low P0 n• and large X/n•). Consequently, it is not surprising to find Boltzmann distributions in the present measurements performed only for J ~ 10, with X/D"' ( 36, and P0 D* ~ 5000 Torr-mm (i.e. 10 to 100 times higher than elsewhere).Experiments with better sensitivity are now under way for detecting transitions with J > 10. Also, new techniques (multiphoton laser induced fluorescence or ionization) will be used soon in free jet and nozzle beam diagnostics.
892
REFERENCES
I. F. Robben and L. Talbot, Measurements of rotational temperatures in a low density wind tunnel, Phys. Fluids, 9 : 644 (1966) ; P. V. Marrone, Temperature and density measurements in free jets and shock waves, Phys. Fluids, 10 : 521 (1967) ; R. G. Sharafutdinov, A. E. Belikov, N. V. Karelov, and A. E. Zarvin, 13th International Symposium on Rarefied Gas Dynamics, Novosibirsk, USSR (July 1982), these Proceedings.
2. E. P. Muntz, Static temperature measurements in a flowing gas, Phys. Fluids, 5 : 80 (1962) ; D. Coe, F. Robben, L. Talbot, and R. Cattolica, Measurement of nitrogen rotational temperatures using the electron beam fluorescence technique , in "Rarefied Gas Dynamics," R. Campargue ed., CEA, Paris (1979); A. K. Rebrov, G. I. Sukhinin, R. G. Sharafutdinov, and J. C. Lengrand, Electron beam diagnostics in nitrogen. Secondary processes, Sov. Phys. Tech. Phys., 26 : 1062 (1981).
3. K. Bergmann, W. Demtroder, and P. Hering, Laser diagnostic in a molecular beam, Appl. Phys. 8 : 65 (1975) ; F. Aerts and H. Hulsman, Population evolution of internal states of Na2 in a free jet expansion, in "Rarefied Gas Dynamics," R. Campargue ed., CEA, Paris (1979); D. H. Levy, Laser spectroscopy of cold gas-phase molecules, Ann. Rev. Phys. Chem., 31 : 197 ( 1980) • ,
4. S. P. Venkateshan, S. B. ·Ryali, and J. B. Fenn, Terminal dist·ributions of rotational energy in free jets of C02 by infrared emis~ion spectrometry, J. Chem. Phys., 77 : 2599 (1982). J. P. Martin, unpublished results.
5. I. F. Silvera and F. Tomasini, Intracavity Raman scattering from molecular beams : direct determination of local properties in an expanding jet beam, Phys. Rev. Lett., 37 : 136 (1976) ; I. F. Silvera, F. Tomasini, and J. R. Wijngaarden, Direct measurement of density and rotational temperature in a co2 jet beam by Raman scattering, in "Rarefied Gas Dynamics," J. L. Potter ed., AIAA, New York (1977) ; H. P. Godfried, I. F. Silvera, and J. Van Straaten, Rotational temperatures and densities in Hz and D2 freejet expansions, in "Rarefied Gas Dynamics," S. S. Fisher ed., AIAA, New York (1981); G. Luijks, S. Stolte, and J. Reuss, Molecular beam diagnostics by Raman scattering, Chem. Phys. 62 : 217 (1981).
6. J. J. Valentini, P. Esherick, aPd A. Owyoung, Use of a freeexpansion jet in ultra-high-resolution inverse Raman spectroscopy, Chem. Phys. Lett., 75 : 590 (1980) ; M. D. Duncan, P. Oesterlin , F. KÖnig, and R. L. Byer, Observation of Saturation broadening of the coherent anti-Stokes Raman spectrum (CARS) of acetylene in a pulsed molecular beam, Chem. Phys. Lett., 80 : 253 (1981).
7. ~ Demaray, C. Otis, K. Aron,and P. Johnson, Laser enhanced collisional effects in the multiphoton ionization of molecules in supersonic expansions, J. Chem. Phys., 72 : 5772 (1980).
893
8. P. B. Scott and T. R. Mincer, Molecular beam rotational temperature measurement, in "Rarefied Gas Dynamics", D. Dini ed., Editrice Tecnico Scientifica, Pisa, Italy (1971) ; B. M. DeKoven, D. H. Levy, H. H. Harris, B. R. Zegarski, and T. A. Miller, Rotational excitation in the electron impact ionization of supercooled N2, J. Chem. Phys., 74 : 5659 (1981) ; M. Faubel and E. R. Weiner, Electron beam fluorescence spectrometry of internal state populations in nozzle beams of nitrogen and nitrogen/rare gas mixtures, J. Chem. Phys., 75 : 641 (1981) ; S. P. Hernandez, P. J. Dagdigian, and J. P. Doering, N2 rotational energy distributions in cold, supersonic beams from electron excited fluorescence measurements of N2+, in press, Chem. Phys. Lett.
9. M. A. Gaveau, J. Rousseau, A. Lebehot, and R. Campargue, Rovibrational state population distributions calculated from spontaneaus fluorescence measurements for CO (v ~ 4, J ~ 1 0) in highly heated supersonic free jets of CO in N2 and Ar, 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy (Sept. 1982),Proceedings tobe published, Plenum Publishing Corporation, Ne~., York.
10. R. Campargue, J. Bouffenie, and A. Recule, A furnace for heating a circulating gas stream and especially for producing molecular beams, Patent 2,179,045 filed July 1974 (France) and Foreign Patents : 4,063,067 (USA), etc.; R. Campargue, M. A. Gaveau, A. Lebehot, J. C. Lemonnier, and D. Marette, High enthalpy generator (P0 ~ 50 bars, T0 ~ 3000 K) for supersonic free jets and nozzle beams, 7th International Symposium on Molecular Beams, Riva del Garda, Italy (May 1979).
11. R. Campargue and A. Lebehot, High intensity supersonic molecular beams with extremely narrow energy spreads in the 0-37 eV range, in "Rarefied Gas Dynamics", M. Becker and M. Fiebig, eds., DFVLR Press, Porz Wahn, West Germany (1974) ; R. Campargue, A. Lebehot, J. C. Lemonnier, and D. Marette, Measured, very narrow velocity distributions for heated, Xe and Ar-seeded nozzle-type molecular beams of He and H2 skimmed from free jet zones of silence ; Xe energies up to 30 eV, in "Rarefied Gas Dynamics," S. S. Fisher, ed., AIAA, New York (1981).
12. S. S. Penner, "Quantitative Molecular Spectroscopy and Gas Emissivi ties," Addison-Wesley, Reading, London (1959) .
13. H. Ashkenas and F. S. Sherman, The structure and utilization of supersonic free jets in low density wind tunnels, in "Rarefied Gas Dynamics", J.H. de Leeuw, ed., Academic Press,New York, London (1966).
14. J. P. Martin ; P. J. Bender, M. Mitchner, and C. H. Kruger, (see Gaveau et al.9 and references therein).
15. C. E. Treanor, J. W. Rich, and R. G. Rehm ; R. E. Center and G. E. Caledonia (ibid).
16. V. F. Gavrikov, A •. P. Dronov, V. K. Orlov, A. K. Piskunov, and V. L. Shikanov (ibid).
894
FREE JET EXPANSION WITH A STRONG CONDENSATION EFFECT
N.G.Gorchakova, P.A.Skovorodko, and V.N.Yarygin
Institute of Thermophysics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk 630090, USSR
INTRODUCTION
Today the investigations of nonequilibrium processes, such as condensation, chemical reactions, vibrational and rotational relaxation in expanding flows remain urgent. The effect of nonequilibrium processes, and especially condensation on gasdynamics of expanding flows is the subject-matter of this paper. Although the role of these processes is qualitatively clear, there is no sufficient number of specific data, especially as applied to low density free jets. The above-said concerns both theoretical and experimental studies. The latter are particularly important under the conditions, where qualitatively differing relaxation processes occur simultaneously, e.g. condensation and vibrational relaxation including heterogeneaus one.
Since for a given nozzle the process of condensation and vibrational relaxation are completely determined by the value of stagnation temperature To and pressure po~ it is natural to study the effect of each of these parameters individually.
In Ref. 1, among other things, the effect of stagnation temperature To on the structure of nitrogen jet behind a sonic nozzle in a flooded space was studied. It was found that a relative position of Mach disk was not changed with T 0 over the range 300-4200 K.
The investigation of the effect of stagnation parameters on C02 expansion behind a supersonic nozzle using the electron-beam diagnostics is more detailed in Refs. 2,3. In both these papers the homogeneaus condensation effect on the co2 expansion is discussed. However, in the conclusions of these works concerning C02 expansion there are some differences. In Ref. 2 the conclusion is made that
895
co2 jet expansion takes place when the specific heat ratio y=cp/cv is about 1.4. In Ref. 3 the value of y for C02 jet is assumed to be equal to 1.3. In both these papers the viscosity effect on the nozzle flow is not considered.
The purposeful experimental and theoretical investigation of T 0 and p 0 effects Cthe nozzles with d*=2 and 2.85 mm were used) on C02 and N2 jet expansion behind a sonic nozzle was made in Ref. 4. It was found that depending on the value of modeling parameters To and pod* the co2 expansion is possible either with y=const=l .4 Cif relaxation processes are frozen in subsonic region CM<I)) or with y<1.4 Cif the freesing takes place in supersonic flow region CM>l)). The present study being an extension of the above work, contains the results of experimental and theoretical investigations of C02, N2 and Ar expansion from supersonic nozzles.
EXPERIMENTAL ARRANGEMENT AND MEASUREMENT TECHNIQUES
The experiments were performed in a vacuum wind tunnel4. A conical nozzle with 0.5 mm throat diameter d*, 2 mm exit diameter da, 10° half-angle was used. The Stagnationparameters To and po were varied over the range 300 to 800 K and p 0 =C0.2-30)•105 Pa. Test gases were of industrial purity, as a rule. In the special experiments with co2 the gas was pre-cleaned by the cryogenic vacuum method.
The peculiarities of nonequilibrium expansion were studied for C02, N2 and Ar jets based on the local density measurements from the intensity of the electron-beam induced X-ray bremmsstrahlung. Using the X-ray spectral region excited by an electron beam for density measurements has some advantages as compared with an optic region, namely, a better spatial localization, the absence of gas temperature and secondary electron effects. A methodic peculiarity of these experiments is that the measurements are made at fixed distances x from the nozzle exit at constant gas flow rate and background pressure and with Stagnation temperature To varying from an initial to maximum value at the given heating rate.
In addition, spectral measurements of C02 jet radiation in a region of 4.3 ~m were performed.
THEORETICAL ANALYSIS OF GAS EXPANDING INTO VACUUM
Tagether with the experiments, the calculations of flowfield in a jet expanding into vacuum from a conical nozzle were performed. The effects of viscosity in the nozzle flow and nonequilibrium processes of condensation and vibrational relaxation were considered. To calculate a steady two-dimensional flow in the nozzle and in the
896
jet b~hind it, the numerical methtid utilizing a natural coordinate system defined by streamlines and Lines normal to them5 was used. Various types of calculations were made as applied to the experimental conditions.
a> Calculations taking into account viscosity effects. Such calculations were performed for C02, N2, and Ar. The gases were assumed tobe perfect, the nonequilibrium processes were not taken into account. The details of a corresponding calculation technique are presented in Ref. 6.
The calculations show that taking into account the viscosity effects Leads to increasing streamline divergence behind the supersonie nozzle exit, that results in decreasing a relative density. This is caused by decreasing both Mach number at the nozzle exit and an effective exit radius due to the displacement action of the boundary Layer.
b) Calculations taking into account condensation4. Such calculations were performed for C02 and Ar. The viscosity and heat conductivity effects were not taken into account. The nonequilibrium homogeneous condensation process is described in the frames of classical nucleation theory. The expressions for thermodynamic values of C02 and Ar used in calculations are given in Refs. 4,7, respectively.
c) Calculations taking into account vibrational relaxtion of C02 4. The viscosity and heat conductivity effects were not taken into account. The relaxation process is described in the frames of relaxation equation for the specific vibrational energy of C02 •
d) Calculations taking into account simultaneous course of condensation and vibrational relaxation of C02 7. The algorithm consists in combining the methods mentioned above in parts b) and c).
The calculations show that taking into account the vibrational energy relaxation Leads to the down-stream displacement of the condensation front, that results in the decreasinp in the condensation process rate.
RESULTS AND DISCUSSION
Effect of To. The stagnation temperature effect on a relative density at the point on the jet axis at x/ra=41 .4 (x is the distance from the nozzle exit, ra is the exit radius) is illustrated in Fig. 1. The calculated results for the experimental conditions are shown by Lines, the experimental results are shown by points. The role of internal degrees of freedom in flow formation is evident, the character of the.p/po(To) dependences in Ar, N2 and C02 jets
897
Ar
10
8
6
4 .300 400
F ig. 1 •
___ _ _ a ' ' ~ Ma
6
5
4 . _,,_"_ "_
500
Effect of stagnation temperature on relative density. a - Ar jet, b - N2 and C02 jets.
differs qualitatively.
Let us now direct our attention to Fig. 1a, which shows the calculated results for the Ar flow with taking into account condensation <solid line), and with y=S/3 (dashed line). The experimental and calculated results for Ar allow gasdynamical aspects of the condensation effect to be revealed. This effect is mainly due to the decrease of Mach number at the nozzle exit, Ma, because of the condensation heat Liberation. The Ma (ToJ dependence obtained in calculations is presented by a chain-dotted Line. It is interesting that although Ma reaches its terminal value for the given nozzle (Mc=5.954) for y=S/3 when T0 =530 K, the effect of condensation in the jet behind the nozzle is essential: the relative density is more than twice lower than the values correspondin~ to the expansion with y=S/3. The viscosity effect on the nozzle flow is insignificant that will be illustrated below (see Fig. 3). Thus, as can be seen from Fig. 1a, condensation can make a significant effect bot h on the nozzle flow and on the jet behind the nozzle. Note that a good agreement between theoretical and experimental data means that the used model for the homogeneaus condensation describes adequetly main features of the phenomenon.
Now Let us consider the peculiarities of molecular gases (N2 and C02> expansion (Fig. 1b). The character of p/ po(To) dependences for N2 jet is mainly due to the viscosity effect on the nozzle flow. The condensation effect under the given conditions is not observed clearly, though the experiments with the nozzle with d* =0.3 mm at po=1 .9·106 Pa have r evealed it definit e ly.
898
The role of internal degrees of freedom in the flow formation is seen in the experiments with C02. Fig. 1b represents the experimental results for p/po(To) for two regimes in comparison with the calculated results for the_ flow taking into account condensation only <Curves 1 and 2), as well as the effects of simultaneaus course of condensation and vibrational relaxation (Curves 3 and 4>. According to the conclusion of Ref. 4 Curves 3 and 4 were obtained for vibrational relaxation time T/4,< being the approximation of data obtained in the shock tube experiments8. Shown in Fig. 1b are also the results of calculations of p/po(To) for vibrational equilibrium expansion without taking into account condensation (Curve 5), as well as the results of calculations with y=1.4 (dashed line).
Analysis of experimental and theoretical data for C02 (Fig.1b) shows that the calculated model simultaneously accounting for condensation and vibrational relaxation describes the experimental behaviour of p/po(To) qualitatively well. 8oth the experiments and calculations show the presence of the maximum in this dependence. The maximum on p/po(To) curve is due to the competition of processes of condensation and vibrational relaxation. At low Stagnation temperatures condensation is a dominating process in the C02 expansion formation. Further the excitation of vibrational degrees of freedom becomes essential with increasing stagnation temperature. At first, it Leads to the decrease in p/p 0 growth in the condensation effect region, and then after passing the maximum, to decreasing a relative density. The maximum region of p/po(To) is characterized by a simultaneaus course of condensation and vibrational relaxation having an opposite effect on p/p 0 • In the region of dominating condensation effect (to the left from the maximum of p/po(T0 )) the calculated and experimental results are in good agreement. In the meantime, in the region of dominating vibrational relaxation effect (to the right) the peculiarities have been revealed which can not be explained now. The most impressive fact is that the relative density measurements in a high temperature region <T0~790 K for po~3·106 Pa) are about 10% below than the corresponding values of a vibrationally equilibrium expansion. This fact can likely tobe explained by the impurity condensation effect. The experiments with pre-cleaned C02 reveal the growth and displacement of the maximum of p/po(To) to the region of higher values of T0 , as well as increasing values of density in the region to the right of the maximum of p/po(To). To the left the experimental results are insensitive to impurities, that is probably due to the basic effect of homogeneaus C02 condensation under these conditions. The measured density differs from the calculated one more sionificantly for the regime with po=7.5•105 Pa in the region to the right of the maximum. This is probably caused by impurity and viscosity effects.
The effect of simultaneaus processes of condensation and vibrational relaxation was also investigated based on the measured
899
Fig. 2.
8 () r" "'683 ~ '§ 6
0 "'600 I)
• -518
~ ()
~ ~4
~ 0
0 0
.s 2 • r-," • •
0 1 2 3 4,16 4,28 4,4 J.,J'm ·6 P0 ·10, Pa
Effect of stagnation temperature and pressure an spectral and integral intensities of C02 jet IR radiation.
spectral and integral intensities of co2 jet infrared radiation in the region of 4.3 ~m (Fig. 2). The analysis of C02 IR radiation data tagether with the density measurements (Fig. 1) allows to determine the effect of the above processes on the vibrational relaxation kinetics with the presence of clusters.
Effect of po. The effect of the modelling parameter po is also studied. The investigations were mainly performed fo r T0 =300 K, although in some experiments higher temperatures were also used. Presented in Fig. 3 are the p/p o(T0 ) variations at To=300 K in N2 and Ar jets. In Fig. 3a the calculated results for the flow taking into account viscosity in the nozzle (solid line) and the data obtained for y =1 .4 (dashed line) are presented. It is evident that the model taking into account the viscosity effect on the flow inside the nozzle describes satisfactory the experimental data. Fig. 3b illustrates the comparison of exper i mental results obtained with Ar with ~he calculation of flow taking into account viscosity only (Curve 1) and condensation only (Curve 2). It may be seen, that at p o > 106 Pa the dominating process is the nonequilibrium condensation, and at p o <5·105 Pa that is the viscosity in the nozzle flow. For C02 jet the similar Situationtakes place.
10 --~-.--.--.-r----.----,---.-,
8
6
4 l__---o~-r~~
P0 ,Pa
2 4 6 8 105 2 4 6 8
20
10
8
6
4 4
JJs _____ -----·10 ~ f 8
6 2
.:i • 41,4
Ar
4 6 8 "o,Pa
Fig. 3. Effect of stagnation pressure on relative density. a - N2 jet, b - Ar j et.
900
..:.cjza =80,75
10 8
6
4
6 P0 = f,ljl·/0 Pa
2 o- ~ = .500 K ct 410 111 5.30
• 600
106 0 20 40 .!1/ZQ
Fig. 4. Effect of stagnation temperature on transverse distributions relative density in co2 jet.
Analysis of transverse density distributions. For a more detai~investigation of the effect of nonequilibrium processes on the jet expansion character, the transverse density distributions were measured. In Fig. 4 an example of the results obtained for C02
jet at constant po and different To are shown. The experimental results are shown by points, the calculated ones - by lines. Curves 1,2,3,4 are obtained with taking into account condensation only for To=300 K, 410 K, 530 K and 600 K, respectively. Dashed line reP.resents the calculations with y=1.4.
The experimental and calculated data being in good agreement evidence that transverse sizes of jet increase with decreasing To. This is due to the condensation effect, which Leads to a strong divergence of streamlines behind the nozzle. A similar condensation effect was observed in Refs. 1,3.
Comparison with Beylich's2 results. The methods of calculation of flow taking into account the effects of nonequilibrium processes were applied to the analysis of the experimental data obtained in Ref. 2. The comparison of the calculated results obtained under the experimental conditions2 for p 0 =8 bar and T0 =290 K reveals that when x/da> 10, the calculated and experimental results are in satisfactory agreement, while, if 1<x/ da<10, the experimental curves are below than the calculated ones. A good agreement of calculated and experimental values of number density in a far field of C02 jet (for x/da>10) allows to make a conclusion that the results of electron-beam measurements of number density in a near-field region
901
of the jet (for 1<x/da<10, i.e. for n>10 16 cm-3 ), have been underestimated.
CONCLUSION
The experimental and theoretical investigations allow to determine the peculiarities of expanding Ar, N2 and C02 jets from supersonic nozzles. The ranges of the dominating effects of viscosity inside the nozzle as well as condensation and vibrational relaxation on the jet expansion have been established. The above results in comparison with the results of similar investigations of the jet behind a sonic nozzle4 reveals a much stronger effect of nonequilibrium processes on the jet behind the supersonic nozzle. This ciscumstance is of princinple and is due to a strong dependence of jet expansion character on gas parameters at the nozzle exit. The sonic nozzle exit parameters are a little sensitive to the effect of nonequilibrium processes, while the supersonic nozzle exit parameters significantly depend on the prehistory of the flow inside the nozzle.
REFERENCES
1. S.Christ, P.M.Sherman, and D.R.Glass, Study of the Highly Underexpanded Sonic Jet, AIAA J., 4:68 (1966).
2. A.E.Beylich, Experimental Investigation of Carbon Dioxide Jet Plumes, Phys. Fluids, 14:898 (1971>.
3. N.J.Kislyakov, A.K.Rebrov, and R.G.Sharafutdinov, Structure of High-Pressure Low-Density Jets beyond a Supersonic Nozzle, J. Appl. Mech. Tech. Phys., 16:187 (1975). -
4. N.G.Zharkova, V.V.Prokkoev, A.K.Rebrov, P.A.Skovorodko, and V.N.Yarygin, The Effect of Nonequilibrium Condensation and Vibrational Relaxation in Supersonic Expansion of Carbon Dioxide, in: "Rarefied Gas Dynamics", CEA, R.Campargue, ed., Paris, 2:1141 (1979).
5. F.P.Boynton and A.Thomson, Numerical Computation of Steady, Supersonic, Two-Dimensional Gas Flow in Natural Coordinates, J. Comput. Phys., 3:379 (1969).
6. P.A.Skovorodko, A Method for Calculation of Nozzle Viscous Gas Flows, in: "Rarefied Gas Dynamics, Proc. 6th All-Union Conference", Institute of Thermophysics, Novosibirsk, 2:143 (1980) - In Russian.
7. P.A.Skovorodko, The Peculiarities of Condensation Process in Conical Nozzle and in Free Jet behind It, This Book.
8. M.Camac, BSD-TDR-64-96, Research Report, 194 (1964).
902
MEASURED DENSITIES IN UF6 FREE JETS
ABSTRACT
P. A. Hoisington and S. S. Fisher
School of Engineering and Applied Science University of Virginia Charlottesville, Virginia 22901
Measurements of gas densities in UF6 free jets expanding from converging nozzles into a vacuum are described. These densities are deduced from measured intensities of laser photons Rayleigh scattered by the gas. The nozzle Reynolds number Re = p a d /~ , where dt is the nozzle throat diameter, ranges from f:tso €o 0 92oo? Axial d1stances from the nozzle as large as 25 throat diameters and radial distances as large as 8 throat diameters are covered. Measured densities are in good agreement with those predicted on the basis of inviscid-flow method of characteristics calculations.
INTRODUCTION
The free jets in this investigation are those expanding from smoothly converging nozzles into a vacuum. Local gas densities are measured by focusing a high-intensity laser beam in the gas and observing, with high spatial resolution, photons elastically scattered by the gas. For these jets, viscous and rarefaction effects are for the most part negligible. Hence, it is possible to compare measured densities with those calculated on the basis of inviscid fluid-mechanics theory. To these writers' knowledge, these data are the first of this type obtained for a gas with a very low ratio of specific heats.
903
APPARATUS
A diagram of the apparatus is shown in Fig. 1. The jets were produced in a small tank which was evacuated by means of a coldtrapped oi~ diffusion pump. The jet-forming nozzle, a cryopump, and a portion of the optics required for the Rayleigh-scattered intensity measurements were housed within this tank.
The cryopump is a small container with vanes attached to its front. This container is cooled by filling it with liquid nitroge~4 Because the vacuum tank background pressure was kept below 10 Torr, the jet gas expanded all the way from the nozzle to where it condensed on the cryopump. No important shock waves formed.
The Rayleigh-scattering light source was a CW argon-ion laser tuned to its 514.5 nm line. The available laser power was approximately 2 W. The laser beam entered the tank as shown in the diagram and was brought to a focus near the tank center. The beam waist diameter was approximately 0.1 mm. This laser beam was absorbed in a light dump located at the bottom of the tank.
904
~----------·------<d=>>---------~
LB I
BD CP DP
FV
\~ N,----+-_ ___;;L~ 1- 1--- .. -----
r~ lx I i I
J nl DP BD
beam dump FV focal volume
cryopump LB Iaser beam
diffusion pump N nozzle
Fig. 1. Apparatus
A fixed fraction of the photons scattered by the gas in the vicinity of the laser beam waist was collected by a large lens located directly behind this waist. This lens produced an image of the waist at a small external aperture. Photons able to pass through this aperture were detected by a photomultiplier tube. The photomultiplier current was measured by means of pulse-counting instruments. These detected photons originated from approximately 0.4 mm along the laser-beam axis.
In order to reduce the flux of stray laser and other photons into the phototube, the inner surface of the vacuum tank and the outer surface of the nozzle were covered with flat black paint. Also, several collimating apertures (not shown) were installed along the laser beam axis inside the vacuum tank. Finally, several discriminatory filters were included in the detection optical and electrical trains.
Although the laser beam waist diameter was quite small, a weak halo surrounding this waist could not be eliminated. As a consequence, density measurements became impractical within 2 mm of the nozzle.
Jet densities will be presented as a function of axial distance x measured from the hozzle exit plane and lateral distance r measured from the jet axis (see Fig. 1). These Coordinates were varied by moving the nozzle. This movement was accomplished by means of a translating mechanism and a special vacuum seal where the gas supply line for the nozzle passed through the vacuum tank wall. Cited values for these coordinates are accurate within 0. 5 mm. These Coordinates could be repeated within 0.1 mm. The coordinate r was varied by translating the nozzle perpendicular to the view shown in Fig. 1.
Three geometrically similar nozzles were employed. Their respective throat diameters were 1, 2, and 4 mm. Their general shape and their precise dimensions are shown in Fig. 2. Each nozzle was attached by means of screw threads to the end of the nozzle supply line.
Because the entrance section of these nozzles was smoothly convergent, the vena contracta effect was kept small. Because this convergence was also rapid, the thickness of the nozzle internal boundary layer was also kept small. These nozzles produced sonic flow along a surface very nearly coincident with their exit plane.
Gas for these experiments was obtained by evaporating distilled, solid UF6 . This gas was passed through a 1 jJm sinteredmetal filter before it entered the nozzle supply line. The stagnation temperature T for the nozzle flow was approximately 300 K. The stagnation pres~ure p for this flow ranged up to 80 Torr.
0
905
Typically, for individual density measurements, photons were counted for several tens of seconds. Photon fluxes were converted to gas densities by means of calibrations carried out in a separate gas cell.
Other details concerning this apparatus and the techniques employed in the experiments are given in Ref. 1.
DENSITIES
Axial Distributions
Density distributions measured along the jet axis are shown in Fig. 3. In this figure, n is the ratio of the local number density for the gas, n, to that corresponding to nozzle stagnation conditions, n . The coordinate x is equal to x/dt. Data are shown for each of ~he three nozzles for a single value of the nozzle Reynolds number Re = p a d /~ = 3450. A second set of data for the smallest nozile, fo:f' t:Re 0 = 9200, is also included. These distributions cover a range o~ x from 1 to 25 and a range of n from 0.08 to 0. 0001. .The representative error limits shown correspond to a 95% confidence level. Within these error limits, with the exception of two data points near the bottom of the graph, the exhibited variations are independent of Re . For such high Re values, this inde-
d · d n n pen ence 1s expecte .
Also shown in Fig. 3 is a predicted Variation of n VS X derived from method-of-characteristics inviscid-flow calculations carried
dt/2 45°
4.~ _1 5.95
l
dt F L R
1.02 0.12 0.20 198} 1.98 0.25 0.40 3.96 mm
3.99 0.50 0.76 7.92
Fig. 2. Nozzle dimensions
906
out by Hasegawa. 2 Hasegawa assumed a calorically perfect gas with y = 1.06, and began bis calculations with a uniform flow at the nozzle throat at a Mach number M = 1.0038. The solid line shown for i less than 10 corresponds to values of M as a function of x actually calculated by Hasegawa. The dashed line shown for i greater than is 10 an extrapolation based on this solid line. Good agreement between measured and predicted densities is seen. Only the two flagged-circle points near the bottom of the graph disagree with this prediction outside the indicated error limits.
Hasegawa's predicted variation for M along the jet axis can be shown to be in good agreement with corresponding method-of-
10-1
10-2
Hasegawa --
MOC, Y = 106
0
d
o.
0
dt(mm) Ren
3450
I 9200
2 3450
4 3450
Fig. 3. Axial density distributions
907
characteristics predictions given by Anderson3 (if one interpolates between Anderson's results for y = 1.05 and y = 1.10).
The n vs x prediction in Fig. 3 may also be shown to be in qualitative agreement with the calculation of Mikami4 for the Variation of n with radial distance for a spherically-symmetric, compressible, inviscid source flow of a gas for which y = 1.062. If the sonic radius for Mikami's source flow is taken tobe 0.75 times the exit radius of Hasegawa's sonic nozzle, quantitative agreement is achieved.
A better estimate for y for UF for the present experiments is 1. 075. This estimate is based on ~ermodynamic-equilibrium values for y calculated as a function of temperature by Fisher. 5 (Because the vibrational-relaxation collision number for UF 6 lies in the range 10-20 (see Ref. 6), and because the present jets are formed at relatively high density levels, there is little need to take vibrational-lag effects into account.)
10-1
x d1(mm) Ren
(l I 4 1150
0 2 2 3450
10-2 0 3 3450 ® 6 9200
'il 10 9200
n 'V' 10 3450
10-3
10-4~-L----~----~------~----~----J-~ -2 0 2 4 6 8
7
Fig . 4. Lateral density distributions
908
From Anderson's calculations, it may be concluded that a 1.5% increase in y (from 1.06 to 1.075) will raise the predicted value of n at any x by approximately 3%. This increase improves the agreement between measurement and prediction for the data in Fig. 3.
Lateral Distributions
Measured Variations of n as a function of the reduced lateral coordinate r = r/dt' for different fixed values of x, are presented in Fig. 4. Here too, data are included for each of the three nozzles. Although not all data in this figure are for the same value of Re , because Re is always large these differences are of little signfficance. Err~r bars in this figure also correspond to a 95% confidence level. Note that these data exhibit the expected jet broadening with increasing x.
The expected effect of reducing Re (below 1000, say) for these flows would be to cause n at any giv~n value of x and r to drop (this drop is quite clearly evidenced in experiments described in Ref. 7). For the present experiments, Re is simply too large for this drop to be observable. n
10-1.--.----.-----,-----.-----.--,
dt = 2 mm
x = 2
Ren= 2300 to
9200
Hosegowa MOC, _} \
r = 1.06
Fig. 5. Measured vs predicted lateral distributions
909
The non-dependence of ii. upon Re for these data is further demonstrated by the results presented iR Fig. 5. The solid curve in this figure is one fitted through data obtained for Re = 2300, 3450, 4600, and 9200. The indicated error bars exhibit fbe extent to which individual measurements deviate from this curve for the different values of Re . The widths of these bars are comparable to the 95% confidence lim~ts for these measurements.
A predicted lateral density distribution is also shown in Fig. 5. This prediction is constructed on the basis of Hasegawa's calculations and the self-similar character of the far-field portion of these jets as described by Sherman. 8 Note that this prediction is in reasonable agreement with the experimental results.
ACKNOWLEDGEMENTS
This research was supported by the United States Department of Energy, under Contract No. DE-AC05-820R209000. The authors thank Messrs. R. H. Krauss and M. G. Rodgins for assisting with the apparatus, and Mr. E. B. Boysen for assisting with the experiments.
REFERENCES
1. P. A. Hoisington, "Rayleigh-Scattering Density Measurements in UF6 Free-Jet Expansions," Masters Thesis, Engrg. Phys., Univ. of Virginia (1980).
2. K. Hasegawa, "Analysis of the Exhaust Flow Near a Centrifuge Gas-Supply Nozzle," Masters Thesis, Nucl. Engrg., Tokyo Inst. Tech. (1977, in Japanese).
3. J. B. Anderson, "Inviscid Freejet Flow with Low Specific Heat Ratios," AIAA J. 10: 112 (1972).
4. H. Mikami, "A Computational Study of the Density and Temperature Distribution in a Freely Expanding Uranium Hexafluoride Gas," Nucl. Sei. Engrg. 67:235 (1978).
5. S. S. Fisher, "Specific Heats and Enthalpies for Gaseous lJl?6 ," Univ. of Virginia, School Engrg. and Appl. Sei., Report UVAER-760-82U (1982).
6. H. E. Bass, F. D. Shields, W. D. Breshears, and L. Asprey, "Vibrational Relaxation of Uranium Hexafluoride: Ultrasonic Measurements," J. Chem. Phys. 67:1136 (1977).
7. M. G. Graybeal, "Impact Probe Rarefaction Effects Measured in C02 Free Jet Expansions," Masters Thesis, Engrg. Phys., Univ. of" Virginia (1982).
8. F. S. Sherman, "Self-Similar Development of Inviscid Hypersonic Free-Jet Flows ;" Lockheed Missiles and Space Co. , Sunnyvale Calii:; Tech. Report"No. 6-90-83-61 (1963).
910
ROTATIONAL RELAXATION OF NO IN SEEDED, PULSED NOZZLE BEAMS
Introduction
H. W. Lülf and P. Andresen
Max-Planck-Institut für Strömungsforschung Göttingen Böttingerstr. 6-8, 34-00 Göttingen I BRD
In the last few years pulsed nozzle beams became an important experimental tool in spectroscopy, especially in connection with the development of pulsed high power lasers. High momentary intensity at moderate pumping speed, good cooling and control of dimerisation by !arge nozzles are only some of the important advantages. Several different designs for pulsed valves ha.3e been realized to produce short gas pulses in the submillisecond range 1- , but still little is known about hydrodynamic properties of these beams. They do depend on the details of the design, and it is known today that ultrashort gas pulses do not give good nozzle beams 1• Theoretical estimates have been made about the time that is necessary to come to a stationary behaviour of the expansion lj.. We want to present a method that gives some insight into the dynamics of these beams. We study in detail the features of the pulsed valve designed by Smalley 2, by looking with LIF at the rotational relaxation of NO in very dilute mixtures with He and Ar. LIF is an ideal probe because the molecules can be detected without any interference with the beam.
Rotational relaxatioi1 in steady state nozzle beams has been studied extensively with a !arge variety of experimental methods. However most of this work was concerned with the investigation of the asymptotic behaviour of the expansion as a function of the source parameters p0 , d and T. Only in rather few experiments were relaxation phenomena also studied as a function of the nozzle distance x/d 5. Whereas the asymptotic studies of rotational relaxation gives essentially data ab out the collision numbers for rota tional energy transfer, the studies along the expansion are sensitive to the detailed rate constants as a function of temperature 6. In this paper we record first data about the rotational state distributions of NO as a function of the nozzle distance at the very first few nozzle diameters, where the important
911
part of the relaxation occurs. NO is a very suitable particle for the investigation of rotational relaxation because (1) it is one of the very few stable particles where LIF <fn be applied and where the rotational structure can be completely resolved (2) at room temperature no vibrational and only a moderate number of rotational states are populated. We seed 1 % NO in He (Ar), so that the rotational energy transfer is exclusively due to NO-He (Ar) collisions, and the main properties of the hydrodynamics should be governed by the carrier gas. By this method, i.e. following the rotational relaxation along the expansion, we obtain a very detailed picture about the rotational relaxation in the expansion of pulsed nozzle beams. It is shown that we do have stationary conditions in the middle of the gas pulse, at least for the case of He.
- 226 nm tunable UV laser
Fig. 1
/ nozzle beam /
to laser power meter
to mass spectrometer or open ion gauge
imaging optics
Experimental setup
The experimental setup is shown schematically in Fig. 1 The nozzle beam is generated by a pulsed 1_ource as described by Smalley, ~ a modified version of the source of Giese • The source operates at 5 Hz, has an open time of lOO)Jsec, and the pressures used are in the range from 300-700 torr. The stagnation temperature T0 is 300° K. The nozzle has a diameter of .7 5 mm, and due to the mechanical setup we have a subsequent slightly divergent channel of 1.7 mm length and a 1 mm hole at the end. Th'; experiment was done with a moderate pumping speed of 2000 I/sec at 10- Torr back-
912
ground pressure in the vacuum chamber. Care has been taken that no mechanical parts in the machine come close to the beam, because interference with backscattered particles can destroy the nozzle beam. The UV probe laser passes through a light baffle and crosses the molecular beam at right angle. The pulsed source can be moved relative to the laser beam, so that we can probe the gas pulse at different distances from the nozzle exit. Proper timing is achieved by a central clock that triggers pulsed source, laser and detection devices. The delay of gas pulse versus probe laser is adjustable so that we can record the time profile of the gas pulse at different positions x/ d. NO is detected by LIF via the 2Tr -2 L transition at 2262nm. The el2ctronic groundstate of NO is split into two multiplet levels 7T1/ 2 and Tl3t 2 which are seperated by 121 cm- • Rotational distributions were measured in both states although the popul:ttion in the 2llj12 state was very small. The lifetime of NO in the excited ~ state was measured at different distances and even at small distances no indication of quenching was found. The rotatianal quantum number N used here is J - 1/2. The tunable UV at 226 nm is obtained by mixing the fundamental of a NdYAG at 1.064 nm with the frequency doubled output of the NdYAG pumped dye laser. At 226 nm we can get up to 1 mJ/pulse and a linewidth of IV .3 cm-1, which is far above the Doppler broadening of the beam but sufficient to resolve the rotational structure. The laser beam diameter is .5 mm and this defines a cylindrical fluorescence volume where the laser crosses the nozzle beam. The fluorescence is observed perpendicular to both laser and molecular beam. In order to look at the fluorescence also at small distances the source was modified as may be seen in Fig. 1. Two razor blades, 15 mm long, are mounted 20 mm away from the beam center line and define a 1 mm slit. This and the imaging optics select only a small part of the cylindrical fluorescence volume, so that only the fluorescence from the centerline of the beam passes the optics. The spatial resolution of the experiment is then given by the diameter of the laser beam and is .5 mm, corresponding to roughly a nozzle diameter. Thus these !arge nozzles, which do not require !arge experimental effort with pulsed nozzles, offer the big advantage that studies of rotational relaxation can be done at these very first few nozzle diameters. The total fluorescence is then measured by a phototube whose DC-ouwut current is recorded. Dividing this current by the Hönl-London factors , gives our final signal. To reduce the level of scattered light, in some cases gated integration with a Boxcar has been used. Most efficient, however, in reducing scattered light is the use of imaging optics. The primary beam intensity can be recorded with an open ion gauge or a mass spectrometer. No formation of dimers has been observed at the mass spectrometer under the above mentioned conditions.
Results
Fig. 2a shows the time profile of a gas pulse for 1 % NO in 500 Torr Helium, taken at a distance of 6.7 mm from the nozzle. The laser is tuned to a specific transition, so that the fluorescence intensity is proportional to the density of NO in a well defined rotational state. Varying the delay of laser versus gas pulse we obtain the time profiles for the rotational groundstate (N = 0) and the excite.d N = 5 state of NO.
913
Fig. 2 Time profiles for the density of NO in the rotational groundstate (N = 0) and in the excited rotational state (N = 5) for 1 % NO in 500 Torr Helium. Distance: 6.7 mm.
The behaviour of the rotational groundstate N = 0 is very similar to the behaviour of the total density. The intensity rises first slow and then fast, reaches a plateau of N 100 psec length and falls off again. The shape of the gas pulse is very different if we probe an excited rota tionc_.Lsta te like N = 5. Note that this curve has been multiplied by a factor of '3o for better visibi!ity. The flat part in the middle of the pulse corresponds to good rotational relaxation, but in the beginning and at the end of the gas pulse we have a higher intensity in this excited rotational state. This implies that the rotatianal cooling in the beginning and the end does not correspond to the cooling in a steady sta te nozzle beam. In the middle however we do find for the case of He a flat plateau for aJI rotational states and at all distances. The data discussed later are all taken by using only this part of the bearn. This beh.::viour suggests that we do reach in the case of Helium a stationary state after "'ltO )JSec and that we expect the sarne features frorn the relaxation as from a continous beam with the sarne p0 and d.
In the same way, we measured the time profiles for 1 % NO in Argon. We obtain Ionger pulses and the plateaus are not as pronounced as in the case of Helium, but we think, that we are close to a stationary state. From theoretical estirnates it is expected lt that it takes a Ionger time for Argon to reach sta tionary conditions and this explains the difference to Helium.
In Fig. 3a-d we see the population of various rotational states NO (N) as a function of the nozzle distance with 1 % NO seeded in Helium and Argon.
914
.s c
.Q .... d :J Cl.. 0 Q. .!
Q)
> .... d Q) •• ...... -
1.7
Fig. 3
!7-\"Q jn He 700\orr
d
1..1. 7.1 9.8 12.5 1.7 1. .1. 7.1 9.8 12.5
0 I S TA N C E [ m mJ
Relative populations of NO (N) along the expansion. The populations are corrected for the density dropoft by normaliz ing the total fluorescence of all rotational states to unity. Paramete rs of the measurements are given in the figure .
915
For these measurements we tuned again the laser to a specific transition probing the density in one rotational state. Then we moved the nozzle relative to the laser and measured the fluorescence as a function of the nozzle distance. In Fig. 3 we plot the density of NO in different rotational states N as a function of the nozzle distance. These data are corrected for the density dropoft due to the decreasing density in the nozzle beam by summing up the fluorescence intensity at a given distance over all rotational states. This corresponds simply to particle conservation in the beam. So Fig. 3 represents the relative population in the rotational states normalized to unity. In order to study the pressure dependence of the relaxation we took three different pressures for 1 % NO in He (Fig. 3a,b,c with 300, 500 and 700 Torr) and to compare the relaxation of NO in He with NO in Ar we took another measurement at 500 Torr for Argon {Fig. 3d). In these measurement the fluorescence intensity varies over several orders of magnitude demonstrating the high sensitivity of LIF. Although it is in principle possible to measure also the higher rotational states at !arger x/d this was not done in these first experiments. The N = 4 state for NO is missing in these plots because there was an overlap with another spectral line that was only partially resolved.
All these figures show very nice the typical behaviour of rotational relaxation. With increasing distance the higher states decay very fast and become less and less important for the rotational redistribution. The rotational groundsta te N = 0 is growing because finally most of the molecules will end up in this state. The intermediate rotational states do first increase and then decrease, demonstrating that the relaxation process proceeds via these intermediate states. A closer look at these curves reveals many details of the rotational relaxation.
Comparing Figs . .3a-c we can study the pressure dependence of the rotational relaxation in the case of Helium. For 300 Torr we see that the first excited rotational state N = 1 is always more populated than the rotational groundstate N = 0. At 500 Torr the N = 1 and N = 0 populations come closer tagether and at 700 Torr the N = 0 population dominates already at large distances. A similar behaviour is observed for the other rotational states: With increasing pressure the population becomes smaller relative to the N = 0 state. Obviously the cooling is more efficient at higher values of p0 d as expected from steady state nozzle beams.
For 300 Torr NO in He we see at large distances that the population in the lower rotational states varies slowly, indicating that the relaxation is almost complete. This is less pronounced at 500 and 700 Torr, because the relaxation extends to larger nozzle distances with increasing pressure.
The first significant difference between the relaxation of 500 torr Ar compared to 500 Torr He isthat the population of the rotational groundstate is dominating already at a few nozzle diameters. The other higher rotational states decay faster in Ar than in He and show that rotational cooling in Ar is more efficient than in He. At large distances the population of the rotatio-
916
nal states varies stronger in the case of Argon; the expansion range extends to !arger distances. Obviously the rate constants for rotational energy transfer are !arger for Argon than for Helium, which in turn explains that the cooling in Ar is better and that the expansion range in Ar is langer.
The lowest line in all these plots describes the relaxation of the NO <27T3/ 2) state. Here we plot only the rotational groundstate because all higher rotational states are eve~ less populated. Thus this state cannot contribute to the population in the "ff112 state, at least not in the range of distances covered by this experiment. Otherwise we would deal with a much pore complic~ted relaxation process where electronic relaxation from the lT312 to the Jli12 state p~ays an important role and the Interpretation of
the aata may become meanmgless.
We see in all cases that already at the beginning of the expansion only a few states contribute to the rotational redistribution. Proceeding further in the expansion, we see that the hlgher rotatlonal states become more and more negligible and the relaxation can be described with very few state to state rate constants. Thus our data should be sensitive to state to state rate constants at very low temperatures, and such data are very hard to obtaln from other methods.
Fig. 4 Boltzmannplot of the data from Fig. 3 for He 300 Torr (-o--) and Ar 500 Torr (-0----) at selected distances.
-o-o-- Ar 500 Torr 2.0 o - -o-o-- He 3CO Torr
z 20
"-11--,~.2 mm
&---..
0
o,.._<t_, L.1mm 0 ~ ......
...... .............
...... o, ...... ......
·~ 65°K
~
...... 0 ...... ......
3 OB ~\13.7mm
\'\ 'o, 28° K
...... .....
o \ 10.8°K
\.,:~\ \
....... ....... " 19.7"K
80 160 2LO 320 LOO L80 550 6LO ?20 BOO 860
J (J • I)
917
In Fig. 4 we plot these data in a Boltzmann plot for several distances (2.2 mm, 4.7 mm and 13.7 mm) for the cases of He 300 Torrand Ar 500 Torr which represent the cases for best and worse cooling. Obviously most of the data can be described to a good approximation by straight lines corresponding to a well defined rotational temperature in this range. A more careful look shows however that already some of the higher rotational states tend to freeze out, but these weak higher sta tes are not observed to the end of the expansion, where we do see a non Boltzmann behaviour (see Fig. 6). Due to the cooling, the slopes of the curves in Fig. 4 become steeper the more we proceed in the expansion from 2.2 to 13.7 mm.
At small distances the slope for Ar 500 Torr and He 300 Torr is the same and later in the expansion the rotational temperature of He 300 Torr is clearly above that for Ar 500 Torr. This demonstrates, that, although the cooling in the He 300 Torr and Ar 500 Torr is very different we do get the same rotational temperature in the beginning. Obviously we are first close to equilibrium of rotational and translational temperature and deviations are found later on.
If we take the rotational temperatures from Boltzmann plots as in Fig. 4, we obtain the rotational temperatures in Fig. 5 as a function of the nozzle distance. At small distances all temperatures are close together, and this again supports the assumption of hydrodynamic equilibrium. With increasing distance however the rotational temperatures deviate more and more in the series Ar 500, He 700, He 500 and He 300 Torr. This can be
Fig. 5
918
60
50
'SC 40 LJ l.u30
~ ,_ ~ 20 l.u
~ ~ I
";{ 10· <: . 0 :::: ~ 0 Cl:: 5
1.0
·~
2..0
~:~ ~--""' -~ '""' ~."·,
~~,:----·-.~ He.300Torr
'\. '·'· · " '-........._ ·~·'- ....._He,500Torr , ""'-,
"-..... He, 700Torr
""'-Ar. 500Torr
3.0 (0 5.0 8.0 10.0 20.0
DISTANCE[mm]
Rotational temperatures for 1 % NO in He and Ar as obtained from data in Fig. 3 in double logarithmic plot as a function of the nozzle distance.
either explained by deviations from translational equilibrium or by deviations between rotational and translation equilibrium. In the case of He 300 Torr we see, at !arge distances, that the temperature varies very slowly indicating that the expansion is almest complete. In the series He 300, He 500, He 700 and Ar 500 Torr the slopes become steeper corresponding to an increasing length of the expansion range.
Fig. 6 shows in a Boltzmannplot the final population in the lowest remaining rotational states for the expansion of 1 % NO in Helium. These measurements are taken in the asymptotic range at 250 mm at different source pressures. The data are normalized to give the same relative population in the N = 0 state. Obviously the rotational distribution can be no more described by a temperature, demonstrating that the higher rotational states are frozen. Nevertheless, if we want to obtain a cross section o-"r t for translatio~~ rotational energy transfer according to the model of Gal~agher and Fenn ' , we can define a temperature by taking the lftio of the N = 1 to the N = 0 population. We then obtain a value of 18.4 1\ for (j'rot' which corresponds to a col!izion number Zrot "' 2.2 if we assume a hard sphere cross section of 40 1\ •
-o.~
-o.,
-·
-l.l
_,
_,.
-o- = 300 torr ~- :4()0 torr ~ .=500 torr
-+- = 550 torr --x-·- = 630 torr
o = 750 torr
-• ·r---...----,-~----.----:'"::---~-.......---.---,---.---.---,--~---r-l 0 I J .Z 6 J 9 ~ l 6) 7 I 9 I 10 4 U .7 ll U .J 1) 6 a6 q 18 .( 19 ~
Fig. 6
J(J+J)
Boltz mann plot of the asymptotic behaviour for the expansion o f 1 % NO in He at different source pressures.
919
Summary
It has been shown in this paper that pulsed nozzle beams, at least from the pulsed valve from Smalley, do correspond to steady state nozzle beams in the central part of the beam. In the beginning and at the end of the gas pulse however unsteady state conditions are found. For Helium it takes
40).lsec to come to a stationary behaviour, for Argon this time is considerably langer. Along the very first few nozzle diameters a complete picture of the rotational redistribution of NO in He and Ar has evolved. The typical features of nozzle beams such as increased cooling with p0 d, extended relaxation range with increasing pressure are observed. Up to 10 nozzle diameters the rotational distributions can be described by a temperature, in the asymptotic range a non Boltzmann behaviour is found. The model of Gallagher and Fenn gives a reasonable number for the cross section for rotational energy transfer.
Acknow ledgements
W e are very thankful to Prof. Smalley for sending us his very detailed plans for the pulsed source. Thanks are also due to many helpful discussions with Prof. E.L. Knuth, Drs. R. Miller and H. Lang.
REFERENCES
1) C.F. Giese, P. Chow, IX International Conference on the Physics of Electronic and Atomic Collisions, Vol. 2, p. 964
W.R. Gentry, C.F. Giese, Rev. Sei. Instrum., 49, 595 (1978)
W.R. Gentry, C.F. Giese, J. Chem. Phys., 67, 5389 (1977)
2) M.G. Liverman, S.M. Beck, D.L. Monts, R.E. Smalley, In "Rarefied Gas Dynamics", Vol. 2, 1037 (1978)
3) J.B. Cross, J.J. Valentini, Rev. Sei. Instrum., 53, 38 (1982)
R.L. Byer, M.D. Duncan, J. Chem. Phys., 74, 2174 (1981)
4) K.L. Saenger, J. Chem. Phys., 75, 2467 (1981)
5) D.R. Miller, R.P. Andres, J. Chem. Phys., 46, 3418 (1967)
F. Aerts, H. Hulsman, In "Rarefied Gas Dynamics'_', Vol. 2, 925 (1978)
6) H. Rabitz, S.H. Lam, J. Chem. Phys., 63, 3532 097 5)
K. Koura, Phys. Fluids, 24, 401 (1981)
920
H. Lang, In "Rarefied Gas Dynamics", Vol. 2, 823 (1978)
T.E. Gough, R.E. Miller, G. Scoles, Appl. Phys. Letters, 30, 338 (1977)
T.E. Gough, R.E. Miller, G. Scoles, J. Mol. Spectr., 72, 124 (1977)
7) H. Zacharias, A. Andres, J.B. Halpern, K.H. Welge, Optics Communications, 1.2, 116 (197 6)
8) R.J.M. Bennet, Mon. Not. R. astr. Soc., 147, 35 (1970)
9) R.J. Gallagher, J.B. Fenn, J. Chem. Phys. 60, 3487 (1974)
R.J. Gallagher, J.B. Fenn, J. Chem. Phys. 60, 3492 (1974)
M. Quah, Chem. Phys. Letters, 63, 141 (1979)
1 0) H. Ashkenas, F .S. Sherman, In "Rarefied Gas Dynamics", Vol. 2, 84 (1966)
P.K. Sharma, W.S. Young, W.E. Rodgers, E.L. Knuth, J. Chem. Phys. 61_, 341 (1975)
921
THE FREE-JET EXPANSION FROM A CAPILLARY SOURCE1
D.R. Miller, M.A. Fineman, and H. Murphy
Department of Applied Mechanics and Engineering Seiences University of California, San Diego La Jolla, California 92093, USA
ABSTRACT
A comparison is made of the free jet expansions originating from an orifice and a capillary by measuring the terminal gas properties. Time-of-flight and intensity data are reported for pure gases (He, Ar, COz) and mixtures of COz/He, together with condensed dimer intensities for Ar and COz, Pitot tube data are reported for Nz. The results suggest that the free-jet expansions are nearly the same, provided the capillary is modeled as a non-isentropic Fanno flow process. The Fanno flow is slightly non-adiabatic, which complicates the analysis. Only the condensation kinetics appear strongly sensitive to the differences in the initial conditions for the supersonic expansion, but, no doubt, any kinetic process relaxing near the capillary orifice exit would be affected.
INTRODUCTION
The isentropic free-jet expansion is well understood and widely utilized as a flow field to study kinetic processes or for spectroscopy. Thermodynamic properties in the free-jet are calculated in a Straightforward manner using the method of characteristics. While most researchers have utilized simple orifices and short converging nozzles as the free-jet source, aerosol or condensation studies are being carried out with capillary sources. Indeed, 1Presented at the 13th International Symposium on Rarefied Gas Dynamics, S-9 July 1982, Novosibirsk, USSR. Research supported by NSF Grant No. CPE-8017871. The use of Professor Clark Brundin's low density wind tunnel (University of Oxford) for the pitot tube measurements is also gratefully acknowledged.
923
for the very small orifices (<30 ~), which some researchers currently use for molecular beam sources, the subsonic section may be more like a short pipe than a sharp-edged orifice. The differences, if any, in the free-jet expansions derived from the two sources, orifice versus capillary, must be due to the different subsonic accelerations to the sonic condition. The subsonic acceleration for an orifice is nearly isentropic, except for a thin viscous boundary layer near the orifice boundaries, as the streamlines converge to the orifice. The subsonic acceleration for a capillary is a nearly adiabatic, viscous acceleration, with straight streamlines, called a Fanno process. In the range of flows used by molecular beam researchers, the capillary flow is a laminar Fanno flow. While there have been some interesting and useful studies of turbulent Fanno flow in pipes, Re> 10~, there seem tobe few of laminar flows, Re < 2000.
We present here some experimental results of the Fanno flow and subsequent free-jet expansion and compare these flows with the free-jet expansions from an orifice.
EXPERIMENTAL
Two experimental facilities were used to obtain the results. The first, used by one of us (DRM) while on leave, was Oxford's low density wind tunnel, under the direction of Clark Brundin. A 101 cm long by 1.1 cm diameter, smooth wall copper pipe was fitted with static pressure taps at -!, -1, -5, -10, and -90 pipe diameters from the exit. The pipe flow (Re ~ 1000) exhausted into the low density tunnel. The flow was probed with a stagnation pressure pitot tube (0.15 cm OD) from -1 to +3 diameters. N2 was the only gas studied in this facility.
The second facility used was the free-jet molecular beam facility here at UCSD. Two free-jet sources, one an orifice of diameter 0.038 cm, the other a capillary 0.039 cm in diameter and 5 cm long, were held in the same metal block at the same temperature inside a vacuum chamber. The flow passed through a calibrated flow meter to either source. The free-jets from these sources were sampled by a conventional time-of-flight diagnostic: a skimmer, of diameter 0.027 cm, kept about 50 diameters from the source; an 8.9 cm diameter chopper, 5.8 cm from the skimmer, gating time· 18 ~sec; and a quadrupole mass spectrometer detector, 55 cm from the chopper. The chopper blade had both narrow slots for timeof-flight measurements and equal on-off slots for lock-in amplifier intensity measurements. The chopper could be moved to affect either type of measurement without modifying the source or detector conditions, and the sources could be interchanged rapidly without modifying the chopper or detector conditions. Time-of-flight data was averaged on a PAR TDH-9 Waveform Eductor then put into a
924
chart recorder, while intensity data was analyzed by a PAR HR-8 lock-in amplifier and read as a voltage. Ar, He, C0 2 , and He/C0 2 results are given below.
DATA AND DISCUSSION
We report here the main features of the results, anticipating that additional details will be published elsewhere. The wind tunnel pitot tube analysis showed that the sonic transition occurred well upstream of the pipe exit and that the centerline exit Mach number was about 1.5. This value depends upon an extrapolation of the static pressure profile and could be off by a few percent. The last static pressure tap was 0.25 diameters from the exit and, together with the total pitot pressure, gave a Mach number at -0.25 of 1.43. If the static pressure does have significant radial Variation within the pipe, these results would be in error. The velocity profile at the exit is curved, but considerably flatter than the parabolic laminar velocity profile for incompressible pipe flow. Figure 1 shows the free-jet Mach number distributions obtained for this pipe source. The Mach numbers were calculated from the total pressure data assuming there were no further viscous effects downstream of the pipe exit, i.e., that P0 remains fixed
~~----~--------------~----~------~------, .&.._ e MEASURED B>' PITOT TUBE
5 + USING- I'IEASUilED STAGNATION PltEJSURE 81./T ASSUMING M•J. AT EXIT
IDEAL l~ENTROI"IC ( Y•/.4)
J.
$.0
f)ISTANCE FROM EXJT IN DIAMETERS
Figure 1. Pitot Tube Measurement of Centerline Free-Jet Mach Number Exiting From a Pipe.
925
at the pipe exit value at M = 1.5. The isentropic, ideal result expected for a flat M = 1 exit profile is also shown. Finally, we show the results which would have been obtained from the pitot tube measurements if we assumed the measured exit total pressure corresponded to M = 1 and not M = 1.5 as measured. The centerline of the free-jet from the pipe flow has a more gradual expansion, starting within the pipe, and appears to remain a bit ahead of the ideal orifice case (sharp-edged orifices have a sonic line curved outward with M = 1 at about +0.25 diameters on the centerline for y = 1.4). The two curves are approaching each other and probably become indistinguishable at larger diameters. The pitot tube could not be taken out further conveniently and would have required probe Re corrections at lower densities.
Such initial differences in Mach number distributions are not expected to alter the properties at large diameters unless the properties are affected by non-equilibrium, relaxation kinetics in the exit regionwherethe gradients arealtered by the different initial conditions. Results from the second facility, used to measure terminal free-jet properties, verify this conjecture, and indicate only slight differences for translational and rotational relaxation, but larger differences for condensation kinetics. To affect the comparison, equal mass flow rates were fixed for the orifice and capillary sources. The velocity distribution and/or intensity of the molecular beam extracted at 50 diameters were then measured and compared. Since the diameters of the two sources are nearly the same, equal mass flow rates provide nearly exit Re, Kn, and effective (P0 D), the parameters normally used to characterize free-jet relaxation processes.
The time-of-flight results can be summarized by the statement that generally the capillary gave consistently higher speed ratios and higher mean velocities. For mixtures, there was less slip for the capillary source. Table 1 summarizes these trends, showing the negligible difference for speed ratio, but significant increase in velocity. Errors in velocity are about 1%, while those in speed ratios may be 5%-10%, depending on magnitude.
Figures 2 and 3 show an example of raw time-of-flight data taken from the eductor and normalized to the peak intensity. It is not clear whether the long tails in the mixture co2 data reflect the true molecular distribution or are due to the characteristics of the electron ionizer on the quadrupole which we used. We ignored these tails for the comparisons we are making here. Despite this problem, it is clear that the molecular distributions are nearly indistinguishable, and Table 1 reflects similar results for all cases we have studied. Our fit to the data using the standard Boltzmann model is shown also in Figures 2 and 3.
926
Table 1. Comparison of Terminal Speed Ratios and Mean Velocities
COz He COz He
in Free-Jets From a Capillary and Orifice at Same Mass Flow.
Gas m x 103 (gms/sec)
Ar 2.70 Ar 5.40 Ar 12.90 He 0.42 He 0.57 He 2.20
CO 2.30 CO 5.90 CO 9.90
(84% He) 0.94 (16% COz) (84% He) 2.32 (16% COz)
J.D
ARGON ÄCAPILLARY e OR.TFICE rh • 5.-fx JO""' G-HS/SEC
0.8 Re •810
0.6
0.4
o.z
0 0.9
ORIF"ICE SOURCE PRESSiJRE•..t-16 7"01/A
Orifice P0 (Torr)
80 146 310 49 62
197 58
139 242
61
131
SR(CAP)/ V0 (CAP)/ SR(OR) V0 (OR)
1.10 1.03 1.07 1.03 1.08 1.02 0.98 1.03 0.99 1.03 1.04 1. 02 1.04 1.04 1.02 1. 02 0.96 1. 02 1. 01 1.05 1.00 1.04 1.06 L01 1.00 1. 03
80L7"Z/'1ANN F"Ir F"OI? S.R•J?"
.t . .t .t.2
Figure 2. Comparison of Time-of-Flight Data for Ar From Capillary and Orifice.
927
LO
0.8
0.6
co. ~ NrxruRE c.t,zcq, ,tH~IEJ .t. CAPILLARY e ORIF"ICE m •Z.'SxJO~GHS/SEC Rt!•45S Ollii'"IC.E SOURCE PRESSURE•.t:S.LTOiill
BOi.TZMANN FIT FOR Jll•l+
• ~
••• • 0o~.B------,o~ . .------L~o~-----.J.,l------~L2~------~------~.+
t/f171u
Figure 3. Comparison of Time of Flight Data for C02 in a 16% C02/84% He Mixture From Capillary and Orifice.
The increase in mean velocity for the capillary source js sufficient to violate an energy balance for the flow. Since adiabatic Fanno flow conserves energy, or stagnation temperature, and T0 = 295°K for these experiments, the maximum mean velocity for argon, for example, is 5.54 x 10~ ern/sec. The 146 Torr Ar run in Table 1 gave 5.50 x 10~ for the orifice and 5.67 x 104 for the capillary. Since our velocity accuracy is 1%, the capillary has about 3% too much energy. As Tab1e 1 suggests, this feature was regularly obtained. We fee1 it is due to heat transfer from the capi11ary walls to the subsonic flow, near the exit where the static temperature drops.
A simple estimate of the heat transfer rate demonstrates the liklihood of this non-adiabaticity. We ask what length of capillary, L, is necessary to add 3% energy to the flow if the öT driving force is ~50°K. (If the capillary walls remain near 295°K and the flow reaches M = 1, then the flow temperature is 221°K for Ar, so that such a öT near the exit is not unreasonable.)
A lower limit of the Nusselt number for fully developed laminar flow is Nu = 3.66 = hD/k. The heat transfer rate q ~ hnDL(öT) = 3.66 knLöT. The Prandtl number Pr = ~Cp/k = 0.67 for Ar so that
928
. q ~ S.S.~CpnL~T. The e~ergy of the flow is.E = mCpTo so that we find 4/E ~ (22/Re)(L/D)(~T/T0). Setting 4/E = 0.03, ~T/T0 = 0.17, and Re ~ 1000, we find L/D ~ 8. Since the capillary length is L/D 128, it is clearly feasible to transfer this magnitude of energy in the last 8 diameters where, in fact, the gas temperature drops substantially. In the orifice flow, the acceleration to M = 1 is fast enough and the main flow far enough from the walls so that the flow is adiabatic and isentropic. For capillary studies, this slight non-adiabaticity may be a problern in future analysis.
Figure 4 shows typical centerline intensity data, measured as number density in the mass spectrometer, again for the same mass flow rates. The capillary maintains a slight increase (3%-10%)
OIUFICE SOUI?CE !'RES SURE (TON/V
:$S 100 1~1l_ 10~ ;HO 300 .1~0
• • • • • • .. • :$0 • • • • • • • • • • .., • ..... .. ~25 • :::, • • >.. • ~ zo ••• • JA. •• ~ •• .... • CQ Cl( JS ~
~ ~ 10
~ ...fQ,._ q
5 4 CAPILI.ANY
e ORIFICE
Re-2.2xJO' m 0 ;z + ' tJ 10 u ~~
MASS FLOW RATE (GRAHS/SEC xiO-')
Figure 4. Comparison of C02 Density in Capillary and Orifice Free-Jets.
in the beam intensity, a consistent result for all cases we have studied. At higher flows, the intensities are no Ionger increasing due to background scattering in the nozzle-skimmer region. Figure 5 shows the most dramatic difference we have observed. Here, the dimer densities are given as a function of source mass flow rate for Ar condensation. A similar result is found for C02 condensation. The capillary flow appears to enhance condensation in the free-jet at a lower m and continues at a higher rate. This is not unexpected
929
0/?IFICE. SOUI?CE PI?ESSURE (TOI?R)
12 AR
~ CAPiüARY • • ~ • ~10 e ORIFICE :::s )._ Q:
l?c.~J.§ x10~,.;, • ~ 8 R • ~ ~ ~
' • ~ ~ • ~ ~ -+
~ • • ~ • ~ z • •
• • • • • 0 o~------2~L-~~~~--~--~,------~r-~~~~--~-.T-------rl
MASS FLOW I?ATE (GRAMS/ SEC yJO~)
Figure 5. Comparison of Ar Dimer Density in Capillary and Orifice Free-Jets.
since condensation is a three-body process, and at these densities the three-body effects relax or "freeze-out" very close to the source exit and are, therefore, most sensitive to the initial differences between orifice and capillary free-jets. Essentially, the capillary source maintains the cold flow near M = 1 at a higher density for a Ionger flow time.
We feel the preliminary data we have presented suggest that the free-jet expansions are essentially the same for orifice or capillary sources when they are scaled to the same exit conditions. Therefore, capillary jet properties can be estimated from literature results for orifices by simply determining the stagnation conditions required for the orifice to yield the same mass flow rate for the same diameter. Only those processes which are relaxing near the exit will be significantly altered in the terminal jet properties. We are proceeding with theoretical calculations, using the measured initial conditions and a method of characteristics approach for this rotational flow to examine these cases in more detail. Experimentally, we are pursuing more detailed measurements of slip in mixtures, condensation, rotational relaxation of H2 , and vibrational relaxation of C0 2 , all of which can depend on the flow near the exit.
930
ROtATIONAL RElAXATION IN HIGH TEMPERATURE JETS OF NITROGEN
R.G.Sharafutdinov, A.E.Belikov, N.V.Karelov, and A.E.Zarvin Institute of Thermophysics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk State University Novosibirsk, 630090, USSR
INTRODUCTION
Nowadays inelastic collisions with the energy exchange of a rotational motion of molecules attract a wide attention of researcher <e.g., see recent review1). Supersonic expansion of gas in nozzles or free jets is one of the most suitable methods of investigation of rotational relaxation that explains the appearance of a great number of experimental works related to this topic. In the experiments on the rotational relaxation in free jets various diagnostics methods are used, however, a more detailed information on rotational relaxation has been obtained with the use of the electron-beam diagnostics.
The authors of some works2,3 evidence that transition from the equilibrium state in a gasdynamical source to the non-equilibrium one at a certain distance from it occurs through the sequence of Boltzmann Levelpopulation distributions, while the others4,5 reveal the disturbance of Boltzmann distribution. The population distribution form is significant in a theoretical description of rotational relaxation.
Investigations of rotational relaxation in jets by the electronbeam fluorescence technique were performed, as a rule, at a room stagnation temperatures, and, consequently, at a low translational temperature in gas flow~ Interpretation of these data is difficult due to a possible effect of small clusters under the conditions of a highly supercooled gas6, as well as the pressure of a background gas7 and uncertainties in the electron beam fluorescence technique8. The above-mentioned distorsions must be decreased with increasing
931
translational temperature of gas. For this purpose the stagnation temperature T0 must be increased.
The aim of the present paper is to investigate the kinetics of rotational Level population of molecules in a nitrogen free jet at high values of local translational temperatures.
EXPERIMENTAL ARRANGEMENT
The investigations were performed in a low-density wind tunnel of the Institute of Thermophysics of the Siberian Branch of the USSR Academy of Sciences. It is equiped with the electron-beam diagnostics for measuring density and rotational Level populations6.
Because of a great velocity of rotational relaxation comparable with that of translational relaxation, and the necessity to obtain prominent deviations from an equilibrium flow at sufficiently high translational temperatures (100 to 300 K),it is required to realize the experiments at low Reynolds numbers determined from the parameters in the nozzle throat. In the present measurements the Reynolds numbers Re* were changed from 50 to 1000. At low Re* the drag and heat-exchange effects in the nozzle and disturbance of equilibrium in a subcritical section of the nozzle are likely to be possible. Therefore determination of the stagnation temperature To demands a particular attention. In the present work the value of To was found by using the discharge method, from the density ratio of heated gas and room temperature gas, the temperature calculated from spectrograms in an equilibrium region of the expansion and the thermocouple data at great Re*. All the methods provided a satisfactory agreement. At low Re* the values of To calculated from spectrograms were preferable.
EXPERIMENTAL RESULTS
Fig. 1 presents the measured line intensity distributions in the R-branch of the band 0-0 in the first negative system of N2 -ion shown as lg(Ikr/I1k') vs x/d*. Here Ikr,I1 are the intensities of the kth line and the first one, respectively; k'- the rotational quantum number B2 E(v'=O) state; x- the distance from the nozzle with diameter d*. The values of Ik' are presented only for odded k'. The measurements were made at To~900 K for stagnation pressures p0 =3.9, 19.8 and 36.4 torr. Solid lines illustrate the values lg(Ikr/I1k'J calculated by Sharafutdinov et al.9 and dashed line denotes lg(Ikr/I1k'J calculated according to Muntz model.
As may be seen from Fig. 1, the differences between the models are insignificant, while the experimental data differ markedly from the equilibrium ones. Consequently, the above-mentioned difference is caused by the Levelpopulation nonequilibrium in X1 E state, i.e.
932
I • lg ..-:i.... x,. k
Fig. 1. The example of measurements of rotational line intensities in the spectrum of firstnegative system along the axis of a free jet. To=900 K; d*=5.12 mm. Equilibrium isentropic flow calculation (y=1.4) by the Muntz model <ök=±1) - dashed line, by the multiquantum model (ök=±1,3,5, ••• ) -solid line.
a less rate of rotational relaxation as compared to translational one at the rapid decrease of translational temperature. The dependences of the results on stagnation pressure po and distance from the nozzle x/d* are in good agreement with the existing tendencies of the rotational relaxation in expanding gas, in other words, a continuous deviation from equilibrium takes place first at high Levels. This deviation begins the more early (inx/d*) the less Stagnation pressure, i.e. the less gas density in flow. The values of lg(Ikr/I,k') obtained under the highest Stagnationpressure are most close to the equilibrium ones.
DISCUSSION
In the jets from sonic nozzles the rotational Level population kinetics is determined by the values of stagnation parameters and
933
proper geometric reference lengths of nozzles. For axisymmetrical nozzles at constant stagnation temperature To the product of stagnation pressure and nozzle diameter, pod*, can be used as a similarity parameter of rotational relaxation4. When pod* and x/d* are constant, the population distribution is the same. In order to compare the results obtained at different Stagnation temperatures, it is necessary to find the similarity parameter of rotational relaxationvalid at arbitrary To.
For a steady state flow the equations of population kinetics accounting for R-T processes only, take the form
(1)
00
where Nk=nk/~nk; u is the mean flow velocity; n=~nk - the gas density; nk- the population of the kth rotational Level. The equilibrium rate constants entering these equations are the averaging of inelastic scattering total cross-sections of Oij over Maxwell distributionsf(v}:
K .. =fvf(v)a .. dv. 1-J 1-J
The constants Kij can be represented as follows:
K •• =V<a . . > .. 1-J 1-J
where v=/BkBT is the mean velocity, kB- the Boltzmann constant, 1Tm m- the reduced mass. After simple transformations and normalization to the limiting velocity, nozzle diameter and density no, the expression (1) takes the form
dNk U V - 2:_ (~) ~ (~) -- (~ )nod x
d(x/d*) - no d* u d* umax d* * (2)
As follows from (2), when efficient inelastic cross-sections of collisions are independent of temperature, the product of density and nozzle diameter, nod*, is the parameter of the problem, and deviation of the Levelpopulation from the equilibrium at given x/d* depends on nod* only. The same product can be used for generalization of the results of rotational Level population kinetics10 and when describing rotational relaxation within the framewerk of the relaxation equation11.
934
Therefore, the comparison of measurements results at different stagnation temperatures is to be made at constant values of nod* . Certainly the question about the independence of collision crosssections on temperature for a wide range of temperatures is disputable. However, as it follows from the further analysis of the results, the use of nod* as a similarity parameter is valid. The comparison of the results at constant n0 d* allows the effect of gas temperature on rotational relaxation in jet to be extinguished.
The measurement results obtai~ed at different values of stagnat i on temperature, such as 93,300,690 and 970 K and n0 d* =2.4·1017cw- 2
are presented in Fig. 2 in the coordinates l g[Ik r/(I , k ']] from k ' (k 1+1] for six values of x/d*, from 1 to 7. The solid Lines show the calculated values of l g[Ikr/(I,k' )] at an isent ropic temperature with y=1.4. They were obtained from9, with accounting for the transitions ßk '=1,3,5, ... The regularities in the Line intensity variations <and, consequently, in the Level population in X1 E state) are equal for all the stagnation temperatures, when the distance from the nozzle throat increases. The most equilibrium stat es are observed
0
I . lg _k_
I 1 · k
0 100 200 300 k(k+1) 100 200 300 k(k+1) 100 200 300 k(k+1)
~\On \ \ o llro .\ \ cv 'Cl
~~
Fig . 2. Rotational Line intensity distributions at different distances from the nozzle. b. - To=970 K; o - To=690 K; 0- To=300 K; 'V - To=93 K. Solid Lines- isentropic calculation for y=1.4.
935
in the vicinity of the throat, and the most nonequilibrium ones take place at a maximum distance from it. However the lower the stagnation temperature, the moreevident the difference.from the Boltzmann Level populations, i.e. the line intensity distribution is more curved. Unlike high stagnation temperatures, where the intensities of all the lines deviate from the equilibrium, at less stagnation temperatures and variations in x/d* a portion of lower Levelsfora long time is in agreement with the equilibrium values. Presented in Fig. 3 are the rotational temperatures of lower Levels Ttr normalized to isentropic values Tis when y=1.4 vs x/d*, for the same conditions as in Fig. 2. At a room stagnation temperature the rotational temperature of lower Levels is approximately equal to the isentropic one (with y=1.4). At the sametime it is higher than the isentropic value for high stagnation temperatures and lower for To=93 K.
It follows from Fig. 2,3 that with decreasing Stagnation temperature, the rotational temperature of lower Levels approaches to the translational one. It is indicative of a stronger connection of lower rotational Levels with translational degrees of freedom at Low temperatures, i.e. the rotational relaxationrate increases with decreasing temperatures for lower Levels.
Within the framework of this assumption it is possible to explain an anomalaus lower-Level population behaviour, when To=93 K. Actually, due to freezing of relaxation process, a portion of rotational energy (first of all, energy stored at higher rotational Levels) didn't release to translational degrees of freedom, and then to the flow velocity. Therefore, Tt in the flow is below than in the case when y=1 .4 (in the Limiting case of frozen flow Tt is equal to the isentropic value if y=1.67). In this case a more effective connection of lower rotational Levels and translational degrees of freedom, the rotational Level temperatures are also lower than isentropic ones when y=1.4). Presented in Fig. 3 are the values of Ti 3 (y=1.67) normalized by Tis (y=1.4) (solid lines).
For upper Levels the situation is opposite. At low temperatures the Levelpopulations deviate from the equilibrium more than at high To. This result is shown in Fig. 2 but it is moreillustrative in Fig. 4, where for constant x/d*=2 the differences of lg[Ik r/(I1k ')] have been obtained experimentally and calculated for isentropic temperatures when y=1.4. The results are presented as a function of k'(k'+l). As shown in Fig. 4, when To=300 and 93 K, the lower Levels are more close to the isentropic values than when To=970 and 690 K. When the number of rotational Levels increases, the Lower the stagnation temperature, the greater the deviations of the measured Line intensities from the equilibrium of lg[Ikr/(I1k ')]. This follows from the obtained results that for upper Levels the rotational reLaxation rate increases with stagnation temperature, i.e., with translational temperature in gas flow.
936
1, 6 IT
I 0
tr 0 - T0 • 970 K
T18(l)•1.4) • 690 • 0 93 0
1,4 t 293 0
~ • ....
c 0 ° .. 1, 2
.. ~ 0
~ .. Q ~· t ~ A. V I",. 1. 4
..... Dtf '! ~ 'f (t Dt9 0
0 ~ 'V= 1j67 x/d• ,...
1,0
0,8 0 2 4 6 8 10
Fig. 3. The deviation of on-axis rotational temperature of Lower Levels (TtrJ from the equi Librium isentropic one with y=1.4 (Ti8 ) for different Stagnation temperatures.
I , , I , /
( lg--!L,) exp -(lg--L ) / I 1 . k' is, ~·1. 4 / I 1 · k / ' y I
I
// 2
1
-- To=
9:~ I ---· 293 K / - ·- 690 K / - ·· - 970 K /
/
I
V I
/ /
_.-/ I /
L ""' ~~ f-··-/ -~ k'(k'+1) . - ---.:::f . . 0
0 100 200 300 400 500
Fig. 4. The deviation of experimentally measured Line intensities (exp.) from the equilibrium (isentropi c) values for y=1.4 (is., y=1 .4) . n0 d*=2.4 1017 sm- 2 ; x/ d*= 2
937
RE FE RENCES
1. J.P.Toennies, The calculation and measurement of cross-section for rotational and vibrational excitation, in: Ann. Rev. Phys. Chem., 27:225, Ann. Rev. Inc., Palo Alto (1976).
2. G.Luiks, S.Stolte and J.Reuss, Molecular beam diagnostics by Raman scattering, Chem. Phys. 62:217 (1981).
3. D.Coe, F.Robben, L.Talbot and R.Cattolica, Rotational temperatures in nonequilibrium free jet expansion of nitrogen, P.hys. Fluids. 23:706 (1980).
4. P.V.Marrone, Temperature and density measurements in free jets and shock waves, Phys. Fluids 10:521 (1967).
5. M.Faubel and E.R.Weinar, Electron beam fluorescence spectrometry of internal state populations in nozzle beams of nitrogen/rare gas mixtures, J~Chem.Phys. 75:641 (1981).
6. N.V.Karelov, A.K.Rebrov and R.G.Sharafutdinov, Population of rotational Levels of nitrogen molecules at noneguilibrium condensation in a free jet, in: Proceed. of the 11th Int. Symp., R.Campargue, ed., Commissariat a l'Energie Atomique, Paris, 2:1131 (1979).
7. R.G.Sharafutdinov, Interaction of background molecules with a low density free jet, in: Proceedings of 7th Int. Symp., D.Dini, ed., Editrice Technico Scientifica, Pisa, 1:563 (1971).
8. A.K.Rebrov, N.V.Karelov, G.I.Sukhinin, R.G.Sharafutdinov and J.-C.~engrand, Electron beam diagnostics in nitrogen: secondary processes, in: Progress in Ast ronauti es and Aeronaut i es, S. S. Fisher, ed., AIAA, N.v., 74:931 (1981).
9. R.G.Sharafutdinov, G.I.Sukhinin, A.E.Belikov, N.V.Karelov and A.E.Zarvin, Electron beam diagnostics in nitrogen: multiquantum rotational transitions, in: This Proceedings.
10. N.V.Karelov, R.G.SharafLrtdinov and A.E.Zarvin, Rotational relaxation in nitrogen free jets in the transition regime, in: Progress in Astronautics and Aeronautics, S.S.Fisher, ed., AIAA, N.Y., 74:742 (1981).
11. R.G.Sharafutdinov and P.A.Skovorodko, Rotational Levels population kinetics in nitrogen free jets, ~: Progress in Astronautics and Aeronautics, S.S.Fisher, ed., AIAA, N.Y., 74:754 (1981).
938
TRANSLATIONAL NONEQUILIBRIUM IN A FREEJET
EXPANSION OF A BINARY GAS MIXTURE
ABSTRACT
Norio Takahashi, Tomio Moriya and Koji Teshima
Department of Aeronautical Engineering Kyoto University, Kyoto 606, Japan
Frozen parallel temperatures and frozen velocities of each species in the free jet expansions of helium-argon, helium-neon and neon-argon gas mixtures have been measured by a molecular beam timeof-flight method in the range of Pod=l-2 Torr·cm, where P0 is the source pressure and d the orifice diameter. We have obtained that the frozen temperature of the heavy species is higher than that of the light species and that the ratio of the frozen temperature of the heavy species to that of the light species increases with increasing their mass ratio. We have also made the numerical calculation for the source flow expansion of the binary mixture using an ellipsoidal velocity distribution function. The calculations have been made for a rather larger value of Pod compared with the present experiments, but the results are qualitatively in good agreement with the present experiments in the dependency of the frozen temperature of each species on the mole fraction of the heavy species and on the mass ratio.
INTRODUCTION
Theoretical treatments for a velocity slip and a translational nonequilibrium in a free jet expansion of a binary gas mixture have been given by Willis and Hamel,l-cooper and Bienkowski,2-Miller and Andres,3 and Soga and Oguchi.4 They have solved this flow problern for the case of a rare gas mixture by the moment method of the Boltzmann equation. Willis and Hamel, and Soga and Oguchi have used the BGK model and predicted that the frozen temperature of the heavy species is lower than that of the light species. However, Cooper
939
and Bienkowski, and Miller and Andres have evaluated collision terms and obtained the opposite predictions. Recently, Chatwani and Fiebig5 have calculated this problem by the direct Monte-Carlo simulation method and obtained that the frozen temperature of the heavy species is higher than that of the light species. Experimental studies3,6,7 have also shown the similar results as Millerand Andres' theoretical calculation and Cooper and Bienkowski's theory. These opposite results on the frozen temperatur~ of each species have made confusing the understanding of the translational relaxation process between the heavy and the light species.
In the present paper, we have measured the frozen parallel temperatures and the frozen velocities of each species in the free jet expansions of helium-argon, helium-neon and neon-argon gas mixtures with a molecular beam time-of-flight (TOF) method. We have also made a numerical calculation of the source flow of the binary mixture using an ellipsoidal velocity distribution function and the moment method of the Boltzmann equation.
EXPERIMENTAL APPARATUS
The experimental apparatus was a conventional molecular beam apparatus connected with a time-of-flight system as shown in Fig.l. A four stage differential pumping system was used: the expansion chamber was evacuated by a mechanical booster pump with a 700 1/s nominal pumping speed, the skimmer-collimator, the flight and the
3xl02' byt•
BuffK Mmlory
Micro COfrJIIJf•r IMPU-MC6«19J
-1 6dO rorr
Colttnilting SJit
,-- ----., I D•l~y I I 8bit Al D J ;=::r=, ~=I::::.., I rrigg.r ....., IO:U word l L--- _J l Buff•r 1
I I I MHntTY I ~--------
-1 15xl0 rorr
Fig.l. Schematic diagram of molecular beam time-of-flight apparatus and data acquisition system.
940
detector chambers were evacuated by 600 1/s oil diffusion pumps and were kept at the pressures of ~lo-3, lo-6, 6Xlo-7 and l.Sxlo-7 Torr, respectively. An orifice with 30 ~m nominal diameter was used, the effective diameters of which were determined from the flow rate measurements because they depended on the source condition. A skimmer with 0.67 mm inlet diameter was used. The time-of-flight system had a slit function of a half width ~tFWHM=36 ~s, a flight length Lt=67 or 117 cm and a detector's ionizer width ol=7 mm. The detector consisted of a quadrupole mass filter and a secondary electron multiplier. The detected signal was stored in a transient memory recorder at a sampling rate of 2 ~s/word. The signal was very weak so that the signal integrating up to a number of 215 or 2l6 was made with a micro-computer to obtain the high signal-to-noise ratio. The obtained TOF signal was fitted to a single Maxwellian velocity distribution function convoluted a chopper slit function in order to determine the stream velocity and the parallel temperature of each species. The TOF measurements were made at several orificeskimmer distances ranged from 40 to 120 orifice diameters and then the obtained values were confirmed to be the frozen values.
EXPERIMENTAL RESULTS
Experiments were made for 1-90% Ar-He, 1.5-30% Ne-He and 20% Ne-Ar gas mixtures in the source pressure range of Po=350-650 Torr. Typical TOF spectra for a 92% He-8% Ar mixture are shown in Fig.2, where subscripts 1 and 2 indicate the light and the heavy species, respectively. It can be seen that deviation from the Maxwellian velocity distribution exists in the higher velocity components of helium. The appreciable deviation was not observed for helium at the mole fraction of heavy species, X2, less than 3%, but above this
r·· ·-r ~-·-. II. !! ---~ ·-- pj
... .. u ·--.. ·~ -
II' M -Jj ·-!!II
: ' I• ~,
II lj >- ,_
:~ II .... I
'~·- .... .... ··-· .... ~ ·=
(a) Helium in He-Ar. T111=11.3 K, (b) Argon in He-Ar. T11z=27.0 K, ul=1.28Xlo5 cm/s and S11'f=5.98. uz=l.2Sxl05 cm/s and s11z=l1.8.
Fig.2. Time-of-flight spectra observed for 92% He-8% Ar mixture at Po=650 Torr, deff=27.5 ~m and To=280 K. 100 ~s/div.
941
value the deviation becomes noticeable with increasing X2· Inversely, the deviation from the Maxwellian was slightly observed in the lower velocity components of argon and neon at xz<3%, but above this value their distributions were in good agreement with the Maxwellian. With increasing xz, the collision frequency between light-light species decreases but those between light-heavy and heavy-heavy species increase. Therefore, the deviation from the Maxwellian for helium becomes appreciable with X2 due to the increase of the collision effect between light-heavy species, while the deviations for argon and neon decrease because of the increase of the collisi on effect between heavy-heavy species. However, we have determined the translational temperature from finding the minimum dispersion between the experimental spectrum and calculated one using a single Maxwellian distribution for all experimental conditions.
The frozen00translational temperature normalized by the source temperature, T111/To, is plotted against X2 in Fig.3 for He-Ar and He-Ne mixtures. It is found that the frozen temperature of the heavy species is higher than that of the light species and that the ratio of Tuz/Tni increases with increasing their mass ratio; 2.0-2 . 6 for He-Ar (m2/m1=10) and 1. 75-2.0 for He-Ne Cm2/meS). These results agree with Miller and Andres' theoretical calculation and Chatwani and Fiebig's Monte-Carlo results. Since the collision effects between light-heavy and heavy-heavy species increase with increasing X2• both frozen temperatures of the light and heavy species decrease with x2 as shown in Fig . 3. The measured speed ratio, s11i_, and the velocity slip, ~u/uisen• are shown in Fig.4, where uisen is a terminal velocit~ achieved from an isentropic expansion of the mixture and ~u=u1-u2· The speed ratio of argon slightly increases with X2• and that of neon is almost constant against X2• while that of helium
0-2 I
T0•2MK
d •JOI'm
f),•650Torr -
4Hf' f ;n H• -Nr
....... oNf
fr 0-1 ':V • •• • Hf ~ -in Ho-Ar
00 • Ar 0 . 0
SA .... .
0 . . " " " o.o i
0 o.s /.0
Molo Fr11ction of H•11•1 Spocl•s
Fig.3. Normalized frozen temperature of each species against the mole f r action of heavy species for He-Ar and He-Ne mixtures .
942
IS.------.---,-------,.------r--.--.------r--.-------,.-----,
... I- "" u; IOJro Oo o o
~ .. ~ !ir.
1 ~·" 9" ~~ s-. ... <>
.. i"' • <> <> • ..
•
•
V•locity slip
J•H•- Ar <>H•- N•
-
-
Fig.4. Speed ratio of each species and velocity slip against the mole fraction of heavy species for He-Ar and He-Ne mixtures.
slightly decreases from the value of pure helium. The velocity slips decrease with X2 for both cases. Both the ratio of Sni_lsui and the velocity slip increase with increasing the mass ratio, m2/m1.
THEORETICAL APPROACH
Millerand Andres, 3 Knuth and Fisher, 8 and Toennies and Winkelmann9 have solved the problern of a freejet expansion of a pure rare gas using an ellipsoidal velocity distribution function and the moment method of the Boltzmann equation and obtained a good agreement with experiments. Therefore, we have tried to extend their idea to the case of a rare gas binary mixture. Basic assumptions of the present theory are to treat an expansion beyond an orifice as a spherical symmetric flow and to use an ellipsoidal velocity distribution function for species i given by
mi 1/2 mi mi 2 mi 2 (2~kT ) (2~kT )exp[- -2kT (v,,.- ui) - 2kT,~ v...L~], (l)
/fi :Li /Ii ~ _,_..._ ...
where v 11i and v.Li are the parallel and the perpendicular velocity components with respect to a stream line, T11i and T~i the corresponding translational temperatures, ui is the stream velocity, mi the mass and k the Boltzmann constant. With eq.(l), the Boltzmann equation for spherical coordinate is given by
a mi -;:-v f - --(v -Clr 1/i i rkT//i /Ii
2 Clf. L (----..!.)
. 1 Clt colli. J= J
943
where (afi/at)collij denotes the change of fi due to the self- or cross-collision. Here, the moment method is used to obtain the basic equations for four different functions of velocity components, i.e., mi, mi~ti• mi(vui2+v~i2)/2 and mi~i2/2. In the present theoretical model the collision terms can be precisely evaluated including the realistic collision interaction potentials. However, since their evaluations require complicated calculations including the six-order multiple integrals, we assume that the perpendicular temperature is equal for each species, i.e., T~1=T~2=TL, for simplification of the calculations. Then the order of multiple integrals on collision terms is reduced to four from six and the number of the basic equations is reduced to seven from eight. Finally, we obtain the following basic equations;
n~ u* r*2 = F1• , 1 i
l 3TI/~ (-- --)
du~ T * 3 1 --1 =-.L-+--c +--c
2 10u~2 1
dr* 5u~r* lOu~ li 5u~ 2i '
l 3T/f~ (-- --) 2 10u~ 2
1
dT//~ __ dr*
dT.L.* _ 2T.L* + C dr* = r* 31
1 1 1
-(1
(2)
(3)
(4)
(5)
(6-1)
(6-2)
(6-3)
where Fi is the reduced flux of the flow for species i and ~·· denotes the collision terms between species i and j. In the1~eduction of eq.(2)-(6), we have used the following reduced parameter: r*=r/d, Ti*=Ti/To, ni*=ni/noi and ui*=ui/umaxi. where d is the orifice diameter, noi the source density of species i and umaxi the maximum stream velocity obtained by the pure gas expansion and given by umaxi=(SkTo/mi)l/ 2 . The assumption of the equal perpendicular temperature has been made by the prediction that both perpendicular
944
temperatures of the light and heavy species would keep an isentropic relation during the expansion. From this assumption, we obtain the following relation;
According to the momentum and the total energy conservation principles, further two relations on collision terms are obtained;
~12[mlvlll] + ~21[m2v112l = 0 ' (8)
ml 2 2 m2 2 2 ~12[2(vl/l + v.ll)] + ~21[2(v//2 + ·~1..2)] = 0 • (10)
Thus, the evaluation of the collision terms is necessary for ~11[v~12 1, ~12[v11 1J, ~12[v1112 l and ~12[vL12l and the others are calculated from the relations of eqs.(7)-(9), which means that the collision term on the heavy-heavy collision is determined from those on the light-light and light-heavy collision as given by eq.(7). With the equal perpendicular temperatures, the collision terms can be reduced to the simplified analytical forms including the fourthorder multiple integrals except ~11[v~12 ], where ~11[v~12 l is the same as that obtained by Toennies and Winkelmann.9 To furthermore simplify their forms, the velocity slip terms in the collision terms are expanded to the first order because the velocity slip is much smaller than the stream velocities. Then the translational nonequilibrium term in the collision terms is expanded into a power series. Finally, we can replace the fourth-order multiple integrals on collision terms with the infinite series of multiplications of four single integrals, two of which can be presented with known analytic functions, and thus we can easily calculate the collision terms. The set of eqs.(2)-(6) is solved numerically with Runge-Kutta-Gill method together with eqs.(7)-(9). The initial conditions were determined from Ashkenas and Sherman's formulalO at any distance from the orifice in the range of r*=l.2-1.8 and the terminal condition of T..L*/T11t < 0.05 was used.
A typical result of perpendicular and parallel temperature changes during the expansion of a 97% He-3% Ar mixture at Pod=7 Torr-em and To=300 K is shown in Fig.5. It shows that the parallel temperature of the heavy species is always higher than that of the light species during the expansion, while the perpendicular temperature almost keeps the isentropic relation. The frozen parallel temperatures, the speed ratios and the velocity slips are plotted against X2 for He-Ar and He-Ne mixtures at Pod=7 Torr.cm and To=300 Kin Figs.6 and 7. It is found that T~i is always higher than ~/i
945
T0 .JOOK
f'bd • 7 Torr.cm
97%H~ -3%Ar
' /~\
11•ntroptc ' #X~fl~IOft \\
r•
\ \
100 300
Fig.S. Calculated parallel and perpendicular temperature changes during a source flow expansion of a 97% He-3% Ar mixture at Pod=7 Torr.cm, where the starting point of r*=l.S was used.
00 00
in the calculated range of X2 although the ratio of T112/T111 decreases with increasing X2· This result qualitatively agrees with the present experiments shown in Fig.3. The speed ratios of argon and neon are larger than that of helium and increase with x2 , whereas that of helium slightly decreases from the value of pure helium, and that
Pod = 7 Torr·cm
T0 •3001<
x He I • Ar in He-Ar
~;He l . m He -Ne o Ne
Mole Fraction of Heavy 5pecies
Fig.6. Dependency of calculated frozen parallel temperature of each species for He- Ar and He-Ne mixtures on X2 at Pod=7 Torr·cm.
946
1? .. --..:-:_ ~
~-~D::---------J I x::----:---~ -~ ·-... "·· ~- - , o ro ~ ~ 4
Mole Fraction of Heavy Species X1 ('!.}
Fig.7. Dependency of calculated speed ratio of each species and velocity slip for He-Ar and He-Ne mixtures on X2·
for He-Ne is slightly larger than that for He-Ar. The velocity slip decreases with x2, and that for He-Ar is always larger than that for He-Ne.
In the present calculation, the starting condition is determined from Ashkenas and Sherman's formula under the assumption that an isentropic relation will be kept up to the starting point, and the velocity slip terms in collision terms are expanded only to the first order. Therefore, we could not solve eqs.(2)-(6) in the range of our experimental condition (Po=l-2 Torr.cm), because the nonequilibrium effects already exist upstream from the starting point and also the velocity slip becomes larger for the smaller P0d vales to make the approximation of the small velocity slip invalid.
DISGUSSIONS
By a comparison of the present experiments with the present calculations, the qualitatively good agreements are obtained in the dependency of the frozen temperature, the speed ratio of each species and the ve~ocity slip on X2 and mz/ml· However, in the calculation the ratio of T11z/T111 appreciably decreases and the speed ratio of the heavy species considerably increases with increasing X2· This is mainly caused by the fact that the collision term on the heavyheavy collision, ~zz[v~ 22], is not ,calculated directly in the present calculations because it can be determined from eq.(7) under the assumption of the equal perpendicular temperature. Namely, this fact means that the difference between ~zz[v~z2] evaluated from eq.(7) and its correct value becomes large with X2 in the calculation.
947
Fig.8.
115
• H•- Ar I 1 '* -N# &".mn•nt~ • N•- Ar
• IÖ3L-~~~~~,_-L-L~~~.-~~
I I
(JJ~/if/111 d( 6C~2/IIJ -"'J-m./'o kTo
Gorrelation between the present experiments for the velocity slip and a slip parameter of Miller and Andres3 together with their theoretical prediction, Chatwani and Fiebig's MonteCario results5 for m2/m1=5 and the present calculations.
The normalized velocity slip is plotted against a slip parameter introduced by Miller and Andres3 in Fig.8 as well as their theoretical prediction, Chatwani and Fiebig's Monte-Carlo results5 for m2/m1 =5 and the present calculations, where for Monte-Carlo results we regarded the gas mixture of m2/m1=S as the helium-neon mixture. In the slip parameter, m is the average mass of the mixture, c61,2 the coefficient of the attractive term in Lennard-Jones (12,6) potential between species 1 and 2. The present calculations are in good agreement with Miller and Andres' prediction, but are !arger than the present experiments. For the smaller values of the slip parameter, the present experiments agree with Chatwani and Fiebig's Monte-Carlo results.
The scaling parameter on the frozen temperature has been first presented by Miller and Andres.3 However, their parameter has not a good correlation with the frozen temperature of the heavy species, i. e., it has a dependency on X2 and m2/m1 for r112. as indicated by Patch,ll because they have not included the effect of the diffusion of a binary mixture in their parameter. Patchll has included this effect and reduced the modified frozen temperature parameter using a sudden freezing model. In Fig.9, the normalized frozen temperatures of the light and the heavy species are plotted against Patch's frozen temperature parameter together with Chatwani and Fiebig's Monte-Carlo results for m2/m1=S and the present calculations. The
948
' I
ul
• H• ·Ar! t H• A "'' EtfHt'llnMfJ • Nt · Ar
~ ..,. H.,d jph~rt
o ~•Awtll moltculr
Fig.9. Gorrelation between the present experiments for the frozen temperatures of the light and the heavy species and a frozen temperature parameter of Pat§hll as well as Chatwani and Fiebig's Monte-Carlo results and the present calculations.
frozen temperature parameter is a complicated function of source density, masses, mole fraction and potential parameters presented in eqs.(46)-(50) of ref.ll. In his parameter, (mi/mj+5) in eq.(47) must be substituted with (mi/mj+5Qij(l,l)*/2Qij(2,2)*), where Qij(l,l)* and Qij(2,2)* are the standard collision integrals that occur in the Chapman-Enskog expression for binary diffusion and viscosity, respectively. It is found that Patch's scaling parameter has a good correlation with the frozen temperature of the light species, but has a slight dependency on the mass ratio for that of the heavy species. The experiments for the light species show a good agreement with the calcuiations, while those for the heavy species show a weaker dependency on the scaling parameter than the present calculations. Chatwani and Fiebig's results show a different dependency on the scaling parameter from the present experiments and calculations.
Since the Maxwellian velocity distributions with respect to the parallel and the perpendicular directions have been assumed in the present calculations, the results may not be applied precisely to the flow where the deviation from the Maxwellian velocity distribution becomes prominent. Nevertheless, the present calculational results qualitatively explain the present experimental results in the dependency of the frozen temperature of each species on t he mole f raction of the heavy species and on the mass ratio.
949
REFERENCES
1. D. R. Willis and B. B. Hamel, Non-Equilibrium in Spherical Expansion of Polyatomic Gases and Gas Mixtures, Proc. 5th 1nt. Symp. on Rarefied Gas Dynamics, Vol.1, C. Brundin, ed., Academic Press, New York, 837 (1967).
2. A. L. Cooper and G. K. Bienkowski, An Asymptotic Theory for Steady Source Expansion of a Binary Gas Mixture, Proc. 5th 1nt. Symp. on Rarefied Gas Dynamics, Vol.1, C. Brundin, ed., Academic Press, New York, 861 (1967).
3. D, R. Miller and R. P. Andres, Translational Relaxation in Low Density Supersonic Jets, Proc. 6th 1nt. Symp. on Rarefied Gas Dynamics, Vol.11, L. Trilling and H. V. Wachman, ed., Academic Press, New York, 1402 (1969).
4. T. Soga and H. Oguchi, Source Flow Expansion of Gas Mixtures into a Vacuum, Proc. 9th 1nt. Symp. on Rarefied Gas Dynamics, Vol.1, M. Becker and M. Fiebig, ed., DFVLR Press, Porz-Wahn, B.3-l (1974).
5. A. U. Chatwani and M. Fiebig, Source Expansion of Monatomic Gas Mixtures, Proc. 12th 1nt. Symp. on Rarefied Gas Dynamics, Vol.74, Part 11, S. S. Fisher, ed., A1AA, New York, 785 (1981).
6. N. Abuaf, J. B. Anderson, R. P. Andres, J. B. Fenn and D. R. Miller, Studies of Low Density Supersonic Jets, Proc. 5th 1nt. Symp. on Rarefied Gas Dynamics, Vol.11, C. Brundin, ed., Academic Press, New York, 1317 (1967).
7. R. Campargue, A. Leb~hot, J. C. Lemonnier .and D. Marette, Measured, Very Narrow Velocity Distributions for Heated, Xe and Ar-Seeded Nozzle-Type Molecular Beams of He and Hz Skimmed from Freejets Zones of Silence; Xe Energies up to 30 eV, Proc. 12th 1nt. Symp. on Rarefied Gas Dynamics, Vol.74, Part 11, S. S. Fisher, ed., AIAA, New York, 1674 (1981).
8. E. L. Knuth and S. S. Fisher, Low-Temperature Viscosity Cross Sections Measured in a Supersonic Argon Beam, The Journal of Chemical Physics, 48, 1674 (1968).
9. J. P. Toennies and K. Winkelmann, Theoretical Studies of Highly Expanded Free Jets: 1nfluence of Quantum Effects and a Realistic 1ntermolecular Potential, The Journal of Chemical Physics, ~. 3965 (1977).
10. H. Ashkenas and F. S. Sherman, The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnel, Proc. 4th 1nt. Symp. on Rarefied Gas Dynamics, Vol.11, J. H. deLeeuw, ed., Academic Press, New York, 84 (1965).
11. D. F. Patch, Application of Free Jet Sources to Relative Crossed Molecular Beam Experiments, Ph. D. Dissertation, University of California, San Diego, 1972.
950
LASER INDUCED FLUORESCENCE STUDY OF FREE JET EXPANSIONS
ABSTRACT
P. Willems, H. Hulsman and F. Aerts
Physics Department, University of Antwerp (U.I.A.) Universiteitsplein 1 B-2610 Wilrijk. Belgium
The first part of a sodium seeded free jet expansion (from the orifice to about 20 nozzle diameters downstream) is illuminated with light from a single mode CW dye laser and the spatial distribution of the resulting fluorescence is measured. Results for different carrier gases (Ar, Ne, He, N2) and different stagnation pressures (1-20 Torr.mm) have been obtained. At the highest stagnation pressures the fluorescence intensity distributions are in good agreement with the prediction from MOC calculations, a qualitative analysis of the deviations at lower stagnation pressures is given.
1. INTRODUCTION
In previous work we studied the first part of a loy ~r3ssure Na/Na2 expansion by means of laser induced fluorescence ' ' . The results mainly concerned the evolution of the internal state distribution of the Na2 molecules in the course of the expansion. The interpretation, however, of the results in terms of the relaxation of internal energy requires a thorough knowledge of the gasdynamics of the f~e~ ~et. Existing calculations, using MOC and "freezing" models ' ' , provide ample information on the near continuum and downstream situations. Our experiments, however, concentrate on the upstream region close to the nozzle exit and on relatively low stagnation densities, where the situation is not so well known. Therefore we decided to conduct a series of experiments using the same fluorescence technique to study the gasdynamic structure of the free jet. This can be done conveniently by
951
"seeding" sodium atoms in very low concentrations to e.g. a noble gas expansion and exiting these using a single frequency CW dye laser.
2. METHOD AND APPARATUS
Sodium metal, contained in an oven, is heated to a temperature of_~out SOOK, corresponding to a vapour pressure of the order 10 Torr. A flow of carrier gas passes over the sodium surface and mixes with the vapour. Then it is heated to the stagnation temperature (T =633K) and expands through a small converging n~zzle (throat0 diameter D=S~) into a vacuum better than 10- Torr (see fig. 1). ·An expanded light beam from a single mode CW dye laser (CR 490) illuminates the first part of the free jet (0-20D downstream from the n~zzle). The ~aser frequency is stabilized to the frequency of the s 112 , F=2 + P312 , F=3 transition of th.e Na atom. (This hyperfine compbnent is chosen to avoid optical pumping). The density of the Na atoms in the expansion is kept so low, that trapping of the resonance radiation can be neglected, while on the other hand the nurober of transitions is high enough to provide a fluorescence intensity that can easily be measured. The excited atoms have a lifetime of 16ns, which corresponds to a transition linewidth of 10MH7. This is larger than the laser linewidth but is very small ceropared to the Doppler shift caused by typical thermal velocities: Atoms which are moving wi!y a velocity component along the laserbeam larger than about 6ms "see" a Doppler-shifted frequency that falls outside the transition linewidth and are not excited. This makes that for the experimental geometry of fig. 1 the fluorescence is confined to a thin plane originating from the nozzle and oriented perpendicular to the direction of the lightbeam. In this plane the fluorescence intensitv at a certain ooint is only dependent on ZoaaZ gas properties since during the excited lifetime the particles will only travel distances of the order 10~m. The spatial distribution of the fluorescence is obtained as shown in fig. 1 (see also ref. 3): Outside the vacuum tank a cameralens projects an image of the fluorescing plane which is analyzed with a movable pinhole coupled to a photon counting system. A formalism relating the observed fluorescence intensities to the gas properties along the line of sight (densities, temperatures and 7 velocities) has been presented in ref. 1. Recently we have shown that this relation can very well be approximated by simple expressions that only contain the temperature(s) and density of the excitable particles in the central plane of the expansion. In the present experiment the excitable particles (Na) are seeded in different (e.g. noble gas) expansions. At high stagnation densities, however, the slip between seed and carrier is minimal, consequently the concentration of the Na-atoms is constant, and their velocity and temperature are as determined by the carrier expansion. In the downstream part of the expansion these quantities are known from
952
PIEZO EL .
. SHUTTER
BEAM EXPANDER
BEAM EXPANDER
DYE- LASER AR+-LASER
CARRIER GAS
PUMP
-lJ.
Fig. 1. Schematic diagram of the experimental set-up. (vertical and horizontal section)
PIN DIODE
953
MOC calculations, e.g. the calculated angular5density distribution in the point-source region may be represented by:
n(r,cj>) E 1T cj> n (r ,o) . cos ('2 cj>PM) (1)
with: E 3 cj>PM 'IT/2 for y 5/3 E 4.42 cj>PM = 2.28 for y 7/5
Under these circumstances the fluorescence, I, measured at a position (y,z) ~n the fluorescing plane (origin at the nozzle exit) is proportional . to:
I(y,z) - n(O,y,z)/ /y2 + z 2
So the angular variation of the density may be determined from transverse measurements of I(y ,z) as a function of z.
0
3. RESULTS AND DISCUSSION
(2)
In fig. 2 transverse fluorescence intensity measurements are presented for a N2 expansion far downstream. The differences between our data ana the MOC representation eq. (1) are of the same order ~s the discrepancy between eq. (1) and the numerical MOC data points • In fig. 3 similar measurements are shown for Argon and
..
:J · .
Na -N 2
10 Torr.mm
- M.O.C.
o ~~~------------~------------~--~ -1 0 z/y
Fig. 2. Transvers.e intensi·ties. ,for Na in a N2 expansion at y0 = 7D. Drawn curve: MOC representation, eqs. (1) and (2).
954
* "' Na- Ne 1 2 Torr. mm
o Na- Ar 20 Torr. mm
cos4
0 ~--~------------~--------------L-~ -1 0 z/y
Fig. 3. Transverse intensities at y 0 =7D together with eq. (3).
, ...
I I
* Na- Ne
12Torr.mm
.. .... '·"
-
··-·· -~--~ -... . -... ...... -... . ............. I I I I
12 Torr.mm
· ...... . . . . . ....... . ............. · .. , ... · ·.... . !! ! !! ! ! ! !!! I 'e
.... ~. ~· • S T~rr.'rr:m~;~~~~~~ 3 Torr.mm / 1.5 Torr.mm
Fig. 4. Axial fluo;rescence :i.ntensities. In. the lower part the horizontal lines indicate downstream averages.
955
• Na - Ar 20 Torr. mm
Na-Ne 12 Torr.mm 0.3
0.1
0 .7 ,--------------------,
0
. . ...
20 Torr.mm
. . . . .. · ... ~-·. . .. ... . : ... . .... . . .. · ... . . . .... ·. .. .. ·. "• . . 10 Torr.mm 2.5 Torr.mm 1 Torr.mm
y!D ...
. .. . . .....
10
Fig. 5. Analogaus to fig. 4 : Fluorescence intensities for Na seeded in Argon. The drawn curve in the upper part corresponds to the experimental data of fig. 4.
Neon expansions. A clearly narrewer distribution is found, in accordance with eq. (1) which results for y = 5/3 in:
I(y,z) = I(y,O).cos4 (Arctg(z/y)) (3)
Intensity measurements along the axis of the expansion are presented in fig. 4 for Neon and in Fig. 5 for Argon. ~2om eq. (2) it is seen that in the point source region, whe~I n - r , the axi-al intensity I(y,O) should be proportional to y In the lower parts of figs. 4 and 5 it is demonstrated that the experimental data very well comply with eq. (2).
In the direct neighbourhood of the nozzle, in the region where the streamlines curve, experimental data are scarce in litterature. The present method, however, is capable to provide measurements with a very high spatial resolution, is very sensitive to the curving of the streamlines (due to its Doppler selectivity) and does
956
not disturb the flowfield. In fig. 6 some preliminary results are presented for an Argon expansion. For several axial distances the transverse fluorescence intensity is compared to the axial 9n7; points of equal fraction are mapped. Although the relation ' between the intensity and the local flow parameters (n, T, u) is rather complicated, we expect the lines of equal fraction of fig. 6 to give a good impression of the flowfielo. A quantitative analysis will be published in due course.
Oeviations from the continuum behaviour are expected at lower stagnation densities as a result of various mechanisms: A. The freezing of the parallel temperature Tll influences the excitation probability (via the Doppler selectfvity) and the flow velocity. As shown in ref. 7 this causes an increase of the fluorescence intensity I(y,z) with respect to the high pressure limit I~(y,z). In the downstream region this increase can be approximated by:
3 3TII - 7;.. I(y,z) = I~(y,z).(l- S ST) (4)
0
B. At low stagnation densj_ties the difference between the masses of the ieed and the carrier partic~es will give rise to separation effects • These depend on the rat~o of the two masses, e.g. in an
100"!. 90"!. 80"!. 70"!. 60"!. 50"!.40"!. I
30"!.
20"!.
10"!.
* Na- Ar 10 Torr.mm
.Fig. 6. .Fluorescence intensity map close to the nozzle exit for Na seeded in an Argonexpansi on (see text).
957
Na-He mixture the (heavy) Na will be concentrated on the axis, while in Na-Ar mixture these same Na particles will diffuse outwards from the axis. 4 c. At low Reynolds-numbers viscous effects cause a diminishing of the discharge coefficient (Total flux/Stagnation density) • This gives lower I(y,z) values downstream (I(y/D) ~ I(y/D ff)), but at the nozzle exit I(O,O) is (almost) unaffected. e D. At (very) low stagnation pressures the supersonic expansion changes into an effusive one. The angular density distribution than evolves from eq. (1) to n(r,<j>) = n(r,O) .cos <j> For the presen-ted data we de not expect this to eccur.
For our experiments the influence of these factors is different fer the axial and for the transverse fluorescence intensities. For the shape of the transverse curves c. is irrelevant. The freezing, A., always broadens the curves, while the mass-separation, B., depends on the ratie ef the masses. One expects fer Helium a streng and for Neon a very small focussing while for Argon some defocussing is expected. This explains the experimental results of figs. 7-9. In the case of Helium (fig. 7) the clear narrowing with respect te the centinuum curve is caused by streng focussing, B., which evercompensates the broadening effect of freezing, A •. In the case of Neon (fig. 8) a small broadening is seen, caused by the freezing, A., slightly cempensated by focussing, B •. In Argon (fig. 9) both A. and B. cause a broadening, so the net effect is !arger than that for Neon.
.. .. . ... . . .
.. Na-He
11 Torr.mm
- cos4
. . · . ·· . . ....
0 ~--~------------~--------------L-~ -1 0 z/y
Fig. 7. Transverse fluorescence intensity for Na seeded in Helium compared to the MOC representation eq. (3).
958
d ~
/,l //
~/ 1.5 Torr.mm /,'
3 Torr.mm /.' /,:' 6 Torr.mm /"'
-:;"' .. ""' 12 Torr.mm
0 -I 0 zly
Fig. 8. Transverse intensities for Na seeded in Neon for several stagnation pressures. The upper part shows the experimental data on shifted scales together with symmetric curves fitted to the data points. The lower part of the figure repeats these fitted curves.
959
Na- Ar
:J Torr.mm
2.5 Torr.mm
10 Torr.mm
20 Torr.mm
0 ~--~------------~--------------~~ - 1 0 z/y ...
Fig. 9. Transverse intensities for Na seeded in Argon (analogous to the lower part of fig. 8).
100%
* Na- Ar
10 Torr.mm 2.5 Torr.mm
80% 60% t.O%
20"/a
Fig. 10. Comparison of the intensities close to the nozzle exit for Na seeded in Argon at two stagnation pressures (cfr. fig. 6).
960
Evidently, one can also study the deveZopment of these transverse distributions by measuring close to the nozzle exit. In fig. 10 distributions for two stagnation densities are compared (cfr. fig. 6).
For the axial fluorescen9r the influence of freezing, A., will always increase the quantity :;T • y/D of figs. 4 and 5 while the viscous effects, c., always r~sult in a decrease. Again the effect of B. depends on the mass-ratio: focussing (as for He) causes an increase and defocussing (as for Ar) a decrease. As seen in the lower part of figs. 4 and 5, the net effect both for Ne and Ar is a relatively small decrease. On the basis of the current data it was not yet possible to quantitatively separate the different contributions A-D, however, for the analysis of our Na/Na2 measurements it is very satisfactory that the differences between tfie various systems remain small.
ACKNOWLEDGEMENTS
We thank J.P. Huysmans and E. De Langhe for expert technical assistence. Financial support from the Belgian science supporting agency F.K.F.O. is gratefully acknowledged. One of us (P.W.) wishes to thank I.W.O.N.L. for a scholarship.
HEFERENCES
1. F. Aerts and H. Hulsman, Population evolution of internal states of Na2 in a free jet expansion,in:"Rarefied Gas Dynamics",ed. R. Campargue ed., CEA, Paris (1979), p. 925.
2. F. Aerts and H. Hulsman, The internal state distribution of Sodium dimers in a free jet expansion, Chem. Phys. Letters 72: 237 (1980). --
3. F. Aerts, H. Hulsman and P. Willems, Laser induced fluorescence study of the first part of a Na/Na2 free jet expansion: Dimer formation and excitation-energy transfer, Chem. Phys. 68: 233 (1982). -- --
4. J.B. Anderson, Molecular beams from nozzle sources in:"Molecular Beams and Low Density Gasdynamics", P.P. Wegener ed~ Marcel Dekker, New York (1974) p. 1 and references cited therein.
5. A. Habets, Supersonic expansion of Argon into vacuum, Ph. D. Thesis, (Technical University Eindhoven, The Netherlands, 1977).
6. C.E. Klots, Rotational relaxation in sonic nozzle expansions, ~· Chem. Phys. 72:192 (1980).
7. P. Willems, H. Hulsman and F. Aerts, On the use of laser induced fluorescence to study free jet expansions, Chem. Phys., in press (1982).
961
XIV. JET-SURFACE INTERACTIONS
EXPERIMENTAL STUDY OF PLUME IMPINGEMENT AND HEATING EFFECT
ON ARIANE'S PAYLOAD
J. Allegre and M. Raffin
Societe d'Etudes et de Services pour Souffleries et Installations Aerothermodynamiques, Paris (France)
J-C. Lengrand
Laboratoire d'Aerothermique du C.N.R.S., Meudon (France)
ABSTRACT
The purpose of the present study is to determine experimentally the heat transfer on the second and third stages of Ariane launeher as well as on its integrated payload, during the separation of the second stage.
As a matter of fact, during the separation, the plumes issued from the second stage's deceleration thrusters impinge upon the adjacent wall of the vehicle and possibly upon the satellite itself, involving interaction effects as far as forces and heat transfers are concerned.
Comparisons are made in the present paper between experimental heat transfer values obtained from measurements performed in the rarefied gas SR3 wind tunnel and theoretical values predicted by simplifi~d semi-empirical models.
965
I . INTRODUCTION
The problern of rocket exhaust plume impingement on adjacent surfaces is of practical interest for various space applications such as launeher stage separation and satellite orbit control. The parasite force applied to the surface by the exhaust plume must be estimated and accounted for in the force balance of the spacecraft. Heat transfer to the surface must also be estimated in order to avoid configurations where the surface (e.g. solar arrays) may be damaged by excessive heating. Both pressure and heat transfer distributions on flat surfaces impinged upon by underexpanded jets have already been investigated by the authors 1 ' 2
The present investigation concerns the separation of the second stage of Ariane launcher. As that time of the flight, deceleration thrusters mounted on the lower part of the second stage are fired and the exhaust plumes impinge upon the wall of the second and third stages, and possibly upon the satellite itself and its surrounding equipment such as solar arrays and antennas.
In the following sections approximate calculated heat transfer values will be compared with direct measurements performed in wind tunnel on small-size models and thrusters. Due to the complexity of the wall-jet interactions, the simple theoretical model of Ref. 2 is not expected to give quantitative correct information. It was used essentially to deduce flight conditions information from wind tunnel simulation results.
2. TEST CONDITIONS
Experiments were performed inside of the vacuum chamber of the SR3 wind tunnel at Meudon. The nitrogen plume simulating the deceleration thruster plume, was generated by an underexpanded nozzle with a nominal exit Mach number of 2.5. Stagnation pressures ranged from 4 to I5 bars and stagnation temperatures could be as high as I260 K. External pressures (vacuum chamber pressures) were maintained as low as a few pascals during the whole thruster operation.
The relative position of thruster, vehicle and satellite was identical to the Ariane L03 launehing configuration with Meteosat satellite as payload (fig. I). The scale of the model was l/I36 except for the nozzle size which was relatively larger. The angle a between the thruster axis and the space vehicle axis was II 0
•
966
20 . 63 m
0 , 35 m
"METEOSA'f" payload
impinged surfaces
Fig. l - Ariane L03 launehing configuration.
3. APPROXIMATE CALCULATION OF HEAT TRANSFER
3 .I Method
Approximate predictions for the convective heat transfer to a flat surface impinged upon by an underexpanded jet under continuum flow conditions have already been proposed. As described in Ref. 2, Maddox' formulation 3 can be used as a guide for finding nondimensional variables which greatly reduce the nurober of parameters involved in correlation of theoretical or experimental results.
The analysis described in Ref. 2 yields
q x2 y2 f (- i, y, 800)
qref h h
with
qref Po UL Re -o,s(r /h) 1' 5c (T -T) c c p 0 w
c-o.s
p0 is the stagnation density, UL, the limiting velocity based on stagnation conditions, Rec, the throat Reynolds nurober based on throat diameter, rc, the throat radius, cp, the specific heat at constant pressure, T0 , Tw, the stagnation and plate temperatures respectively, C, the Chapman-Rubesin constant.
From the above analysis, the nondimensional heat flux q/qref depends on geometric variables such as the local surface coordinates X2, Y2, and the surface angle i with respect to the jet axis : q/q ef depends also on the specific heat ratio y and on the cone half angle in which the gas should be ejected at the nozzle exit, assuming a naught value of the external pressure and uniform exit conditions (8J.
967
A similitude which should respect the geometry (X2/h, Y2/h, i) and the values of y and 800 should entail, at the corresponding points, to identical values of q/qr~f· The influence of all others parameters is included in qref express1on.
It is worth pointing out that the ratio of the nozzle size_rc to the flow field characteristic length h may be different for s1mulation and flight conditions because this ratio is included in qref and has no influence on f.
3.2 Calculated Heat Transfer on Ariane Second and Third Stages and on Meteosat Payload
The characteristi~of the jet issued from a real deceleration thruster are the following :
- stagnati~n.pressure and temperature : p0 = 123.5 bars, - throat rad1us and throat Reynods nurober : rc= 20.55mm, - nozzle angle of divergence : a = 6° - exit Mach nurober (deduced from nozzle
geometry and boundary layer calculation) - specific heat ratio and Prandtl nurober - mass flow and thrust : m = 10.58 kg/s, F
: Me = 3.62 y = 1.21, Pr
28.78 kN.
The three impinged surfaces considered are :
T0 = 3150K Rec= 4.6 106
0.82
the wall of the second and third stages : h = 0.35m, i=-11°, O<X2/h<46 the side wall of Meteosat : h = 0.57m, i=-11°, 36.2<X2/h<38.6 the base of Meteosat : h = 20.63m, i= 79°,-ü.l<X2/h<O,
I
Heat transfer values are summarized on the upper part of Table I. The calculations were first performed by assuroing plane and infinite walls, then corrections were made in order to take into account both the finite dimension of the base of the satellite and the bluntness effect.
A multiplying factor of 6 for bluntness effect was deduced from fig. 6 of Ref. 2 and applied to the value of the heat flux on the sidewall of Meteosat. Concerning the correction for the finite dimension of the satellite's base it can be shown that a multiplying factor of (I+h/R)0.5 should apply if the satellite was a sphere of radius R located at distance h on the jet axis. For the actual satellite, an equivalent sphere radius was estimated and the above expressions yielded a multiplying factor of 2.6.
The values of Reh= Rec rc/h characterizing the rarefaction degree are very high and correspond to a continuuro flow regime in the three cases. Consequently, the function f = q/q f was calculated in conti-
• d" 'f re nuuro reg1me accor 1ngly to Maddox ormula.
968
Table I - Calculated heat flux for flight and simulation conditions.
wall of the 2nd side wall of base of and 3rd stages Meteosat Meteosat
Reh 2. 7 105 1.7 Io5 4.6 103 Q)
.-4 CJ
qref (kW/m2) ·~ 236I II36 5.2 ..c:: Q)
> .-4 f f<8.66 10-2 1.3 10-4<f<I.49 10-4 0.53<f<0.57 CIS Q)
'"' q (kW/m2) max 204 O.I7(x6)=I.02 3(x2.6)=7 .8
Reh 2.4 104 I .5 104 404
='"" o:><: qref (kW/m2) 460I 22I4 10.I6 .~ 0 .j.JQ cu-
.-4 ..... ::l II f f<O.II8 I.4 10-:-4<f<I.6 10-4 0.82<f<0.86 a o .~ E-1
Cll '-'
q (kW/m2) max 543 0.35(x6)=2.I 8.7(x2.3)=20.0I
qreal/qmodel 0.38 0.48 0.34
3.3 Calculated Heat Transfer on the Reduced Model (scale I/I36)
A similar treatment was applied to the simulation model as used for the experiments under the following conditions:
- stagnation pressure and temperature : pQ = I5 bars, T0 - throat radius and exit radius : r = 0,8mm, re = I.3mm
throat Reynolds number : Rec = 7.5 Io4 - nozzle angle of divergence : ae = I0° - exit Mach number (deduced from nozzle geometry
and boundary layer calculation) : M = 2.47 - specific heat ratio and Prandtl numGer: y = I.4, Pr - mass flow and thrust : m = 3.61 g/s, F = 4.48 N.
IIOO K
0.72
Considering the same surfaces as previously defined, calculated heat fluxes on the model are given on the second part of Table I.
969
970
0
qref.= 2361 kW/m2 (flight conditions)
qref.= 4601 kW/m2 (simulation conditions)
\~ -<measured (simulation conditions) ' " T = 1100 K \ 0
\ . ' '· ',, ~.
', ....._. ' ..... ' ' ' ' /',
flight conditions
10 20 x2/h
calculated
si~ulation conditions
30
Fig. 2 - Heat flux distribution along the launeher wall.
148 mm --------·-------------------1
Fig. 3 - Wall heat flux distribution on the model (p0 = 15 bars ; T0 = 1260 K).
3.4 Seale Effeet on Heat Transfer Values
The values of y and 800 for the model differ from the eorresponding values of the full seale vehiele. But the eombination of Me, ae and Y retained for experiments on the model leads to a funetion f = q/q_ref very e1ose to the one eorresponding to the full seale spaee vehiele (fig. 2). In order to transpose the experimental data on the model to the real flight eonditions, the experimental heat flux values were multiplied by :
qreal
~odel
qref real f real .-f--
qref model model
where the seeond faetor is elose to unity and eorreets for non per-feet simulation ; values of q 1;a are given in the lower part of Table 1. rea 'lllodel
It should be notieed that the large uneertainty in estimating the eorreetion for bluntness and finite size of the satellite has negligible influenee on q 1/a •
rea 'lllodel
4. MEASURED HEAT TRANSFER ON THE SECOND AND THIRD STAGES AND ON THE SATELLITE
4.1 Experimental Set-Up
The launeher and satellite model were mounted inside the vaeuumehamber of the SR3 wind tunnel and tested under the previously deseribed eonditions. Heat transfer measurements were performed by means of thermo-sensitive paint deposited on an insulating eoating whieh eovered the model surfaee. Preliminary ealibration of the paint was earried out using thermoeouples imbedded in a thin-wall plate. The rnodel, initially shielded from plume heating was introduced into the plume within a fraetion of a seeond. A movie eamera reeorded the displaeement of the l~nes of equal temperature at the surfaee of the model. Proeessing the reeord yielded quantitative information on heat flux. Forthoseexperiments p0 was equal to 15 bars and T0 to 1100 K and 1260 K.
4.2 Normal eonfiguration
Experimental results are given in figs 2 and 3. For both values of T0 the maximum heat transfer on the launeher wall is higher than 200 kW/m2 and obtained at 5-10 mm downstream of the thruster exit plane. For this eonfiguration, the heat flux on the satellite itself is lower than 8 kW/m2, whieh is the lower limit for the heat flux to be deteeted by thermo~sensitive paint.
971
972
I I I _______ l - I 1-
~--------~,' .~·-~~-----l--+i : I I' 1
-- 1 I \ -
I 1 1\ 1 I I I I I I
-l I I \ - -~-- 1 -- ~-:_-_-~-:_ ~--I-+
I - f - - f I --~ -1 ---- 1 - ---~ - -' - r I I I I . H
10
--- "" "" ~ \ I 1: ~- ~ ~ .__ __ 1~-~~llll~ll -~~~--'-·-Fig. 4 - Wall heat flux distribution on the payload model
(p = 15 bars ; T = 1100 K). 0 0
Fig. 5 - Longitudinal flow visualization over the model (p0 = 4 bars ; T0 = 1100 K; Poo = 2.67 pascals).
4.3 Modified Configuration With Direct Impirtgement of the Plume Upon the Satellite (fig. 4)
This experimental eonfiguration is more eon.sistent with the assumptions used to ealeulate the heat flux on the satellite. As a matter of faet, it was supposed that the satellite was direetly impinged upon by the plume issued from the thruster, without any external disturbanee due for example to the presenee of the launeher wall. Consequently for this partieular experiment, the thruster and the satellite bad the same relative position but the eylindrieal envelope of the seeond and third stages was removed from the plume flowfield.
The measured heat flux distributions are presented in fig. 4 for the base and the side wall of the satellite. These maximum values are presented in Table 2 together with the eorresponding values transposed to flight eonditions and the ealeulated values.
4.4 Diseussion
As it ean be seen in fig. 2, the ealeulation overestimates the heat flux on the launeher wall in the peak region and underestimates it far downstream. The error is less than a faetor of 3 and ean be attributed mainly to wall eurvature.
In normal eonfiguration, the satellite is shielded by the launeher wall and suffers negligible heat transfer. This is eonfirmed by an in-flight value of 1-2 kW/m2 measured during the 3rd Ariane's flight on a sensor loeated near the satellite base.
Table 2 - Experimental and ealeulated heat flux on the satellite. '" , .... - -- -··-
Normal Modified Caleulation eonfiguration eonfiguration
Satellite Satellite Satellite
side wall base side wall base side wall base
~ exp. X < 8 (kW/m2) < 8 25 36 ~ 2 ~ 20
~ax exp. transposed
< 4 < 3 12 12 I to flight ~ ~ 8 eonditions (kW/m2)
973
In a modified configuration which is in better agreement with the calculation assumptions, discrepancies between experimental and theoretical values of heat flux on the satellite side wall and on the satellite base are observed. They.can be attributed to uncertainties in bluntness and finite-size corrections, respectively.
The ratio qreal/qmeasured is obtained by applying the same theoretical treatment both to Simulation and flight conditions and comparing the corresponding results. Thus it is much less sensitive to imperfections of the calculation than the heat flux itself. In particular, the large uncertainty concerning the corrections for finitesize and bluntness have negligible or zero influence on this ratio. Thus the above theoretical treatment is a satisfactory means of transposing experimental simulation results to flight conditions. Furthermore it predicts the correct order of magnitude of the heat flux when its basic assumptions are satisfied.
For illustration purpose, a flowfield visualization obtained using the electron beam fluorescence technique under test conditions is represented in fig. 5.
5. CONCLUSION
Quantitative information relative to heat flux distribution on the Ariane launcher's wall and on its satellite payload has been obtained experimentally on a small-size model in the vacuum chamber of the SR3 wind-tunnel. The configuration corresponded to the time of the second stage separation. The satellite appeared to be shielded by the launeher wall. The heat flux values obtained have been transposed to flight conditions by applying to both simulation and flight conditions the same approximate theoretical treatment. With some restrictions related to bluntness and finite-size effect, this treatment gives the correct order of magnitude of heat flux values.
ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of the
"Centre National d 'Etudes Spatiales".
REFERENCES
2
3
974
J-C. Lengrand, J. Allegre and M. Raffin, ~nt~racti?n of Unde.rex:- 11
panded Jets with Adjacent Flat Plates. ~ Raref1ed Gas Dynam1cs , Progress in Astronautics and Aeronautics; ~. part 1, PP· 447-458, J.L. Potter ed., AIAA, N.Y. (1977)
J-C. Lengrand, J. Allegre and M. Raffin, Heat Transfer to a Surfa~e Impinged Upon by a Simulated Underexpanded Rocket.Exhaust Plume , in "Rarefied Gas Dynamics", Progress in Astronaut1cs and Aeronautics, 74, part 2, pp. 980-993, S.S. Fisher ed., AIAA~ N.Y. 8(1981)
A.R. Maddox, Impingement of Underexpanded Plumes on ~dJacent urfaces, J. Spacecraft and Rockets, ~. pp. 718-724 (June 1968)
THE INTERACTION OF A JET EXHAUSTING FROM A BODY VIITH A
SUPERSONIC FREE FLOW OF A RAltBFIED GAS
I.N. Larina
Computing Center of the USSR Academy of Seiences IJoscow 117967 USSR
DTTRODUCTION
Supersonic flow past a sphere with a jet exhausting from a nozzle in the direction of the axis of symmetry opposite the free-stream flow is studied on the basis of the model kinetic equation. At Knudsen numbers less than unity screen effect of free-stream impulse is found.
EQUATION AND BOUNDARY CONDITION
We shall consider the problern of a sphere of radius R and surface temperature Tw in a supersonic flow of a monatomic rarefied gas. Near the stagnation point of the sphere there is a nozzle of radius rs from which in the opposite flow direction a jet is exhausting. Gas of the jet is of the same sort as in the free-stream flow.
The state o~ gas is assumed to govern by a model kinetic equation
~i ~ =V [F- f] axi
c2
v = _4 __ 1 _ _E___, F = ___!L_3/2 e-T [l- q. a.(T)] Sy'TI Kn ~ (T) (rrT) t I
975
4 c· 5 2 a. =- - 1- (-- L) Kn = "A 112R
1 15 nT 2 2 T ' oo
n = Jfd~, nu. = f~.fdt C 1. = ~ 1.- U1. . 1 . 1
tT= ~ !4- fd~, qi=Jci 2_2 fd~, p=nT
Here n, ui, T, qi are the density, the macroscopic velocity components, the temperature and the energy flux components respectively. When reducing the equations to a nondimensional form the free-stream reference quantities were used, namely,_the density noo , the temperature Too and molecular veloc~ty V0 •
The viscosity variation with temperature corresponding to a Lennard-Jones intermolecular potential is taken in the form2
1-l (T) = T213 q;(B) q;(t) = 0,767 +0.233t-116 exp [-1.17(t- 1)] q;(BT)'
B = Too /T* , T* is characteristic temperature.
The boundary conditions are formulated as follows • .At infini ty
-3/2 - -2 foo = rr exp [- (~ - S) ]
S is the reduced upstream veloci ty (S = Uoo/ V0 , V0 = - -·····--T-- v2kTOO Im).
For molecules emitted by surface
lT 1/2 n = -2 ( - ) r f s: d~= w T ~n ~
w (~ ii)<o
976
For molecules exhausting from a nozzle
Here S8 , n8 , T8 are the average velocity, the density and the temperature of gas particles exhausting from the nozzle.
lJUMERICAL SCIIEI.IE
For solution to the kinetic equation the technique of integral iterations with the use of the numerical scheme described in2 is applied.
The model equation is integrated along a charac• teristic line
f ( x , ~. ) = f0 ( x . - b . y, ~ . ) A + B 1 1 1 1 1
~ - --A = exp [- .l_ r V (y) dy ], ~ == ve + e + e
~ o I 2 3
Ys Y B == 1_ f v (y) F(y) exp [-l f v (y')dy']dy
~ 0 ~ 0
bi are components of unite velocity vector, Ys is a disto.nce to a boundary, f 0 is the boundary value of the distribution function.
For every coordinate point v1e have to calculute the follorling integrals
(n) - (n) - (n) (n) 1 i = r r <p i d~ = r r <p . d~ + r <p. r d~ + r <p . r d~
M 1 M 1 M 1 M"., w s
2 2 <p. == n, ~ , ~ , ~ ~ 1
1 J J
The domains LI"" , I.Iw and 1.18 are formed by the tra-
977
jectories of partielas coming to coordinate point (r,a) from the infinity, the sphere surface and the nozzle hole respectively.
Instead of integrating over the rather complex domain Mw it is possible to integrate over the domain M0 = MW + M8 with the use of a fictitious boundary function
f , , ( T )-3/2 2/ _, w= nw rr w exp [-~ Tw], ~ C Ms
where ~ is defined from no net mass flow condition. Then
I cp. fd~ = I (CD A + B)cp. d~ +I (fs- f' )Acp. d~ 1 w 1 w 1
M2 MO MS
~ c M w , CD w = f~
The characteristic feature of the applied numerical method2 is that the mesh of the velocity space is not fixed in the coordinate space, but it is chosed individually for the every coordinate point. This makes it possible to take into account the distribution function discontinuities and then to calculate the triple integral with a small number of mesh points due to choosing the weight functions based on the analysis of the specific features of the distribution function behaviour.
COMPUTATION PARAI;;iETEHS
Computations were carried out for flows with the following parameters: relative nozzle radius r 8 = = sin 0.125, S = 14, 71, S8 = 1.8, n8 = 10, 10, 20, 301, B = 0.5, Kn = 110, 1, Oo41.
The free-stream conditions were set at the distance 3R from the sphere surface; for coordinate step size the values 1tr = 0.1 and t:w = 0.0625 were chosen. Computation errors were of 3-5 % for macroparameters and 2-3 %· for the sphere drag.
COMPUTATION HESULTS
The computation results give the possibility to
978
evaluate the effect of the jet intensity and flow rarefication on the field of macroparameters, sphere drag and energy flux to the sphere.
To illustrate a flow pattern when a nozzle jet interacts with a free-stream flow vector velocity field is shown in Fig.1 and Fig.2 shows isochores in the case S = 7, ns = 30, Kn = 0.4, Tw = T8 = 4.5, S8 = 1.8.
Present isochore behaviour is qualitatively agree with the results ofJ, where the same problern has been solved for the case of continuum media.
Fig.3 illustrates profiles of the temperature T, axial velocity component u and the density n at the axis of symmetry at front of the body. As it is seen largeintensity jets (n8 = 30) exhausting into incoming flow first accelerate over a certain distance and only then begin to decelerate.
s
----- ------~~---- ----~---~ "" -"'5-~=-----0 '\ ,,---_ .... -__
'\ '~ ~ \,-----___ _ I 1//1 ~ ...._----.:..... ------ .."".~ ------'------~-- -- -. - --
1 2 r
Fig. 1. Velocity vector field for S = 7, n8 = 30, Kn = = 0.4, Tw = T8 = 4.5, S8 = 1.8.
979
1 r
Fig. 2. Isochores for flow with S = 7, ns = 30, Kn = = 0.4, Tw = Ts = 4.5.
In Fig.4 the drag sphere Cn as a function of the gas density of a jet ns is presented for S = 7 and various Knudsen numbers. As it is seen from Figo4 for flows near free-molecular ones the reaction jet force is greater than the screening action of it on the body, however for small Knudsen numbers the screening effect on the free-stream flow essentially decreases the sphere drag.
The exhausting jet provides certain thermal protection of the body surface. Figure 5 shows the distribution of the energy flux E to the sphere surface for S = 7, Kn = 0.4, n8 = 10, 10, 20, 30! (Figs. 1-4 respectively).
The results of calculations are iu a qualitative agreement with the experimental data4,J obtained for large Reynolds numbers.
980
-5 2 3 1 r
Fig. 3. Distribution of macroparameters along center streamline ahead a sphere: S = 7, Tw.= T8 = = 4.5, Kn = 0.4, n8 = 110, 15, 301 (1-~gures 1-3 respectively). -- - temperature, velocity, - · - · - density.
cn 2 .• 4
2.2
2.0
1. 8
2_ -
- - .._3 2 .........
1 • 6 ·'------'------'------' 0 10 20
Fig. 4. The drag sphere Cn as a function of the jet gas densi ty n8 for S = 7 - ·--- , S = 4 - - - •
981
200
100
0 0.5 0.75
Fig. 5. Distribution of the energy flux E to the sphere surface for S = 4, Kn = 0.4, n8 = 10, 10, 20, 301 (figures 1-4 respeotively).
REFERENCES
1. E. M. Shakhov, On generalization of the Krook kinetio relaxation equation, Fluid ~namios (Izvestija Akademii Nauk SSSR, Mekhanl:a zhidkosti i gaza), 5: 142 (1968) -In Russian.
2. I. N. Larina, V. A •. Rykov, and o. s. Ryzhov, Numerioal analysis of diatornie gas flows in the framework of model equations, in: "Rarefied Gas Dynamios", R. Campargue, ed7; Commissariat a l'Energy Atomique, Paris (1979).
3o o. M. Belotserkovskii and Yu. M. Davydov, Numerioal experiments for supersonio and hypersonio flows, Acta Astronautioa, 1:1467 (1974).
4. P. ~inley, The flow of a jet from a body opposing a supersonio free stream, !!.• Fluid~·, 26:337 (1966).
5. v. A. Suohnev, Investigation of the flow past a sphere with a jet exhausting opposite supersonio rarefied free stream, Fluid Dynamios (Izvestija Akademii Nauk SSSR, Mekhanika zhidkosti i gaza), 6:166 (1968) - In Russian.
982
MODELLING CONTROL THRUSTER PLUME FLOW AND IMPINGEMENT
ABSTRACT
H. Legge and R.-D. Boettcher
DFVLR Institute for Experimental Fluid Mechanics Göttingen, Federal Republik of Germany
To accurately predict the effects of control thruster plume impingement on spacecraft structures, an existing analytical plume flow model was extended to deliver allrelevant flow quantities like Mach number, mean free path etc., and to include free molecular plume flow by defining a freezing surface. The impingement is treated by simple local models for continuum, transition and free molecular flow-surface interaction. A comparison of computed impingement torques to inflight data from the Orbital Test Satellite shows reasonable agreement, supporting the basic results of the investigation, but stressing the importance of knowing the gas dynamical quantities of the nozzle expansion flow as exactly as possible, especially the adiabatic exponent.
INTRODUCTION
Small rocket engines are used on satellites to perform attitude control and station keeping maneuvers. Due to the impingement of the thruster free jets on the satellite structures, severe problems arise for the exact performance of these maneuvers. Many theoretical and experimental investigations have been devoted already to the basic phenomena of the nozzle flow, plume expansion flow, and impingement. The purpose of the present work is, to extract from previous work simple models, which can be combined in a computer program to calculate the forces and moments (and heat transfer, which is not described here) for mostly encountered satellite geometries.
This work has been done in the context of an ESTEC contract (No 4154/79/ NL/AK). Several reports are summarized in1 •
NOZZLE FLOW AND PLUME MODEL
The whole flow system can be divided into nozzle flow, plume flow and impingement. The plume flow and the impingement can be subdivided (rather independently from each other) into the continuum and rarefied regimes.
983
From a literature survey the widely used Simons model2 was adopted with some additional changes and extensions. The density distribution in the plume flow field is described by
_g_ f(8) K-1 cos
f(8) f(8=80 )
1T8
p(r,e) = p(r,8=0)·f(8)
p(r,8=0) r* 2 .<:....>..::C...Z..:...*:;-"-'- = ~'r-l
p
e ~ e 0
28lim isentropic nozzle core
-c (0-8 ) p 0 e < e ~ 81im e 0
nozzle boundary layer
( 1 )
(2)
(3) expansion
(4) expansion
with r,e polar coordinates originating from the center of the nozzle exit, r* nozzle throat radius, index E nozzle exit condition, index o stagnation condition, K ratio of specific heats,
984
8 0
turning angle (for ~)
Ma = Mach number, v = Prandtl-Meyer angle
81im
u*/(2u1 . ) l.m
J r(el sin e d8 0
u* = nozzle throat velocity
K-1
8 • [ 1 _ ~( 215E)K+1 J ll.m 1T rE
K-1 1/2 -
A ( K+1) 2Ulim (~)K+ 1 p K-1 ulim 2oE
1/2 (~RT) K-1 0
plume constant
streamline corresponding to edge of nozzle boundary layer
limiting velocity
mean boundary layer limiting velocity
nozzle boundary layer thickness
( 5)
(6)
(7)
(8)
(9)
(10)
( 11 )
nozzle exit Reynolds number. ( 12)
lE = nozzle length, ~ viscosity.
2 We found that the density decrease (p ~ 1/r ) in the vicinity of the nozzle is better described, if the distance r from the nozzle exit center is replaced by r+r (0 is still counted in the original polar coordinate system). Fig. 1 giv~s an example comparison with results of Vick and Andrews 3 •
Ma
Fig. 1
20
15
10
5
gas a1r p0 = 165.5 bar T0 = 305 K
r ' =1588mm rE:71.1.mm e E =l5°
MOC : Vkk & o experiment l " Mae.up = 4·79 Andrews (1966) o Moe = 5 - - Simons (1972)
-- present
3L-~~--------~------~~------~~------~. 0.1 10 100 1000
Axial Mach nurober distribution. Comparison of plume models and experiments and MOC (method of characteristics) results.
Very often the plume impinges at small angles of attack, so that hypersonie approximations, especially in free molecular flow, fail. Therefore not only the density and velocity have to be known, but additional flow quantities must be specified. Knowing the plume density, the stagnation conditions, and the ratio of specific heats, the Mach nurober etc. can easily be determined by applying the one-dimensional isentropic flow equations.
For non-isentropic nozzle flow the concept of effective stagnation conditions is pursued.
This concept is outlined for the nozzle boundary layer in the following. The expansion of the boundary layer is assumed to be isentropic outside of the nozzle. For a certain plume stream tube characterized by the polar angle 0 , the effective stagnation conditions are determined from the static flow properties in the nozzle exit boundary layer at the radial distance r corresponding to 0 in the plume.
The following set of equations is used, where index CL denotes the plume center line:
985
~· (0) -c (0-0 ) l.m u 0 0 < 0 ~ 0 e
~im,CL 0
T ff(0) 2 -2c (0-0 )
o,e = clim(0)) u 0 e T ulim,CL 0
Po,eff = -c (0-0 ) po o e
Po
The constants c and c u po are determined by the equations
01im
~im = ~im f Po,eff( 01im)
Po
-c (0-0 ) e u 0 d0
1 )
( 13)
( 14)
(15)
( 16)
( 17)
(18)
together with Eqs.(10),(15). The integration limit 0lim in Eq.(17) and the exponential dec~ laws are based on experimental and theoretical results, but additional experiments would be helpful for an accurate determination.
The angular momentum flux distribution is shown in Fig. 2. The importance of the ratio of specific heats K is demonstrated in this figure. The Prandtl M:ye~ angle, and ther:by the turning angle 0+im of th: flow.at ~he nozzle ll.p l.S a streng functJ.on of K • An obstacle J.n a certaJ.n regl.on l.n the plume for example at 0 = 40° will experience several orders of magnitude different forces whether the flow expands at the nozzle lip with K = 1.67 or 1.4 or 1.26 • Experiments have usually been performed only with well defined gases as ni trogen 4 •
For real thrusters using hydrazine for example, the effective ratio of specific heats at the nozzle exit is most important. Condensation and relaxation effects can influence this value and the flow properties.
FREE MOLECULAR PLUME MODEL
To model the transition and free molecular flow regimes of the plume, the sudden freeze model was applied. To define the freezing surface the freezing parameter of Bird5 was used,
p .)! • .l 1~1 v p ds ( 19)
which gives for the far plume flow field
p = 2u (20) vr
986
1 P u2
- -2-p UcL
Fig. 2
10.-------------------------------------,
10"'
10"2
10"3
10"'
10' 0
·~-=---·~ , ................ .. ''· ···.
\ ' ....... . ... \ '· ........ \ ' ··.,
\ ., ' ...... \ ., ...... , \ ., ·· .......
\ . ···. ' ·· ..
Ar
Hydrazine
CF,
\ . ' \ ' · ······ ...
\ ' · ........ \ ' ············ ....
'··
e Angular momentum characterized by Ar: K 1. 67 , CF 4 : K = 1 , 17 •
flux distribut i on for different gases the ratio of specific heats K ;
N2 : K = 1 • 41 , Hydrazine: K = 1 • 26,
where v is the collision frequency and s the distance along a streamline. The flow is assumed to be i n the continuum regime for P < 2 and in the free molecular f or P > 2 , t hus defining a free zing surface and freezing distance on a streaml ine rf • An example calculat i on for an ERNO 0 . 5 N t hruster is given in Fi g . 3 . In addi tion to the line P = 2 , the lines of constant Mach nurober and mean free path are shown. The expansion is calculated for K = 1 . 4 . The mean free path A i s cal cul ated by the equation
(w-l)
~0 = ~0 • ( ~J 2 (21)
where w = 0.75 lS taken i n Fig . 3 .
In the free molecular flow regime an ellipsoidal distribution function is assumed .
I MPINGEMENT
The criter ion whether the impingement takes place in the cont inuum , transition , or free molecular regime , is given by a Knudsen nurober Kn = A/1 . The reference length l is chosen to be the width of the impinged struct ure , for example the width ~ of a plate , when it is smaller than the continuurn
987
100
1 50
Fig. 3
~try r":0.3mm1 re=2.38nm1 1\=15•
OPfl'Oiing data: Po= 11.4 bar, T0 = 1344 K ; ll = 1.4 1 M = 14.5 kg~-mot
---------------.........
- A(cm) --- Ma
x/re -----
Lines of constant mean free path A , Mach nurober Ma , and freezing surface P = 2 , for ERNO 0.5 N hydrazine thruster.
plume flow field characterized by rf ; when the continuum plume is smaller, the distance r of the point in question from the nozzle exit is taken as reference length:
1 = d p
if
Impingement ~n the continuum regiroe
1 r for r < d f p
For the impingeroent only locally applicable laws are used. In the continuum regime the Newtonian pressure law was adopted:
c p 1 2
2 Pu
(22 )
where radial
y is the angle between surface normal and flow streamline which is the line from the nozzle center to the point in question.
Free molecular impingement
The free molecular impingement of the continuum plume is modelled by the well known free molecular formulas given for example by Schaaf and Chambre 6 •
For the frozen plume flow t he forces (and the heat transfer) were evaluated for an ellipsoidal distribution function 7 •
988
Using the well known accommodation coefficients
a n
, (23) ; T . - T ~ r
T. ~
(24)
the forces can easily be calculated for a surface element 6 ' 7 •
An extended version of this paper containing the complete formulas for ellipsoidal free molecular flow and impingement is available on request.
Impingement in the transition regime
The transition regime is covered by bridging relations for the reduced pressure coefficient
C I
p
c (Kn ) p p
- c p,c
- c p,c
and the reduced shear stress coefficient
C I T
c (Kn ) T T
- c T c
- c T,c
f(Kn ) p
c (Kn ) T T
cT,FM = f(KnT) ,
(25)
(26)
where the indices FM and c denote free molecular and continuum values. The bridging function f is chosen for c 1 and c 1 to be
p T 1
f(Kn) = 2 ( 1 + log10 (Kn)) , 0.1 ::> Kn ~ 10 , (27)
where the Knudsen number, Kn = KnT , for the shear stress coefficient cT 1 is different from the Knudsen number, Kn K~, for the pressure coefficient cp 1 :
Kn = Kn° = >.0 16 1 ~(To) . .l (28)
p 1= 5 Poo f2n RT 1 l 0
KnT Kn° (29) 2 0.2 + 0.8 cos y
This splitting is based on experimental results and theoretical reasoning that the shear stress coefficient starts to rise earlier than the pressure coefficient when leaving the continuum regime, and reaches the free molecular value also earlier, especially at small angles of attack a. The resulting cp and c1 values are plotted in Fig. 4 for different angles of attack. The drag of s1mple bodies in the transition regime is modelled quite well by the above approximation 1 as shown by the example calculation for a sphere, see Fig. 5.
APPLICATIONS
The outlined simple plume and impingement models have been incorporated in a computer programme. The impinged geometry is modelled by basic shapes, which are approximated by plane surface elements. Shock formation in continuum flow and multiple molecule reflections on concave surfaces are neglected. Surfaces shaded from the plume by other surfaces are omitted.
A comparison between the modelling and experiments of Vick and Andrews 8
is shown in Fig. 6 for the pressure profile on a parallel flat plate at a reduced distance z/rE = 8. The agreement is reasonable,but the experimental results are always above the theoretical modelling, maybe because of the shock system and condensation effects. An adiabatic exponent slightly smaller than the nominal one, K = 1.35 instead of K = 1.4, assumed because of condensation, gives better agreement.
989
1 c' p
Ct
990
1.5
1.0 s~v go• 75. so· 45.
0.5 Jo• c' _ Cp -Cp,t 15. p-Cp,FM- Cp,t
Kn"
Fig. 4 Pressure and shear stress bridging between continuum (Newtonian) and free molecular flow.
6. Kinstow,
I nilragen 10.5 <Mo< 10.8 0.08<Tw/ T0 < 0.17
Pott er
0 Boitey o ir 8 <Mo< 12 Tw/ T0 " 0.1 oir 8.5 <Mo< 13 Tw/ T0 "1 O Legge,
Koppenwallner I
1.0
1 0.8
0.
c ' D 0.
0.2
0
Fig. 5
tiJ = co- co ••
cD,Fiol - cD,c
0.01 0.1 1.0 10
Kn'-----
Comparison between bridging relations and experimental results for sphere drag.
Fig. 6
!. ---rE
80
Comparison of experimental and computed pressure profiles of plume impingement on a parallel flat plate.
15 . .----------------------------------. o mcosurcd po1nts lrom
doys t90 . 218 ond 260
Nm · 103
10~·
-M,
6 'K. : 14 } s. vanablc o ll = I 24
0
190• 200" 210° 220° 230° 240° 250° 260° 21o·
~ -----Fig. 7 Torque. due to OTS solar paddle impingement as function of
the paddle rotation angle. Comparison between inflight data and calculations.
991
Fig. 7 compares inflight data of the moments exerted by plume impingement on the Orbital Test Satellite (OTS) solar arrays with the theoretical modelling. Evidently the moment MX depends strongly on the choice of the adiabatic exponent K • The assumption of frozen vibrational degrees of freedom for the gas molecules corresponding to K = 1.4 seems tobe justified. In addition, a pre-impingement on the main satellite body and the interaction of the two plumes of simultaneously fired thrusters above and below the near paddle has been modelled by a Mach number limitation, resulting in good agreement between in flight data and modelling.
CONCLUSIONS
Combining simple plume and impingement models for conditions between continuum and free molecular flow, a computer programme has been developed allowing the approximate analysis of impingement effects on a variety of spacecraft geometries. Main pending problems are the insufficient knowledge of the gas dynamic quantities for real thrusters and plume deformation by plume-plume interaction and near field impingement.
HEFERENCES
1. R.-D. Boettcher, G. Dettleff, G. Koppenwallner, H. Legge, "A Study of RocketExhaust Plumes and Impingement Effects on Spacecraft Surfaces". DFVLR IB 222-82 A 11, DFVLR Göttingen, 1982.
2. G.A. Simons, "Effect of Nozzle Boundary Layers on RocketExhaust Plumes". AIAA J., Techn. Notes, Vol. 10, No. 11 (1972), pp. 1534-1535.
3. A.R. Vick, E.H. Andrews, "An Investigation of Highly Underexpanded Exhaust Plumes Impinging Upon a Perpendicular Flat Surface". NASA Technical Note, NASA TN D-3269 (1966).
4. J.-C. Lengrand, J. Allegre, M. Raffin, "Interaction of Underexpanded Jets with Adjacent Flat Plates". Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 51, Part 1, edited by J.L. Potter, American Institute of Aeronautics and Astronautics, New York, 1977, pp. 447-458.
5. G.A. Bird, "Breakdown of Continuum Flow in Free Jetsand Rocket Plumes". Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 74, Part II, edited by S.S. Fisher, American Institute of Aeronautics and Astronautics, New York, 1981, pp. 681-694.
6. S.A. Schaaf, P.L. Chambre, "Flow of Rarefied Gases". Aero. Paperbacks No.8 (1961); High Speed Aerodynamics and Jet Propulsion, Vol. III, Sec. H, pp. 687-739, Princeton: Princeton University Press, 1958.
7. H. Legge, "Auftrieb, Widerstand und Wärmeübergangsgrößen in freier Molekülströmung mit elliptischer Verteilungsfunktion". DFVLR-AVA report 68 A 40, Göttingen, 1968.
8. A.R. Vick, E.H. Andrews, "An Experimental Investigation of Highly Underexpanded Free Jets Impinging Upon a Parallel Flat Plate". NASA Technical Note, NASA TN D-2336 (1964).
992
DPINGEMENT OF A SUPERSOBIC, UNDEREXPAlrnED RABEriED Jm
UPO!l A FLAT PLA.TE
Luk:yanov G.A. , Sokolov E. I. , and Shatalov I.v. Mechaaical Institute Leningrad, 19800.5, USSR
The impiDgement of a high density jet upon a flat is lalown 1 to produce a nuaber of regim.es the type ot which dependa on toe aggregate. of geometric and gaso~c parameters. The processes accomp~iDg the tree je~ rarefaction growth bave also been extensivelT studied.2 lhen the plate is brought iDto the jet, its iDtluence and that of rarefaction tells on the jet simultaneously, and the processes, described 1D 1 and2, are superimposed one upon another•
In thia report the resulta ot experimental study of the effects of jet rarefaction upon the flow patterD before an iDttnite flat plate normal to the jet a%is are presented• ~he ~et rarefaction variationa were obtaiDed directly by aeana ot total densit,r variations for ' nuaber of ttxed geometric and gaso~c parameters. The investigatios were conducted in a vacuum chamber and included visualization of the flow by means of smolder spark and highly sensitive tlow direction vanes and measurements ot the static pressure
993
along the plate. Besides, the local density in tbe entire flow field was measured by means of electron-beam probing. The electron X-r~ method o! density measurements based upon the characberistic radiation allowed to make m'asurements in the close Ticinity to the plate surface.
The &2Periment were conducted in the following range o! parameters: the ex:l. t Mach number /'14 = { -T 4. 5' , the expansion ratio n=f'4 /f'•=2+10if( f'a ,,P .. - exit and ambient pressure), the stagnation temperature 7; = 2901(, the nozzle tbroat diameter d*= 09t:57111lJ., the
ratio of specific heats 1.4 (air) and 1.67 (argon). The Reynolds number Re. , calculated with throat flow Parameters and temperature 7;, , was variated from 102
till1~ and ooszle-plate undimensional distance h -from 0 till 2. (All linear dimensions $re related to 1'14 d 4 v'jii. d 4 - nozzle exi t diam.eter).
The impingement of a supersonic jet on a nor.mal flat plate is cbaracterized by the appea~ance of a central shock wave before the plate (see fig.1). The intersection of the central shock and the oblique turning shock of the first jet cell at the point T results in the development of a ~eflected shock and surface of tangential discontinuity. The latter separates the subsonie flow region behind the central shock from tbe supersonic flow region behind the reflected shock.
Depending on the value of the parameter II , there exist several regimes of interaction of a dence jet on a flat plate.1 When 1z. <111 , the flow in the vicinity of the shock l~er ax:l.s between the central shock and the
994
flat plate is independent of the expansion ratio. In the range iz l::: fl <n., shock lqer ga.s flows radially from the centre along the plate, whereas in the range ~~< ~ <IS1 it appears a central circulation zone in the subsonic region of the shock layer. In Cl!lse 1'1> 1&3 a second central shock develops before the plate9 the shock wave structure of a first jet cell is undisturbed. J'or example, tor a Mt:L= ( jet n,= 0. (GI 11,-=- a 7G, hJ= 1.9{.
The growth. of rarefaction results 1D considerable changes in the structure an~ geametry ot shock waves in
a get, iapinging on a plate. The visualization of a wave structure made it possible to establish that the central shock decreases in diameter ~pproaching the plate with the increase of rarefaction. The variation ot a central shock position is also seen in fig.1, where the curves ot.the density distribution along the axis are presented. This results make it possible to get an evidence of the rarefaction intluence on the thick:ne.ss and structur~ of the shock wave before a boqy in an non. UDifo.t'lll. flow.
The criterion Re~ calculated by the essential parameters of the shock layer was introduced to describe the intluence of the rarefaction on the analyzed pro-cess: Re»= .l's "i dr//'11 ( ~s 1 ~ - density and velosity on the jet axis just dow.astream of the central shock1
dr - central shock diameter, viscosity ./'• = ./" ( 7;) ). In the case ot the infinite recession of the plate the criterion Ae», to within the constant coetficient, coincides with the simila;ity parameter of rarefied free jets ße~ , described in~3 In conjunction with the geometric parameter of similarity ot impinging dense jets
995
~ ~------~----------~----r---~r.-, .f'-
0,8
0,2
1 A'e1t JOO J -•-<fO J -n-fO
03 as- as (J7
Fig. t L_
X=084 x~ags :r~{(J7 ~ {
996
fig. E Ha.= I, lz = ( 33, 1-ße0 = 10~ tE- ~e,;- 4~ & - ~eZJ = 10
__ß.. R;- 200 !e~~- 40 !eD~ 10
1"'---.... b .. ~''''''' in"''' 411//,4~'''" 'N'''''''' I t t\\\\\' ,,, +t\\~~\\\\ '~''''''''' lt\\\\\\\\\'\' ~ ",,, ~ .. _,,,,,,,~''''''"'''' f\\\\ ,,,,,.. ,, IIJIJI/JI;JI/1''' ,,,,,,,,,,,
Fig.3. t1a = 1, f=l.6~ lz=f.?J3
n ~ complez ßeR &llOWS to generalize the data OD COD
cerniDg the evolution of the shock wave structure and to 4escribe the bou4ar1es of flow regimes, observe4 before the plate with the rarefaction growth for a w14e range of parametras 11~~. 1 n 1 lt. 1 lle.
~ig• 2 shows the results of the 4ensity measurements iD tbree cross sections of a shock layer between the central shock and the plate. ~hese data give an iDfor.mation on the process of the growth of the mizing layers thickDess along the ring t~gential 4iscont1Du1t,r origiDatiDg from the po1Dt T (fig.1 )-. ~ig. 3 illustrates the results of the flow Tisual1zat1on before the plate by means of flow direction vanes. lhen the total density 4ecreases, the circulation zone, thoroughJ.1' 1Dvestigate4 iD the denae ~ets,1 alters, "goes up" above the plate, an4, f1Dall.7, disappears. ~he pecularities of the flow alteration before the plate iD the /'14 > {
~ets result from the transition ~o X-wave configuration with the 1Dcrease of rarefaction. lD this case the 4isappearance of the circulation zone is preceded by a sharp decrease of its cross dimension. ~he comparison of these results with the density measureaents data Shows that the necessar.y condition for the 41sappearance of the circulation zone is the clos1Dg of ring ab:i.Dg layer, origiDatiDg from the triple poiDt (fig•1), on the jet axis. At the same time the periphe~al maxi•a of the density distribution disappears too. ~he iDflueace of the rarefaction on the static pressure distri~tion along the plate is similar to that shown 1D f1g.2a when the total density decreases, the peripheral aaximua of pressure, co.rrespond1Dg to the ci.;culation flow reattachment poiDt ß , is saoothed down. lD the case
997
ot radial tlow along tbe plate tbe onl7 mazimwa ot pressure 1D tlle cent.re of the ... plate 1s obse~ed. In tbe case ot x .. ave contiguratioD tbe density and pressure CurTes haTe one central waxi•ua whicb by tar surpasses tbe levels observed behiDd tbe central sbock.
Tbe bounda.ries ot tbe uiD tlow .regi.mes 1J:t. the sbock 181'er tor tbe N",e.!f jet are sbowed in .f1g.4. Region 1 cor.responds to tbe undisturbed tirst jet cell, .regioD II - to tbe tlow rith tbe "gone up" circulatioD zone, .region III - to tbe central circulation zone,
. IY - to tbe radial .tlow along tbe plate, !rhe .tlow in tbe vicinity ot tbe ax1.s 1s independent o.t the exterio.; condi tions when paramete.rs P,e D and 1t a.re 1D region Y~ !rhe .region ot dit.te.rent .regimes to.r the M4 > 1 jets han sillilar boundary contigurations but tbe tlow 1D tbe tirst cell and beto.re the plate may take place both with tbe torming ot tbe IBch disk and tbe X-waTe contigu.ration. In thia case it appea.ra.a t.ransition .region troa one wan st.ructure to anothe.r• !rhe bounda.ries of this region get narrow with the. 1Dcrease of MQ, and shitt towards tbe field ot greater P,eR values.
!he e'ftluations of the cent.ral shock thiclmess '! according to the .results of the density.measureaents (f1g.4) showed the ~lidit7 ot the empi.rical corellatioD 'l = 9.5! establiahed earlier fo.r the sbock wave in a unifo.tta flow as well as to.r tbe kch disk ot a tree ~tt3 ( l - aolecule .tree path length in a shock l~er). ~bis corelation 1D conjunction with the .tormulae to.r the central ahock position be.tore tbe plate and tor tbe .tlow: paraaeters in a jet allow:ed to obtain the semiempirical co.relationa tor 'i • !be shock wave
998
lz, 0
ti
I
----- -----------------------------~ ~----------~------------~L-----------~
tfeD 50
Ft:g. -4.
40
~------------L-------------L-----------~ 0 2
999
thickness before the plate depends on botb the jet rarefaction and the relative distance (fig.5).
Thus, the present investigations allowed to establish a number of qualitative features and to give some quantative evalutions of the processes taking place with the growtb of the rarefaction in a supersonic underexpanded jet, impinging upon a flat plate.
REFEREN.CES
1. I.P.Ginzburg, E.I.Sokolov, and V.I.Uskov. The types of wave structure at interaction of underexpanded jet with an infinite flat plate, Journal of applied mathematics and technical pb;ysics, 1:45 (1976) -In Russian
2. V.V.VQlchltov, A.V .. Ivanov, B.I.K,ysliakov, A.X.Rebrov, V.A-.Subnev, and R.G.Sbarafutdinov. The low density jets of high expansion ratio behind a sonic nozzle, Journal of applied mathematics and technical pb;ysics, 2:64 (19?3) - In Russian
3. V.V.Volchkov, A.V.Ivanov. The thickness and the iDner structure of ortogonal shock wave, developed in expansion of highly underexpanded jet in a low-density space, Kechanics of liquids and gases, 3:160 (1969) - In Russian.
1000
SOME PECULIARITIES OF POWER AND HEAT INTERACTION OF A LOW
DENSITY HIGHLY UNDEREXPANDED JET WITH A FLAT PLATE
E.N. Voznesensky,V.I. Nemchenko. and A.V.Sokhatsky
Moscow Physical Technical Institute 141700 Dolgoprudny, USSR
INTRODUCTION
The problern of impingement of a highly underexpanded jet with the flat plate parallel to the jet axis or at some angle with it, practically, does not lead itself to theoretical consideration and precise computation. In this task the most important thing is to define heat and power loading distributions over the plate and to determine the form of the impingement shock wave. The results of investigating the power interaction and the shock wave form 1- 3 are obtained for dense jets. As far as the heat exchange of the highly underexpanded jet with the plate is concerned, the situation hereisstill more complicated. For want of the ways of generalized presentation of experimental data and for lack of generally accepted calculation methods for a heat flux, it is difficult to compare the results of measuring the heat loading distributions obtained by different scientists. For instance, the method of calculating the heat flux with "pseudo"-stagnation point" 4 leads to overstating local heat flux. The methods of criterial presentation of max heat loadings5 based on theory of Cheng6 are under way.
EQUIPMENT AND METHODS OF THE INVESTIGATIONS
The experiments have been made in a vacuum tube of stationary action with air jets under the following conditions: stagnation pressure P0 = 4.13 • 104 -...,. 6.67 · 104 Pa, Stagnation temperature T0 = 395-780 K, wind tunnel static pressure P""=1,33-13.3 Pa, N = P0 /P"" = 0.5·104 -4·104, Mach numbers in the nozzle exit defined without regard to the boundary layer Mae = 1.0-3.92. The exit part of all supersonic cooled nozzles is conical with halfapexangle of cone of 10°. The distance from the plate to the jet axis expressed in the radii of the nozzle exit h = H/re changes in the range of 4-20. In the experiment the numberof ReL=Re*/JN, where Re* is Reynolds number calculated by means of the nozzle throat parameters of the gas varying from 14 to 160, that is, there are much less magnitudes of Re - ÜP + 104 which are characteristic of the results in 1-3.
""
1001
The model of the thennostated flat plate on which most results have been obtainedis a square copperbody with dimensions 1.5·10- 1 x l.5·Hi 1m, on which a layer 0,25·10-3 m thick made of lavsan is glued on the side tumed to the jet. The lavsan layer is intended for measuring local heat fluxes by the method of auxiliary wall 7 • To visualize heat fluxes a polymer layer containing liquid crystals in glued on an analogaus model on the auter side of the lavsan layer. The body of the model has three cavities: a narrow longitudinal central measuring cavity for taking Pw pressure on the plate surface and two lateral cavities with inputs for thennostated liquids. The model wall over the central cavity is drained along the axis line by several orifices. In every measurement only one orifice is used, the rest being deadened. The measuring cavity is connected with calibrated manometric sensor protected by the cooled screen-mask from heat action of jet. Cooling of all the screens, mask, nozzles and model is made by distillated water which is supplied from athennostat at room temperature. The total error because of influence of the degasation, conducti vi ty of manometric sy stem, thenno-tran spi ration and pseudotranspiration does not exceed 3<;'11 The total angular error of the model with respect to the nozzle axis does not exceed 3'. Thanks to this, for measuring pressure distributions on the plate it been possible to do with comparably a small number of the drainage orifices at the expense of using longitud,inal and transverse displacements of the model. At these displacements the Coordinates of Pw max on the plate are defined with high accuracy of 0,1-0.2 mm. The analogaus method was used for measuring heat flux distributions qw and defining max qw position. In order to study geometry of the wave structure of the jet its visualization is made in a symmetry plane surface (a surface, going through the nozzle axis perpendicularly to the plate surface) by means of electron beam technique.
RESUL TS OF THE EXPERIMENTS
An example of a wave picture of the j et over the plate at a gi ven plane where h = 4, M1e = 3.92, N = 1.1·10 4 , Re* = 7990 is shown in Fig. 1 (a). The shock wave over the plate is curved bothin the longitudinal and transverse directions, the disturbed stream after it is three dimensional, and the loading distributions over the plate surface must be of peaky character. A typical pattems of such distributions are shown in Fig. 1 (b) and 1 (c) for conditions of the following parameters: Mae = 1, h = 4, P0 = 5·104 Pa, T0 = 535 K, N = 1.61·10 4 + 2.95·104 ; Re*= 5340: where x = X/re; z = Z/re non-dimensional Cartesian coordinates of the points of the plane extending through the plate plane.
The abscissae axis is the line intersection of the plane, going through the nozzle axis perpendicular to the surface with the plate surface. Reading of x is from the nozzle plane shear. The meaning of the coordinate of absolute max pressure on the plate xm with regard to the "local" jet focus, that is, the value xm- f, where f = F / r ( F is abscissae of the proj ection of the jet focu s on the plate), can be predicte~ by formula (6) from Kononov and Leites paper 3:
X -f=0.5hj 11 ; where 11=(1+~) [l+ 2 rYJ m 1-11 KMa (K-l)Ma 2
e e
where K = cp/cv adiabat gas index. However, at a !arger h in the experiments with low density jets a morefront position of the maximum Pw results for Mae = 1 in comparison with the calculation made by the fonnula mentioned above. The fonn
1002
Fig. l. The example of interaction. (a) impingement shock wave; (b) heat flux and (c) pressure distributions on the plate and (d) form of equallevelline (see the text).
?f th~ equal.pressure lines. for2 ~w/Pg. distribut~ons repre~ented in Fig. l(c) shown m Ftg. l(d). 1- (Pw/P0 ) 10 -0.7 , 2-1.02, 3-1.27, 4-1.90.
All the equal pressure lines near max Pw = Pwm compressed along x axis. As Mae increases the equal pressure lines begin stretching along the abscissae axis. Analogaus data for the heat flux and pressure distributions at other Ma e numbers were received (see Fig. 2) Comparison for the measurements considered of xm coordinate in terms of the formula (6) from 3 for the numbers of Mae and Re* are given in Fig. 3(a). The notation corresponds to the following conditions: 1-5 Mae = 1, Re* = 5340, h = 4-20; 6-8 Ma ~ = 1.675, Re* = 5390, h = 4-12; 9-12 Mae = 2.30, Re* = 3020-6820, h = 4-9.35; 1.5-15 Mae = 2.88, Re*= 2800-6300, h = 4-8; 16 Mae = 3.92, Re* = 8440, h = 4.
While Mae = 1, at Mae~ 1.675 experimental meanings of xm for the lowdensity jets sistematically exceed those of calculated by 8-10 r~ but they, tagether sith the latter, obey linear dependence. Any definite relation be t ween xm and Re* in the range of Re* = 2.8 ·103 - 8.4 ·103 is not revealed. However, any change of the Re* number affects the value of the induction pressure: is the vicinity of the maximum the pressure on the plate dimini sh wi th some decrease of Re* •
All the above results are obtained for streams coming from the nozzle with a cooled wall, the T0 temperature of which is equal tothat of the model body Tm.
1003
In order to find out the influence of the heat conditions of the nozzle on power and heat loadings on the model surface some experiments have been performed with three different temperatures of the nozzle wall t 0 = Tn /T0 = 0.52-0.53; 0.68; 1.0 for Ma e = 1 and Ma e = 2.88. The temperature meanings of the model wall have been kept constant in the range of 0.54-0.55. In this case the nozzle temperature is gi ven by means of the second thermostat. The nozzle is covered with thermostatic mask the temperature of which corresponds to that of the model. The results of measuring pressure distributions along the model axis parallel to the jet axis at different meanings of tn are shown in Fig. 2 for Mae = 1 and Mae = 2.88. It is clear that changes of tn practically don't affect both the distribution of power loading and their magnitudes.
The criterial presnetation3of the distribution measurements of induction pressure on the plate along the abscissae axis shows that in the large range of 2.5 ·103 ~ Re ~ 107, 1 .~ Mae ~ 3.92, 2 ~h ~ 20 the observed scatter of points with regard to the approximating dependence suggested in the mentioned above paper i s 20-60 %- Bett er generalization was achi ved when was used this nondimensional presentation of results for K = 1.4:
'10 = ppw (h2 + 1.3h + 0.2)(1 + ~ )/9(1 -I/'2~ ~0= 1 +X -~mf 2..LII 0 vRe* Xm 11
Here I11 is meaning of I 1 at Mae = 1. Such presentation of the power loading results in decrease of the data scatter in the vicinity of the coordinate ~0 = 1 up to +10% (see Fig. 3(c) where symbols 1-4 are refered to dense jets2•3 , the rest to low density jets). Points in Fig. 3(c) are satisfactorily approximated by the dependence:
0 - 5.28 (~0 + 0.41)2 '1 - 5.80 + (~0 + 0.41)4.5
Configuration of the front of the shock wave over the plate in the symmetry plane of the stream picture (visualization plane), upon the outcomes of treating photos at close Re* in terms of the variables of paper 3 • In addition to this, the shock wave profiles over the plate at low density of the jet are divided into layers according to Mae number. By introducing variables (yO/h) = (y/h)(l-35/Jffe*) and ~ 0 , as int can be seen in Fig. 3(b), a satisfactory generalization of all groops of the data into a singlerelationship is achieved (the symbols of 1 - 6- are dense jet3 , the symbols of 7- 14- are low density jets).
The character of local heat flux distribution of the model is analogaus quantitatively to the character of distribution of pressure. lt is seen, for example, in Fig. 1 (b), where distribution of q is shown upon the plate for conditions similar to those of Fig. 1(c). w
The distributions of local heat flux- at z = 0 in the case of the nozzle cold wall (Tn =Tm) can be represented by analogy with axis pressure distributions in a generalized form (Fig. 3(d)). using the similarity coordinates of
~0 = 1 + ( X - Xmq .!.L_ q xmq- f lll
1004
Fig. 2.
Mae•f h:fQ Po=4.BliO"fJa
7ö=5"3K
20 I
Mae•2.BB h=4 Po·5.01/D4 [Ja
~·.538K
•• tn=Q53 0 " tn-=068 oo tn: f
fO X
The influence of temperature factor of the nozzle on heat and power loadings. (a) Mae = 1, (b) Mae = 2.88.
where xmq is a coordinate of qwm max heat flux. The local heat fluxes are approximated by the following dependence
0 ~ = 3.67(~9 + 0.35)
qwm 3.36 + (~0 +0.35) 4 9
We must say about position of the heat flux max xmq that it can be other than the max pressure position. At first this factwas invention in the paper 8 for the case Mae = 1, h = 12, Tn "'Tm, moreover it turn out that xmq > xm. In cooling the nozzle walls to the temperature of the model body (Tn,.. Tm) a similar situation is observed in all investigated instances. The dependence of non-dimensional distance t:n = ßX/re between max qw and Pw from h and Mae is given for tn .. 0,55 in Fig. 3 (e). The distance between the max is seen to increase as Mae of the jet on the nozzle exit and the distance of h from the jet axis to the plate get !arger.
Configurations of several equal pressure lines and isotherm (lines of qw Ievel) in the vicinity of max interaction based on the measurement results and corresponding the conditions of the Fig. 1 (b) and (c) are shown in Fig. 1 (d). Unlike the equal pressure lines the lines of q Ievel in the example concerned do not differ greatly from circles. The equal pressure line forms investigation on the model au xiliary wall conducted wi th the u se of a thermoindi cating layer containing a composition of liquid crystals glued to the model extemal surface Ieads to analogaus conclusion.
Increasing the temperature factor of the nozzle wall at fixed meanings of Mae, Re* and h results in increasing heat flux through the whole of the model and in displacing head loading maxima towards the nozzle at invarient meanings of xm and Pwm. At tn,.. 1 the meaning of q w.:nin 1.5 - 1. 7 tim es greater than qwm at tn"' 0.5-0.7 (Tn"' Tm) and the max of the heat flux seems tobe closer to the nozzle than pressure maximum (see Fig. 2). Maximum meanings of heat fluxes upon the experiment results (Mae = 1 and Mae = 2.88) are gi ven in a non-dimensional form in Fig. 4, where Sty number and Kj paramo~ter depending on Reynolds number are calculated with allowance for a skip angJe9 y and parameter of a non-disturbed
1005
Im! • f 0 2 ~ 3
10 " I{ ID 5 X 6 ... 7
5
0
Fig. 3.
1006
• g V 1/J 6 ((
• (2 • f! CJ '~
@ II 15 • 16
5 10asn·f/!i {- f
J...o h
z
0
z J
Ir ~.a(JJOi-.f.J.It/ltlJ #-(l$~.id·IL #
® ll I 42'
Dependences between non-dimensional parameters. (a) xm position; (b) shock wave form. (c) Pw di s tribution; (d) qw distribution; (e) tn = x - x
mq m •
o tv •O.SS; tn"'/154 o tw =0.4~ ; in~/152
GI tw rz/J.~· in"'O.Jl Q
5 9 tll t:0./0/ t"-:.1114
J ~ 10' /! K.,/ r
Fig. 4. 2
St (K ) dependence for different t w and t y y n
2 jet. Theoretical dependences of Stanton number from Ky for an obliquely streamed cilinder for two meanings of y are given in solid lines.
The surface of the plate in the vicinity of qwm has been initiated5 by a cylinder, the axis of which is bent to the jet at an angle (90° - y ), equal to the slope angle of the stream line of an undisturbed jet coming to the point of Xmq to the plate surface. The radius of this cylinder (entering K2 ) has been calculated in the same way the radius of an equi valent sphere 10 Y
R eq = h/(1 + e )
where e - is the distance of the shock wave going away from the obliquely strained cilinder, which is referred to its radius. All the points obtained under conditions of T0 "' Tm "' 285 + 293 K lie below the theoretical curves, the discrepancy increasing with the model approximation to the jet axis. However the data referred to the meaning of t 0 oo l lie near and over the theoretical dependence St (K~) for y = 45°. Moreover changing the model temperature factor of lw results to aJditional exfoliation of the results.
Thus the experiment results show that presentation of max heat f!uxes in the :arm of s~.,/K~_) should be essentially added for taking the influence of tw, t0 , T0 mto cons1derat10n.
1007
Üianging of the heat flux maximum position and dependence of heat flux values on temperature factor of the nozzle wall are conditioned by influence of the nozzle boundary layer when jet expand into the low pressure space.
HEFERENCES
1. E.
2. V.
3. Yu.
4. A.
5. V.
6. H.
7. V.
T. Piesik, R. R. Koppang, D.J. Simkin. Exhaust impingement on a flat plate at high vacuum, Joum. Spaceraft and Rackets, 3 : 1650 (1966). A. Zhokhov. Calculation of pressure distribution in a gas flow about flat plate, Uchenie zapisky TsAGI, 4: 14 (1973)- In R..1ssian. N. Kononov, E. A. Leites. Flow parameters in component jets, Trudy TsAGI, 1721: 1 (1975)- In Russian. R. Maddox. Impingement of underexpanded plumes on adjacent surfaces, Joum. Spacecraft an Rockets, 5 : 718 (1968). M. Antokhin, Yu. I. Gerasimov, V. A. Zhokhov, A. A. Khomutsky. Heat interaction of free expanded jet on a flat plate, lzvestiya Academii Nauk USSR, Fluid and gas mechanics, 4 : 119 (1981). - In Russian. K. Cheng. The blunt-body problern in hypersonic flow at low Reynolds number, lAS. Paper, 92: 1 (1963). K. Aslanyan, E. N. Voznesensky, V. I. Nemchenko. Application of the gradient technique for measuring substatially nonuniform distributions of local heat fluxes, Journal of Engineering Physics, 35 : 29 (1978) -In Russian.
8. E. N. Voznesensky, V. I. Nemchenko. Action of a highly underexpanded jet of heated air on a flat plate. Trudy MPhTI, Airmechanics, 1 : 8 (1973)-In Russian.
9. R. J. Vidal, C. E. Wittliff. Hypersonic low density studies of blunt and slender bodies, RGD, 3d Int. Symp., New-York-London, Acad. Press, 2 : 343 (1963).
10. V. I. Blagosklonov. Method of calculation of free expanded jet interaction with plate Uchenie zapisky TsAGI, 4 : 99 (1970) - In Russian.
1008
XV. CONDENSATION IN FLOWS
NONEQUILIBRIUM CONDENSATION IN FREE JETS
ABSTRACT
Alfred E. Beylieh
Technische Hochschule Aachen, 5100 Aachen, West Germany
The cooling of mixtures of sr6 and other gases in a free jet expansion is investigated. A model for the nonequilibrium condensation process is developed from mass and energy balances for the individual cluster groups. First sample calculations show good agreement with experimental data1 .
INTRODUC TI ON
Cooling of a model gas such as sr6 in a gasdynamic expansion can be achieved by mixing sr6 with a gas having only few degrees of freedom. In this case the expansion may lead to a state far beyond the saturation line, and one can observe the effects of condensation which may destroy the cooling effect, unless the process is performed in such a way that, due to nonequilibrium effects, the phase transition is essentially frozen. This does not necessarily mean that one has to perform a rapid expansion, since then, on the other band, one faces the problern of freezing of internal molecular energy due to relaxation.
The search for an optimum between condensation and relaxation, however, is then further complicated by other unwanted nonequilibrium effects, such as temperature difference, velocity slip, and background penetration. These phenomena are to be investigated and, if possible, minimized, if one wants to achieve a proper gas cooling.
In the present study the problern of condensation is investigated. The model for the condensation developed here is a cluster
1011
approach which also considers an energy balance for each cluster group.
THEORETICAL CONSIDERATIONS
In the classical approach 2 to the problern of-condensation an isothermal quasi-equilibrium nucleation rate is used in connection with growth laws for individual (supercritical) clusters. In the present work a pure cluster approach3 is attempted which considers mass and energy balances for cluster groups. Such an approach may be desirable for a regime where condensation effects just start; it is always limited to a finite number of groups and, therefore, not so well suited for problems where very large cluster sizes are to be expected.
It is assumed that the expansion on the jet axis may be described by a stream tube with cross section A(x). Since velocities change very little in a free jet expansion, the assumption of one speed u(x) for all species and clusters may be not unreasonable. One may start with a mass balance for the cluster density n (i.e. g moleculeslcluster)
g
dn _ß: = I - I dt g-1 g
(1)
with the cluster flux
I = a c n1 ng ß - a. ng+1 g 1,g g+1 (2)
where the incoming rate is
ß1,g = (8nkT) 112 o~g I 112 ]J ig (3)
with 1 + (J ) (J1g = 2 (o1 g (collision diameter)
3gm1 1/'!J (J = [4Tip J g s
(spherical cluster shape)
]J1g = g m1 I ( l+g) '
(reduced mass)
Here, o1 is the molecular diameter (SF6 ), p is the mass density of the condensed phase, and n1 is the snumber density of the condensing species {SF6 ). The outgoing flux may be described by
1012
with
2 s = 4 1r a
g g
ng = t:00 [ 1 - ( 1
n ng exp [ - k ~ ) J ( 1 + kT) , ( 4)
g g
(cluster surface)
(law of surface potential).
(5)
(6)
Adjustment of the condensation probability a can be made at the saturation line (g-+ 00 ). It should be noted thgt the incoming flux. is of bimolecular and the outgoing flux of monomolecular nature. One should keep in mind that Eq. (4) is valid only for very large clusters ( g + oo); for small clusters the exponent should be replaced by a relation of the type4
Tl 3g-1 ( 1 - 3g~T) (7)
which goes towards exp [ - n /kT ] for large g and the secend term small. One may "correctg" g this deficiency of a by artificially rising t: 2 (the potential of the dimer; € 00 is g the "flat surface" evapgration energy at 0 K). More complicated relations may be used .
At least the group g=2 needs a special treatment, since dimers can be stabillzed only by an additional collision which has to follow shortly after the buildup of the complex (X- SF6 )*. We consider the following processes
(8)
where X may be any one of the molecules. For the buildup of the activated complexes * a simple relation is used nj1
* 3 j = 0,1 (9) nj1 = 1f0j1 n. n1 J
For the total mass balance one obtains then
1013
(10)
It follows from the extremely streng dependence of ~ on the cluster temperature T that an energy balance is impBrtant. For the cluster group g Bne may write
+ ng+1 ~g+1 (eg+1 - Eg-+1 ) - n e [ a n ß + "' J g g c 1 1,g ~g
- {(1-a) a n ß1 (2 + ~11.) + a n ß (2 + ~i) } c e1 1 ,.g eo o o,g o
n k (T - T)' g g (11)
with the inner energy of the cluster
f (T ) e = { g (3 + V g ) - (3 - ! 0 ) } kT - E (12)
g 2 2 g2 g g
with the potential energy
€2 -.185 E = g € 00 [ 1 - (1 - 2'E) (~) ]
g 00
the energy of the incoming molecules
(13)
and a relation for the outgoing energy, when a molecule is ejected
Here, f (T ) is the nurober of active degrees of freedom of the inner vYbr~tional modes of the SF6 molecule, (a realistic value for the six normal modes is used out quantum effects for the vibrations between the molecules of the cluster are here still neglected), and 2~: is the total nurober of inner degrees of freedom of species Jj. The last term of Eq. (11) takes care of the cooling effect of non-condensing particles (a . are the coefficients ofthermal accommodation). el
1014
For g = 2, a simple relaxation equation is used
This set of Eqs. (1,10,11,15) has to be completed by the gas conservation equations:
Continuity
~x (A u p) = 0
with
Momentum
du dx u p + dpt = 0 •
We assume that Pt ~· p = (n
Energy
d 2 d m1
dx (~ + h) = - dx (-2 p
with
A L
g=2
0 + n1 ) kT
= )
ph = n m h + n1 m1 h1 0 0 0
A and - = L n (hg - gh1).
g=2 g
g n • g
.
(15)
(16)
(17)
(18)
For the description of the vibrational relaxation, one single equation is used
dT V
e*v(T) - e (T ) V V
dx =-------u c "b (T ) T
Vl V V
with '[ V
as relaxation time6 •
(19)
1015
After introducing relative densities ng = ng/n10 , one may transform
!!._ (n"' ) ~ d ("" ) dt g ~ u dx ng · (20)
When Eqs. (1,10,11,15) are nondimensionalized using the throat radius, r*, an~ equilibrium spread of sound at the stagnation chamber, a = y p /p , essentially three para-eo eo oo oo meters appear
K = n r* 0 2 ( 8 7f m ) 1/2 1 oo 1 yeo m1
(21)
which is a reciprocal Knudsen number,
(22)
which appears in the balance equation of the dimers, and
(23)
which governs the monomolecular process of evaporation. For all interesting cases K is very large (typically of the order of 104), and therefore tqs. (1-15) are extremely stiff. Stiffness decreases with increasing g and, during expansion with decreasing n1 and T.
Initial conditions are obtained fr>om Eqs. (1,10,11,15) requiring steady state and zero flux (i.e. I = 0 ):
g
Then, the right-hand sides of Eqs. (1,10,11,15) become a system of coupled algebraic equations which can be solved by iteration. It may be of interest to discuss some properties of the temperature distribution T (which is not equal T ). For small g we obtain approximat§ly
e 1 +E%-e g- g
This is only possible, if
1016
T g
> T g-1
(25)
and the increase in tempe-
rature is considerable. For critical g* (i.e. a n1 ß1 = a ) c ,g g
a [ e + e - 2 e + E+ - E J g g-1 g+1 g g+1
I +_ß.
n g
[ e - e + E+ + E- 1 ] g-1 g+1 g+
- w (T - T) = 0 g g
(26)
If g* is large and the flux I ~ 0, then T "'- T, but if T is large this is no longer the cas@ and T > T.g g
g
SAMPLE CALCULATIONS
Calculations have been performed with a maximum group number of A = 200 for 90 % Ar - 10 % sr6 mixtures. During the iteration the convergence for the small sizes g was very quick (for the first 40 groups about 3 - 4 iterations for three leading digits); it decreased for larger g. The reason for this effect can be seen in the fact that the far end of the "tail" ( i. e. g large) has to sweep over orders of magnitude for a little step ~x downstream. In Fig. 1 the development of the cluster distribution for a volume element moving downstream a1ong the jet axis is shown. Note the fast rise during the first steps, the development of a maximum, and the final freezing of the distribution when the local Knudsen number is of order one.
Fig. 1
1. 50 100
g 150 200
p0 0:400mbar·2mm
go•t.Ar .1o•t. sF6
-\0 10
I X:2.060 t.x:.3050
Development of cluster distribution axis for D = 2 mm, p0 = 400 mbar, T0 10 % SF6 , ac = .5 , E00 = 3795 K.
n along the jet =g 293 K, 90 % Ar +
1017
Fig. 2
1.
t
.1
2
..---- 800mbar·lmm "--
/ -- 400 ·2 f/ 111 "- 200 ·4
I / tl I tfl
I
0
0
0
0
0
5 10 20 50 - xtO
1.
'Xe
t
.1
.01 100
Scattering intensity the jet axis. Points p = 400 mbar, D = 2
I = (Igas + Icluster) I Igas along are from experiments
0 p D = 800 mbar mm. 0
mm. Degree of condensation X for c
In Fig. 2 the degree of condensation, X , is plotted along the jet axis, and the calculated scattering fntensity is compared with measurements. There are still some discrepancies in the initial stages, which can probably be removed by adjustment of the unknown coefficients.
REFERENCES
1. A more detailed paper may be obtained from the author. 2. R. Becker and W. Döring, Kinetische Behandlung der Keimbildung
in übersättigten Dämpfen, Ann. der Physik 24: 719 (1935). 3. A.E. Beylich, Über Kondensationsvorgänge in Gasströmungen von
geringer Dichte, in: "Festschrift 65. Geburtstag Prof. SchultzGrunow", Institut~. Allgem. Mechanik, Technische Hochschule Aachen, 15/1 (1972).
4. M. Polanyi and E. Wigner, Über die Interferenz von Eigenschwingungen als Ursache von Energieschwankungen und chemischer Umsetzungen, physikal. Chemie (A), Haber-Band, 439 (1928)
5. E.R. Buckle, A kinetic theory of cluster formation in the condensation of gases, Transact. Farad. Soc., Aberdeen 65: 1267 (1969).
6. W.D. Breshears and L.S. Blair, Vibrational relaxation in polyatomic molecules: SF6 , J. Chem. Phys. 59 : 5824 (1973).
1018
CONDENSATION AND VAPOUR-LIQUID INTERACTION IN A REFLECTED SHOCK REGION
ABSTRACT
*) s. Fujikawa, I. Mizuno, T. Akamatsu and V.V. Zhurin
Department of Mechanical Engineering, Kyoto University, Kyoto, Japan
*) Computing Center, Academy of Sciences, Moscow, USSR
Nonequilibrium condensation processes of methanol and water vapours have been studied on an end wall of a shock tube behind a reflected shock wave. This problern involves not only the adsorption process of vapour molecules on their condensate but also the thermal accommodation process. An optical diagnostic technique has beendeveloped to measure condensation and thermal accommodation coefficients of vapours. A theoretical analysis also has been made on the reflection of a shock wave from the end wall of phase-changing and heatconducting liquid. The experiments and theoretical analyses indicate that energy transfer to the liquid surface and condensation strongly affect the pressure, temperature and density of vapours in the reflected shock region, and that there is some correlation between condensation and thermal accommodation coefficients. These coefficients have been found to be considerably larger than previously measured values.
INTRODUCTION
Condensation kinetics of vapour has been studied by numerous investigators, theoretically from points of view of the kinetic theory of gases, the theory of rate processes, the irreversible thermodynamics, and experimentally by means of Laval nozzles and shock tubes. Recently, further researches have been required in connection with many problems in engineering, e.g., cavitation, boiling, twophase flow, combustion and heat pipes.
Previous works on condensation can be classified into two categories: (l) kinetic behaviour of vapour molecules including the ad-
1019
sorption process of them on the liquid surface and (2) thermal accommodation process between vapour molecules and the surf'ace. The former has been one of the main subjects of the kinetic theory of gases. Schrage presented a formula for the condensation rate under nonequilibrium conditions by taking account of not only the diffusion of vapour molecules but also their convective motion[l]. An unknown parameter which is called " condensation coefficient " in this formula has to be determined experimentally or theoretically. The condensation coefficient represents the ratio of vapour molecules sticking to the liquid surface to those impinging on it. Therefore it depends on the physical properties of the vapour and the surface. Although many experimental studies have been made to determine the condensation coefficients of various substances, accurate values of them have not been yet obtained. The theoretical approach has not succeeded in determining condensation coefficients, either. " Thermal accommodation coefficient " expressing the efficiency of energy transfer between the gas and the surface is also.introduced as a parameter. Very few data appear to be available on thermal accommodation at the liquid surface. Alty & Mack~ conjectured from measurements on evaporating water drops that the reflected vapour molecules attained thermal equilibrium with the surface before leaving it[2); that is, the thermal accommodation coefficient was near unity. Hill also derived the same conclusion as Alty et al's from the experiments on condensation in supersonic nozzles[3]. Their conclusions do not seem to be reliable in the sense that the coefficients were not measured directly.
By using a shock tube, the present authors designed experiments to control the condensation and thermal accommodation phenomena in order to accurately reproduce the events both in location and time [4,5]. The condensation and thermal accommodation coefficients were measured using an optical diagnostic technique and flow fields of vapour near the liquid surface were elucidated by numerical calculations. In the present paper, these coefficients have been measured more accurately and the effects of a noncondensable gas on condensation have been clarified. Same useful equations have been derived describing the gas dynamics near the phase-changing and heat-conducting surface by means of the technique of matched asymptotic expansions.
CONDENSATION IN A SHOCK TUBE
Figure 1 shows a shock wave advancing towards and reflecting from the end wall of the shock tube in the vapour, where x is the space coordinate normal to the liquid surface and t the time. The reflected shock wave produces a region of stagnant high temperature vapour near the end wall. However, the wall itself is almost kept at an initial room temperature because the heat capacity of it is much larger than that of the gas. As the thermal accommodation proceeds between the hot gas and the cold wall, an unsteady thermal
1020
Liquid Film
Thermal Ooundary Layer
Solid
' X
lncldent Shock
Figure 1. Shock waves, thermal boundary layer and condensation in a reflected shock region.
boundary layer develops into the vapour. The vapour begins to condense on the end wall if the following condition for condensation is satisfied:
p(t,o)
lfT(t,o) < 0 (1)
where Tt(t,o) and T(t,o) are temperatures of the liquid and the gas at the surface respectively, p t is a saturated vapour pressure e-sa valuated at Tt(t,o) and p(t,o) an actual vapour pressure at the sur-face. Under appropriate conditions(e.g., initial vapour pressure, shock Mach number, cleanness of glass surface) the vapour is found to condense in the form of a uniform liquid film after the reflection of the shock wave[4,5]. By measuring simultaneously (l) the rate of growing of a liquid film on the end wall and (2) the temperature change of the vapour at the liquid surface, we can determine the condensation and thermal accommodation coefficients of the vapour.
THEORETICAL ANALYSIS
We assume that condensation and thermal accommodation processes on the shock tube end wall, subsequent to the reflection of the shock wave, are aaequately modeled by considering a sudden and uniform change in the temperature and pressure of a semi-infinite expanse of vapour in contact with the liquid. Here, mist formation within the thermal boundary layer is not taken into account although it may be prominent after a period of the shock reflection(Fig.l). The validity and the applicable limitation of the present theory will be discussed in RESULTS AND DISCUSSIONS. The liquid surface is located at
1021
x = o and the change in temperature and pressure occurs at t = o. The liquid occupies the half-space x < o. Four dependent variables, pressure p, temperature T, density p and velocity u satisfY the equations of mass, momentum and energy conservation as well as the equation of state. In order, they are:
ap a at + ax(pu) = 0
~pu) + a!(pu2 + p)
p = pRT
a 4 au = ~~ax)
(2)
(3)
(4)
( 5)
where ~ is the shear viscosity, k the thermal conductivity, y the ratio of specific heats, Cv the specific heat at constant volume, C the specific heat at constant pressure and R the gas constant. TEe temperature of the liquid satisfies the thermal diffusion equation,
(x < o) (6)
where D is the thermal diffusivity, the subscript ~ represents the liquid. The initial and boundary conditions are:
1022
t = 0, X ~ o: U = 0, P = P00 > P = P00
X < o:
t > o, X= o:
T = T ~ 0
p(t,o)u(t,o) aM { psat
= 1(2nR) IT~(t,o)
p(t,o)
IT(t,o)}
diS - - p ~dt
p(t,o)u(t,o)L
2-a k = .--!(--) ~t,o)
aT pCpa T=T ax 0
(7-1)
(7-2)
(7-3)
(7-4)
( 7-5)
X -+ - oo; T = T !1, 0
(7-6)
where " aM " is the condensation coefficient, " aT " the thermal accommodation coefficient, a the sound speed, o the thickness of liquid film, the subscript o represents the initial state and oo the reflected shock heated state.
The equations (2) through (6) can be solved under conditions (7-1~6) by using the technique of matched asymptotic expansions[6]. The pressure p(t,o) and temperature T(t,o) of the vapour at the liquid surface areexpressedas follows:
p(t,o) = p + ßp(t) 00
(8)
where,
ßp(t)
y<jl 2-aT a~(poo -psat) - (l+<jl)/(Re•)[(l-S)poo+ ~ (l+<jl)CpTool(2ny) }
x exp(~) erfc( I~) T '
T(t,O)
2tjJT ß p ( o ) l t ;.t + YP {l+lji) (l - WT)} exp(~) erfc( ~)
00
(9)
where Re, w, e, <P, 1jJ and-r are defined as follows:
t a 2 P a T k D 2wM 2 00 00 ( a a)M 0 = ~(~)
r Re -- w = e = T' <P 1jJ = l+M 2' D ' -;;-;- r• k D '
00 00 00 00 00 !1, r
where Mr is the reflected shock Mach number, t 00 the reference time and the subscript a represents the state behind the incident shock wave.
1023
8
Pressure Gauge
Figure 2. Schematic diagram of experimental setup
OPTICAL MEASUREMENTS
The present method is based on the dependence of reflectivities of light beams on the liquid film thickness and on the local refractive index of the gas at the liquid surface. The arrangement of the optical apparatus is shown in Fig.2. The shock ttibe utilized in the experiments is a vertical type and has stainless steel driver sections which are about 2000 mm long respectively. The inner diameter of the tube is 32 mm. The end wall consists of an optically flat BK-7 crown glass(flatness, a tenth of the wavelength of the light) in order to make it possible to observe condensation and thermal accommodation phenomena.
Liquid film thickness
The ·thickness of the liquid film can be directly obtained from the measurement of the reflectivity of the light beam 11 A 11 at the normal incidence on the glass. The reflected light from the liquid film, passing through a half mirror M1, is brought to focus on a PIN photodiode P.D.l by a lens L1. The electronic response time of the detection circuit is less than 0.5 ~s. The reflectivity R~ of the light beam is given by[5],
R~ = 4n no 2n
1 _ ------------------~8~~~-v~------------------- (10)
where Ais the wavelength of light(6328 A), n the refractive index (subscripts s for glass, J/, for liquid, v for vapour). Though the refractive index n of the vapour at the liquid surface changes with
V
1024
the time, its effect on RL is so small as to be considered as a constant value(just unity) in the present optical configuration. Therefore, the liqui.d film thickness ö can be expressed as follows:
1 S V S V [ { (n +n )2 (1-R.l)-4n n }0.5 l
s in- nR. -( -n s-2...:_:..n_R....:.2-) -( n_R._2 ___ n_v_2 _)(...:1:..-...:R_L_) (11)
Temperature discontinuity
The vapour temperature at the liquid surface can be obtained from the measürement of the reflectivity of the light beam 11 B 11 at an inclined incidence on the glass. To measure the relative change of the reflectivity, a beam reflecting from the outer surface of the glass is used as a reference and is attenuated by means of a knife edge K to obtain an intensity equal to that of the main beam. The intensities of main and reference beams are detected by means of PIN photodiades P.D.2 and P.D.3. These signals, being led to a differential amplifier, are recorded on a wave form recorder. Now, neglecting the effect of internal reflections of the light in the thermal boundary l~er, the reflectivity RA , for polarization parallel to the plane of incidence, is given by[5],
r 2+r 2+2r r"vcosß sR. R.v sR. ~ RA = -=~--~----~~~----l+r 2r 2+2r r cosß sR. R.v sR. R.v
(12)
where r = tan(e -9~)/tan(e +SR.), r = tan(ei-e )/tan(ei+e ), ß = 4nnR.öco~~R./A, ~d 9 is th~ angle orvrefract1onvand var1esvwith the refractive index n according to Snell's law. The light of parallel polarization is used for the reason that it is more sensitive for refractive index variations than the light of perpendicular polarization. The RAdepending both on 6 (i.e., n) and ö, it is almost independent of ö for a sufficientlyvlarge angle of incidence. Here, an angle of 9 = 40° is chosen. Then the relative variation Y of the reflectivity ~an be approximately expressedas follows:
Y = S (n - n ) 0 V 0
where,
s 0
(13)
1025
1.0~
= Spontonoouo ~-S.turot..t eo.-noatton ~ Vopour Prouure Begins To Oc:c:ur, ',
-- Theory( P1•454.7Po.T1•29Q2K.~2.575)
---<>--- E•perlment
'"-.,, lnftlet ',
Prouuro~
0o~-~.o~~2~0-~30~~.~o_,1~8-0--8~0~~.oo I ( J.1 S)
Figure 3. Time-histories of the vapour pressure at the shock tube end wall behind the reflected shock wave.
The refractive index n is coupled to a vapour density p (t,o) by the Lorentz-Lorenz equXtion: v
n = 1 + Kp(t,o) (14)
where K is the Gladstone-Dale constant(K = 3.85 x 10-4 m3/kg for methanol vapour, 3.08 x 10-4 m3/kg for water vapour). Thus, given the pressure of the vapour, a vapour temperature at the liquid sur~ face can be obtained using the equation of state.
RESULTS AND DISGUSSIONS
The optical glass used in the experiments has been cleaned in a solution of K2Cr 0 in sulfuric acid and then in boiling ethanol. The glass treate& fn this way is reasonably free of contaminants.
The pressure, temperature in the reflected shock region and the r eflected shock velocity in a vapour are expected to deviate from those in the case where no condensation and thermal accommodation occur. Experiments have been carried out under conditions where the condensation is not so strong in order to accurately compare experimental results with theoretical ones; under the condition of strong condensation, optical measurements(especially, measurements of thermal accommodation coefficients) become difficult due to internal reflections of the light wi thin a thermal boundary layer. To avoid this effect a methanol vapour, whose condensate has a high saturated vapour pressure, has been used mainly. Figure 3 shows the calculated
1026
time.-histories of the vapour pressure at the end wall of the shock tube behind the reflected shock wave. Initial conditions are: pressure p1 = 454.7 Pa, temperature T1 = 290.2 K and incident shock Mach number Ms = 2.575, which will be used throughout the present paper with some exceptions. Calculations have been performed for condensation coefficients aM = O, 0.5, 1.0 and thermal accommodation coefficients aT = 0, 0.5, 1.0. The results show that the pressure is considerably influenced by the value of aM but completely independent of aT. The larger is the value of aM(i.e., the more rapidly the vapour condenses), the lower value the pressure drops to(see Eq.(8)). The pressuredrop.takes place because the density does not become high owing to the condensation at the liquid surface. For example, the pressure decreases to 90 per cent of a Rankine-Hugoniot value for aM = 0 in the case of aM = 1.0. Regardless of values of aM and aT, the pressure becomes almost constant after 5 vs from the instant of the reflection of the shock wave. In Fig.3, a theoretical result is compared with an experimental one(pl = 371.9 Pa, T1 = 285.3 K, Ms = 2.447). Both results are in good agreement until about 15 vs if aM = 0.1 is chosen. Later this value of aM will be compared with an optically measured one. The measured pressure at the end wall is kept at an almost constant value during about 15 vs, but then it drops rapidlY within 100 vs below the initial pressure. This pressure drop may be caused by the spontaneaus condensation of the vapour within the thermal boundary layer developing on the end wall because the vapour.very adjacent to the wall becomes supersaturated after a period of the thermal accommodation with it. In the present experiment, this period is about 10 vs and further 5 vs later the spontaneaus condensation occurs; a slight pressure drop, which causes a further temperature drop, at 10 vs after the reflection of the shock wave may be caused by condensation at the side walls. Once the spontaneaus condensation takes place, condensation zone spreads with the time into the thermal boundary layer and the pressure drops increasingly. In fact, Garen et al have pointed out this phenomenon by velocity measurements of incident and reflected shock waves in a saturated vapour[7]. Their experiment shows that the spontaneaus condensation takes place at the end wall and generates rare~action waves, which overtake the reflected shock effecting the deceleration. It is therefore found that the meaningful comparison between the experiment and the present theory has to be made within about 10 vs before the pressure drops owing to the spontaneaus condensation in the thermal boundary layer.
Figure 4 shows measured time-histories of the thickness of a liquid film growing on the shock tube end wall. They are compared with theoretical results within the range of a linear growth behaviour. The liquid film initially grows in proportion to the elapsed time after its formation and then deviates from the linear behaviour. According to the equation (7-3), the condensation rate is principally controlled by the actual vapour pressure at the liquid surface and the saturated vapour pressure corresponding to a surface temperature of the liquid because the vapour temperature at the surface rapidly
1027
1500,----------------------------,
CH30H Q'M:0.22 , ß : 0.53%
0
0 H20 1000 Q'M : 0.28
0 Q ß :1.18 %
0 Q
Q •
• CH30H
• Q'M :0.12
ß =2.69 %
ß: Mess Fractlon
50 t ( ,US)
100
Figure 4. Time-histories of the thickness of liquid film growing on the shock tube end wall.
drops to the liquid temperature and is then kept at a nearly constant value. In the early stages of condensation process, since these pressures are nearly constant, the liq~d film grows proportionally with the time. As condensation and thermal accommodation proceed, the spontaneaus condensation takes place within the thermal boundary layer to cause increasing pressure drops at the end wall ap already inspected in Fig.3. The saturated pressure, on the other band, increases with the time because the liquid temperature rises owing to the release of the latent heat of condensation. The difference between the actual pressure and the saturated one(i.e., the driving force for condensation) decreases and consequently the condensation rate also does. However, in this stage, the rate of growing of the liquid film does not decrease probably because numerous droplets, which have been formed by the spontaneaus condensation within the thermal boundary layer, contribute to the growth of the liquid film. The liquid film continues to grow with the time even after the actual vapour pressure at the surface drops below the saturated one; the pressure decreases just to the saturated one at 37 ~s after the shock reflection(Fig.3). In this stage, the growth of the liquid film may be due to only the sticking of the liquid droplets on the surface. Now, confining our attention only to the initial stages of the liquid film growth and trying to estimate the value of the condensation coefficient aM from the comparison between the experiment( • ) and the theory, we oötain aM = 0.12. This value is nearly equal to the one evaluated from the pressure measurement at the end wall in Fig.3.
1028
O.Sr--------- -------- -,
... ~.= I I ,...,.!'
00 -- Or = O. I
- · - clr = 0 .5
----- Or = I .O
0 .3
0 .1
-- --- -o L---'---'--L---'-_ __.~_ _ _.____. _ _._ _ _.____J
0 5 10 I I JJ s )
Figure 5. Calculated time-histories of the vapour temperature at the liquid surface.
This value is 3.4 times as large as the previously measured one(0.035) by the authors[5] and 2.7 times the theoretically predicted one(0.045) by Mortensen et al[8]. The discrepancy between measured values is due to the presence of a noncondensable gas(air) in the vapour. To investigate this effect in more detail, experiments have been done by changing the ratio of the air to the vapour(mass fraction ß = 0.5 ~ 3 %). The smaller is the mass fraction, the larger becomes the value of aM. The result( o ) for ß = 0.53 % is shown in Fig.4 and gives a maximum value, aM = 0.22. As to the discrepancy between the measured value and the predicted one, the latter based on the theory of the rate processes seems to be doubtful and to have to be modified. The result( Q ) for a water vapour is also shown in Fig.4 and gives a value of aM = 0.28(ß = 1.18 %). If vapours arefurther purified, condensation coefficients may become larger.
Figure 5 shows the calculated time-histories of the vapour temperature at the liquid surface. The temperature is drastically altered both by condensation and thermal accommodation effects. As expected, the larger is the value of aT, the lower value the temperature drops to. This is the centrast to that the pressure is completely independent of aT. For the same value of aT, the larger is aM, the higher value the temperature increases to. The dependence or the temperature on aM can be explained as follows. Behind the reflected shock wave, a vapour flow is induced towards the liquid surface due to condensation and consequently prevents the thermal boundary layer from developing into the gas. Therefore, the temperature gradient and the temperature at the liquid surface increase for the larger values of a . In the case of a = 1.0, the interfacial temperature rises witM the time because tMe thermal boundary layer becomes thinner, as the time elapses, due to the high speed
1029
QS r-----------------------------------~
-- Theory
,:-1,:-1 I
.... .!
0 [0 P1• 523.7Pa.T1 289.6K.Ms• 3.765
• • Experiments e P,• 454.7Pa ,T1-290.2K.M5- 2.575
0.3
0 0 0
0.1 ll'M=0.12, r=•.o
'-/__ • • • 5 10
t ( JJS)
Figure 6. Measured time-histories of the vapour temperature at the liquid surface.
vapour flow towards the liquid surface. Here, it should be noted that, for the larger values of aT, temperature curves are very close with each other and therefore the pressure and density have to be meas ured accurately for the accurate estimate of aT.
Figure 6 shows the measured time-histories of the vapour tem• perature at the liquid surface. They are compared with the theoretical results. The aM being obtained independently of aT, the latter can be uniquely determined by curve-~itting between the measured values and the theoretical results. The best-fit values o~ a are 0.05 and 1.0 for aM = 0.02 and 0.12 respectively. However, t~e value of aT = 1.0 does not seem to be so accurate because temperature curves are very close with each other between aT = 0.3 and l.Oas discussed in Fig.5. Some errors must be included ln data processes (pressure and density measurements). The true value may be between aT = 0.3 and 1.0. Judging from the present results, there seems to be some correlation between aM and aT; that is, in the case where aM is large, then aT is also large and vice versa. This s eems tobe r ealistic and does not contradict the author 1 s previous result(aM = 0.035, aT = 0.03). The reason for this fact is not clear at the present. Further quantitative investigations have tobe made along this line. As to the thermal accommodation coefficient of a water vapour, the authors are now making efforts for measuring them. The measurement is extremely difficult in comparison with cases of other vapours because a saturated vapour pressure of water i s very low at a room temperature and consequently effects of condensation on the reflectivity of the light beam become prominent. However, there are some possibilities t hat the measurement may be successful. By heating the shock tube or using a weak shock wave, the condensation can be subdued. To the present, fairly reasonable results have been
1030
obtained although the analysis is suffered from random noises. The preliminary results show that the thermal accommodation coefficient is unity.
CONCLUSIONS
Shock tube experiments have been performed for studying nonequilibrium condensation and thermal accommodation processes at a vapeurliquid interface. Condensation and thermal accommodation coefficients have been measured by means of an optical diagnostic technique. It has been found that the values of these coefficients are strongly influenced by the presence of noncondensable gases, and that there is some correlation between these coefficients. Thermal accommodation coefficients of methanol vapour were 0.05 and 0.3 ~ 1.0 in the case where condensation coefficients 0.02 and 0.12. A maximum condensation coefficient was 0.22. For a water vapour, these coefficients were 0.28 and close to unity respectively. For further understanding of condensation phenomena, experimental and theoretical studies should be made by taking account of the spontaneaus condensation within the thermal boundary layers both on the end and side walls of the shock tube.
HEFERENCES
[ l] Schrage,R.W., " A Theoretical Study of Interphase Mass Transfer", Columbia University Press, New York, 1953.
[ 2] Alty,T. and Mackay,C.A., " The Accommodation Coefficient and the Evaporation Coefficient of Water ", Proceedings of the Royal Society(London), Vol.l49 A, 1935, pp.l04-ll6.
[ 3] Hill,P.G., "Condensation of Water Vapeur during Supersonic Expansion in Nozzles ",Journal of Fluid Mechanics, Vol.25, 1966, pp.593-620.
[ 4] Fujikawa,S., Akamatsu,T., Yahara,J. and Fujioka,H., " Studies of' Liquid-Vapour Phase· Change by a Shock Tube ", Applied Scientific Research, Vol.38, 1982, pp.363-372.
[ 5] Fujikawa,S., Akamatsu,T., Yahara,J. and Fujioka,H., " Condensation and Thermal Accommodation Processes of a Vapeur by a Shock Tube ", Japan Society of Fluid Mechanics, Vol.l, 1982, pp.l65-179.
[ 6] Clarke,J.F., "The Reflexion of a Plane Shockwave from a HeatConducting Wall", Proceedings of the Royal Society(London), Vol.299, 1967, pp.22l-237.
[ 7] Garen,W., Brudi,K. and Lensch,G., "Velocity Measurements of Incident and Reflected Shock Waves in Various Gases and in Saturated Water Vapeur ", Proceedings of the 13th International Symposium on Shock Tubesand Waves ", Buffalo, 1981, pp.l67-175.
[ 8] Mortensen,E.M. and Eyring,H., "Transmission Coefficients for Evaporation and Condensation ",Journal of Physical Chemistry, Vol.64, 1960, pp.846-849.
1031
HOMOGENEOUS AND HETEROGENEOUS CONDENSATION OF NITROGEN IN
TRANSONIC FLOW
R. M. Hall,a E. H. Dotson,b and D. H. Vennemannc
aAerospace Engineer, NASA Langley Research Center Hampton, Virginia bcraduate Student, GWU, NASA Langley Research Center cEngineer, DFVLR-Cologne, Cologne, West Germany
INTRODUCTION
Being able to predict the onset of condensation effects in nitrogen gas has taken on new importance with the advent of transonic cryogenic wind tunnels, which achieve high unit Reynolds number by cooling the nitrogen test gas down to cryogenic temperatures. The lower in temperature the tunnel can be operated, the greater the increase in unit Reynolds number capability. Since the onset of condensation limits the minimum operating temperatures and, consequently, the maximum Reynolds number capability, the NASA Langley Research Center has undertaken a program to study the effects of both homogeneaus (condensation on seed particles created by the gas itself) and heterogeneaus (condensation on preexisting seed particles1 nucleation in the Langley 0.3m Transonic Cryogenic Tunnel (TCT). Recent experiments have resulted in the determination of the onset of homogeneaus nucleation in a higher range of local pressure {up to 1.5 atm) and temperature {up to 72K) than has been found in the nitrogen literature. The data can be compared to calculatons which utilize not only the standard classical liquid droplet theory (CLDT) but also corrections to CLDT for surface tension and for differences in the energy states between the droplet and the bulk liquid. Other experiments involving static pressure measurements in the 0.3m TCT have resulted in estimates for the size and number of background seed particles in the flow. This information is crucial in order to predict whether the pre-existing seeds can cause effects before homogeneaus nucleation is predicted to occur.
1033
NOMENCLATURE
cP c D g AR J K k t M p q r T AT t V w X a rrep A p a
static pressure coefficient, see Eq. 7 airfoil chord, 0.152m Tolman constant, m, see Eq. 3 mass fraction of condensate difference in entha}pies ~ftween gas and condensate, J/kg nucleation ra~e, (m_1sec) rate term, {m "sec) , see E~~ 1 Boltzmann constant, 1.38x10- J/K mean-free path, m Mach number pressure, atm (1 atm=101kPa) dynamic pressure, atm radius of droplet, m temperature, K supercooling, K time, sec specific volume, m3/kg term representing energy barrier, see Eq. 1 linear dimension along airfoil chord, m angle of attack, degrees correction factor to nucleation rate, see Eq. 4 thermal condu§tivity, J/m"K"sec) density, kg/m surface tension, N/m
Subscripts and Superscripts
t liquid properties sat saturated conditions on vapor-pressure curve t conditions in settling chamber of tunnel * critical radius ~ freestream conditions in test section
THEORETICAL BACKGROUND
The necessary ingredients for a mathematical model to predict condensation effects are a description of homogeneaus nucleation, an idea of the size and number of pre-existing seed particles, and a droplet growth equation.
Homogeneaus Nucleation
The first attempts to predict homogeneaus nucleation centered around the CLDT. (CLDT is also known as the capillary approximation.) A complete description of the CLDT can be found in
1034
Wegener and Mack, 2 Kotake and Glass, 3 and Reiss. 4 The general form of the resulting equation for the rate of formation of the nucleating sites,- or seeds, can be expressed as
-W/kT JCLDT = Ke
(1)
where K is a rate term which takes into account how many molecules are striking the forming cluster per unit time, and W represents the height of the energy barrier to the creation of a * "critical-sized" droplet. A "critical-sized" droplet of radius r is in metastable equilibrium with the surrounding vapor. (If one additional molecule joins the droplet, then the droplet will grow, or if a molecule is lost, it evaporates.) When homogeneaus nucleation is occuiring, it is generally assumed that droplets of critical radius r are formed and are subsequently given a chance to grow. The form of the term W is
a3
W = Constant x I~~-;-~:{;/;:::>]2- (2)
A key assumption of the CLDT is that the properties of the critical-sized droplets, which typically contain only 50 molecules for the experiments discussed herein, can be described by their bulk, or macroscopic, values. Whether this assumption is tr~e for the surface tension a is open to question. With the a term in W, an error in a of only 10 percent may lead to
an error in JCLDT of 6 orders of magnitude and may lead to an error in the predicted onset temperatures for homogeneaus nucleation of at least 3K for cryogenic tunnels. One proposed correction to a to account for the small size of droplets is due to Tolman5 who calculated that for small droplets the correction
0 o (3) a = ~-~-~~
r should be applied to a , the value of surface tension for a
0 planar surface of liquid. The constant D is now known as the Tolman constant. This correction, however, is controversial, and even Tolman sounded a note of caution in using Eq. 3 to predict the surface tension of droplets as small as those normally associated with critical-sized droplets.
A second major assumption of CLDT is that there are no differences in the energy states of a critical-sized droplet surrounded by vapor and a critical-sized droplet surrounded by the bulk liquid state. To correct for this assumption, the CLDT nucleation rate must be multiplied by a correction factor,
1035
rep J • r JCLDT (4)
There are disagreements in the literature concerning the estimation of the terms in rrep • Lothe and Pound6 estimated rrep to be on the order of 1217 in their 71assic paper of
1962. Later comments by Reiss and Kikuchi suggest that the Lothe and Pound correction is only appropriate for condensate in the solid phase, which would be expected if nucleation takes place at some temperature below the triple point temperature. For liquid droplets, Reise and Kikuchi both come to the conclusion that the replacement factor can be approximately represented by
(5)
if one does not take into account curvature effects on the surface tension.
Heterogeneaus Nucleation on Pre-existing Seed Particles
Not only is homogeneous nucleation important in predicting the onset of effects in cryogenic wind tunnels, but heterogeneaus nucleation is important as well. At transonic speeds and at low temperatures, the flow velocity is only on the order of 180m/sec. Thus, if seed particles are present in sufficient numbers, there may be enough condensate on the seeds during the flow over !arge models to influence the aerodynamic data.
Droplet Growth Equation
A growth equation is needed to describe the growth both on the pre-existing seeds as well as on critical-sized dropl§ts which result from nucleation. The growth equation by Gyarmathy is designed to be applicable for all droplet-size regimes, free molecular and continuum, and is used in the current calculations. For nitrogen, it is given by
dr dt
).tl-r,r' ----~-----~---- x (T (p)-T) pt ßH(r+2.71 i) sat .
where Tsat (p) is the saturation temperature at the gas pressure.
1036
(6)
COMPARISON OF THEORY AND EXPERIMENT
Estimate of Seed Size and Number
In order to detect the influence of condensation on preexisting seed particles in the 0.3m TCT, solid top and bottom walls were installed in the test section so that the differences in static pressure gradient down the contraction cone and through the test section could be compared for condensation-free tests as well as for tests with varying amounts of condensation. By employing a one-dimensional flow analysis such as that presented by Wegener and Mack, the actual distribution of liquid mass fraction g down the tunnel could be determined. The Gyarmathy growth equation was then employed to match the observed growth in the test section to that calculated for estimated values of seed size and number. Preliminary results indic~~e that the radius of the seed particles is ar~roximately 0.25x10 m and the number density is about 3.0x10 seeds per kilogram of the gas.
Condensation Over the CAST-10 Airfoil
In order to observe condensation effects due to homogeneaus nucleation, a 0.152m CAST-10 airfoil was tested in the 0.3m TCT. (The airfoil was at Langley during a cooperative program between the West German DFVLR and NASA.) The tunnel test section was outfitted with slotted top and bottom walls to reduce wall interference. The freestream Mach number M was held constant at
00
0.65, and the airfoil was set at an angle of attack of 6 degrees. With these conditions, the maximum local Mach number generated over the airfoil was approximately 1.4 for the range of total pressures tested. (This local Mach number combined with experimental total pressures of up to 5 atm allowed the onset of homogeneaus nucleation to occur above the triple point. Data above the triple point are necessary for the Reiss and Kikuchi theory comparisons.) Baseline, or condensation-free, pressure coefficients Cp over the first 35 percent of the chord distance x/c are shown in Fig. 1, where Cp is defined as
p-poo c =-
p qoo (7)
This initial supersonic region over the airfoil will be used for comparison to theory because of the one-dimensional nature of the analysis, which models the flow about the two-dimensional airfoil with the isentropic area distribution of the streamtube just above the airfoil surface. When condensation occurs in this streamtube over the airfoil, the area distribution changes to
1037
Fig. 1.
-2.2
-2.0
-1.8
c -1.6 p
-1.4
-1.2
-1.0
-.8
0
0
0 0 0 0 00
.1 .2
xlc
0
0
.3 .4
Pressure coefficients in region of interest for CAST-10. No effects present. M ~ 0.65 and a = 6°.
avoid choking at Mach 1 due to the effective heat addition. In the one-dimensional model, however, the area distribution is not allowed to change, and the calculated flow will choke at any sonic condition downstream of the condensation. Consequently, the theory and experiment can most easily be compared in the supersonic region of the airfoil aft of the sonic line but before the recompression shock. This area limitation of the analysis restricts the utility of the one-dimensional model to only qualitatively predicting the magnitude of effects once condensation is actively occurring. The actual total pressure and temperature at which onset begins, nevertheless, should be well approximated.
The effects of condensation will be shown by plotting only the differences in pressure coefficient AC caused by condensation. These measured differences in pressßre coefficient will be compared to predicted differences from one-dimensional calculations u!fnizing CLDT; CLDT as modified by Tolman with D = 0.25x10 m, labelled CLDT-T; CLDT as modified by Reiss and Kikuchi, labelled CLDT-RK; and CLDT as modified by Lothe and Pound, labelled CLDT-LP.
The first data comparison with theory is shown in Fig. 2. For these data, the total pressures Pt of the test were near 4.0 atm. The total temperature Tt was varied from values which did not permit saturation in the supersonic region over airfoil down to those temperatures shown. For Tt = 96K, the data exhibit
1038
CLDT-LP\
.15 I Pt= 3. 7 atm
Tt = 96 K
fl.T = 18 K .10
fl.C 05 p .
I I
J
. 10 .15 .20 .25 .30
x/c
Pt= 3.6 atm
Tt = 94 K fl.T = 20 K \./CLDT -T
7
I?'~ClDT-RK _9,.;/ ...
0 .05 .10 .15 .20 .25 . 30
x/c
Fig. 2. Comparison of data (circles) and theory. a = 6°.
M =0.65 and 00
little or no effects even though the value of supercooling AT at the maximum local Mach number of the flow is 18K, where AT is defined as the difference along the expansion isentrope between the static temperature at which the flow is saturated and the static temperature at which the flow attains the maximum local Mach number over the airfoil. At 96K, the calculation for CLDT-LP clearly overpredicts any possible trend in the data. At Tt = 94K where the supercooling has only increased from 18K to 20K, there are very definite experimental effects. While the CLDT shows no effects (the line does not appear above the zero-effects axis), both CLDT-T and CLDT-RK show effects of the same order of magnitude as the data. Again the CLDT-LP calculation clearly overpredicts the magnitudes of the effects. Since in this example the condensate occurs above the triple point and is in the liquid phase, the overprediction of the Lothe and Pound correction is consistent with the already mentioned Observations that their correction may only be applicable to solid condensate.
A major question concerns what influence the existence of seed particles might have on the above calculatons for the onset of homogeneous nucleation. Fig. 3 shows the calculations for CLDT-T repeated at both 96K and 94K assuming there are 3.0xl012 seeds per kilogram of the gas and that these seeds are 0.2Sxl06m in radius. There are very small predicted effects at 96K with seeds present and, in fact, a slight reduction of effects at 94K Since the magnitude of these additional effects due to seeds is slight and within the experimental uncertainty, it will be assumed that the pre-existing seeds do not influence the onset of homogeneaus nucleation for the present tests.
1039
.15
.10
h.Cp. 05
p1 = 3. 7 atm
T-96K t-
SEE~S 1_ o~~~~~rF~~~-
- . 05 IL_...J,'-,---l-=-~'-=-~'---'' 0 . 05 .10 . 20 . 25 . 30
x/c
p1=3.6atm
T = 94 K ,: t d
No €h SEEDS (;\~;;; """'-- SEEDS
~"'
0 .05 .10 .15 .20 .25 .30 x/c
Fig 3. Influence of seeds on calculated effects using CLDT-T. r ~ 0.25x10-6m, N = 3.0xl012 seeds per kg of gas.
The next comparison, shown in Fig. 4, is for a test of the airfoil at M~ • 0.65 and at a constant value of Pt = 5.0 atm. At Tt.• 101K with AT= 18K, there ·may or may not be effects in the data--it is difficult to judge because the differences shown are typical of the experimental uncertainty. Both CLDT-T and CLDT-RK predict small effects. At Tt • 99K with AT • 20K, effects are obvious as are the deviations predicted by CLDT-T, CLDT-RK, and CLDT. Once again, both CLDT-T and CLDT-RK are close to the magnitudes of effects. The fact that the deviations in the data do not continue to grow at the rate predicted by the one-dimensional model should be expected, because of the limitations of the one-dimensional model in predicting effects over the two-dimensional airfoil. The CLDT-LP comparison is not shown in this example because the condensate is again in the liquid phase.
.15
.10
h.C 05 p .
0~~~~~~~~~
-.05 0 .05 .10 .25 . 30
CLDT-T\
\=99K / /-cLDT- RK ßT = 20 K I I 0
,"'/§ 8 /,g CLDT\
0 .05 .10 .15 .20 .25 .30 x/c
Fig. 4 Comparison of data (circles) and theory. a • 6°.
M = 0.65 and ~
1040
SUMMARY OF RESULTS
Onset of both homogeneaus and heterogeneaus nucleation for nitrogen gas has been determined in the Langley 0.3m TCT. Homogeneous nucleation occurring above the triple point has been measured for a DFVLR CAST-10 airfoil and is used to evaluate the classical liquid droplet theory and several proposed corrections to it. Good agreement with data is found when combining the classical theory with a Tolman constant of D = 0.25x10-10m, or when using a correction derived by both Reiss and Kikuchi for differences in translational energy states. Poor agreement is found using the Lothe and Pound correction and is attributed to the liquid phase condensate.
Onset of heterogeneaus nucleation on pre-existing seed particles in the flow has been studie~6 and prelimiy~ry estimates of seed radius and number are 0.25x10 m and 3.0x10 per kilogram of the gas. It is shown that this pre-existing seed population is insufficient to influence the observed onset of homogeneaus nucleation.
REFERENCES
1. R. Hall, Onset of Condensation Effects in Cryogenic Wind Tunnels. Paper presented at 1980 Workshop on High Reynolds Number Research, NASA CP-2183, 1981.
2. P. Wegener and L. Mack, Condensation in Supersonic and Hypersonic Wind Tunnels, in: "Advances in Applied Mechanics," Vol. V, H. Dryden and Th. Von Karman, Eds., Academic Press, 1958.
3. S. Kotake and I. Glass, Flows with Nucleation and Condensation, in "Progress in Aerospace Sei.," Vol. 19, Pergamon Press, Ltd, 1981.
4 H. Reiss, The Replacement Free Energy in Nucleation Theory, in: "Advances in Colloid and Interface Sei.," Vol. 7, Elsevier Sei. Pub. Co., 1977.
5. R. Tolman, The Effects of Droplet Size on Surface Tension, J. Chem. Phys., Vol. 17, No. 3, March 1949, pp. 333-337.
6. J. Lothe and G. Pound, Reconsiderations of Nucleation Theory, J. Chem. Phys., Vol. 36, No. 8, April 15, 1982, PP• 2080-2085.
7. R. Kikuchi, Statistical Mechancs of a Liquid Droplet, in : "Advances in Colloid and Interface Sei.," Vol. 7, Elsevier Sei. Pub. Co., 1977.
8. G. Gyarmathy, Condensation in Flowing Steam, in: "TwoPhase Steam Flow in Turbines and Separators," M. Moore and c. Sieverding. Eds., Hemisphere Pub. Corp., Washington, 1976.
1041
INVESTIGATION OF NONEQUILIBRIUM HOMOGENEOUS GAS CONDENSATION
A.L. Itkin, U.G. Pirumov, and Yu.A. Rijov Moscow Aviation Institute 125871, USSR
INTRODUCTION
In various technological processes nonequilibrium homogeneaus condensation may take place.in a rather wide range of initial temperatures and pressures.For practical needs it is of interest to study nonequilibrium condensation starting from the Cl ... i tical point and furti1er to the temperatures considerably lower than that at the triple point.Evidently,rnarked departures of real gas thermophysical parameters from those of an ideal gas occur under these conditions.
THE INFLUEHCE OF REAL GAS PROl'EH.TIE.S
Considerations of real gas properties may influence the cri tical nucleus radius, the rate of j_ts formation, the level of supercooling end the mean radius of clasters significantly.This is the objective of investigating the influence of theTinophysical characteristics upon the rate of critical nuclei formation.The results of calculations done for water5 vapor by raeans of the Frenlcel' -Zel' dovich formula a-re compared to those obtained from the formulas considering the influence of real gas properti,s)Figure 1 shows the dependences c:; = C3(T8 ) useä in - (curves 1,2).It demonstrates o..lso
the dependences I= I( G' ( T,"), T) for each of the c9.ses, where ~ -surface tensionQcoefficient,I-rate of critical nuclei formation, T -the saturation tempe1 ... ature and T-the e;as temperature. 8 0ne sees that while in the
1043
Supersaturation region s=4(s=P/P8 ,where P-pressure, P -saturation pressure) the values of I differ by an o~der of magnitude,at s=2 they differ by three orders of magnitude.The same figure shows the water vaporll~quid saturation curves (curves 4,5) borrowed from ' and the rates of nuclei formation I=I(P (T ),T).The values of I differ from an order of magni~ud~ at s=4 up to 8 orders at s=2.Curves 6,7 present the critical radius r* as a function of ternperature calculated by rneans of Thornson's forrnula using the ideal gas chemical pot~ntial and the real [;as chemical potential calculated in .The name paper gives the corresponding values of I.'l'he raaximurn difference in I reaches 10 orders of magnitude.The e.uthor' s calculations show also that e, 5% error in detel'raination of liquid's density at T=JOOK results in I varying 7 fold.Cu:cve 3 presents the dependence of the ratio of densities obtained from the state equations of ideal and real gases on the saturation line.
A COilllZCTIOH MULTIPLIER
The above results show that the error in determina.tion of thet·mophysica.l pt:u~allleters influence the rate of nuclei formation and the mixture enthalpy and the value
es [dyne/cenfimete-z] Ji.cleoej Jt e ~tO~ I=-8.0·10~ t'A[A]
8 I•/,3-/016 :--s::2;S' _ -----• : I•l.O·IO 1~ I, OS ~
~' __ '!..., ___
I .: S=2 I I
62 'b :;;--, T 0,95
~ 0 1-J (J V 0,65 52 \) ....... "' ~
E 1'1" •...) 40 0,75 42 .,.., ...... 6 tl c: ........ ~ I ~65 32 " ~ I ~ .... Ql I 'W c: ,ss 22 t: ~ 5 20 I ~ I
~ '-' I ~"'s 12 a.."' I ,.....
Llt 4
I
420 -170 520 STO
1044
of maximum supercooling and the mean condensate particle size significantly.As a matter of fact,the maximum difference is attained at low values of supercooling when the absolute value of I is small.
It is of interest to evaluate the influence of error in determining the critical size nuclei formation rate upon the value of supercooling and the mean size. of the conden,aee particles.In order to do this,the technique of ' has been used to calculate homogeneaus condensation in nozzles while the classic formula for I was4supple~ented with a multipliert which varied from 10- to 10 .Figure 2 shows the dependences of the gas temperature,the value of supercooling and the mean radius of condensate partiales upon ~ .One sees that changing p by two orders of magni tude resul ts in the change of A T by 4-5K and the change of rm by 25-200A. An agreement o~ experimental and calculated'data may be obtained by choosing the value of ~ and/or by introdueine a correction multiplier into the enrgy of cluster formation based upon a series of some reference experimen95!1ifuch an approach has been realized,for instanse, in .However i t :cesul ts in an incor:rect determination of the mean radius of condensate particles o.nd the distribution function of the condensate particle's sizes.
t.7oo[ K}
.SV
40
20
10
T{k) ,.f
- .c.Too=7$(P)-7{P)
sso .f50
2tro
{!iC
rmfAJ ~~-
' t • ..,
·~ (,
){t.=O '('-.::: -lcm c. F~ =1,4cm2
Mi=2 Tc. :.3001(
e" p. = 28 2Q-5338 " , -T · qi =-0 '
P [dyne/c.rn2) o~~~~---~~~~--~~~------~so~+-~
3,t/ ,4 F [c~niilnete..,_f.J
. Jig. 2. The dependences of t!le gas tempe1·ature T, the value of supercooling T and the mean radius rEl of condensate particle upon a multiplier.
1045
In order to obtain agreement of calculated and experimental data,a correction multiplier must be introduced into the condensation coefficient too.
THE DETERMINING PARAMETERS OF NONEQUILIBRI~l CONDENSATION
The problern of choosing determining parameters is extremely important for parametric,numerical and experimental studies of nonequilibrium condensation.Let us used a system of stationary equations describing the process of homogeneaus condensation in the first approximation in a certain coordinate system.According to the author's calculations,the influence of nonstationartty of the process of expansion upon the flow parameters vanishes in 200J"s;thus,such a system of equations proves to be valid for quite a number of real systems(see Fig.J).Let gas be expanding in accordance with a certa~n law P=f(T.} while for isentropic expansion P"" T ,where n=k/(k-1) and k-adiabatic exponent.Then,the gas density is a function of T only.Let us further assume that condensation starts to influence the gas dynamic parameters considerably at the point where the condensate's mass fraction q =const.The latter point will be considered to be. point of condensation initiation.Then,in the interval from T8 to T0
dq0 /dt=(dq0 /dT)(dT/dt)=F(T)
dq /dT=T-1F(T) c
( 1)
Let us still further assume that the interval7from T9 to T is small for the dT/dt grate change and dT/dt is equal to dT/dt at the dew point T8 (dTidt is the ~ate of coolihg,t-t~me).Then,from Eq.(1 ) one gets
• 1 T • 1 q =T- ° F(T)dT=T- F1(T9 ,T0 )
c s T s s and,hence,
• AT =T -T =F2(q ,T ,T ) s s c c s s
(2)
Similarly, i t may be shovrn that . r =l!, (q T T )
m 3 c' s' s Since the process of condensation is assumed to start at the point where q =const,it follows from Eqs.(2,3) that the saturation temperature T9 and the rate of
1046
P[d
yne/
cm2)
d ifJ
!t
10 ·~~$·a.s-
ro -~
';O
O
(ofT
fro
OJJ
M?S
.S" ,,
-100
AT
vo[K
] '1,
31
1;19
1,2
1 1,
25 1
,23
1,2
4 4o
~ ~,.
-~~~
~""-
~~-.
-/ ...... :::'
_L _-:
:-:._
30
20
fO
W.·
=48
400
' cm
fs
~ ~
i )(
1i,=
37.?
k P
:=7·
/0S
' (.
qc=
o o r
~--=-
>
F~ •
1,4
em2
T[K)
I
~ 3
4 J
es 7
a 9
xc;m
) T
[l{}
1-
t=O
-(i
..ni
.tl.
<:~e
afp
"U>
x.i.m
o·
2-t
= -/O
O~s
~c:on)
3-/:
.:t~
200f
S ( s
lof.
«xea
~ /fo
....r)
ro,#
K=
l,a~
'o
,.~-
-,
1 l'o
~4·t
o .. j.s
-4lf,6
j 3
0,81
(4 f -
---X
Ft
4.
10
?
10 7
29
o
' '
I I
• D
'
' .
...
1 2
3 ."!f
:. I
1 ~
~ 4
s 6
7 8
!I
x [c
m]
Pig
. 3
. T
he
dep
end
ence
s o
f th
e v
alu
e
of
sup
erc
oo
lin
g
up
on
th
e ad
iab
ati
c e
xp
on
ent
and
th
e
stag
nati
on
~
tem
per
atu
re
and
th
e
dep
end
ence
s o
f th
e
gas
......
tem
per
atu
re
up
on
th
e
tim
e.
cooling T at the dew point T are the determining parameters Öf the process of n9Hequilibrium condensation. It can be easily shovvn that T is a function of stagnation temperature T ,initial c~oss-section dirunete~ d and adiabatic expoHent k and ~epends on T0 and k sli~htly only (see Fig.J).Thus,T and T (or d ) ar~ the determining parameters of the ~rocesssof noHequilibrium condensation.It sho~~d 1 ~e stressed that a BU~2er of similarity relations in ' of the tyne PT ad =const,at r_= iconst follow directly froQ ~he 9oHdi~ion T =const,m T =co~st.For monatomic gases it ~~llows froM the condi-~!on T =const that a=k/(k-1).In the importance of f is ~tressed which is equivalent to the importance of Ts.Thus,the number of experiments necessary to determine ~ T and rm at various T0 and P0 may be made much smal
ler.8
:m3ULTS
The authors have been carried out parsmetric studies of nonequilibrium condensation for water vapors at various values of the determining parru~eters T and T .The results are presented in Figure 4.Their anRlysis si1ow that the growth of T is accompanied with diminishing of 6T ,while r grÖws at the point of condensation in the ~ange of Waturation temperatures from 270 to 420K.However,the growth of T results in a strenger influence of condensation upon ~he flow pararaete1·s. Thus, if at T0 =400K the calculetions give a pressure jump at the poifit of condensation,at T =300K condensation proceeds without pressure growth Rt the condensation zone. The growth of 1!:. T which accompanies the drop of T may be explained as fÖllows:the drop of T results in tliminishing of the mixture' s densi ty and a growth of the f:('ee molecular path;thus,time needed for a nucleus to attain a given size,becomes larger.On the other hand,a drop of T at a fixed supercooling is accompanied wi th a gro;vth of the c~itical radius of the condensate particles which also resul ts in the time neces~al1Y for a nucleus to attain the size and,hence,for T =const the value of the maximum supercooling 1:. T8 , becom~ng larger •
• While T grows at T =const, I::J. T also t>ecomes lar-
ger in accor~ance with t:He l1 elation 8 T -T 1/B. lJ:Ihe meli-n radius of the condensate partic~es s diminishes whil-e Ts grows.It should be pointed out that variation of Ts is equivalent to varying ' (see Fig.2).
1048
rmfRJ
F'i.8.4 ' 1to t·832rs
NO 5
(20
/00
2 3 4 . .s 400 400 400 100
3 3 3 .3
0 0 0 0
2,7 1,0 o,s Ft.8.s O,.OS" 0,02 o_ol
4 6 ß 10 12 N 16 -18 ~0 AToo[K]
The dependences of the max:imum value of supercooling, the mean radius r and the gas te:.-nperature upon the determini!He parrunete:;.~s T and s Ts.
Fig. 5. Expansion of a gaseaus sphere in a com~dinate system x-t and the dependences of the supercooling upon the time.
1049
COlJDJ~NS.'\.TIOH IN A RAHEFACTIONS WAVE
The numerical techniques and computer developed by the authors allow to study a vapor condensation in the subsonic and supersonic parts of a nozzle.Investigation of vapor condensation in1 th~ 5transonic part of a nozzle has been carried out in 4, .However,condensation at small Mach numbers has been studied insufficiently.To do it the present work deals with the problern of expansion of a gaseaus sphere initially at rest into vacuum. The vapor condensates in a rarefactions wave travelling to the center of the sphere.Such an expansion in the x,t-plane is presented in Figure 5.In the course of gas expansion into vacuum,the spherical boundary which will be further called a piston moves with a constant velocity u=(2/(k-1))a ,where a is the speed of the sound of the gas at rest.~he condeHsation zone originates at the piston and propagates towards the center of the sphere.The intensity of condensation sho9k is lower towards the center of the sphere,because T is lower.But the condensation autooscillations invest~gated in 14,15 is absent in aut~or's calculations.
REFE:aElifCES
1 •
2.
3.
4.
6.
7.
1050
L. H. Davidov, Investigation of nonequilibrium condensation in supersonic nozzles and jets, Information of the USSR Academ of Seiences "Mechanics o Liguid and Gases", 3: -in Russ1an.
11.. V. Chirikhin, The method of calculation of supercooling of water vapor flow in a supersonic nozzle, High-temperature Heat Physics, 1:132 (1980) -in Russian.
N. B. Vargaftik, V. N. Volkov, and L. D. Volyak, About international tables of H20 surface tension, Heat-and-Power Engineering, 5:73 (1979) -in Russian.
A. A. Alexandrov, and z. A. Ershova, Vapor-liquid curve equation for H20 and n2o, Journal of Engineering Physics, 5:854 (1981 ) -in Russian.
L. E. Sternin, "The Principles of Gas Dynamics of Two-Phase Flows in Nozzles", :Mashinostroyeniye, Moscow (1976) -in Russian.
s. L. Rivkin, and E. A. Kremnevskaya, H 0 water and vapor state equation for machine calEulations of the processes and electrostation equipment, Heat-and-Power Engineering, 3:69 (1977) -in Russian.
U. G. Pirumov, and G. s. Hoslyakov, "Gas Flows in :Nozzles", Moscow State University, Moscow (1978)
-in Russian. 8. V. u. Gorbunov, A. L. Itkin, u. G. Pirwnov, and
Yu. A. Ryzhov, The nonequilibrium homogeneaus gas condensation, in 12th Rare.fied Gas Dynamic, USA, July 7-11, Book o.f abstracts, 20 (1980).
9. L. M. Davidov, and U. G. Pirwnov, Some questions o.f nonequlibrium homogeneaus condensation o.f gases in high speed .flows, Information o.f the USSR Academ.y o.f Seiences "Mechanics of Liquid and Gases", 6:81 (1978) -in Russian.
10. M. E. Deich, and G. A. Philippov, 0 Gas Dynamics o.f Two-Phase Atmospheres", Energia, Moscow (1981) -in Russian.
11. A. V. Kurshakov, R. A. Tkalenko, and G. A. Saltanov, Theoretical and experimental investigations o.f condensation in center rare wave, Journal o.f Applied Mechanics and P~ysics, 5:117 (1971) -in Russian.
12. 0. F. Hagena, Surface Science, 101 (1981) 13. A. A. Vostrikov, A. K. Rebrov, and Semyachkin, Con
densation of SF6,CF?,C12 ,co? in expanding jets, Journal of technicai Ph~sics, 11:2425 (1980)in Russian.
14. G,.A. Saltanov, and R. A. Tkalenko, Investigation of transonic nonstatyonary flow with phase transition, Journal of ABplied Mechanics and Physics, 6:42 (1975) -in Russ~an.
15. D. Barschdorf, and G. A. Philippov, Analysis of some special condition of Laval nozzle work with local heut input, Information of the USSR Acade~y of Seiences "Power En . .,.~neer~n- und Trans ort", 3: 4 970 in Russian.
1051
THE PECULIARITIES OF COJDEISATIOI PROCESS
IN CONICAL NOZZLE AND IN FREE JET BEHIND IT
P.A.Skovorodko
Institute of Thermophysics, Siberian Branch of the USSR Academy of Seiences Novosibirsk 630090, USSR
INTRODUCTION
Conical nozzles find increasing favour in experimental investigations of condensation in expanding gas flows. Beylich's investigation made on the basis of the cluster scattered light intensity measurements show that, the intensity distributions in the direction perpendicular to the jet axis have the peaks at a certain distance from the axis. The position of the peaks as well as their value depend in a complex manner on the stagnation conditions (p0 , T0 ), the peaks being not observed in a certain region of p0 , To• -
The measurements of the mean cluster size (N) in a molecular beam s~pled from the free jet were made by Hagena and Obert • It was found, in particular that, the data for Ar jet behind a con!cal nozzle reveal a reproducible stepwise structure of I(p0 ) dependence. The corresponding dependences for jet behind a sonic nozzle are of smooth monotonaus structure.
Up-to-date the above-mentioned effects have not been elucidated sufficiently.
This paper presents the calculations of CO and Ar flows as applied to the experimental conditions~ Based on the comparison of computed and experimental results an explanation of the above peculiarities is suggested.
COMPUTATIONAL METROD
To compute a steady two-dimensional flow in a jet expanding into vacuum from a conical nozzle, the numeri-
1053
cal method utiliz.ing a natural coordinate system defined by streamlines and l~nes normal to them, proposed by Bounton and Thomson , was used. The following two types of calculations was made.
a) Calculations of flow with condensation taking into account viscosity effects. Such calculations were performed for CO and Ar flows. The nonequilibrium homogeneous condensation process is described in the frames of classical nucleation theor.y. The method consists in combining4the method for viscous heat-conducting perfect5 gas flow together with that for flow with condensation. The following expressions for Ar thermodynamic values were used (See notation in Ref. 5): 6 (dyn/cm)= 17.943 - 0.0541·T, P- (torr) = exp(16.11 - 827.2/T), 9~ (g/cm) = 1.9183- 0.0060224·T, ~ = 5/3, v = 1, L•1.72io9 erg/g. The corresponding values for CO are given in Rer. 5. The 6 (T) dependences were obti!ned properly choosing from the condition of the best agreement of calculated results with the available experimental data for free jets behind the sonic nozzles.
b) Calculations taking into account simultaneaus course of condensation and vibrational relaxation of co2• Such calculations were performed without taking into account viscosity and heat-conductivity effects. The method cons~sts in combining the method for flow with condensation 5together with that for flow with vibrational relaxation • The problem is solved in the simplest formulation, i.e. vibrationally excited molecules are assumed to be condensed in the same manner as unexcited ones, and the presence of condensate does not change the vibrational relaxation time.
RESULTS AND DISCUSSION
Analysis of experimental data obtained by Beylich1• According to the experimental conditions the calculations of CO? flow were made for a conical nozzle with 0.56 ~ throat diameter d*' 3.542 mm exit diameter d and 11 half-angle 'I • The stagnation parameters we~e varied over the rangea 4 "- p0 (barg '= 12, 288~T0 (K)~390. Computed distributions of I = n•r were compared with experiment, n being the number density and r a mean radius of the clusters. Three series of calculations have been made.
a) The flow inside the nozzle was assumed to be one-dimensional (conical source), two-dimensionality began in the jet behind the nozzle exit. It was expected that the gradients appearing in the rarefaction wave can result in nonuniform distributions of I. But it did not happen, corresponding distributions were of smooth, bellshaped profile.
1054
100 r--J ----.-----o-~,&.-:-,-"r.,-,K--, 12 JQO
ll 28! 12 288 8 JOO 8 28! 8 28!
10
- tltro,~ -e- •rp•rtm.nt
.:c/d", • 2.~6
Fig. 1 Distributions of scattered light intensity.
b) In the flow pattern the wall boundary layer effect was taken into account, but in this case the peaks do'nt appear either.
c) It was assumed that, in the nozzle throat, from which the calculations started, streamlines were parallel to the symmetry axis, and the nozzle wall contour represented a junction of a circular arc with a straight line. The nozzle flow in this case is not one-dimensional. The computed intensity distributions obtained in this case, shown by solid lines for some regimes in ~ig. 1, are in evident qualitative agreement with the experimental ones (points).
Analysis of computed results permitted to establish causes leading to the appearance of peaks in the intensity distributio~s.
It is known that, at the junction of the throat profile and the cone an oblique shock wave originates in the conical nozzle flow. The shock reaches the axis downstream, reflects from it, and the nozzle flow is thus characterized by a complex shock wave structure. The shock wave intensity is low and the difference of gasdynamic parameters of such a flow from the parameters of conical source flow does not exceed several tens per cent?
Figure 2 in x/r - y/r* coordinates shows the nozzle contour and the position of an incident and reflected shocks (dashed lines). Solid lines represent the conden-
1055
2 'd/t;,
0 2 3 4
Fig. 2. The position of shock waves and condensation fronts for conical nozzle flow.
sation front positions (points where the nucleation rate in the given streamtube reaches its maximum) for four regimes, T0 = 288 K and p0 (bar) = 12, 8, 4, curves 1, 2, 3, respectively, and T0 = 339 K, Po = 8 bar - curve 4. Clearly seen is the salient point on the condensation front when it crosses the incident shock.
It happens because the behaviour of t~e streamtube area changes at the moment of shock passage. This phenomenon is illustrated in Fig. 3, where the dependence of A/A* (relative streamtube area) vs x/r* is presented for the nozzle axis (curve 1) as well as for the streamlines bounding 20% and 52% of gas flow rate ( curves 2 and 3, respectively). On the flow axis where disturbance focusing takes place, the oscillating behaviour of A/A*(x/r*) is observed at the moment when the incident shock reaches the axis. Away from the axis the change in the gas expansion degree at the moment of shock passage becomes more smooth. The decrease of the streamtube area expansion rate up to the streamtube convergence in the nearaxial regiqn leads to the decrease of nucleation rate, the latter being di:f:ferent ·for various streamtubes. This
3 A/ A.,.
2
2
1 - o'/. 2 -207. 3 -52/.
X/r,. 3
Fig. 3. The behaviour of relative expansion degree for various streamtubes.
1056
0 Q2 0 .. 4 0.6 0.8
Fig. 4. N, q and r distributions at the nozzle exit.
results in a highly nonuniform transverse distribution of the number of drops per unit mass of the vapour-condensate mixture (N), formed at the condensation front. Behind this front the supersaturation is eliminated mainly due to the increase of cluster size, but not by the formation of new nuclei. Therefore, the nonuniformity of the N distribution among the streamtubes developed at -the condensation front will be invariable downstream both inside the nozzle and in the jet behind it. On the other band, the condensate mass fraction (q) is almest constant in the transverse direction both in the nozzle exit and in the jet because its value depends mainly on the flow expansion degree but not on the condensation process kinetics. This means that, the nonuniformity of the N distribution causes the nonuniformity in the r distribution, since q~N·r3. This situation is illustrated in Fig. 4 where the distributions of N, q and r values at the nozzle exit for Po = 8 bar and T0 = 288 K are shown. In the whole region, except the boundary layer one, these distributions have the above features, and the maximum value of r takes place just inside the streamtube corresponding to the salient point on the condensation front (see Fig. 2).
Presented in Figs. 1 - 4 are the computed results obtained without taking into account CO? vibrational energy relaxation effect. As may be seen from Fig. 1 the calculations at T0 = 300 K are in better agreement with the experiment at T0 = 288 K. It was assumed that this is due to the effect of CO vibrational degrees of freedom. The calculations perfo~ed taking into account simnltaneous course of condensation and vibrational relaxation show that, the vibrational energy relaxation
1057
300 350 400
Fig. 5. The mean cluster radius at the nozzle exit center veraus stagnation temperature.
effect is to a high accuracy equivalent to some pre-heating (A T0 ) of vibrationally nonrelaxing (or nonexciting) gas in the stagnation chamber.
Similar conclusion can be made from the analysis of computed r( T0 ) dependencies at the nozzle exit center for Po = 8 bar, shown in Fig. 5. Curve 1 is obtained without taking into account the vibrational energy relaxation, curve 2 - with taking into account this effect. Nonmonotonaus character of r ( T0 ) dependencies is explained by the above-mentioned causes, namely, by the effect of nonmonotonaus A/A*(x/r*) behav.iour on the nozzle axis. As may be seen, curves 1 and 2 are quite similar, the value of equivalent pre-heating .ö. T0 being about 12 Kat T0 = 288 K and about 30 Kat T0 = 360 K.
The presence of the region of T0 in r ( T0 ) dependence, when the mean cluster radius increases with increasing T0 is unexpected but, in accordance with the above consideration, in the case when with increasing T0 the condensation front moves through the local streamtube convergence region, just such a behaviour of r ( T0 )
must take place inside the given streamtube. Due to the same causes, the situationa when the mean cluster size decreasea with increaaing p0 , are posaible.
It should be noted that, under experimental conditiona the boundary layer effect is small. The computationa performed without taking into account the viacosity and heat-conductivity effects lead to practically the aame results.
Thua, the non-uniformity of tranaverae diatributions of cluster scattered light intensity in the jet behind the conical nozzle ia caused by the ahock paaaage through the nucleation zone. From this point of view practically all the effecta observed by Beylieh can be explained:
a) The poaition of the peaks in the jet flow corres-
1058
ponds to strealine position. b) The decrease in the distance between the peaks
with increasing T0 and decreasing Po is caused by the condrnsation front movement downstream, which results in approaching to the axis the salient point on the condensation front (see Fig. 2).
c) The decrease of peak values up to their disappearance at further increasing T0 is due to the fact that the condensation front is dieplaced behind the point of the incident shock reflection from the axie and is placed in the region of reflected shock whose intensity is much weaker. (In the experiments the peaks disappeared at T0 = 339 K for Po = 8 bar, in the calculations under the same conditions slight peaks are observed, but at T0 = 350 K they disappeared).
Although the qualitative agreement between co!puted and measured data for the distributions of I = n·r is evident, it should be noted that, the absolute values of computed data are systematically 3 - 4 times less than the experimental ones. This is likely to be connected with an approximate nature of the used CO condensation model as well as with a possible effec~ of drop coagulation process.
Anal~sis of experimental data obtained ba Hagena and Obert • According to the experimental con itions the calculations of Ar flow were made for a conical nozzle with d* = 0.14 mm, d = 4.427 mm and ~ =5°. The regime with T0 = 419 KB was considered, thl stagnation pressure being varied over the range 3000~p0 {torr)~ 20000.
Figure 6 represente the compari~on between computed results for the mean cluster size ( N ) veraus p at ~he nozzle exit (curve 1) and experimental data for ~N/Z) ,
~m· ~
2 ~·48K
Ar
~ 4 6
4
2
m~ ~~
4 w• at
2 • 6 2
Fig. 6. The mean cluster size and mass condensate fraction veraus stagnation pressure.
1059
obtained in molecular beam (curve 2). Both the measured and computed dependencies have a stepwise structure, though the computed results are systematically 2 - 4 times below than the experimental ones.
The anal~sis of computed results shows that, the structure of N ( Po ) dependence is mainly due to the strong displacement action of the boundary layer under the experimental conditions - the displacement thickness at the nozzle exit achieves a half of the nozzle exit radius and even exceeds this value at low Po• The displacement action of the boundary layer influences in a manner being dependent on Po upon the streamtubes areas behaviour over the whole region, including the condensation front region, that predetermines nonmonotonaus N ( Po ) dependence.
The computed dependence of the mass condensate fraction at the nozzle exit veraus Po (curve 3) is monotonous and smooth that confirmes the above statement about the conservative nature of this value.
CONCLUSION
The CO and Ar condensation model used in this paper is of2approximate nature, inspite of the fact that the applicability of classical nucleation theory for description of condensation process in conical nozzle flows is undoubtable. In this model the dependence of surface tension on nucleus radius is Lot taken into account. Such processes as crystallization, coagulation and drop fragmentation are not considered also. Calculation without remembering the drop-size distribution function does not allow the drop evaporization process be correctly taken into account. There is a number of phusical processes which are to be accounted for, or to be specified.
Inspite of approximate nature of the model it is safe to say that a good qualitative agreement between computed and experimental results is not accidental and the main features of the condensation process obtained in accordance with this model, are correct.
The performed investigation allow to suggest the explanation of the peculiarities of the condensate parameters behaviour observed in the experiments with conical nozzles. These peculiarities are caused by an extremely high sensitivity of nucleation process to the dependence of flow expansion degree along the streamline. When a smooth flow expansion typical, for example, for free jet behind the sonic nozzle is disturbed by some reasons, such as the shock passage or displacement action of boundary layer, irregular dependence of condensate parameters on determining parameters (p0 , To) should be expected.
1060
For the nozzle or jet gasdynamic flowfield these peculiarities arenot very important. Such value as the mass condensate fraction, determining a thermal effect of condensation on flow parameters, is insensitive to the above peculiarities.
However, in the problems when a mean cluster size is investigated, it becomes necessary to account for them. Care should be taken when transferring the results of the cluster mean size investigations from one conical nozzle to another one using sealing criteria, since in a conical nozzle there can appear the conditions when the mean cluster size increases with increasing T0 and decreasing Po• A correct account of the displacement action of the boundary layer requires a particular attention.
REFERENCES
1. A.
2. o.
3. F.
4. P.
5. N.
6. H.
1. N.
E. Beylich, Condensation in carbon dioxide jet plumes, AIAA Journal, 5:965 (1970). F. Hagena and W. Obert, Cluster formation in expanding supersonic jets: effect of pressure, temperature, nozzle size and test gas, J. Chem. ~·· 5:1793 (1972). P. Boynton and A. Thomson, Numerical computation of steady, supersonic, two-dimensional gas flow in natural coordinates, J. Comput. Phys. 3:379 (1969). A. Skovorodko, Method for calculation of nozzle viscous gas flows, in: "Rarefied Gas Dynamics", Proc.· 6th All-UnionConference, Institute of Thermophysics, Novosibirsk, 2:143 (1980) - In Russian. G. Zharkova, V. V. Prokkoev, A.K. Rebrov, P. A. Skovorodko, and V. N. Yarygin, The effect of nonequilibrium condensation and vibrational relaxation in supersonic expansion of carbon dioxide, in: "Rarefied Gas Dynami es", CEA, R. Campargue, ed., Paris, 2:1141 {1979). M. Darwell and H. Badham, Shock formation in conical nozzles, AIAA Journal, 8:1932 (1963). V. Drozdova, Yu. G. Pirumov, G. s. Roslyakov, and V. P. Sukhorukov, Supersonic flows in conical nozzles, in: "Some Applications of Mesh Method in Gasdynamics", G. S. Roslyakov and 1. A. Tchudov, eds., Moskow University Press, Moskow, 4:129 (1974) - In Russian.
1061
INVESTIGATION OF NONEQUILIBRIUM ARGON CONDENSATION IN SUPERSONIC JET BY MASS-SPECTROMETRY, ELECTRON DIFFRACTION AND VUV EMISSION SPECTROSCOPY
E.T.Verkhovtseva, E.A.Bondarenko, V.I.Yaremenko and Yu.S.Doronin
Physico-Technical Institute of Low Temperaturee UkrSSR Academy of Sciences, Kharkov, USSR
INTRODUCTION
Even the first studies of VUV emission spectra from an electron excited supersonic argon jet discovered an important peculiarity, namely,a strong dependence of the jet spectrum composition and intensity on the stagnation pressure Po and temperature T0 {Fig.1).The investigation of the dependences I/po {T0 ) of atomic and molecular emissions at various Po {I is the intensity of a separate emission of the spectrum, po the gas density at the nozzle entry) revealed a nonmonotonic behavior of I/}b vs Ta [1] • According to the peculiarities observed, the curves I/}b {Ta) of all emissions are divided into two groups shown in Fig.2a as the 106.7 nm resonance line and the continuum with maximum at 127 nm.Group I includes the dependences I/po{To) of ipi--•s0 and 3 P1 -- 1 S0
resonance lines as well as the 107.5 nm short-wave and the 192 nm long-wave continua emitted by molecules during their transition from the ~ghJvibrational levels of the exci ted electron state 3 : into the ground re-pulsive state 1~g{Fig.2b). Group I emissions are radiated by the jet in processes {1)-{5){Fig.2b) the most important of which are both dimer excitation and light cluster dissociation by an electron impact and the cathodoluminescence of clusters containing a few tens of atoms[1,2].The curves I/fl0 {T0 ) of group I feature T1 and Tt maxima and their shift towards higher temperatures at higher pressures.Group II comprises the "main" 127nm lon~nu~ emitted by molecules under the same transition :ELL- ~gbut from the lowest vibrational levels of the
1063
Fig. 1. VUV spectra of supersonic argon jet at various P0 and T0 •
excited state8 Such molecules arise within rather large clusters where the vibrational relaxation occurs throughout the lifetime in the 3:E: state [1 ,2)_. Unlike the curves I/~(T0 ) of group I, the curves of group II display a peculiar sharp kink T3 in a low temperature range.The analysis of I/po(T0 ) curves suggests that their features display changes in the jet phase composition and structure during nonequilibrium gas condensation. This gave an impetus to our studies of condensation of argon issuing through a supersonic nozzle into vacuum. Below presented is the discussion of the results.
BXPERIMENTAL
The dependences I/po (To)(P0 =const) and I/foo (P0 )
(T0 =const) for the argon 106.7nm resonance line and the 127 nm continuum were studied. Unlike the experiments described in Ref.1, ours were made 30 mm off the nozzle exit to ensure the local measurements. Simultaneously, the jet mass composition was measured with a mass-spectrometer, the molecular beam method being used, while· cluster structures, avera.ge size and temperature were
1064
2
100
(a)
il= t06.7nm
D~f.1Pa
D3MR:, i.AzCS.)•e- fi2'('8.1R,,) +e 2.A{('P,)•2Az('Sa)-ll2;(32..'!)•f1z('S.,)
3 f/2 (':E...'~•e/ At(P, .'8.2)•/iz(S.)•t> ' 91 "'A2; ('·'L:;r)+t>
.:, Az •e / li2*CR.'P..,)•A2n_,+., . " "'-.Llz:C·'L.:)+fl2n-z +e
Tempezatuee To,K 5.Cathodotumill<?SCMCe of
cfustezs
(a) Curves I/po (T0 ) at various P0 for the 106.7 nm line and the 127 nm continuum. (b) Curves of potential energy of ground and lower excited states of Ar2 molecule, and processes of VUV spectrum radiation.
determined by the electron diffraction technique. The results obtained were compared with the two-dimensional calculation data on the argon jet issue with condensation carried out at the Institute of Ther.mophysics, Siberian Branch of the USSR Academy of Seiences D].The nonequilibrium condensation was described in terms of the classical nucleation theory.
The experimental setups are shown in Fig.J. Their description is given in Ref.4,5. The setups comprise a vacuum chamber in which gas jet 2 is formed by supersonie conical nozzle I with a throat o.34mm in dia, a
1065
cone angle of 8.6 and the area ratio 36.7. The jet gar<· was condensed onto liquid hydrogen-cooled metal surfacc 3. The gas temperature was varied within 150 to 550 K by heat exchanger 4. The gas pressure was varied from 0.02 to 0.5 MPa.At the argon flow rate of 50 crn3 /s the .vorking pressure in the chamber did not exceed 10-3 Pa.
In the spectroscopic studies the jet was excited by an electron beam of 1 keV, 20 rnA.The bearn diarneter was 3 mm.The jet VUV radiation was dispersed by a SP-68 mo~ochromator and recorded by a FEU-64 photomultiplier coated with sodium salicylate.The mean deviation of thc maximurn and rninimum positions in the curves I/.Po(To) is
(a)
Fig. J. Setup diagrams: (a) spectroscopic; (b) electron diffraction.
±4 K and in the curves I/po (P0 ) i t is ± 10-2 MPa. In the mass-spectrometry studies a molecular beam was produced by a conical skimmer with an orifice 0.5 mm in dia which was placed 30 mm off the nozzle exit. The molecular beam composition was analysed by a radio-frequency rnonopole mass-spectrometer. A diaphragm with the 2 rnm orifice was located in front of the mass-spectrometer transducer. The mass-spectrometer recorded the argon clusters up to Ars inclusive.In the electron diffraction studies the jet was crossed by an electron beam (0.2 rnm in dia, 60 keV) 5 mm off the nozzle exit.The diffraction patterns were recorded photographically. The rneasurements were te.ken. between 0.2 and 0.5 MPa and 150 and 210 K.
1066
The jet contained fcc microcrystals with more than 800 atoms per cluster. At Po< 0.2 MPa, T0 > 220 K there were some difficulties in the measurements due to large electron scattering by the gas atoms. An average cluster size z was determined by diffraction rings broadening, and in the temperature and pressure ranges studied varied between 2.0 and 3.5 nm. An error in the ~ value was ± 55%. An increase in the lattice parameter of 2 nm microcrystals as compared to that of bulk crystals is as low as o.2%~] .Therefore,the temperature dependence of bulk crystal lattice parameter was used to determine the temperature of clusters of 2 ~ 2 nm [7] •
RESULTS AND DISGUSSION
The experimental dependences I/Po (T0 ) (P0 =0.1 M.Pa) and I/Po (P0 )(T0 =275 K) for the 106.7 nm line and 127 nm continuum are shown in Figs.4a and b. A change in the experimental geometry (the jet is electron excited at the distance of 30mm from the nozzle exit instead of 10 mm as was done in previous experiments ~D causes no Variation in the dependence behaviors.The position of temperature peculiarities, however, changes a little when the distance from nozzle exit is 30 mm. The analysis of curves I/po (P0 ,T0 ) and their comparison with mass-spectrometry, electron diffraction and two-dimensional calculation data confirm the assumption that the curve peculiarities reflect the changes in jet phase composition and structure during gas condensation. A high-temperature plateau in the curve I/po(T0 ) for the line indicates that at these issue regimes the jet has an atomic composition and the 106.7 nm line intensity is dictated by process I (Fig.2b). In this case a mass-spectrum contains only the peak of Ar+ monomers.An increase in the 106Q7 nm line intensity with a reduction in temperature from Ti to T2 (the increase correlates with the temperature dependence of dimer signal*) is typical of the earlier stage of gas condensation associated with generation of dimere and small clusters and the increase in their concentration.The line intensity is mainly determined by the contribution of dimer and trimer dissociation by electron impact and the cathodoluminescence of larger clusters.With further reduction in temperatureTo clusters of critical size -"condensation nuclei"- appear in the jet and the developed gas condensation begins. The latent heat liberation causes an increase in the gas temperature and therefore a decrease in the concentration
*Signals from Ar; , Art , Ar; and Ar~ are characterized by similar temperature dependences.
1067
of dimere and small clusters.A decrease in the 106.7 nm line intensity between T2 and T3 which correlates with variation in the dimer signal is due to the fact that the dimer and small cluster concentrations lower and the interaction of electrons with new larger clusters makes a ernaller contribution to the line intensity.The T2-
Fig. 4e
(Q)
IOD 300 500 To,K Q/ 02 03 Ot;P.,MPa.
ec;P. 3.3
3.1
29 \----
Mach nwn&rz
21; 2.5 2.6 ~ T. 0.1 0.2 0.3 Qi; f?,MPa
(a) Curves I/Po (ToHPo=0.1 MPa). (b) Curves I/~(P0 )(T0 =275 K). (c) Positions of T2 , T~, P, T5 peculiarities as a function of T0 and Po• (d~ Two-dimensional calculation data on condensate fraction,vapour temperature and Mach number.
maximum position which marks the onset of developed gas d t ' ' d 'b d b th 1 t' n T -2·7 ±0.2 con ensa 10n 1s escr1 e y e re a 1on .r0 0 =
const which is in agreement with mass-spectrometry data on small cluster concentrations given elsewhere~J.The growth of "condensation nuclei" with decreasing the tem-
1968
perature T0 4 T ,produces the appearance of larger clusters in the je\. This fact is confir.med spectroscopically by the increase of the 127 nm continuum intensity in the T2 -T3 range. As stated above, the continuum is intensively radiated only when electrons collide with rather large clusters.Vigorous heat liberation during the phase transition that causes changes in the thermo- and gas dynamics parameters of the system produces a sharp change in the j et structure and the appearance of maxima and minima in the curves I/po (T0 ) at To=T~ and I/po (P0 ) at Po =P4 • For T0 > T3 , the luminescent jet bulk is a fairly homogeneaus cone. In the T3 -T~ region the jet boundary is separated and a second cone of a larger cone angle but of a lower luminancy appears. The calculation da~a on the flow parameters with gas condensation (Fig.4d) confir.m generally our conclusions on T~ and Pf peculiarities in the curves I/~(To,Po)• The pressure and temperature at which the peculiarities T~ and P1 are observed are related by P0 T;~=const.A jumplike decrease in the continuum intensity at low temperatures (the T6 -minimum) is due to the changes that occur in large clusters, as there is no such kink in the curve for the 106.7 nm line. The T5 -position is pressure independent wi thin the experimental error :t 4 K. The T 6-minimum seee to be due to phase transition in clusters. As show.n by electron diffraction measurements, for the pressures studied (0.2 to 0.5 MPa) and temperatures below Ts, the jet contains argon fcc microcrystals of the 2-3.5 nm average size. The microcrystal temperature is (40 ± 6)K and insensitive to size within the experimental error.
Figure 5 shows experimental and theoretical radii of clusters as a function of P0 and To which are approximated as e "'P0o.s and Z,... T0 - ~.Lt • The clusters in the jet are of the same average size at P 0 and T0 related by P0 T0 -.2.?±0.i =const which agrees with the data from
/
',, Po=03MPa Tö=I70K/"' ~3 ~ ~3 // 0
c:. 0 ', r:: / 0 leJ.. .............. ,~.. / 0
2 o 0....<>-. 2 o/ 150 170 190 210 T;,f. ~at::-.2---'--a~3_.__,~--;-.4...,...o,MPa
Fig. 5. Dependences 2 (T 0 ) and ~ (Po): experimental ( ), theoretical (-----).
1069
Ref.9. The correlation between theoretical and experimental curves ~ (P0 ) and z (T0 ) indicates that the classical theory of condensation (at a once selected dependence of the surface tension coefficient on the drop temperature [3) used in our calculations) is appropriate for a qualitative treatment of the developed gas condensation in jet.
Thus, our studies provide support for the assumption that the I/po(P0 T0 )-peculiarities of the argon jet VUV emissions are due to changes in the jet phase composition and structure under gas condensation. A detailed treatment of some peculiarities requires further experimental and theoretical studies•
REFERENCES
1. E.T.Verkhovtseva, On VUV radiation spectra of electron excited supersonic rare gas jets,in: "Advances in Spectroscopy",Moscow,1:87 (1978)- in Russian.
2. E.T.Verkhovtseva and E.A.Bondarenko, On specific features of VUV radiation spectra of argon clusters, VI All-Union Conference on Physics of VUV radiation and radiation-matter interaction, in: "Abstracts of papers",Moscow,85 (1982)-in Russian.
3. P.A.Skovorodko, The peculiarities of condensation process in conical nozzle and in free jet behind it, 13 Intern.Symposium of rarefied gas dynamics. Book of abstracts, Novosibirsk, 2:357 (1982).
4. EoT.Verkhovtseva, V.I.Yaremenko, P.S.Pogrebnyak and A.E.Ovechkin, A small-sized version of the VUV gas jet source, Exp.Instr.and Techn. 4:210 (1976) - in Russian.
5. E.A.Katrunova, A.P.Voitenko, G.V.Dobrovolskaya, V.I.Yaremenko and E.T.Verkhovtseva, An electron diffraction equipment for investigation of supersonie jets issuing into vacuum, Exp. Instr. and Techn. 3:208 (1977) - in Russian.
6. R.Shuttleworth, The surface tension of solids, Proc. Phys. Soc. A63:444 (1950).
7• o.G.Peterson, D.N.Batchelder and R.O.Simmons, Measurements of X-ray lattice constant, thermal e*pansivity and isothermal compressibility of argon crystal, Phys. Rev. 150:703 (1966).
8. D.Colomb, R.E.Good, A.B.Bailey, M.R.Busby and R~Dawbarn, Dimers, clusters and condensation in free jets, J. Chem. Phys. 57:3844 (1972).
9. o.F.Nagena and WeObert, Cluster formation in expanding supersonic jets: effect of pressure, temperature, nozzle size and test gas, J. Chem. Phys. 56:1793 (1972).
1070
XVI. CLUSTERS AND NUCLEATION KINETICS
THE MICROSCOPIC THEORY OF CLUSTERING AND NUCLEATION
R. Dickman and W. C. Schieve
Center for Studies in Statistical Mechanics The University of Texas at Austin Austin, Texas 78712
I. Introduction: Metastability and the Classical Theory
The fundamental theoretical interestoin nucleation is that it is an important example of metastabili ty. '" It is a transi tion phenomenon from an unstable state to a stable one, which may be· understood from a "potential" in one dimension shown in Fig. 1. The source of this potential need not concern us here but may be viewed as the potential determining the s~eady solution to a Fokker-Planck equation, or on a macroscopic-cohtinuum level a Ginsberg-Landau free energy. The maximum at x2 is a saddle point in a space of more than one variable. Due to stochastic behavior the system may leak over (or through) the barrier. The state at x1 is globally unstable. The steady rate of transition from region x1 to region x3 is determined by the barrier height which depends on external parameters. As these parameters are varied (the barrier may decrease, disappear, or increase in height), the rate of1transition varies dramatically. A simple formula due to Kramers , illustrates this in terms of the decay time,
(1.1)
-1 The flux of probability is proportional to (T) • The valid-
ity of such a formula depends on the rapid relaxation to ~ equilibrium at x1 • Then, a quasi-steady flow of probability occurs across the barrier. The state at x1 is not a true steady state, and it is its description from the microscopic point of view which we want to focus on in the next sections. Such metastability is an important phenomenon in physics and chemistry ranging from particle decay2 to chemical kinetics3. Homogeneaus nucleation phenomena in a supersaturated vapor and the transition of a ferromagnet upon
1073
ro ~ c 2 0 0.
/ /
/
/
/
' /
/
' ' /
/
'
/ /
/
p* ----- .....
' ' \ \G ' I
I I I I
' I I I
u
Fig. 1. Qualitative dependence of the free energy of formation G, and of the resulting potential U in a Fokker-Planck description of cluster growth.
field reversal are more familiar examples. 4 ' 5 For !arge clusters of molecules the free energy of formation, G(p), may be expected to be
G(p) = (~L-~ )p + scr (p/p)l-1/d G s
·r(l. 2)
where p is the number of particles in the cluster, ~ ,~ the chemical potential of the liquid and gas respectively, cr theLsu~face tension, d the dimensionality. The maximum in the poten~ial barrier is at x =p* and is due to the surface tension. Note there is now no globafly stable state since the dominant (bulk) contribution to G approaches - 00 as p~, as seen in Fig. 1. The droplet of size p* is the rate-~imiting size (critical nucleus) for the formation of drops of p>p*.
Classical nucleation theory 7 estimates the transitionrate by use of a steady state kinetic picture assuming a monomer addition hypothesistt
t~ -~ =-k Tjn(P/P') ; (P/P') being the Supersaturation ratio. L G B
ttcomments have been made by Zurek and Schieve on the validity of this assumption; w. H. Zurek and W. C. Schieve, J. Chem. Phys. 84, 1479 (1980).
1074
A p
+ A
k(r) ~1
1 (ft k
p
A 1' p+
and the Szilard boundary condition7 , A = 0 for p>p >p*. Utilizing simple kinetic estimates and detailed Balance, c
k(r) N p+l p+l
k(f> N p p Nl '
one obtains by macroscopic ther.modynamic arguments
- -N = Nl exp (-G(p)/k T) p B p~* (1.3)
and the steady state particle flux J, across the barrier at p=p* d2G
(f)- -, -~ J = k * N * (~ 2 */2~kBT) (1.4) P P ap P
J is determined by properties of k(f), N , and G"(p) near p* by the rate limiting assumption earlier. P P
Agreem~n~ between theory and experiment is remarkable but not consistent. ' t Clearly this macroscopic approach for p~l0-100 is open to serious question theoretically. An indication of the conceptual difficulty is seen in subsequent "corrections" imposed on the description. The capillarity approximation involves the assumption of spherical droplets of the bulk liquid phase, so that a is independeng of p. Many attempts have been made since t~e4e~r~y work of Talman to take into account the finite size effect.' ' Calculations for Argon indicate that a may not even be a monotonic functio~ of p, and for small clusters, may be 50% !arger than its bulk value.
A conceptual revision (controvi0sial!) was introduced into the classical theory by Lothe and Pound who suggested that translational and rotational degrees of freedom of the droplet center of mass should increase the phase space, leading to a correction
(1.5)
YrrY and y being the translational, rotational and so-called "rep~acement" partition functions. y ,y account for the Brownian motion in the gas and y represents tfie ~ibrations and rott7ions of a cluster in the bulk fluid. This produced a factor of 10 change in the cluster current, and generated enormaus argument which5is interestingly reviewed in the two volumes edited by Zettlemoyer . These enormaus factors are not easy to distinguish experimentally since they
1075
iead to critical Supersaturation pressure diffrfences of 5-10. Water seems to favor Eq. (1.2) and ammonia Eq. (1.5) .
The point of this controversy, in our view, is that what is needed is not a quasi-phenomenological theory, but rather a statistical mechanics approach which will describe the unstable state (at least approximately). This we will turn to in the next si2tion where we consider the partition function theory of condensation . In the last section we discuss some recent universal scaling results we have obtaini~' which suggest unsuspected simplicities in a microscopic approach .
II. THE PARTITION FUNCTION DESCRIPTION OF CONDENSATION AND CLUSTERS
The object is to utilize the partition function to construct a description of the local equilibrium state, The system is imagined as a mixture of non-interacting clusters. The grand partition function, ~. is then
ln ~ = pV - E Q z P 1
k T - I p· 1 B p
( 2 .1)
where Q 1 is the configurational partition function of all possible clusterptopologies of p 1 particles. Q is our main interest and will be discussed in detail in the ne~ section. z=exp(V/k T) where ~ is the chemical potential.t Of course, it is expected (h~ped!) that ln Q ~G(p) of Eq. (1.2), in agreement with the macroscopic droplet m8del. For dilute cluster densities, it seerns reasonable to neglect the cluster-cluster interactions and write Eq. (2.1) as an approximation to the Mayer series. Little has been done with the Mayer theory itself for supersaturated vapors other than to suggest that the ß integrals must be cut off 16 for p>p* leading to a cluster distrigution analogaus to the classical result.
Hill has reordered the series in terms of physical clusters 17 •
Strogyn and Hirschfelder 18 calculated dimer (p=2) distributions, and Zurek and Schieve 8 compared dimer and trimer (p=3) distributions to molecular dynarnic calculations 19 • Recently, S. M. T. de la Selva et al. have calculated distributions through p=5 for hard-core square well (HCSW) gas utilizing the proximity and negative energy definitions of a bond20 •
How can we expect that ~ can lead to metastability? As has been aptly said, if we calculate a Van der Walls or rnetastable loop, we have made a mistake 14 • The theorern of Yang and Lee 21 states that
tFor reviews of the droplet model, see the article by w. J. Dunning in Ref. 4 as well as Ref. 14 and 15.
1076
in the thermodynamic limit the pressure and density are monotonic increasing functions of ~nz and are continuous except at isolated values of ~nz. In addition, for positive llll=ll-]J • (c is the con-:densation point) the series diverges, 1 ~' 22 exhibftinq an essential singularity. This is just the region of growing droplet size.
Physically, we are describing local equilibrium, so it seems reasonable to cut off the series for p>p (p >p*) and avoid all these difficulties. Langer23 ' 15 has ex~inea this singularity in some detail for llll:o. Using the droplet model assumption ln Q ~G(p) for large p, he identifies a saddle point corresponding to p*.P He then analytically continues ~ from a negative llll to positive llll in the complex l1 plane, ~ having a branch point at llll=O. The discontinuity along the cut is evaluated, and the metastable contribution to ~ is identified. This term is the partition function with an appropriate cutoff near p*. Newman and Shulman2 ~ have considered the continuation of the Ising free energy which is conjectured to follow from the analytic properties of eigenvalues of the transfer matrix • In mean field theory, the critical droplet is a global excitation and Penrose and Lebowitz 25 have shown that it grows to infinity more slowly than the volume.
To evaluate Q in Eq. (2.11) entails an arbitrary definition of a physical cluster~ This point was made early by Hill and Stogyn and Hirschfelder in their work. on dimers 18 , It is our view that such a definition is largely conventional and a matter of convenience. We adopt a pair-wise proximity definition. In Ref. 19 and 20, we found, for few-particle clusters (p<S), that the negative energy and proximity differed appreciably only in the uninterestinW high-temperature regime (B*=E/k T<<l,E being the HCSW well depth) 2 • There, the proximity definitionBis larger than the negative energy by a factor of three for pentamers.
Complementary to the point of view to be discussed in the next section is the program of McGinty and Haare, Pal and others 26 to compute numerically the cluster free energies of a limited topological set of Argon-like L-J potential microclusters. The cluster partition function, Q , is calculated for restribted solid like motions (localized!). Logal minima in the many-body potential (T=O) are found. Low energy structures in ··the L-J potential for p<20 are noncrystollographic and based upon the p=l3 icosahedron. The motion for T>O is assumed to be harmonic. Using detailed balance, and assuming an equilibrium cluster mixture, the nucleation rate is
pc J = aP/(2~mkBT)~ E1 (An )-l (2.2)
p= p p
where a is the sticking probability, A the cluster area, and p
1077
n p
p (-§JE.L) k T exp k T '
B B
G(p) being obtained from the restricted topologies as discussed. No assumption concerning the surface vs. bulk contribution or of a critical nucleus is made. The free energy does have the shape (with oscillations!) of Fig. 1. The results agre~asonably well with the experiment of Wu et al 27 on Argon condensing in Helium, whereas the classical theory is many orders of magnitude incorrect. However, the surface tension of solid argon is not available, so a readjustment of the classical theory utilizing Eq. (1.2) is possible.
The agreement of this approach with this experiment is impressive. However, many questions are raiseg. Are the inclusion of a few topologies sufficient for calculations near the condensation point? This would seem not to be so12 except at low temperature. Also, the assumption of harmonic motion seems simplistic since the L-J potential is very non-harmonic.
III. Universality and Scaling in HCSW and Lattice Gas Models of Clusters.
In this section we briefly review some recent work 13 on simple models of clusters which reveals the scaling behavior of the cluster partition function, Qp· In the lattice gas models, exact partition functions for clusters of up to about 10 particles may be found through direct enumeration of graph embeddings. Analysis of regularities in the early terms yields estimates of asymptotic behavior, much as in the series expansion method in critical phenomena~ 8 We also consider the HCSW model potential
V(r) { :: o,
(3 .1)
In the lattice gas there is an attractive potential -E between particles occupying neighboring sites, and two particles are forbidden to occupy the same site. Particles i and j are considered bound if V(rij) < 0. This pairwise proximity definition is reasonable at low temperature and density. A cluster is a set of particles connected through proximity bonds. Thus the bonding structure of a p-particle cluster is isomorphic to some connected graph with p points. The cluster partition function may be written as
( 3. 2)
where N is the nurober of lattice sites (or the volume in the HCSW
1078
model), Ais the thermal de Broglie wavelength, and ~ = exp(E/~T). k indicates the number of bonds, and k (p) is the maximum numßer
of bonds in a physically realizable clusrer. In the square lattice, for instance, k (p) = [2(p - lpij • The inner sum is ~ver all connected graph t~pologies with p points and k lines. C is the numbTr of distinct labellings of the graph T. In the lattice models, a is the number of embeddings (strong or weak) of T in the lattice. For the HCSW model, aT is an integral over all particle configurations consistent with the bonding structure T.
We write the partition function as
(3 .3)
where
T T a =E ac. p,k t(p,k)
(3.4)
our results concern the scaling behavior of CJ k' Assuming the existence of the cluster free-enerqy per partic~~. i.e. lim~- 1 lnQP, a simple argument shows that
CJ k ~ exp(CJp f(k/p)) p, ( 3. 5)
as p+oo, where CJ is a constant. To account for surface effects, the scaling form must be generalized, for finite p, to
CJ k ~ exp(CJp f(x(p,k)) (3.6) p,
where limp+oox(p,k) = x(k/p). For the square lattice we take
x(p,k) = k - p + 1 p-u'P + 1 ( 3. 7)
so that x is, approximately, the ratio of the excess nurober of bonds over particles and the maximum possible number of excess bonds. Note that when p ~ p* = n2 , and k = k*(n) = 2n(n-l), x(p,k) = 1, independent of n. For these values of p and k there is a unique cluster (a perfect square) and CJ * k* = 1. Thus f(l) = o. A similar construction is possible inpany lattice, and in general x(p,k) has the asymptotic form
k - p + 1 x(p,k) qp-xpi-I/a + 1 ( 3. 8)
where q and X are lattice-dependent constants.
Eq.'s (3.6) and (3.8) lead directly to a "droplet" type asymptotic behavior for Q
p
lnQ = Kp- ~pl-l/d + Op1- 2/d + Olnp (3.9) p
1079
where K and f.l are temperature-dependent quantities re1ated to the bu1k free energy and the surface tension. Thermodynamic arguments regarding the c1uster specific heat 1ead to the fo11owing conc1usions ~out the sca1ing function: (i) df/dx +-oo as x + 1; (ii) d f/dx2 < 0; and (iii) f(x) has a maximum somewhere in the interva1 0,1).
We have determined dings in the square and tabu1ated in ref. 29). ior of cr k we define p,
cr k for p < 12 for strong and weak embedt~lang1e 1attices. (Numerica1 va1ues are To investigate the finite-p sca1ing behav-
( 3 .10)
> where cr 0 (p) = supk1ncr k It is found that for p 10, cr 0 (p) ~ crp - o, where cr and oP' are 1attice-dependent (for strong embeddings in the square 1attice, cr~1.4, and 6~4.7). Figure 2 is a p1ot of the known va1ues of f (x) for strong embeddings in the square 1attice. The curve is oRr estimate for f(x). It is given by
Fig. 2.
1080
f (x)
flxl
.8
.4
0
• p:: 9 10 11 12 13-15 16-19
• 20-26
(3 .11)
\ .4 .8
X
f (x) for p>9 for strong embeddings in a square 1attice. The sg1id 1ine ls given by Eq. (3.28) with X =.03, n=1.52 and
0 Y=2/3.
where Y=2/3, n=l.S, and x =.03. For other lattices, fp(x) can also be fit closely by functio~s of this form, with somewhat different values for x and n. However, it appears that the exponent y is 2/3 for strong e~eddings in all two-dimensional lattices.
Values for f (x) for the two-dimensional HCSW model (A =1, A2=1.96) are plot~ed in Fig. 3. These results were obtainea via Monte Carlo integration. The solid line is again given by Eq.(3.11) with Y=2/3, n=l.S and X =0. The scaling function for a is thus insensitive to the loca~ structure of the model. This R~parent universality suggests that some fundamental combinatorial-geometrical principle underlies the scaling law.
Using the scaling law embodied in Eqs. (3.6), (3,7), and (3.11), we can evaluate the cluster partition function, and obtain cluster properties and abundances. Wehave compared our results with those of a Monte Carlo simulation of the square lattice gas performed by Binderand Kalos 3 D. Their system consists of a fixed number of particles, N, in a lattice of S sites, with N/S~.Ol, at a (lattice gas) temperature of k8T/E=l/3. Under these conditions the equilibrium state consists or a single large cluster of p* particles surrounded by a vapbr of monomers and small clusters .
Fig. 3.
.8
.4 o p= 5
6 7
• 8
X
f (x) for p~S for the two-dimensional HCSW mqdel, Eq. (3.1), ~th A1=1, A2=1.96. The solid line is given by Eq. (3.28) with X =0, n=l.S, and Y=2/3.
0
1081
To calculate <p*> we use the canonical ideal cluster mixture approximation
partition function in the
~1 ~2... ~N ~
N
~SQP) p
N ! p
( 3 .12)
The sum is restricted by the condition I'pN =N. We retain several hundred terms whose contribution is >s%Pof Pthe maximum term. This approximation gives a good estimate of the probability distributions P(p*), and P(N1). In Fig. 4 we compare our results with those of the simulation, for N = 120 and S = 3600. We find <p*>=l06, and <N1>= 10.6. The Monte Carlo simulations give <p*>=l02, and <N1>=12, in very good agreement.
.05
ptp"l
05
2D •• •• dj . .. liP".,~"''tt&~ 0o 2 • 1!1
" '• ... 80 -· •
I 4
~---100~----'-------o,::.,o.--------'--''- P"
Fig. 4. Comparison of theory and experiment for clustering in a lattice gas. Upper graph: Probability distribution, P(N1), of the nurober of monomers in a lattice gas of N=l20 part1cles in a region of 8=3600 sites at temperature T*=l/3. <N1>MC is the estimate from the Monte Carlo simulation of Binaer and Kalos. The inset shows the time evolution of N1 in a typical run, (t is in units of 10 5 spin exchanges), redrawn from ref. 30 . Lower graph: Probability distribution, P(p*), of the size of the large cluster in the lattice gas with N, S, and T* as above.
1082
The simulations 30 also give results on the derivative of the surface free energy with respect to cluster size, given in the Simulations by
c c (]1-].l ) (1-p /p )
c G L (3 .13)
h . h . "1 c c w ere ]l 1s t e coexistence chem1cal potent1a . p and pL are the coexist~nce gas and liguid densities respectively.G
With the surface free energy of a p-particle cluster defined as
Fs = Kp- lnQ p p
we compute 3Fs/dpl ~ ~(Fs -Fs ) . A comparison of theory and experiment is show~ in fi~:1s.p-lThe solid line is the prediction of the scaling theory. The dashed line is the prediction of the droplet model using the capillarity approximation with bulk and surface free energies obtained .from exact lattice gas solution. The failure of the macroscopic droplet model and the success of the scaling theory utilizing the asymptotic form, Bq. (3.28) is apparent.
Fig. 5.
5
.5
.1
10 20 50 100 200 500 p
Camparisan of scaling theory and Monte Carlo results for (k T) - 1 (3Fs /3p) I . The dots are Monte Carlo results of Bi~der andPKalos~ The solid line is the prediction of the sca ling theory, and the da shed l i ne the predi ction of the droplet model.
1083
Let us briefly comment on how the scaling theory outlined here can be extended to more realistic models. The application of the theory to three-dimensional HCSW or lattice-gas systems is Straightforward, but computationally formidable. A more fundamental conceptual generalization is required if we wish to apply the theory to continuous-potential models (e.g. Lennard-Jones) • For such a system there is no meaning to "k", the band number, and thermodynamic arguments indicate that we should instead expect a scaling law to apply to g (E), the p-particle density of states:
p
g (E) ~ exp{crpf(E/E)}, p 0
where E is the p-particle ground-state energy. 0
ACKNOWLEDGEMENT
We wish to thank s. M. T. de la Selva and c. Canestaro for their work on the Monte Carlo calculation of the HCSW cluster integrals.
REFERENCES
1. H. A. Kramers, Physica 7:284 (1940). 2. A. I. Baz, Ya B. Zel'dovich, A. M. Perelomov, "Scattering
Reactions and Decay in Nonrelativistic Quantum Mechanics." 3. I. Oppenheim, K. E. Shuler and G. H. Weiss, "Stochastic
Processes in Chemical Physics," M.I.T. Press, Cambridge (1977). 4. "Nucleation," A. c. Zettlemoyer and Marcel Dekker, ed., New York,
(1969); "Nucleation Phenomena," A. c. Zettlemoyer, ed., Elsevier, New York (1977).
5. F. F. Abraham, "Homogenous Nucleation Theory," Academic Press, New York (1974).
6.
7.
8.
9. 10. 11. 12.
13.
1084
S. Glasston, K. Landler and H. Eyring, "The Theory of Rate Processes," McGraw-Hill, New York (1941).
M. volmer and A. Weber, z Phys. ehern. 119:227 (1926); L. z Phys. Chem. 125:236 (1927); R. Becker and w. DBring, Phys. 24:719 (1935); J. Zeldovich, Z Exp. Teor. Phys. (1942).
Farkas, Ann. 12:525
R.
K. J. H. J.
R.
c. Tolman, J. ehern. Phys. 17:333 (1949); seealso J. G. Kirkwood and F. P. Buff, J. ehern. Phys. 17:338 (1949). A. Nishioka, Phys. Rev. Al6:2143 (1977). Lothe and G. M. Pound, J. Chem. Phys. 36:2080 (1962) L. Jaeger et al. J. Chem. Phys. 51:5380 (1969). Frenkel, J. Chem. Phys. 7:538 (1939); W. Band, J. Chem. Phys. 7:324 (1939); seealso J. Frenkel, cap. VII, in: "Kinetic 'rheory of Liquids," Dover Pub., New York (1955). Dickman and w. c. Schieve, The Asymptotic Form of the Cluster Partition Function in a Two-Dimensional Lattice Gas, Physica 112A:51 (1982).
14. M.
15. J.
16. J. 17. T.
18. D.
19. w.
20. s.
21. c. 22. A. 23. J. 24. c. 25. o. 26. J.
E. Fisher, "Lectures in Tb. Physics VIIC," 1, Univ. of Colorado Press, Boulder (1965); K. Binderand D. Stauffer, Adv. in Phys. 25:343 (1976). s. Langer, in: "Lecture Notes in Physics 132," Sityes (1980), L. Garrido,-;d., Springer-Verlag, Berlin (1980). Y. Parlange, J. Chem. Phys. 48:776 (1968). c. Hill, Chap. 5, in: "Statistical Physics," McGraw-Hill, York (1956). E. Strogyn and s. H. Hirschfelder, J. ehern. Phys. 31:1531 ( 1959) • H. Zurek and w. C. Schieve, J. Chem. Phys. 73:4061 and 4663 (1980).
M. T. de la Selva, R. D. Dickman, W. c. Schieve and c. Canestaro, J. Chem. Phys. (to appear). N. Yang and T. D. Lee, Phys. Rev. 37:404 (1952). F. Andreev, JETD 13:1415 (1964). s. Langer, Ann. Phys. (N.Y.) 41:108 (1967). M. Newman and L. s. Schulman, J. Math. Phys. 18:23 (1977). Penrose and J. L. Lebowitz, J. Stat. Phys. 3:211 (1971). J. McGinty, J. Chem. Phys. 55:580 (1971); M. R. Hoare, P. Pal, and P. P. Wegener, J. of Colloid and Interface Science 75:126 (1980). See also a review by M. R. Hoare, in: "Advances in Chemical Physics XL," I. Prigogine and s. Rice, ed., Interscience- John Wiley, New York (1979) pg. 49; N. G. Garcia and J. M. Torroja, .Phys. Rev. Lett. 47:186 (1981).
27. B. J. c. Wu, P. P. Wegener, and G. D. Stein, J. Chem. Phys. 68:308 (1978).
28. C. Domb and M. S. Green, eds., "Phase Transitions and Critical Phenomena," vol. 3, Academic Press, New York (1974).
29. R. Dickman and w. c. Schieve, to appear. 30. K. Binderand M. H. Kalos, J. Stat. Phys. 22:363 (1980).
1085
KINETICS OF CLUSTER FORMATION AND GROWTH IN THE PROCESS
OF ISOTHERMAL CONDENSATION
B.F.Gordiets, L.A.Shelepin, and Y.S.Shmotkin
P.N.Lebedev Ph7sical Institute of the USSR Academ1 of Seiences Moscow 117924, USSR
In recent 1ears much interest has been shown in the investigation of proceases that incorporate clusters. Clusters pla1 an essential role in various nonequilibrium processea incorporating high densit1 and low temperature gases, and the1 are ot particular importance in the kinetics ot tirst-kind phase transitions. The theor,r of homogeneaus condenaation seems to be most elaborated now. This theor" is inseparabl1 linked with the stud1 of kinetics of cluster tormation and growth in the process of condensation.
The homogeneaus condensation is known to include two stag-es. The first on comprises the tormation of condensation centres, i.e. clusters, which are bigger or equal in size to some critical value and are capable of further growth. The second stage coaprisee the growth itself, that ultimatel1 results in a decreaae of the gas-phase molecule concentration and phase equilibrium. As a rule, tor theoretical description ot the homogeneaus condenaat-ion there has been applied a classical approach based on the liquid model for clusters [11 • However, certain difficulties arise, which are caused b1 the necessit1 to describe the cri-tical size clusters, containing a small number of molecules at a high degree ot supersaturation, as a luquid drop and to appl1 to it such macroscopic notions aa surtace tension6 and densit1 ~ • In some latest works the elementar.r processes that lead to tormation and destruction ot the condensed aubstance particles are treated as chemical reactions involving molecular cluatere. Such treatment allows to use a ta1r11 well developed theory ot monomolecular reactions in dealing wi th the condenaation process [2]. Thus, the article [3] contains numerical stud1 ot eTolution of the distribution tunction ot the condensed partielas (clusters) ot metals b1 the aumber ot atoms in them with the aupposition that these clusters are tormed through an activated caaplex. However, due to the ditticulties connected with numerical inte-
1087
tegration of a set of aany kinetical equations for cluster condensation, only the initial etage of the condeneation hae been investiga~ed at which the eupersaturation degree hae not yet decreaaed. A calculation of tempor~l eupereaturation development in condensing the metal vapour hae been pertormed in paper (4] , however, the condensation hae been described with the help of a simplified kinetic model aeeuaing that vapour moleculee are critical particlee. Such an aseumption is not valid in many cases.
Ae follows from the above said, the study of condensation kinetics is still an important issue. Special significance is attached to the analyais baaed upon the detailed evaluation of elementary acta involving clusters as well aa obtaining (within the bounda of such an approach) of analytical expressions for temporal evolution of the condensed particle diatribution function and for temporal Variation of the Supersaturation degree. The present paper deale with the above-mentioned probleas for isothermal conditiona of condensation.
When describing the kinetics of a new phase formation from the supereaturated vapour we proceei froa the quaai-chemical model, i.e. the aupersaturated system is auppoeed to consist of the gas-phase moleculea of a condensing substance (monomere) and molecular aggregates (clusters) composed of two or more monomere. Clusters are treated as polyatomic complexes characterized by a definite value of dissociation energyE~, that depends on the number of monomere ~ in the clustere. The cluster growth occurs due to the addition of single moleculee to them (aasociation), and the destruction meana their dieaociation. Based on euch a model the equatione for concentratione Nn of the cluater conaisting of/L monemers take the following form:
dlv'n _ .., _ ,., _ dC - c/IL-1 J ·~ ) ( 1 )
where
(2)
i~ the cluster flux in the epace of their eizes; ~4r~> and t~r~J are conatanta of the association and dissociation ratee
reepectively that can be expreseed in the following form (4] : f.4 (n.) = ft.t_. I,d tz._, {1'1]), {cJ(n.)-= fc;;. ·I ,<(tt-.t, [Ml), (3)
where lc; and /:,; are the rate constante at high pressures; J~(n,[Ml) is the Cassel's integral that depends on the cluster
eize n and the total concentration of the gaa-phase molecules [141.
Using (2), (3) one can easily obtain quasistationary values ~o for the flux
(4)
where ~4 is the quasiequilibrium distribution function in the absence of the flux. Its recurrent relation haa the following
1088
form:
(5)
Should the expressions of a simplified k~etie model (e.g. see C51) be used for rate cogstants A:.;;. and t.ft. one ean get the fol-lowing expression for t1lh. : ll.-L
NIL":: fr't. e;xp { ;tr [ f Ei.•.t - (tt-t)Eoo} + (ll.-1) fu S}' ( 6 )
where Eoo is the energr of molecule release from the macroseopie volume; 5 is the supersaturation degree.
A rather general approximation expression of type (51 oan be propoaed for En :
"' ""'l /'L';};i., En.+J. = Ero - A E [n - (ll.-1.) ote (o, i). (7)
Note that when AE = E.QJ formula ( 7) is similar to those of [4J.. while forcl.-= j and AE=411"6·(3m./4t'.P)Z/?l formula (7) is olose to classical expression for the energy of a molecule with mass ~ released from a drop having the surface tensiond and the density J' ·• Applying equation (7) in (6) we receive the follow-ing: ~
.1•- A/. exo[- ~E·(n-.t) + rn-1-)f!n. 5} /Vn - :t..' I I( T • ( 8)
For supersaturated systems distribution mu.m at aome point 12.-if which is equal to:
(8) has ita mini-
( 4E oL )..1.-!l."' = KT ·e;;s ..,_d. + i (9)
-4' The value of n is just corresponding to the critical size af the cluster. Using (8) after approximate calculation of the integral Ne derive from (4) the following expression of the main kinematic characteristic for the homogenous nucleation process:
'Jo =-Nt· f;. · c~. ~: · T 1<. (tz..., [Hl) 7 (10) ...2 [2cl. (1-tL) A E d. -•i!.Tt where Ci!. is the Zeldovich t'actor equal to 2~. =- ~ 'K-f {h") J
Formula (10) differs from the similar classical formular. The difference is explained by the use of approximation (7) for the cluster dissociation energy. Another possible reason for this difference is conneeted wi th the additional mul tiplier I IL ( tt , [ Ml), h..:.tt...,. in the equation (10). This multiplier can be appreciably less than 1 for small values of 11-* •
Let us consider the formation of the supercritieal particle distribution function (~~n*) assuming that the supersaturation occurs instantly and is maintained further on a constant level. It is possible to show that the kinetic equation system within tt ~ n* is equivalent to one partial differential equation
..1;. /t. · ?-t [l:tt(ll)·Ain.] + ~ [ ~.dn) ·N/1.] = 0, f'l. 7 n.1f Nt . (/l.) V U ( 11 )
1089
Here ft/,o is the gas-phase molecular concentration maintained constant.
The equation (11) is solved as follows [51 :
.1 { I ftz dn. ) t:. '>' 0 , IVIL = .;o 1c (n.). Jn..,(t- Mo P-- Ctz) n•< n .c: nt_(l:) (12)
/V1 • 4 1 fL ft 1[;Q '
where :J tt"' is the v alue of the flow through the cri tical point, and n C,(tJ iJl the U.Pper boundary value derived from the condition: i: - };;ö J f.H.J ~ == o.
' ,._ * rc;;rn.> If we appl;y one of the approximate solutions [ 61 for Jn•
':/11.• = :Jo·e-x?[-~o/l], (13)
where t o is the setting time of the quasistationar;r value for the flux at the point ~~,then the function (12) will be trans-formed into: [ t. 0 } -t:. 7 0
• ~ = L-zr-: . /jo. e"X p ~( " f tJ.E:._ "'" ' {V t.. ft/,". fL4 (fL) ..e_ - ;v,a rz.._ i:a. (rt) fZ. < n < fZ.f.ffJ.( 1 4)
According to (14) the distribution function of condensing s;rstems, where the supersaturation is instantly established and then kept constant, increases from zero to a stationar;y value in the process of formation. Aa it follows from (14) the characteristic time of establishment of a stationar;r value of the distribution function at the point n is the following:
f n. Jn fll. =io +- ;z .. ·J ~ (15)
I #t.1r .fl.ll
Now we'll study the condensing s;rstem evolution considering the change of the gas-phase molecule concentration in the process of condensation and assuming that quasistationary conditions are characteristic of subcritical area during the whole condensation process. Equation of temporal variation of the gas-phase molecule concentration is of the following form [5]. 00
dtl. ==- 'j.(n .. +-1.) - M· f llf.cfl.)·tl,..dn. dt h."'+f (16)
where ~ is the quasistationar;y value of the flow determined by the instant gas-phase molecule concentration value.
Exact calculations show that the process of condensation is of avalanche-type. The regime of abrupt change of supersaturation degree is preceded b;y that with no essential change in the gaa-phase molecule concentration. It's important to note that when the decrease of the supersaturation degree is still negligibly small, no further increase of the total number of the condensed oarticles is observed. It is accounted for the fact that the condensed particle formation is determined b;y the flux ~ moving through the critical point. Besides this flux decreases rapidly when the aupersaturation degree goes down.
1090
The above said allows to simplify the analysis and solve the problem analytically. For this purpese we'll consider two time interval., ( o" tilr.</. ) and (tih.J.1 oo ) , where the induction time tw. is the time required for t -!old decrease of the gas-phase
molecule concentration (the value of i will be determined later). Suppose that at the initial stage of the process a supercritical particle distribution function of the type (14) is formed where ~· is the initial molecule concentration. Applying (14) ln (16) after approximate integration we derive:
.N, ~ N;"- /,:!/, •. ( S ~ttv'/t)fJ t ctittJ. <n> Here ~o is the flux quasistationary value corresponding to the initial stage of super-saturation. When equation (17) was composed it was assumed that the Cassel's integral 'Nas equal to 1 in the supercritical area and the association rate constants were approximated by the following expression of a simple kine-tic model: T 1.. ~ ~ z.
l t ~)i!. gm,)l -~a.Ctr.) = fet- /l.'l = .,Jr5. ~~'Jt't1L • ~ . tz...s ' ( 18)
where E is the adhesion factor of the molecule to the cluster.
The obtained expression (17) allows to determine duction time I!. 04•1z -t
3 ·[(f-"t}·Kt· f/J.f fq,_cl. = A+ .;o M
• 1'-f'fVf ·vo
the in-
( 19)
It is convenient to carry out the condensation process analy:sis for -t :> t ind. in new nondimensional variables: 1::: .. t: I't,V:,tft and 'fL = tft· {:rJ:rY.n.. Consequently, at the time twl. there will be formed a distribution function o! supercritical partiales, the total number of which will not change considerably. The evolution of this distribution function is determined by equation (11), provided the condensed particle distribution at t "lit-ul. is taken as the initial distribution. In the result we obtain the following:
./ -:! .. {V /1. : Al, 0.. /(_: '['- 7:' ;"_J. ( 'it < 1_" (20)
where 'l';"J ::: ,41,".1/·. i:.~d.
Distribution (20) spreads within the molecule number range in the partielas along 1l-axis. It is defined in the interval
tt t::. t IL1, llz.] , where the boundaey numbers /Z.~ and rz 2. are determined from the condition n 1 ,.,_, dtz 1. r z. a!L
t'-?:ind, ::- lcl s _,_.,. ' t'.::: ~/ J ~-t (21) h. .. ~,.. 1&."" 1'-n..
The distributions (20) and (16) make it easy to determine the Variation in time of the gas-phase molecule concentration in variables~ , [5]
1091
~' ,,. '3. · ( r)f (22) fVt ~ l"f - ll./11'/'. .3
Equation equation:
(22) is equivalent to the following differential
,J~ - 1.... J.l: f- 3:. f - -f.c. ' (23)
1 1· where J:.::(f-t44/~·)•={t-S/So)"; S.and S are the initial and eurrent Supersaturation degrees in the system; -t.c. is the eharaeteristie time of the proeess:
J - J_ ·[ ~l-(#.")Z.Jf l: c. - 1..,. ./o ':J
ILt 1V1 o • (24)
By integrating (27) we derive the following transeendental equation to determine Alt or S :
.f e,_ f=..3: + IVCC fq ~ :::- §: t tZ 1.,.:): 0 "'c. (25)
Fig.1 shows the ealeulation results by formula (25). As one ean see the eharaeteristie time of eondensation is equal to tc . • Within this time the moleeule eoneentration will reduee
~n half. Note that usi~g (19) and (2:4) one ~an find the corelat~on between t e.. and -i:. .:nJ. : t. c . ..: t:.<11d.· (t- lf) ~ • The time protiles of Supersaturation degree tor various _ condition.s of water vapor eondensation, determined from (25) and fig.1 are in good agreement with ealeulation result [71 , illustrated by tig.2.
A number of important eharaeteristies of the condensation proeess ean be still found without solving the integral equation (22). For example, one can determine the total number of partielas tormed in the proeess of eondensation by integrating the flux over the time interval o < I:< t.i!J. For this purpose we should allow for an approximatetime variation of :1 due t_o deerease of 5 • One can prove [5] that when <So- S)/So .<< 1
s ) ~z.:~J. ~ = -:J •• (So • (26}
Applying (20) and (26) for the Maximum eoneentration of eondensed partielas we obtain the following expression:
ft,/, rco1 ~ J ::1 •• ·[ fct-r,t1•Je]l <21> r t,+:N, n. •. !1. •
In this ealeulation we also approximately determine parameter ~ as well as the eorresponding induetion time i int;f.
"" = i- !,.. tiM. ;:: L: M. ·[ -lt ~w,~.e I' (28) ,.. 7 · 'IL't · r n.0 • o •
It is eonvenient to study the evolution of other important eharaeteristios of the condensation proeess in terms of such time dependant parameter as the condensation degree 'l = c"'·-~)/IY,0
1092
i .O -0.5
r-0 0.5 f .O -t/tc.
Fig.1. Timeprofile of aupersaturation degree S of condenaing vapour. S .. ia the initial Supersaturation degree; -tc. is the characteristic condensation time
30
20
10
0 0.04 0,013 -t.' s ca::-.
Fig. 2. Time variation of supersaturation degree S of0water vapour in the process of condensation at T•21J C for two values of initial supersaturation. So •37 (curve 1),
1093
So= 34 (curve 2). Dotted lines - calculation by formulae (~i), (~5). Solid lines show the result of numerical Solution (11 •
Thus, the evolution in the condensation process ot the meansizeclusters < tt"'J in tenns of ~ takcs the following form [5] :
<tr:> _ { (fl.o"") -t/~ • 12 34 , t .Ci. Uu/., ~ < Ji~ < ft /Q) - e t > t: ~·~ ;, .. .t. b < 29 >
' ." -c r ll! where <n>.., ie the maximum size of the cluster equal to:
<tL>Q> = (nJ )/ . [ l:t (']?~] f (30)
The obtained analytical expressions of such experimentally observed characteristics of the condensation process as the maximum size <. !L"lcn of clusters and the characteristic time of the process ~c. depend on the initial flow ~0 , assocation rate constants, and consequently, on parameters af: and rL which determine f~ . This permits us to determine the cluster dissociation energy E~ by comparing the theory and the experimental data. As follows
from (30) and considering (8) - (10) ~<tt~~ is practically a linear function of (tit S'.)-oC./(1-Ii} • The slope of the straight line, which expresses graphically this dependence, should uniquely determine .<1 E when <t. known. Such linear deoendence is contirmed by fig. 3 which shows the processed experimental data obtained in ref. [8] devoted to the investigation of isothermal condensation of iron vapours in the inert gas atmosphere. The best fit of theoretical and experimental data is provided for cl.- = o, 1 ~ and
A E = 1, 2. 6· -10- 12. ag. • One can also detennine the cluster dissociation energy by evaluating the temperature dependance of the critical Supersaturation degree. Using the obtained expression for the characteristic time of the process (see (24))and taking into account that for the critical supersaturation degree this time ia equal to some characteriatic time of the condensed particle destruction, one can derive the following equation to determine Sc_ (T), [5J :
~ 1-d.. (<i. AE )~. 1 "" - Öz. Sc. + ET C = 'f - d- ~ T (en. 'l> e.) 1-.r 1<. ., (31)
where C is the function of the characteristic time of the condensed partic1e destruction slightly depending on d E, T, eL , Sc. At proper choice of C, AEand eL equation (31) describes wen the experimental temperature dependance Sc. (T) [9] (see fig. 4). For lead the parameters ol.. and A !,;: have the following numerical ex:oression: ct: ~ and A E-= 1"'r5·f0-1~ elrf!.
1094
Fig. ).
0 0.05 0.10 0.1S
Maximum mean particle size < rz..>e10 veraus the ini tial supersaturation degree SD in the procese of iron vapour condensation at T=1700°C. Points correspond to experimental data (8].
Sc.
Fi g. 4. Tamperature dependence of the critical Supersaturation degree Sc. (T) for P~ • Solid lines represent calculation
1095
b;y fo:rmula {.3 7) when C=20 and ot "' ! ( curve 1 ) , d.. :: i ( curve 2). Dotted line - calculation [9], points - experiment t,J.
Heferences
1. Prenkel Y.I. Kinetic theo~ of liquide.- M.-L., USSR Academy ot Seiences Publishers, 423 (1945} - In Russian
2. Robinson P., Hallbrook K. Monomolecular reactions. - M., Mir Publishers, 380 (1975) - In Russian
3. Bauer S.H., Prurip D.J.Homogeneous Nucleation in Metal Vapors. 5. Self-consistent kinetic modal, J.Chem.Ph;ya., 81:1015 (1977)
4. Zaslonko I.S. Monomole~ular reactions in shock waves and energ;y exchange of highl;y excited molecules. Dias. of Doctor of Ph;ys. math. sciences. Moscow, Institute of chemical physics, USSR Acad. of Sc. Publishers, 502 (1980) - In Russian
5. Gordiets B.F., Shelepin L.A., Shmotkin Y.S. Kinetics of isothermal processes of condensation and carbon-black fo:rmation. Preprint 78.-M., PlAN 72 (1982) - In Russian
6. Lushnikov A.A., Sutugin A.G. Modern State of Homogeneous Nucleation Theor;y, Uspekhi Khimii, 45:385 (1976) - In Russian
1. Courtne;y W.J. Kinetics of condensation of water vapor, J. Chem.Phys., 36:2018 (1962)
8. Freund H.J., Bauer S.H. Homogeneaus Nucleation in Metal Vapors, 2. Dependance of the Heat of Condensation on Cluster Size, J.Chem.Ph;ys., 81:994 (1977)
9. Frurio D.J., Bauer S.H. Homogeneaus Nucleation in Metal Vapors, 3. Tamperature Dependance of the Critical Supersaturation Ratio for Iron, Lead and Bismuth, J.Chem.Phys., 81:1001 (1977)
1096
RELAXATION PROCESSES DJ A MOLECULAR DYNAMIC MODEL OF
CLUSTER FROM THE LENNARD-JONES PARTICLES
S. P. Protsenko and V. P. Skripov
Department of Physico-Technical Problems of Energetics, Ural Science Centre of USSR Ac. Sei., Sverdlovsk, 620219, USSR
An information about cluster properties on a molecular level is of great interest. To study microscopic processes in clusters by experiment is, however, rather difficul t. One of the means of investigo:ting clusters as systems of interacting particles is computer simulation. The molecular dynamic (MD) methodi-~, which consists in a numerical integTation of the equations of motion of N particles interacting between each other by a well-known law, has been used for the simulation of relaxing clusters.
The objective of this work is to study the cluster reaction to the perturbation of density simulated by a sudden fonnation of a cavity in an equilibrium configuration of particles, to describe the microscopic mechanisrn of relaxation of the structure, the behaviour of therrnodynarnic f'unctions in time, to evaluate the characteristic relaxation tirnes of various properties, and to compare the relaxation of a cluster with that of an extended rnolecular systern.
Each of the·clusters under investigation presented a set of N particles with a fixed centre of mass interacting pairwise with the Lennard-Jones potential (6-12) located in a spherical reflecting shell of radius R. The model is described at greater length in 4 • Further non-reduced quantities have been obtained on the basis of the potential parameterB e and 0 , and the molecule mass, denominate quantities - for the case of Ar (€. = = 1.65 . 1o-2.i J' 6 = 3.405. 1o-1o m).
Each of the initial clusters prepared for relaxation had 256 particles. The sizes of the two cells
1097
Table
. T R N tre te trc te trn r1 r
K R c ps ps ps ps ps m/s m/s
20 4.2 225 11.5 2.7 9 • .3 ,3.9 11 82 69 1.9
40 224 9.6 1.6 6.0 ,3.2 14 82 69 55 2.30 7.0 1.1 6.0 2.6 11 118 6.3 70 2.30 5.5 0.9 4.5 1.5 14 190 125 90 2.31 6.5 1 • .3 5.0 1.6 18 168 107 90 ~ 198 9.0 10.0 19 298 125
used in the calculations (R = 4.2 and 5 • .3) were chosen in such a way that at N = 256, the average numerical density in the first cell (n = 0.8,3) was close to the density of liquid Ar in the triple point, and that in the second cell was two times less.
At a certain moment we withdrew all the particles from a sphere of radius Re< R, the centre of which coincided with the centre of the cell. In small systems (R = 4.2) in the sphere R = 1.9 there were 25-.32 particles, in a cell with R = 5 • .3 from the sphere R = 2.6 58 particles were withdravm. In order to fix the mass centre of the summary particle rnomentum component, the systems were reduced to zero.
The relaxation of molecular modele after the onset of a cavity was realized under isothermal conditions (see the Table) for which the temperature of the systemwas reset to the initial value of To by correcting the particle velocities. Such a notorious idealization of the model allows us to separate in a pure form the inertial stage of the relaxation, but excludes the thermal one.
For the sake of a rigorous quantitative evaluation of the properties of relaxing systems, it is necessary to carry out some calculations of the model from different initial microstates and to average the instantaneous values by the ensemble of the experiments. Here we have obtained rougher results by averaging the pro-
1098
Fig. 1. Sequence of the particle confit:,>Uration projections after the formation of a cavity. The partielas are shown with circles of' a single diameter. The projection of a cavity of initial radius 2.6 is shown with a dashed line.
perties under investigation by time intervals of length 50~t or 100~t with ~t = 0.01 ps where the evaluated properties did not change appreciably.
How do relaxation processes proceed in a cluster? Density disturbance upset sharply the mechanical and the thermodynamic equilibrium of the cluster. In a hydrodynamic approximation, in a condensed phase there formed a cavity with zero pressure into which the substance flew due to the difference in the outside and the inside pressure. 'l'he average pressures in the cells before the formation of cavities P0 were 0.6, 3.0, 10.4, 18.8, 42.6, 2.5 MPa at T0 = 20, 40, 55, 70 and 90 K, respectively. The latter value refers to a cell with H = = 5.3.
At the molec~lar level, the process of filling the cavity may be described as follows. In a homogeneaus system all the partielas were in equilibrium with their neighbours, with the minimum configuration energy possible at a gi ven densi ty. After the formatj_on of a cavi ty partielas situated close to its boundary turned out to be without several of their nearest neighbours, which led to an increase in their potential energy. The necessity to decrease the latter was realized by attraction through the cavity and a possibility of further movement inside it, without repulsion. The frontier part-
1099
icles carried along all the particles of the cluster. The movement of the particles acquired a primary direction into the cavity preserving chaotic heat jumpe that are easy to observe in nonnal directions at the particle movement trajectories. A similar particle motion could be observed in a hydrodynamic flow.
\-le may specify the microscopic pattern of the molecul~r system evolution by observing the sequence of projections of molecular configurations. Fig. 1 shows the projections of a cell with a thickness of (0.3-0.7)•2R along the axis OZ. For a system relaxing at 90 K, R = = 5.3, the photography moments are given in picoseconds. We may draw a conclusion that in the process of relaxation there occurred an all-round contraction of the cavity boundury, i.e. the filling of the cavity looked like a collective flow, close to a spherically-syrarnetric one, directed to the centre of the cell. During the cavity evolution in simulated systems no activation particle-by-particle molecule evaporation has been explicitly detected.
The deter.mination of spherical symmetry of the filling process allows us to make use of the for.malism of the function of density distribution by the cell radius n(r). New results have been obtained when investigating the time variations of n(r). The instantaneous values of n(r) were averaged by quasi-equilibrium time intervals of 0.5; 1 ps and approximated by polynomials. The sequence of profiles obtained at 55 K is shown in Fig.2. We may subdivide the process of redistribution of the local density in a cluster with.a disturbed structure into several typical stages: (1) smoothing of the initial stepped profile at the first steps of relaxation, (2) advance of the smoothed front towards the centre of the cell, (3) growth of the density in the vicinity of the cell centre up to the values exceeding (n}, (4) density drop in the central region of the cell due to a slight collective reverse motion or alowing down of the particles and a more uniform diatribution of them in the cell volume, (5) establishment of a stationary, alightly nonuniform profile. The stages (3) and (4) repeated themselves several times, i.e. after the filling of the cavity with moleculea,· the establishment of stationary density diatribution was preceded by a space-time oscillation densification-rarefaction process. We could distinguish the typical stages mentioned above in all the cases of relaxation.
Some quantitative results have been obtained from time dependences of the internal energy of the clusters E(t), the number of particles N (t) in a cavity of radius R , the poaition of the cavity boundary r (t).
1100
1
0,5 0 i
0,5 0 1
;:::::; 0,5 .._ ~ 0
1
0,5 0 1 0,5 0
0 RO R 0 RO R
Fig. 2. Evolution of the density radial profile after the fo1~ation of a cavity. A straight line shows the average density in the initial system and the thickness of the particle layer at the initial moment of the evolution. T = 55 K, R = 4.2, Re = 1.9.
Fig. 3 shows the results of calculating E(t). After the initial stage of the evolution t 0 , that lasts (2-4) ps, E(t) and Nc.(t) are easy to describe with the relaxation function f'( t), that satisfies the equation
1- (f(t)- f(t 0 ))/((f)- f(t 0 )) = exp(-(t-t0 )/'t),
where (f> is the averae;e value of f(t) in an equilibrium state after relaxation, 't is the drunping decrement of the approximative function shovm j_n Fig. 3 wi th a solid line. A reliable judgment on the regularity of the variation of f(t) in the first (2-4) ps is not easy to form on the basis of isolated experiments. The relaxation time t~ was evaluated graphically by the moment of f(t) leaving into the vicinity of (f). The relaxation times of internal energy t-re. and the number of particles in the cavi ty t~e , the time of establishing a stationary density profile t'l'l\. and the corresponding darnping decrements are given in the Table. In systems of the same size we observe a tendency towards decrease of tre. and t'I'C. wi th increasing temperature. It would be inte-
1101
-5 ~ <~""""
e e e~
,o /" z;;; e • e e 1 • 6 , .... - o- i
o - 2 ~ ..-... e• 3
~ -4 ..-• .",.--... • • := 4 t .,. •••• •••• "_5 g :,r"""&' •• • •-6
~ -3 8 ~ I X - w y S C
.. w
••• • • • ••••••• .. . ..
-2 . • • • ••
o~------~~o~------~20~~
t,ps Fig. J, Time variation of the internal energy of clus
ters after the formation of cavities. The relaxation is at the following temperatures: 1 - 20 K, 2 - 40 K, J - 55 K, 4 - 70 K, 5 -
90 Kin a cell with R = 4.2, Re= 1.9, 6 - 90 K, R = 5.3, Re = 2.6.
resting to know that the least relaxation time has been obtained at 70 K. We may assume that in our series of experiments at a temperature exceeding 70 K the thermal motion makes the process of filling chaotic and decelerates it.
A direct observation of the molecular configuration sequence showed a change in the size and the form of the cavitywithin a period up to 5 ps, and further we could observe but changes in the external form of the cluster. It took almest the same time to restore the thermally-equilibrium movement of the atoms situated close to the cavity boundary. The data given in the Table enable us to form a more precise idea of the relation between the relaxation times for different model characteristics. In particular, energy relaxation in the cases of evolution we have considered came to an end immediately after the filling of the cavity, with a lag of only (1-J) ps.
The filling of the cavity and the stabilization of internal energy did not signify completion of the
1102
relaxation processes in a model system. The complete relaxation came to an end with detailing of the microscopic structure, the setting time of which t~~ was identified with the setting of the stationary values n(r). According to the calculations, t~n/tr~ tends to growth wi th increasing T : trn. /tl"e ~ 1 at T = 20 K and ~ 3 at 90 K. The internal energy of the system proved to be of little sensitivity to the equilibriurn microscopic structure of the substance. As a whole, the consequences of a perturbation that is large by the ·scale of the model - formation of a re gion of a sharp densj.ty variation wi th a size ~o .1 of the system volume, were elirninated wi thin periods of time comparable by the order with the time of balancing of homogeneaus configurations, which is known from rnany of the model calculations.
Fig. 4 shows time dependences of the position of the cavity boundary rc.(t). The rate of the cavity radius decrease defined as was evaluated by the slope rc and up to the ter mination of filling did not change greatly, with the exception of the initial stage of (0.5-1) ps, wi thin which the rate defined as rc1 exceeded by (20--50) m/s the rate at any other time for syst ems in a cell with R = 4.2. The values of average rate (r~> and r~ 1 are given in the Tablc. The decrease of t he system average density by a factor of 2 and a simultaneaus twofold increase in the perturbation volume made the i nitial stage of collapse appreciably qui cker at T = = 90 K (see the Table). In that case, the rates of the
t.., 10 0
10 '• ~OK •••• 0
. .. . ,.
10 ·' .!· 55K .. .... 0 .,
10 ··' 70K 1 •. 0 : .
10 . ·-=· Rt 90K ,• 0
.. ,.
90 K 10 '• .· . lt•5.3 R . . . . 0 0 2 't
Fig . 4. Time dependences of t he cavity r adii in relaxing clusters.
1103
linear sections of filling practically coincided. In conclusion, it should be emphasized that in the
process of MD simulation of relaxing microscopic clusters there have been observed some qualitative traits of the evolution of cavities that allow us to call it conventionally a "microhydrodynamic" flow. However, despite the formal resemblance to the collapse of' cavitation bubbles in macroscopic volumes of liquid, the evolution of a cavity and a microcluster, as a whole, is a phenornenon of a purely microscopic order and, to a great extent, is conditioned by the infinitesimal and the constancy of the nurnber of particles in the system. Evidently, due to these circumstances, the rate of the bubble radius decrease practically does not change with time, which corresponds neither to the Rayleigh inertial solution5, nor to more precise modern solutions, for instance6, that take into account the heat-transfer processes during the evolution of a bubble.
REFERENCES
1. B. J. Alder, T. E. V/ainwright, Studies in Molecular Dynami es. I. General Method, .1[. Chem. ®:s. 31:459 (1959).
2. A. Hahman, Cerrelations in the Motion of Atom in Liquid Argon, Ph;"L§.. Rev. A, 136:405 (1964).
3. A. M. Evseev, .lVI. Ya. Frenkel, A. N. Shinkarev, The Molecular Dynamics Method in the Theory of Equilibrium States and Irreversible Processes, Vestnik IvloskovskogQ Universiteta. Hirniya, 11:154 (1970) - In Russian.
4. S. P. Protsenko, V. P. Skripov, Molecular Dynamics Investigation of the Evolution of Density Perturbation in the Small Lennard-Jones System. Relaxation of the Molecular Structure, Zhurnal Fizichesk.QY., Himii, 55:2481 (1981) - In Hussian.
5. Rayleigh, On the Pressure Developed in a Liquid durj_ng the Collapse of a Spherical Cavi ty, Phylos. Mae.;_. Ser. 6, 34:94 (1917).
6. R. I. Nigmatulin, H .s. Habeev, The Dynami es of Vapour l3ubbles, Izv. AN SSSR. Mehanika Zhidkosti _i_Gasa, 3:59 (1975)- In Hussian.
1104
* QUANTUM-CHEMICAL STUDY OF PROCESSES WITH CLUSTER ISOMERISM
Zden~k Slanina
The J.Heyrovsky Institute of Physical Chemistry and Electrochemistry,Czechoslovak Academy of Sciences, Machova 7, 121 38 Prague 2,Czechoslovakia
INTRODUCTION
Remarkable progress has recently been made in the application of quantum-chemical1methods combined with the cluster concept ~n the study of real gases3 , of the interactions of gases with solids , and of the liquid state • The importance of clusters for quantum chemistry is based on the fact that these clusters represent a useful model that can be used in contemporary numerical quantum chemistry and permits description of at least some of the important characteristics of these systems. Although these clusters represent simple models, contemporary quantum chemistry is not able to give an exhaustive description of their ene~getics, i.e. cannot yield the complete potential energy hypersurface (in the framewerk of the Born-Oppenheimer approximation). In general, the energy hypersurface is rather mapped only in5terms of localization and characterisation of its stationary points • This systematic investigation of the energy hypersurfaces of cluster systems (in the framewerk of various quantum-chemical approximations or molecular mechanics methods) has frequently demonstrated that the given system is not represented by a single species (as has so far been assumed in the interpretation of the observed data) but rather that the given hypersurface contains a greater number of nonequivalent local energy minima that are relevant for the given system, i.e. that cluster isomerism ~~9involved. Thus isomerism has been ~heoretically demonstf8ted e.g. 10 for (H2 > 2 ~ (N0) 2 , (cl2)?il (co2) 2 , d~mers of formaldehyde 1~_ygetone , 1,1,1-tr~-fluoroet~,n~S , some heteromolecular clusters , and model clusters ' that can be used for study of the interactions between
*Part XVIII in1She series Multimolecular Clusters and Their Isomerism; Part XVII Ref. •
1105
gases and solids, as well as for activated complexes19 involved in cluster (auto)isomerisation. Discovery of cluster3i28merism is not21 limited to dimers alone, see e.g. isomers of water ' and methanol oligomers. The nurober of isomeric structures generally increases with an increasing nurober of particles, as has been ~2~~gstrated in a study based on very simple empirical potentials
While in the framewerk of the concept of representation of energy hypersurfaces in terms of their isolated points alone, the individual isomers are clearly distinguishable (as a result of this reducing approximation), in experimental studies the possibility of distinguishing the individual isomers becomes important only under special conditions and when using suitable observation techniques (e.g. microwave and radio-frequency spectroscopy). Generally however, the cluster system involving isomerism will act as a whole as an effect of the Observation technique and conditions employed. The results of this experimental study are then interpreted assuming that a single cluster structure was present. As the energy barriers are essentially low, the formation of inter-isomer equilibria can be expected under the experimental conditions. Consequently, every structurally-dependent characteristic of the cluster system involving isomerism measured by a technique that does not distinguish between individual isomers (even if all the structures are active in this technique) will have a convolutional nature resulting from the contributions of all the isomers to the final overall value. On the other hand, theoretical treatment yields primarily the partial values of this characteristic corresponding to the individual isomers. If the theoretical and experimental values are to be correctly compared, then it is necessary to carry out an adequate weighting treatment leading from the partial to the overall values. This work is concerned with the problern of weighting for the standard (or activation) enthalpy and entropy terms of processes involving cluster exhibiting isomerism.
WEIGHTING AND ILLUSTRATIVE EXAMPLES
Consider equilibrium process (1) in an ideal gaseaus phase
(i= l, .... ,m) (l)
11~~ing to formation of the i-th isomer of cluster c(i) of particles A , where vk designates the stoichiometric coefficients. tsrume that all the eigenstates and the corresponding eigenvalues e.~ of the t7fh cluster are known (the ground state energy is desigrtated as e ~ ). Then the partition function q. (2) ofthisisomer can be
0 ~
q. = 4 exp [- (e ~i) e (i) ) /kTJ (2) ~ J J 0
1106
constructed. The magnitude of the contribution of this isomer to the overall terms will depend26- 28 on the weights w. (3). The simple
~
W. ~
q. exp (-e(i) /kT) ~ 0
(j) j~lqj exp (-e0 /kT) (3)
configurational factors wi (4) will also be introduced, that are
w~ ~
exp (-e_(i)/kT) 0
exp ( -e - ( j) /kT) 0
(4)
sometimes recommended as a useable approximation to the w. values. The usefulness of these configurational factors is given By the fact that knowledge of the depths of t~e)individual local minima on the potential energy hypersurface, e- ~ , alone is sufficient for their construction. 0
26-28 If the resul ts of the general isomer.ic problern are employed , then Eqs (5) and (6) can be shown to be valid for transition from the
0 111 0 lls = . .l...l w. (lls . - R ln W. )
T J= ~ T,~ ~
o m o da = .L1 w. dHT . T J= ~ ,~
(5)' (6)
0 0 partial enthalpy and entropy terms AH . and llsT . to the corresponding overall values AH0 and lls0 • In order f6~effectiv~Iy test some of the hypotheses employ~d in th~ literature, differences ÖXT E and ÖXT G' given in terms of Eq. (7) , (deficits of the "best" single configuration)
AXO T
(X = H or S; Y = E or G) (7)
can be introduced, where indices E and G designate the isomers that are mst stable at zero temperature and at temperature T, respectively. The indices correspond to determination of the stability order on the basis of the potential energy scale and the Gibbs function scale,resp.
It will be noted that, if the appropriate partition function of the adsorption complex is considered (suppression of translational and rotational motion, co~~i~§ration of the distinguishability of surface sites- cf. Refs ' ), then the above formulae can also be used for weighting in gas-solid interactions.
In the study of the weighting behaviour and to test some symplifying variations recommended in the literature, several systems have been selected for which recent theoretical studies have demonstrated cluster isomerism and that have also yielded sufficient structural, vibrational and energy information on these systems so that weighting could readily be carried out (provided the rigid rotator and barmenie
1107
oscillator approximation is considered satisfactory for partition function q.). The following systems were selected: (82) 2 , (N0) 2 , HF-8Cl, 8F~ClF, (820) 2 , and o2+graphite. The descr~p~7~~s 1~or (82 ) 2 and (82o) 2 were at the äb initio SCF CI level ' ' 7 ' 29 .fgr30~e f6m~In~~g gas-pfiase complexes at the ab initio SCF level ' ' ' '
' ' and, for the physical adsorption of 0 on gri~hite, the description was based on the empirical potentiaf (Ref. ) . The (820) isomerism was not that of a stable associate, but rathi9 of activate~ complexes leading to autoisomerization of water dime6s • The (82) 2 systemwas found to c~§ti~n four isomeric structures , the (82o) 2 and o2+graphite three ' , and the others two.
The temperature dependence of the weight factors w. and simple configurational factors w~ of the considered systems in1a broad temperature interval are depicted in Figs la-f. It is readily apparent from these results that, even though there are systems for which w~ would be an acceptable approximation to w. for at least some tempe~ rature regions, there are also situations1in which this procedure cannot be used at practically any temperature. A good example is the HF-ClF system (Fig.ld). The usefulness of w~ is dependent on the cancellation effects in Eq. (3) for which no §imple rule can readily be proposed.
The relationships between the partial and overall enthalpy and entropy terms are given for the (N0) 2 , 8F-8Cl, 8F-ClF, and o2+graphite systems in Figs 2a-d, condensed as d1fference diagrams indicating the temperature dependences of the öx E and öx functions. Provided that the most stable structure in theTtemperatut~Ginterval from zero to the upper temperature limit chosen is that lying lowest on the potential energy scale, then the courses of functions OX andO X are essentially identical and continuous. 8owever, when t~~re is T,G a critical point at which a change in the most stable structure on the Gibbs function scale appears (Figs lc,d), an interesting situation occurs, leading not only to different courses of functions ÖX E and OXT , but also to discontinuous character of function öxT' . Apparentfy, if the discussion is to be limited to a single "b~s~" structure, then, rather than selecting the most stable structure on the potential energy scale, the structure that is most stable at the given temperature on the Gibbs function scale should be chosen (provided that these structures are different). Nonetheless, the results given in Figs 2a-d clearly indicate that at least at some temperatures deficits 8K attain values such that replacement of the overall term by theTpärtial values corresponding to either of the two considered "best" single structures is unacceptable. The possibility that function 8K need not necessarily increase with increasing temperature (e.g. Pi~.2a) is interesting.
CONCLUSIONS
It has been demonstrated that, provided that quantum-chemical
1108
0 CO
Fig
.l.
I 1 J
I ~~ Q5 0
0 0
100
a
B
d
B
. ..
A
A ](
)()
T,K
50
0
'·,·,,,<
; ',
T
---10
0
b e
~
f
--___
c; "
B
----
----
----
---
I
100
lOO
T
,K
500
Tem
pera
ture
dep
en
den
ces
of
weig
hts
w
. (-
--)
an
d
sim
ple
co
nfi
gu
rati
on
al
facto
rs
w:
(--
-).
(a)
Pla
nar-
recta
ng
ula
r (R
) ,
lin
ear
(t),
p
lan
ar-
ort
ho
go
nal
(0),
an
d n
on
-pla
nar-
ort
ho
gon
al
(N)
(H)
; (b
) T
ran
s (T
) an
d cis
(C
) (N
O)
; (c
) H
F.H
Cl
(A)
an
d H
Cl.
HF
(B
);
(d)
HF
.Cl F
(A
) an
d Cl~.SF
(B);
(e
) P
lan
ar-
lin
ear
(P),
cfo
sed
(C
),
an
d b
ifu
rcate
d
(B)
acti
vate
d c
om
ple
xes
in th
e
(H2o)
2 au
tois
om
eri
zati
on
; (f
) A
dso
rpti
on
o
f o 2
m
ole
cu
le at
the cen
tre o
f a
gra
ph
ite
hex
ag
on
(~),
at
a carb
on
ato
m
(B),
an
d a
t th
e cen
tre o
f c-
c b
on
d
(C).
0
Fig
. 2
.
a a
100
300T
50
0 ,K
-
m
300T
50
0 ~
: ' ' 1)(
)
c c 300
T K
500
~
1)(
)
d
d 30
0 T
K .
500
----
-!--
Tem
pera
ture
dep
en
den
ces
of
fun
cti
on
s O
HT
, O
S (-
--)
an
d
OH
E
' O
S E
(--
-)
for
sele
cte
d c
luste
r fo
rmati
on
s.
(a)
(N0)
2;'
Yb
) Hr~~Cl;
(c)
HF-clr~
(d)
t~e
so
rpti
on
o
f o 2
o
n g
rap
hit
e
(cf.
F
igs
lb-d
, f).
studies indicate the existence of at least two different isomeric structures of comparable stability for a cluster system, it is generally necessary to apply weighting with weights w .• Only then does it become possible to carry out correct comparisoh with the observed data, which are generally of an overall, convolution nature. Simultaneously, it has been demonstrated that the two hypotheses used frequently in quantum-chemical study of clusters are not generally acceptable. These are the general inapplicability of the simple configurational factors as satisfactory replacements for w. and the fact that the overall terms cannot generally be satisfa~torily approximated in terms of the partial ones corresponding to a single structure, especially for the entropy. Simultaneously, a further logical step in the development of the weighting treatment would appear to be the necessity of improving the quality of weights w. in transition from harmonic to anharmonic vibrational motions. ~
REFERENCES
1. z.Slanina, Consequences of isomerism of n-particles clusters for confrontation of their quantum-chemical characteristics with Observable quantities, Collect.Czech.Chem.Commun. 42:3229(1977).
2. R.C.Baetzold, Application of molecular orbital theory to catalysis, Advan.Catal. 25:1(1976).
3. E.Clementi, "Liquid Water Structure" ,Springer-Verlag,Berlin(l976). 4. R.F.W.Bader and R.A.Gangi, Ab initio calculation of potential
energy surfaces, Theor.Chem.2:1(1975). 5. J.W.Mciver,Jr. and A.Komornicki, Structure of transition states
in organic reactions, J.Am.Chem.Soc.94:2625(1972). 6. E.Kochanski, B.Roos, P.Siegbahn, and M.H.Wood, Ab initio SCF-CI
studies of the intermolecular interaction between two hydrogen molecules near the van der Waals minimum, Theor.Chim.Acta 32:151(1973).
7. S.Skaarup, P.N.Skancke, and J.E.Boggs, Structure and force fields of the isomers of N o2 , J.Am.Chem.Soc. 98:6106 (1976).
8. J.P:issette and E.Koc~anski, Theoretical study of the (cl2) 2 d~mer, J.Am.Chem.Soc. 100:6609 (1978).
9. N.Brigot, S.Odiot, S.H.Walmsley, and J.L.Whitten, The structure of the carbon dioxide dimer, Chem.Phys.Lett. 49:157 (1977).
10. D.J.Frurip, L.A.Curtiss, and M.Blander, Characterization of molecular association in acetone vapor, J.Phys.Chem. 82:2555 (1978).
ll.L.A.Curtiss, Ab initio calculations on hydrogen bonding in alcohols: Dimers of CH30H, CH3cH20H, and CF3cH20H, Int.J.Quantum.Chem., Symp. 11:459 (1977).
12. Z.Slanina, Automatie optimization of the space arrangement of two molecules controlled by the Buckingham potential, Collect. Czech.Chem.Commun. 39:3187 (19741.
13. B.Jönsson, G.Karlström, and H.Wennerström, Ab initio molecular orbital calculations on the water-carbon dioxide system:
1111
Molecular complexes, Chem. Phys.Lett. 30:58 (1975). 14. J.Prissette,G.Seger, and E.Kochanski, Theoretical study of some
ethylene-halogen molecule (Cl2 ,Br2,r2) complexes at large and intermediate distances from a5 in~tio calculations, J.Am.Chem. Soc. 100: 6941 (1978).
15. P.Hobza, M.M.Szcz~sniak, and Z.Latajka, HF-HCl: Stationary points on the SCF energy hypersurface and thermodynamics of formation, Chem.Phys.Lett. 74:248 (1980).
16. P.Hobza, M.M.Szcz~sniak, and Z.Latajka, HF-ClF: Minima on the 4-31G and 4-31G* energy hypersurfaces and thermodynamics of formation, Chem.Phys.Lett. 82:469(1981).
17. Z.Slanina, Adsorption-complex isomerism and quantum-chemical study of gas-solid interactions: A model example, Theor. Chim.Acta 60:589 (1982).
18. Z.Slanina, Adsorption-complex isomerism and quantum-chemical studies in heterogenous catalysis, Int,J,QuantumChem. (inpress),
19. Z.Slanina,Isomerism of the activated complex in water dimer interconversion,Advan.Mol.Relaxation Interact.Processes 19:117(1981).
2o. J.C.Owicki,L.L.Shipman, and H.A.Scheraga, Structure,energetics and dynamics of small water clusters, J.Phys.Chem. 79:1794(1975).
21. L.A.Curtiss, Molecular orbital studies of methanol polymers using a minimal basis set, J.Chem.Phys. 67:1144 (1977).
22.D.J.McGinty, The single configuration approximation in the calculation of the thermodynamic properties of microcrystalline clusters, Chem.Phys.Lett. 13:525 (1972).
23. J.J.Burton: The configurational contribution of the free energy of small face centered cubic clusters, Chem.Phys.Lett. 17:199 (1972).
24. S.H.Bauer and D.J.Frurip, Homogenous nucleation in metal vapors, J.Phys.Chem. 81:1015 (1977).
25.M.R.Hoare, Structure and dynamics of simple microclusters, Advan. Chem.Phys. 40:49 (1979).
26. Z.Slanina, Equilibrium processes with reaction components of isomeric composition, Collect.Czech.Chem.Commun.40:1997 (1975).
27. Z.Slanina, Isomerie structures, weights and effective characteristics of clusters, Advan.Mol.Relaxation Interact.Processes 14:133 (1979).
28. Z.Slanina, Chemical isomerism and its contmporary theoretical description, Advan.Quantum Chem. 13=89 (1981).
29. Z.Slanina, Isomerism and energetics of (No) 2clusters and their role in real gas phase nitric oxide, Collect.Czech.Chem.Commun. 43:1974 (1978).
30. Z.Slanina, The role of the "less stable m~n~mum-energy structure" in evaluation of the characteristics of the HF-HCl van der Waals system, Chem.Phys.Lett. 82:33 (1981).
31. Z.Slanina, HF-ClF isomerism: The consequences regarding system thermodynamics, Chem.Phys.Lett. 83:418 (1981).
32. Z.Slania, Breakdown of the conventional formula for the partition function of free internal rotation, J.Phys.Chem.86 (1982) (in press).
1112
THE HOMOGENEOUS NUCLEATION AT THE CONTINUOUSLY CHANGING
TEMPERATURE AND VAPOUR GONCENTRATION
A. G. Sutugin, A. N. Grimberg, and A. S. Puchkov
Karpov Institute of Physical Chemistry, Moscow 107120, USSR
In most cases the modelling of disperse phase formation due to homogeneaus nucleatian of a supersaturated vapour is based an the steady-state solutions of the classical Volmer-Weber-Becker-Daering theory (Sutugin et al., 1976). However, the condensation process takes place usually with the simultaneaus cooling of condensing system. The steady-state nucleation rate, ] 0 , depends on the time via the dependences of a temperature, T (t) , and monomer concentration, X~(t) • The temperature drop being fast, the rate of alteration of J 0 becomes too big to allow the use of the steady-state values, even if the correction for nonsteadiness is made in the usual form J = J 0 { 1 - exp ( t:/r)} • This difficulty can be avoided by using a numerical description of a mass transfer along the axis of the cluster size ( 9 -axis) in the precritical and near-critical ranges of cluster size and af the condensation growth of the supercritical particles. The application of the combination of an eulerian description of mass transfer in the ini tial regian of 9 -axis wi th lagrangian descriptian in the supercritical regian has previausly been demanstrated far the case of rapid type nucleation by Sutugin and Fuchs (1970) and Sutugin (1978).
It has previously been shown (Sutugin et al., 1981) that the microkinetic description of aerosol formation through the spontaneaus condensatian leads to essentially different results in comparison with the calculations based on the steady-state solutions of the nucleation theory. It has been shown for a single case of the condensation that after the beginning of the super-
1113
saturation drop the nucleation rate decreases considerably slower than the steady-state model predicts.
The present work was carried out to explore the spontaneous condensation process by means of numerical experiments at varied concentrations of vapour of the same substance. The alteration of vapour concentration means that of growth rate at the approximate constant clusters evaporation rate. So, the numerical experiments carried out for varied concentrations of vapour permit to study the role of the ratio of rates of the clusters growth and evaporation for non-steady nucleation during the Supersaturation drop.
The concentration of clusters, X~ , consisting of 9 molecules is defined with the kinet~c equation:
( 1)
where X.1 is the monomer concentration, ß s is the kinetic coefficient of cluster growth, and ds is that of evaporation of cluster. d-3 and ßs is expressed by means of the conventional liquid drop approximation for the free energy of cluster formation.
The time evolution of the size distribution of supercritical particles is described by numerical solution of a set of equations for several groups of averaged sizes:
(2)
where K ( (;.p, Kn) is the constant of rate of the monomer molecule collisions with particle containing Gr, molecules, K~ is the Knudsen number, .S is the supersaturation, and Se. ( G..,) is the dimensionlese equilibrium pressure of vapou~ over the particle of size Gp • The size distribution function for supercritical particles is supposed to be presented as a histogram of n groups with sizes G ' r = 1,2, ••• n . Equation (2) describes the kinetics of particle growth in a free molecular regime and takes account of Sherman's (Fuchs and Sutugin, 1969) and Kelvin's corrections.
The vapour exhaustion is accounted for by means of solution of the proper equation simultaneously.
1114
.....
U1
Tab
le
1.
Par
amet
ers
of
pro
cess
un
der
calc
ula
tio
n a
nd p
hy
sico
-ch
emic
al
ones
o
f co
nden
sed
sub
stan
ce
Ou
tflo
w t
emp
erat
ure
En
vir
on
men
tal
tem
per
atu
re
No
zzle
dia
met
er
Ou
tflo
w r
ate
Init
ial
vap
ou
r co
nce
ntr
atio
n
the f
irst
case
the s
econ
d ca
se
Oct
adec
ane
den
sity
Oct
adec
ane
surf
ace
ten
sio
n
T1
=
453
.6°K
T2.
=
32
2.5
°K
d =
1
mm
\50
=
109
m/s
'X~=
10
18 c
m-3
0
18
-3
X 1
= 0
. 6 •
1 0
cm
f =
0.7
62
-
0.0
00
69
(T-
32
3)
g/cm
3
CJ
=
( 2 •
9 58
2 ~
) 4
dy
n/ c
m
Oct
adec
ane
equ
ilib
riu
m
vap
ou
r p
ress
ure
Lo
3l'o
= 6
. 795
7 -
1367
.12
4/T
-5
45
92
8/T
2 m
mHg
The most common condensation conditions are turbulent mixing of the flows with different temperatures and adiabatic expansion. The decrease of the vapour concentration is accompanied with the temperature drop in these cases. Dilution or expansion processes are taken account of by means of solution of mixing kinetics equations (Sutugin and Grimberg, 1975).
On the whole the calculation scheme looks as that described previously by Sutugin et al. (1981). However, somewhat another procedure was used to descript the evolution of the size distribution. New procedure allowed to get more detail representation about the form of the particle size distribution.
The method was applied to the calculation of the spontaneous condensation of octadecane vapour in the free turbulent hot jet of nitrogen issuing into cold, stagnant air. Parameters of process are presented in Table 1. Their values corresponded with those of the previous experiment (Sutugin et al., 1978). The literature data for & , p and p0 used were the same as in Situgin et al. (1978).
So, the calculations enable to follow the time evolution of temperature, monomer concentration, supersaturation, composition of precritical and near-critical clusters population, and also evolution of size distribution of the disperse phase formed.
The values of nucleation rate and the nucleation rate to monomer concentration ratio are shown in Fig. 1
Fig. 1.
1116
i' ...,
<I) ',; ." 12 V
'E r..) .... ~ 11
)<.
cn ~ 19 ,3 0'"1
0 10 18 "._j
6 17 s 5.22 -logt, s
Nucleation rate J and vapour concentration X-t as functions of Supersaturation S and time 18 when initial vapour concentration X~ is 10 molecules per cu cm (the first case).
16 20 24 28 32 4.4
18~ 11 E 16 .;
)(
r:T" 0 ~
Fig. 2. Nucleation rate J , vapour concentrationX1 , and J/ )(1 ratio as functions of Supersaturation S and time t 18 when initial vapour concentration is 0.6·10 molecules per cu cm (the second case) •
..
and Fig. 2 as functions of supersaturation S in increasing and decreasing S • The monomer concentration corresponding to the same time moments is presented also. It will be seen that when the initial vapour concentration is ernaller the "nucleation hysteresis" is found out. To put it another way for the same values of S the nucleation rate is higher in decreasing rather than increasing supersaturation. As distinguish from J hysteresis is more explicit with ratio 1/X.~ • Ratio J / x1 is choosen as illustration to exlude the role
1
10 30 0 70 9
Fig. 3. Kinetic coefficient of cluster growth to one of cluster evaporation ratio ~9/~g as function of the number of molecules 9 in cluster for the two cases (curve 1 refers to the. first case and curve 2 does to the second case).
1117
of preexponential factor in the expreaaion for nucleation rate. When Supersaturation ia maximum for the two conaidered17caaea the vapour concentrationa are 8.8·1017 and 3.5·10 moleculea per cu cm, temperaturea are 421 and 385°K reapectively. So, when initial vapour concentration is smaller the nucleation begins later and it reaches maximum later. Evaporation rate of clustera is therefore ernaller for the second case. The kinetic coefficient of growth to one of cluster evaporation ratio is shown in Fig 3.as function of cluster aize provided that the Supersaturation is maximum
ßv'o{9-=- .) exr (- A~ g-tt3) (3)
Occurence of the nucleation hysteresis will be seen to involve greater value of ratio ~s/d9 . At the same time the nucleation hysteresis will be noted as being universal phenomenon. In the second case of present work the quantity of nucleation hysteresis ia smaller as compared with the previous work (Sutugin et al., 1~81). Apparently i t involvea uaing the factor tS - S~ rather than .S -1 in equa tion ( 2) and that for the vapour exhaustion (Sutugin et al., 1-981). The latter allowa for the isothermal distillation of vapour from small partielas to large onea. The occurence of the nucleation hysteresis is correaponded with ernaller rate of the monomer concentration decrease aa the Supersaturation drops. It is aeen from Figs. 1, 2.
Thus satisfactory agreement and essential discrepancy (t.i. occurence and lack of nucleation hysteresis) both can be observed in comparison the steadystate nucleation rate with that of nonsteady process. Occurence of the agreement or discrepancy is defined by the ratio of the kinetic coefficient of cluster growth to that of cluster evaporation and also by the capture of precritical clusters by partielas of the condensed phase.
The nucleation rate for the nonsteady system becomes much lower than that of the steady-state system from some moment. Thus very abrupt nucleation collapse takes place as compared with the steady-state nucleation process. This is so due to capture of precritical clusters by the growing particles competing with the migration of clusters along the axis of their size.
1118
HEFERENCES
Sutugin, A. G., and Fuchs, N. A., 1970, Formation of Condensation Aerosols at Rapidly Changing Environmental Conditions, J. Aerosol Sei., 1:287.
Sutugin, A. G., and Grimberg, A. N., 1975, Vapeur Condensation under Cooling Jet Submerged, Teplofiz. Vysok. Temper., 13:787.
Sutugin, A. G., Lushnikov, A. A., and Kotzev, E. I., 1976, Methods of Calculation of Spontaneaus Condensation Processes, Teor. Osnovy Khim. Tekh., 10:400.
Sutugin, A. G., Puchkov, A. S., and Lushnikov, A. A., 1978, Spontaneaus Condensation in a Submerged Turbulent Jet, Kolloid. Zh., 40:285.
Sutugin, A. G., 1978, New Principle of Classification of Spontaneaus Condensation Processes, Kolloid. Zh., 40:1017.
Sutugin, A. G., Grimberg, A. N., and Puchkov, A. S., 1981, The Multistate Kinetics of Transient-Type Homogeneaus Nucleation, J. Aerosol Sei., 12:399.
Fuchs, N. A., and Sutugin, A. G., 1969, Properties of HDA, in: "High Dispersed Aerosols", V. V. Bondar, ed., VINITI, Moscow.
1119
MOLEOULAR CLUSTERS AS HETEROGENEOUS CONDENSATION NQCLEI
Zagaynov V.A., Sutugin A.G., and Lushnikov A.A.
Karpov" Institute o:f Physical Chemistry
Moscow, USSR
The study o:f the gas phase nucleation arouses consider~ble interest at present time on the part of many modern investigators. This is caused on the one hand by complexity of the nucleation process and on the other hand by absence of a un1que theory, describing the variety o:f the embryo :for.mation processes.
The most simple case o:f the heterogeneaus nucleation is so called rapid type nucleation (Lushnikov and Sutugin, 1976), when the growth of the critical embryo is restrioted by a collision efficiency with the monomeric vapour molecules. Since this efficiency affects essentially the embryo formation rate the very important question arises concerning the particle size be~ning with whioh its value becomes comparable with unity.
The main goal o:f the present paper is to study the intluence of the heterogeneaus condensation nuclei and to find out the critical sizes of these nuclei after having achieved which they grow effectively due to the vapour condensation. More precise~ this means that a monomeric molecule joins the con~nsation nucleus with the probability not less than 10- •
A molecular cluster is rather interesting but yet. not well studiedobject.Molecular clusters were necessary for injecting them into a silver particle generator in which rather high temperature is settled, hence the
1121
question concerning the thermostability of the clusters was very important.
The molecular clusters o! Fe 0 were obtained as a result of a thermodestruction of fe~CO)c;• To this end gas contain~ng diluted Fe(C0)5 vapour was heated aharply to T=600 K in preaence o2• Arising due to ther.modestruction molecules of Fe20~ were joined into clusters containing several molecuies. The dynamics of the cluster formation could be changing by regulating the dilution of cool air. The smallest clusters thus obtained contained 3-4 molecules of Fe?O~. The dispersed composition of the clusters was de~er.mined by the diffusion dynamical method. The clusters having passed diffusion batteries were enlarged in a two step system of particle amplifiers. In parallel with the study of the cluster dispersed composition the kinetics of the cluster collisions with solid surfaces was investigated. In passing the gas flow contaminated with the molecular clusters through the cylinder channel the penetration (the output and input concentration ratio, Characterizing the rate of the diffusion deposition of clusters on the channel wall) is occurred to depend not only on the geometric parameters of the tube; but also on the material of which the tube is made out. This means that the sticking probability of the cluster to the wall differ.from unity and depends on the material, so the traditional diffusion dynamics method (Lushnikov and Sutugin, 1976; Zagaynov et. al., 1976) is not valid. It was extended by us (Lushnikov et. al., 1977) to the case of nonunity sticking probability.
To this end the equation was solved describing the diffusion of clusters to the walls
(1)
where D is the diffusion coefficient, n - cluster concentration, v is the flow velocity. The boundary condition to Eq. (1) was chosen in the form:
fn{r}J3 - (r-J)v,.n(r)/5 -::O (2)
'..Che parameter J' appearing in Eq. (2) is related to the probability of a cluster to stick to the wall as a result of a single collision. The solution to Eq.(1) with the boundary condition (2) was found by Lushnikov et. al., 1976. The solution may be approximated as
1122
n(at) - ~: .z.1 (;lt.) nt~J = e .,/
at z1~ ~K ,where ~"- eigenvalues o! Eqs (1 1 2), z1 = Dz I 2r0 Vo.i z is the dista.nce along the tube, r 0 I.s channel radius, V0 is the mean flow velocity, r 1=r/r0 •
Next step is to settle a connection between the sticking probability and parameter (see Eq. (2)). This connection had been !ound by Lushnikov et. al., 19?7),by means o! expressing the recoil probability q=1-p in terms o! particle !low to and !rom the wall and Boltzman distribution function within the first approximation on the concentration gradient. The result is o! the !om:
$p{1-J') r ~ y +! _"; (4)
withfo = p0 /r0 and theß0 value has the order o! the mean free path. As is seen from Eq. (4) .the dependence p on f contains also a small multiplier wbich is evaluated as 10-/ under ambient conditions. Since in our experiments a considerable change o! n(z)/no (depending on the di!!usion channel) was observed the conclusion was ·achieved5that the sticking probability ~ is ver,r small (p~1o- ). As is seen !rom Eqs. (1,2,4) the penetration depends on the radius, the e!fect which could be never observed i! the boundary condition corresponds the the complete sticking n8 =0.Theoretical dependence o! the penetration on the tube radius r 0 agrees !airly well with experimental data. The latter con!i~s the conclusion concerning·the very small value (10- ~ o! the sticking probability of·the clusters o! (Je20~)~ - (Je~~3)4 to the solid sur!ace. As a re!erence difrusi~n batter,r a carbon (graphite) tube was used, in which the least penetration was observed, i.e. the sticking probability p was maximal one. As can be concluded !rom above consideration I =0 means just p ~p rather than p=1• It should be emphasized that the above !or.malism is applicable not only !or dispersed analysis o! molecular clusters but also for studJing adsorption phenomena, molecular collisions with clusters and coarse aerosol particles.
To investigate the influence o! molecular size clusters on rapid type nucleation a special installation was built up allowing one to study the formation process of Ag aerosols in presence of molecular condensation nuclei.
1123
The experiment was arranged as !o~lows. The Ag aerosol with the mean particle size 5-20 A from a ther.mocondensation generator was obtained and its condensational nuclei concentration and dispersed composition was measured by the di!!usion d:namical method (Fuchs et. al., 1962) without accounting for the corrections related to the nonunity to ßticking probability. Then in the evaporation zone o! the generator the clusters Fe2o3 were injected. The concentration and the mean size of the clusters were carefully controlled. Then again the concentration and the mean size of the Ag aerosols were measured. In this way the dependence was found of the changing the concentration of the Ag aerosol a n on the cluster concentration n~. This dependence is s~own in !ig. 1. In fig. l the dependence is presented of Ag aerosol mean size rk after the cluster injection on the mean size r of pure Ag aerosol. The interpretation o! the 4ata thuH obtained was done along following scheme. Let n be the observed concentration of the injection moleculir clusters,~n2= n -n1 , where n~ is total observed concentration, n -obs~rved c~ncentration without injection of moleculat clusters, Eo - development efficiency o! Ag aerosol in amplifier, E~ - development efficiency of autogeterogeneous nuclei in Ag vapour. Then it can be easy shown that
p'l.
11 Yll. ::: E(;' ;~ n/ (5) -o
Comparing the results (5) with the experi~enta~ dependence shown in !ig. 1 one concludes that 1ti ~ 3Eo (which means that the cluster development in the Ag vapour is more effective than in vapours of ampli!ier. It is not quite surprising since the supersaturation o! Ag vapour in the zone o! formation of Ag particles is much higher than in the ampli!iers used.
Now let us consider the growth kinetics o! initially monodispersed !oreign nuclei in a vapour containing monomere of unit mass. Let g0 be the mass of the nucleus (in monomeric mass), cK(t) - the concentration of partielas containing g monomers, c1 - the vapour concentration and K the rate of joining the vapour monemers to a gmer. ~hen the equations govering the time evolution of the mass spectrum takes the form:
at c1 =- K; t, c1 (t}c, [t) + J<1_, f1_, e1_, {6} c, [t}
ßt ~ 1 = - E K.J h c1(t)c, (tj (6) 1=1·
These equations should be supplemented with the initial
1124
0
2.1o4
Fig. 1. Ag aerosol concentration changa as !unction of condensation nuclai concentration
0
10 rk(A) / 0
/
/
"' 5 /
/ /
/ /
/ 0
0 rb(A)
5 10
Fig. 2. Terminal aerosol particle radius veraus initial aerosol particle radius
1125
~~~ditions: cg (t=O)=cg0 f;,,, , c1 (t=O)=c1• New variab-
1:-=ftK lt)c lt)/tl, i= l§e#C!I(c) • ]}:::jo1:J.((ti}o/t:' (7) 0 -f(l ft~ ~tf z:!, c (e-)) d'
!=!· # have been introduced the solution to Eq. (6) takes the form:
f)l-9o , cl (~t): c;. {J-?o/.1 r24f (- 'P) / Ct(ff): C/ ( 1- -?.o fJ) _
f
(8)
The results (8) have some important consequences. ~?~ the aerosol formation regime described above (K =g which corresponds to the free molecular regimego! condensation) the time dependence of the average mass is
1 {
t = J(~O {j "3- /o T) (9) I .f
On introducing respective particle radii rg and rg0 one gets
- - -( () t !p = fjt) f- 3 K" t 1 r; (10)
where r 1 is the monomeric radius. Besides, Eq.(8) allows the information to be obtained concening the collision efficiency. Assuming the presence of condensation nuclei wi th masses differing b;r /J g0 molecules, then their final masses a!ter having gro\tn in the vapour would differ b;r the value:
r= '1" "r;_ (11) at; Comparing this with the experimental dependence show.n in fig. 1 allows the conclusion the value1 d-' have not too differ from unit;r within the range 10- • This follows from the fact that the experimental dependance rb on rk is ver,y close to the bisetor of the first quaarant.
Experimental data and theoretical results lead to several statments.
Molecular clust;ers (Fe2o ) - (Fe 0 ) take intermediate state between gas phas~ ~olecul€s3~d aerosol partiales. It means this clusters to possess both gas molecule properties and coarse partielas ones• On the o~e hand their sticking probabilit;r to solid surface is 10- , on
1126
the other hand they are condensation nuclei for rapid type nucleation of Ag vapour and they are thermally stable,
The dif!usional dynamical method of dispersed analysis was extended for random sticking probability of aerosol partielas to the walls of diffusion battery. On the other hand this method can be used for evaluation of this sticking probability.
It was shown rapid type nucleation to be autogeterogeneous process if initial ~~omer'c concentration is relatively small (lass than 10 cm- ), Molecular clusters (Fe 0 ) - (Fe 0 ) are condensation nuclei for rapid t~e3nJcleati6n3o~ Ag vapour. Collieion efficiency between vapour monomere and tbis cluster is comparable with the collission efficiency between autogeterogeneours nuclei ~d monomeric vapour and takes valua not less than 10 •
References
1. Lushnikov A.A., Sutugin A.G. Contemporary state of homogeneours nucleation, Advo Chem., 45:385 (1976) -in Russian. 2o Lushnikov A.A., Zagaynov V.A., Sutugin A.G• On boundary condition to the diffusion eqation, Chem. P~s. Letto, 47:578 (1977), 3. Fuchs N.A., Stechkina ~.B., Starrosselsky V.A. On the determination of the particle size distribution in polydisperse aerosols by diffusion method, Brit. J. Appl, Phys., 13:280, (1962). 4. Zagay.nov V,A,, Sutugin A.G,, Petrianov-Sokolov I.V., Lushnikov A.A. On the sticking probability of molecular clusters to the solid surfaces, J. Aerosol Sei., 7:389 (1976).
1127
XVII. EXPERIMENTS WITH CLUSTERS
THE PHOTOCHEMISTRY OF SMALL VAN DER WAALS MOLECULES AS
STUDIED BY LASER SPECTROSCOPY IN SUPERSONIC FREE JETS
Donald H. Levy
James Franck Institute and Department of Chemistry The University of Chicago Chicago, Illinois 60637, U.S.A.
We have used a seeded supersonic free jet expansion to prepare van der Waals molecules consisting of one or more rare gas atoms weakly bound to a seed molecule (the substrate) by attractive van der Waals forces. The electronic spectrum of these van der Waals molecules appears as a satellite to the electronic spectrum of the uncomplexed substrate molecule, and we have used laser induced fluorescence spectroscopy to study both the structure and photochemistry of these molecules.l
This paper will be primarily a review of our sturlies of the photochemistry of small van der Waals molecules, rare gas atoms bound to small polyatomic substrates. When the molecule absorbs visible or UV light it is electronically excited. At the same time it is possible to vibrationally excite vibrational motion in the substrate, and if this is done the molecule may be unstable or metastable with respect to dissociation into free substrate plus free rare gas atoms. Since van der Waals bonds are weak, only a few (frequently one) quanta of vibrational energy in the chemically bound substrate will contain more than enough energy to break all of the van der Waals bonds in the molecule. This process, the flow of energy from the vibrational storage mode to the dissociating van der Waals stretching modes, is known as vibrational predissociation, and our work is directed toward a description of the details of this process.
In favorable cases we are able to measure the lifetime for this process, that is the time required for energy to flow from the storage mode to the dissociating mode. By the uncertainty principle the finite lifetime of the complex produces a broadening in the fluorescence excitation spectrum. We have used this broadening to measure
1131
the dissociation lifetimes in a simple triatomic van der Waals molecule I 2He. 2 The results are shown in Fig. 1. The lifetimes are long compared to the vibrational period of the iodine stretch (~0.4 psec) but short compared to the radiative lifetimes of the iodine molecule (~1 psec).
In the case of iodine van der Waals molecules, it is possible to excite the storage mode with many quanta of vibrational energy far in excess of what is required to break the van der Waals bonds. It is then of interest to determine the distribution of excess energy among the various degrees of freedom of the products. There are three degrees of freedom available: the diatomic vibration, the diatomic rotation, and the relative translational motion of the recoiling fragments. Since the fluorescence lifetime of the iodine is long compared to the dissociation lifetime of the complex, essentially all of the fluorescence that is observed following excitation of the complex comes from the free product Iz* produced by the dissociation reaction. Therefore, we can probe the product state distribution by analysis of the dispersed emission spectrum of the product Iz*.3
We have measured the vibrational state distributions of the product Iz* produced when complexes of the form IzNez dissociate;4 the results are shown in Fig. 2. In the smaller complexes the dissociation process is dominated by the most efficient dissociation channel, one quantum per rare gas atom being drained from the storage mode. As the complexes get !arger the dissociation becomes less efficient and channels requiring more than one vibrational quantum per rare gas atom become more important. This is interpreted as evidence that as the complex becomes !arger, the dissociation process is governed less by small molecule considerations such as the strength of intermode couplings and is directed more by statistical factors.
We are also able to measure the binding energy of small complexes by analysis of the vibrational product state distribution. 5 In the small complexes we interpret the first observed channel as being the first channel that is energetically open. For example, the observation of emission from Iz*(v'=17) and the absence of emission from Iz*(v'=18) and Iz*(v'=19) following excitation of IzAr (v'=20) is taken to indicate that the binding energy of I 2Ar is greater than two and less than three I 2 stretching quanta. A tighter bound on the binding energy can be measured by increasing the initially excited vibrational state. Because of the anharmonicity in the Iz potential, the vibrational quantum contains decreasing amounts of energy for higher vibrational states. Thus when IzAr is excited to v'=30, the first observed state of the product Iz* is v'=26, a four quantum process. This brackets the binding energy of the complex to within the arharmonicity of the Iz stretch.
In the case of van der Waals molecules of nitric oxide (NO)
1132
Iu 3.5 Q)
V> 0~
'Q
~ ~
c: 0
:;::: u
2 u 0
Q)
"' c.
"' "' .", Q)
0..
Q)
2 E
Av +-Q)
".:::: 0 c:
_J
c::::> +-
~ .0
>
0
v'
Fig. 1. The vibrational predissociation rate (and lifetime) of I2He as a function of the vibrational state of the I2 stretching mode that was excited. Points are experimental measurements, and dashed and solid curves are the best least squares fits of the functional form Av2 and Bv2 + Cv3.
1133
1.0 • v' '21 eXCIIOhon
12Ne, Z' l t;~ v"22 ewlahan ... <= <= 0 0
..<=
] u E 12Ne2 ,Z' 2 :>
b c I 0 :::> <:T ""' ' '] <=
~
I 12Ne3,z'3
>
"" = Ia ~ ~ "I" u 1.0 ~ "0 ... Ci.
0 <= ~
] ö 12Ne5 ,Z'5 :;
Q. 0
I 11 Q.
0 ~ • <=
'] ~ ü 12Ne6, z' 6 .§
~ ~ " ~c
' n= z z• I z-2 z•3
Fig. 2. The relative cross sections for the process r 2Nez(v')* + I2* (v' -n) + zNe for the complexes I2Ne-I2Ne6 excited to v'=21 and v'=22.
1134
and rare gas atoms, the excited state van der Waals potential is shifted to larger distances, and the Franck-Condon allowed optical transition is from a bound level of the lower electronic state of the complex to the repulsive wall of the upper electronic state.6 This bound-free transition results in the direct photodissociation of the van der Waals molecule, and the resulting fluorescence excitation spectrum is shown in Fig. 3. The spectrum of free NO results in sharp spectral lines, but the bound-free transition of the complex produces a_ broad, weak feature which is blue shifted with respect to the uncomplexed molecule.
Once again, since the dissociation reaction is fast compared to the fluorescence lifetimes of the substrate, all of the fluorescence is from the product NO* produced in the dissociation. The dispersed emission spectra of these reactions is shown in Fig. 4, and at the time of writing the analysis of the rotational product state distribution was in piogress. The fluorescence spectrum of the nitric oxide dimer was also observed.
Photodissociation can also be studied in van der Waals molecules consisting of rare gas atoms weakly bound to a polyatomic substrate. In this case the photochemical behavior can be more complicated since there is more than one storage mode in the substrate. Moreover, the molecular fragment produced by the photodissociation reaction can accommodate excess energy in a number of vibrational modes in addition to the storage mode. We have studied the dispersed fluorescence spectrum that is produced when the van der Waals molecule tetrazine-Ar is prepared in a supersonic free jet and laser excited to the electronically excited n*+n state.7 (Tetrazine is a 6-membered aromatic heterocyclic ring with formula H2c2N4 .) Studies of the rotational fine structure in the fluores~ence excitation spectrum of tetrazine-Ar have shown that the argon atom sits on top of the tetrazine ring on the axis perpendicular to the ring and passing through the ring center. The distance between the argon atom and the ring center is 3.4si.
The most prominent vibrational mode in the absorption spectrum of tetrazine and its van der Waals complexes is V6a• an in-plane stretching motion produced when the two C-H units move away from the ring center and the four nitrogen atoms move toward the ring center. In Fig. 5 we show the dispersed emission spectrum that is produced following excitation of tetrazine-Ar to the v6a= 1 level of the n*+n state, i.e. electronic excitation to the first excited singlet electronic state plus one quantum of vibrational excitation (703 cm-1) in the v6a mode, all other vibrational modes remaining in their zero point levels.
There are three prominent features in Fig. 5 corresponding to three distinct dynamical processes. The bluemost band at 18,074 cm-1 is the result of resonance fluorescence from the initially
1135
(..)
0
)
2% N
O 1n
Al
P 0 ,
3
Fig
. 3
. T
he f
luo
resc
en
ce ex
cit
ati
on
sp
ectr
a o
f N
OA
r (t
op
),
NO
Ne
(mid
dle
),
and
NO
He
(bo
tto
m).
T
he
gas
mix
ture
use
d
to p
rep
are
thes
e sp
ecie
s are
in
dic
ate
d.
The
to
tal
lack
ing
pre
ssu
re,
Po,
is
giv
en in
atm
osp
her
es.
The
ab
scis
sa i
s d
isp
lace
men
t fr
om
the A
2I:+
(v'=
O)
+x2
rr1;
2(v
"=O
) tr
an
siti
on
of
unco
mpl
exed
NO
. T
he
shar
p st
ructu
re a
t sm
all
6v
is d
ue
to r
ota
tio
nal
stru
c
ture
of
the u
ncom
plex
ed N
O ba
nd.
The
sh
arp
st
ructu
re n
ear
6v=
470
is a
se
qu
ence
of
unco
m
ple
xed
NO
.
excited state. In this process energy is absorbed to produce the electronically (Tr* + n) and vibrationally (v 6a = 1) level; in some fraction of the molecules the energy remains in this state for the fluorescence lifetime (0.5 nsec), and fluorescence is produced by transitions from the initially excited state to several vibrational
Pr2
R.1~o21
01 •P21
PI
NO He (285 cm-1)
712 3/2 ' r'
912 312
sJ6n I~ ~/2
IOOcm-1
-v
___ ..,. ........ , • .l'"' '
15/2 I
312 I
1712 3/2 ' ' 17/~ .. t/2
112 17/2 ' ~
Fig. 4. The dispersed fluorescence spectrum produced when NOHe and NONe are excited. Frequency shifts noted above each spectrum are displacements of the exciting frequency from the band origin of uncomplexed NO. A stick diagram of the six rotational branches of the NO* emission spectrum is drawn below each trace. The spectral resolution is 6 cm-1.
levels of the ground electronic state. For example, the 18,074 cm-1 band is produced by a transition from the initially excited state to the v6a = 1 level of the ground electronic state.
The second prominent feature in the spectrum, the band centered at 18,048 cm-1, is produced by emission from the uncomplexed tetrazine molecule that results when the complex dissociates. In this
1137
w
(X)
Fig
. 5
.
1805
0
T
16
a:
EX
C I
TE
T -A
r 6o~
FREQ
UEN
CY
<CM
-1)
T-A
r
16o~
Hig
h r
eso
luti
on
dis
per
sed
flu
ore
scen
ce
spec
tru
m o
f te
trazin
e-A
r fr
om
the 6
a1 le
vel.
T
he
inst
rum
en
t fu
ncti
on
is
2
.2 c
m-1
FW
HM.
The
te
trazin
e b
and
1
6af
(d
isso
cia
tio
n p
rod
uct
) is
b
road
ened
by
acq
uis
itio
n o
f ro
tati
on
al
ener
gy
du
rin
g d
isso
cia
tio
n.
The
max
ima
of
the P
an
ö R
-bra
nch
es are
ap
pro
xim
atel
y P
(15
) an
d R
(15
).
The
16
a~ t
etr
azin
e-A
r ba
nd is
bro
aden
ed
to 3
.7
cm-1
FW
HM b
y th
e sp
read
of
van
der
Waa
ls v
ibra
tio
nal
freq
uen
cie
s be
twee
n th
e g
rou
nd
an
d ex
cit
ed
sta
tes.
case the dissociation reaction produces both vibrational and rotational excitation of the product tetrazine. The 18,048 cm-1 band comes from the vl6a = 1 level of the excited electronic state of tetrazine, Vl6a being an out-of-plane twisting motion that is excited by the recoiling argon atom. The broad side lobes surrounding the central sharp feature are the P- and R-branch rotational envelope that is produced by rotational excitation of the tetrazine. The maxima in the P- and R-branches corresponds to ~=15. No bands produced by vibrationally cold tetrazine molecules have been observed.
The third prominent band in the spectrum, the 17,962 cm-1 band, is produced by the originally excited molecule tetrazine-Ar, but it is not due to a transition from tbe initially excited vibrational state. In some fraction·of the initially excited molecules intramolecular vibrational relaxation has taken place prior to emission to produce this band. In this process energy is transferred within the complex from the initially excited level v6a = 1 (703 cm-1) to another vibrational level vi 6a = 2(510 cm-1). The extra vibrational energy (703-510= 193 cm-1) is taken up by excitation of the three new low frequency vibrational modes produced when the complex is formed, that is one stretching and two bending motions of the argon atom against the tetrazine molecule.
These three processes, resonance fluorescence, photodissociation, and intramolecular vibrational relaxation, have been observed following excitation to a number of initial vibrational levels of tetrazine argon. The relative intensities of the fluorescence bands produced by these processes and the known fluorescence lifetime of tetrazine have allowed us to measure the lifetimes for photodissociation and relaxation. Analysis of the dispersed fluorescence spectrum has allowed us to measure the vibrational and rotational product state distributions.
This material is based upon work supported by the National Science Foundation under Grant CHE-7825555.
References
1. D. H. Levy, van der Waals molecules, in "Photoselective Chemistry, Advances in Chemical Physics," J. Jortner, R. D. Levine, and S. A. Rice, eds., Vol. 47, Wiley-Interscience, New York (1981), Part I, pp. 323-362.
2. K. E. Johnson, L. Wharton, and D. H. Levy, The photodissociation lifetime of the van der Waals molecule I2He, ~· Chem. Phys. 69:2719 (1978).
3. W. Sharfin, K. E. Johnson, L. Wharton, and D. H. Levy, Energy distribution in the photodissociation products of van der Waals molecules: Iodine-helium complexes, {· Chem. Phys. 71:1292 (1979); J. E. Kenny, T. D. Russell, and D. H. Levy,
1139
van der Waals complexes of iodine with hydrogen and deuterium: Intermolectilar potentiale and laser-induced photodissociation studies, ~- Chem. Phys. 73:3607 (1980); K. E. Johnson, W. Sharfin, and D. H. Levy, The photodissociation of van der Waals molecules: Complexes of iodine, argen, and helium, ~- Chem. Phys. 74:163 (1981).
4. J. E. Kenny, K. E. Johnson, W. Sharfin, and D. H. Levy, The photodissociation of van der Waals molecules: Complexes of iodine, neon, and helium, ~· Chem. Phys. 72:1109 (1980).
5. J. A. Blazy, B. M. DeKoven, T. D. Russell, and D. H. Levy, The binding energy of iodine-rare gas van der Waals molecules, ~- Chem. Phys. 72:2439 (1980).
6. P. R. R. Langridge-Smith, E. Carrasquillo M., and D. H. Levy, The direct photodissociation of the van der Waals molecule NO-Ar,~- Chem. Phys. 74:6513 (1981); E. Carrasquillo M., P. R. R. Langridge-Smith, and D. H. Levy, The direct photodissociation of van der Wa~ls molecules, in "Proceedings of the VICOLS Fifth International Conference an Laser Spectroscopy," Springer-Verlag (in press).
7. J. E. Kenny, D. V. Brumbaugh, and D. H. Levy, Nonstatistical behavior in van der Waals photochemistry: Tetrazine-Ar,
1140
~· Chem. Phys. 71:4757 (1979); D. V. Brumbaugh, J. E. Kenny, and D. H. Levy, Vibrational predissociation and intramolecular vibrational relaxation in electronically excited s-tetrazineargon van der Waals complex, ~· Chem. Phys. (submitted).
DIAGNOSTICS OF CLUSTERS IN MOLECULAR BEAMS
ABSTRACT
K. Sattler
Fakultät für Physik der Universität Konstanz W. Germany
Miereclusters from 30 different materials have been investigated: Metal clusters, ionic clusters and Van der Waals-clusters. The particles are ionized by electron impact and mass analysed by electronic timeofflight spectrometry. Studying the growth processes the transition from successive nucleation to coagulation is observed. The spectra give multiple information about magic numbers of stability of neutral and singly charged clusters, and the critical sizes for Coulomb-explosion of multiply charged particles. Cu- and Ag-halide clusters show a size dependent ionie-eevalent transi tion.
1. I NTRODUCTION
Agglomerates of a few (n) atoms or molecules are usually devided into small particles (n > 1000) and microclusters (n < 1000). Formation mechanisms
- 1 -for small particles are known and a great number of experimental and theoretical papers appeared during the last decade {for a review see ReL2),
In the size range of microclusters information is much scarcer because
of the lack of suitable preparation techniques, The particles have mainly been studied in inert gas matrixes 3, in zeolite 4 or on solid state supports 5 Few studies have been reported on clusters in molecular beams 6•
1141
Generationmethods were sputtering 7, laser evaporation 8, field desorption 9, ion-atom association reactions 10 , direct or Knudsen cell evaporation 11
h.f.-spark evaporation 12 , Penning source evaporation 13 or free jet adiabatic expansion 14 . As the size distributions usually decreased by many orders of magnitude, agglomerates bigger than the dimer could scarcely be studied.
Mierecluster properties have theoretically been approached from many different sides. In this paper we restriet our discussions on theories which are directly related to our recent experiments, A more comprehensive discussion of the literaturewill be given elsewhere 15 •
During the last few years the interest in microcluster research has considerably grown. This is because new methods for cluster generation have been developed which allow systematic size dependent studies. Furthermore, mass spectrometers were developed towards a higher mass range which is essential to the cluster detection.
In this paper the generation and investigation of microclusters from 30 different materials is reported. The formation methods are (i) inert gas condensation, for metal clusters and ionic clusters, and (ii) supersonic expansion for Van der Waals clusterso The particles are ionized by a pulsed electron beam and their masses are analysed by time offlight spectrometry. Size distributions have been measured from (i) metal clusters Bin, Sbn, Pbn, Inn(ii) ionic clusters (NaF)n, (NaCl)n, (NaBr)n, (Nai)n, (CuCl)n' (CuBr)n, (AgCl)n, (iii) inert gas clusters Xen, (iii) hydrocarbons (C2H4)n' (C2H6)n, (C3H6)n, (C3H8)n, (C4H6)n, i-buten (C4H8)n, 1-buten (C4H8)n, (C2H60)n and (iiii) halocarbons (CHF3)n, (CF3Cl)n, (CF3Br)n, (C2H3F2Cl)n, (C2F4Cl 2)n, (C2H4F2)n. Particles up to n = 6000((C02)n) with masses up to 215000 amu ((CuC1) 2100 ) have been detected while individual mass peaks have been resolved up to n ~ 150 (20000 amu for Xen). Multipleinformation has been gained about the conditions for cluster growth, the nucleation process, fragmentation in the ionization procedure, magic numbers for stability and critical sizes for Coulomb explosion of multiply charged particles. The high intensity observed over a wide size range (1 ~ n ~ 6000) henceforth allows a systematic sizedependent investigation of microclusters and the study of the development of the collective phenomena of the solids.
1142
2. GENERATION AND DETECTION INERT GAS CONDENSATION
A suitable method for generation of metal clusters and ionic clusters is the inert gas condensation (Fig.1). With this technique the vapour is emitted from a heated Knudsen cell (ON), is cooled in an inert gas atmosphere, whereby the vapour gets supersaturated. Thus, condensation to clusters occurs (region C). In order to avoid growth of very large agglomerates ("small particles", containing more than about 1000 atoms per particle) the condensation process has to be interrupted. This is done by transforming the laminar flow in the condensation region into a molecular flow, entering the high vacuum chamber of the detection part of the apparatus. 20 mm above the exit of the source the cluster beam is ionized by a pulsed electron beam and the mass of the cluster ions is analysed by time of flight spectrometry 16 •17 •
SUPERSONIC EXPANSION
The cluster source has been developed for gases which have a high pressure at room temperature. The gas expands through a 20 mm long, 0.2 mm thick capillary into vacuum. Behind the nozzle the gas passes two conical collimators and a molecular beam is formed 18•
A source for the expansion of metal vapour in a carrier gas is described in Ref.19. The mixture of metal vapour and gas is expanded through a 30 mm long, 0.2 mm wide capillary into vacuum. A cylinder in the gas inlet system collects the back stream of the metal vapour. High dimer concentrations in beams of Zn have been achieved with this source.
MASSANALYSIS
The sources for cluster generation are connected with an electronic timeofflight mass spectrometer 16 • Fig.2 shows a scheme of the apparatus with the inert gas condensation source. The neutral cluster beam, vertically
effusing from the condensation cell (left hand side of the drawing) is ionized by a pulsed electron gun (EO in Fig.2; 200 nsec pulse width, 1 msec repetition time, electron energy 10- 300 eV). The intensity of the neutral
cluster beam is measured by a film thickness monitor (SO).
The ionized clusters are extracted from an ion optics (10), accelerated
to 2 keV and focussed to a copper plate (P) at the end of the 1.7 m long
1143
drift space, being postaccelerated (by the aperture N) to 7 keV. At the
copper plate electrons are emitted by the impinging ions and collected ih
a channeltron (CH). The single ion detection unit is connected with time to
digital conversion electronics. The digital signals are given to the memory
of a multichannel analyser which then monitorfs complete mass spectra.
Double puls technique or the electron gun and the ion extraction field is
applied to achieve high mass resolution.
3. METAL CLUSTER
The clusters Pbn, Inn, Bin and Sbn are generated by condensation of the
atomic (Pb, In) or molecular (Bi 1, Bi 2, Sb4) vapours in cold inert gas 17 •20
The spectra (Figs. 3-6) are resolved up to high masses {for example up to 20-22 Sb240 ~ 26000 amu) •
From the structure in the Pbn-distribution the magic numbers n* = 7,10,
13 and 19 are deduced which partly are explained by sphere packing models 21
The magic numbers are defined according to ref.18: The intensity drops at
n* amount to a factor of 2 whereas the average changes between neighboured
mass peaks are a few percent. The observed sequence of n* is found to be
independent on the variation of the condensation parameters: oven temperature,
kind of the inert gas, temperature and pressure of the inert gas and the
length of the condensation zone. Furthermore, no change in this sequence
has been observed by variation of the ionising electron ene1·gy between 10 eV
and 300 eV. Fragmentation processes which occur at Ei ~ 20 eV, do not in
fluence the sequence of magic numbers 23.
Sphere packing models for Van der Waals clusters, two body LennardJones potentials applied, yield n* = 7,13 and 19 18•24 • The corresponding
structures have 5-fold symmetry with x7 being a pentagonal bipyramid, x13
an icosahedron and x19 an icosahedron plus a pentagonal cap of 6 atoms.
Of course, the interatomic potentials for lead clusters are different
from the Lennard-Jones type, which may explain why n* = 10 and 17 have not
been predicted from the calculations, mentioned above.
The Sbn-spectrum shows a sequence of peaks at n = 4,8,12,16,20 ••• because
of the condensation of Sb4-units. The structure in this mass distribution
will be explained in section 7 22
The intensity of the In-cluster beam was very low. Therefore at low
masses the cluster peaks could not be separated from the background of the
residual gas. The spectrum extends up to In500 , but only the resolved part
is shown in Fig.5.
1144
Bismuth clusters have been generated up to Bi 280 , resolved peaks up to Bi 70 (Fig.6) are recorded. In the low size range, a trimer sequence (6,9,12, 15) of antimagic numbers attracts notice, whose interpretation is open to question so far.
IONIC CLUSTERS
With the sodium halides all peaks appear at masses Nanxn_1• This ~s because the electron which is removed during ionizating has been localized at the halogen ion. The halogen is neutralized and splits off.
In the mass distributions(Fig.7) maxima are found at n* = 5, 14, 23 and 38. Binding energy calculations of the ionized NanCln_ 1-clusters yield that ionized clusters with these numbers of molecules are especially stable 25
The same intensity anomalies have also found in SIMSexperiments on Csl, where the sputtered positive ions have been analysed in a mass spectrometer 26 .
The observed sequence of numbers indicates that the ionized ionic clusters have simple cubic geometries:
n+•* cubic structure, n x n x n array of atoms
5 3 X 3 X 1 14 3 X 3 X 3 23 3 X 3 X 5 32 3 X 3 X 7 or 4 x 4 x 4 38 3 X 5 X 5
It is open to question why n* 5 has not been found in the SIMS-experiments.
If the counting statistics is increased a sequence of maxima at n = 5,8, 11,14,17 etc. is tobe seen (fig. 8, Ref.25). Calculations 25 show that neutral ionic particles are most stable in geometries where rings are sticking together. These rings contain 3 molecules each and therefore a sequence of numbers 3,6,9, 12,15 ... has been predicted to give maxima in mass distributions.We believe that the clusters with n = 5,8,11,14 etc. are the charged fragments of the highly stable neutral clusters with n = 6,9,12,15 etc. The observed structure in the spectra is a Superposition showing both highly stable neutral and highly stable charged clusters. Transitions from Multiring structures to cubic structures seem to occur in the ionization process.
1145
The spectrum of (CuBr)n(Fig.9) shows a sequence of maxima at n = 15,18, 21,24,27 etc.,It is not understood why the CuCl cluster spectrum (Fig.10) does not show a structure similar to the CuBr-cluster spectrum.
As mentioned above the peaks in the sodium halide spectra beleng to the non-stochiometric particles (Nan x n-1)+, because of the predominant ionic binding character of these materials. If the degree of covalency however is higher (Cu- and Ag-halides), a transition of peaks with n/n-1 to n/n composition is found. We believe that a size dependent ionie-eevalent transition occurs. The critical numbers for this transitions are nie : 7 for CuCl and CuBr and nic = 22 for AgCl (Fig.11).
An interesting feature is to be seen in the CuBr-mass spectrum. Above the threshold nie where all peaks are found at masses corresponding to n/n, the 14/13-peak appears in the spectrum. The reason is that the cubic 3 x 3 x 3 cluster is highly stable. The high intensity at n = 13 is open to question.
5. MOLECULAR CLUSTERS
Applying supersonic expansion technique we have generated clusters from various molecular gases (Fig.12). The Stagnationpressure was~ 1000 mbar, the nozzle temperature some degrees above the boiling point of each gas and the energy of the ionizing electrons was 30- 40 eV.
The probability for fragmentation in the ionization process is found to be high for the monomers. For CF3X (X= H, Br, Cl) the intensity of the fragment ion CF3+ is even higher than the intensity of the parent ion CF3x+. Clusters of these molecules however show much lower probability for fragmentation. In the cases of CF3Br and CF3Cl the ions (CF3X)n+(n ~ 2) are observed even with higher intensity than the fragment ion (CF3X)n_1 • CF3+.
A material which shows this effect most clearly is (CF2Cl 2)n(Fig.13). All possible fragments of the monomer are observed in the mass spectrum. The intensity of the ion CF2cl+ is 300 times higher than the parent monomer ion. Already for the dimers this intensity ratio is reduced to 2 and for bigger clusters the intensity of (CF2cl 2)n_1• CF2Cl+ is lower than of (CF2cl 2)n+· The intensity of the other fragment ions is for n > 2 even less than 20 %
of the parent ion intensity.
Fig.12 additionally shows strong differences in the fragmentation probabilities of c2H4- and c2H6-clusters. The double bond in CH2 = CH2 is scarcely broken.whereas the single bond in CH3-cH3 is broken with high efficiency.
Fig.14 shows a spectrum of (CHF3)n where the fragment intensity is always higher than the intensity of the parent ion. The explanation for one fluorine
1146
atom Splitting off from each cluster is the same as for ionic clusters (see
chapter 4). The charge distribution of the F-bonding is highly nonsymmetric,
the binding electron being localized at the halogen ion which is neutralized in the ionization process.
This effect is even more pronounced with SF6-clusters (Fig.15). All peaks
in the spectrum belong to the masses of (SF6)n·SF5+.
Most mass spectra of molecular clusters do not show irregularities (for
example the spectrum of c2H4, Fig.16a). From all investigated molecular
materials magic numbers n are only found for SF6 (n* = 13, Fig.15) and
c2F4cl 2(n* = 13, 19, Fig.16b).These numbers have also been found tobe magic
for Pbn(Fig.6) and Xen (see next chapter) explained by sphere packing.
The magic numbers 13 and 19 cannot be correlated with one atomic arrange
ment alone. In principle the x13- and x19-clusters can have icosahedral-,
hcp- or fcc-structure. However, in the case of SF6, all these geometries are
different from the solid state structure. The bulk of SF6 has bcc-structure
(plastic phase) and every molecule has 8 nearest and 6 next nearest neighbours.
For bcc-packing n* = 9, 15, 27 etc. would be expected, which is not observed.
Therefore we conclude that SF6-matter begins to grow in structures different
from their final solid state structure, a fact which is also been found for
ionic- and inert gas matter (see chapters 4 and 6).
6. XENON CLUSTERS: MAGIC NUMBERS FOR SPHERE PACKINGS
If Van der Waals systems of a few atoms (n ~ 1000) with simple inter
action potentials (Morse- or Lennard-Jones potential) are concerned, geometri
cal considerations 27 andcomputer simulations28 •29 show, that noncrystalline
structures give the best packing conditions. Packings with n* = 13, 55, 147, 309, 561 •.. spheres are especially favorable because the corresponding icosa
hedron structures allow the most dense arrangement.
A mass spectrum of Xe-clusters is shown in Fig.17, Ref.18. Maxima or
steps are found at the predicted numbers n = 13, 55 and 147. Further observed
maxima can be explained based on icosahedra. From both sides pentagonal caps
containing 6 atoms each can be put on to the 13-atom clusters, yielding
n* = 19 and 25. The same procedure, applied to the 55-atom cluster, yields
n* = 71 and 87, because here the pentagonal caps contain 16 atoms each.
Variation of source parameters as the Stagnation pressure or the tempera
ture before the expansion as well as variation of the electron ionizing
energy did not change the structure in the Xen-spectra.
Mass spectra from Ar-clusters 30-32 did show one intensity step alone,
1147
from n = 19 to n = 20. The differences in the structures of the Xen- and
Arn-mass spectra are not understood at present. Computer simulation experi
ments, considering differences of the interaction potential in Ar- and Xe
clusters could solve this problem.
In early SIMS-experiments, where sputtered Agn-- have been mass analysed,
besides an odd-even effect in the relative cluster intensities, in the se
quence of particles with odd numbers of atoms maxima are found at n = 7,13
and 19 (Hortig and Müller, Ref.7). This is remarkable from two reasons,
(i) sputtering is a bond breaking process and therefore fundamentally
different from successive growth, however yielding in both cases the same
magic numbers. (ii) The analysis of negative ions gives the same magic numbers as the ana
lysis of positive ions. This confirms our results that concerning the ob
served magic numbers fragmentation in the ionization process can be neglected
for metallic clusters and that the mass spectra show intrinsic properties
of the neutral particles.
In the very small size range, where the ratio of the number of surface
atoms to the number of bulk atoms is high, minimalization of the surface ener
gy seems to be more important than other energy contributions. Therefore,
atomic arrangements with the most dense packing, having icosahedron structures,
are favoured. Sphere packing considerations as a first approximation there
fore yield the observed magic numbers even for materials with different birt
ding character: SF6(n* = 13), Ar(n* = 19), c2F4c1 2 (n* = 13, 19), Ag (n* = 7,13,19), Pb(n* = 7,(10),13,19), Xe(n* = 13,19,25; 55,71,87,147). At present
it is open to question why every material chooses a part of the complete set
of suitable packing numbers. Different thermal excitation or the differences
in the symmetry of the atomic wave functions could explain this point. If at
a certain n different isomeric structures are energetically equivalent, this
number will not be observed to be magic. A cluster changing parmanently its
geometrical structure can be regarded tobe in the liquid state.
7. TETRAHEDRONPACKINGS: OBSERVATION OF MAGIC NUMBERS FOR CLUSTERS FROM
ANTIMONY TETRAMERS
Antimony vapour consists mainly of Sb4-particles 33 , which have tetra
hedran structure. As these particles are the building units for condensation
to microclusters, a sequence of peaks Sb4, Sb8, Sb12 , etc. is to be seen in
the mass spectra (Fig.4). Peaks in between this sequence are caused by frag
ments from the electron impact ionization (see chapter 9). The size distri
bution furthermore shows steps from n = 8 to 12,36 to 40,52 to 56 and a maxi-
1148
mum at 84. We conclude that clusters containing 2,9,13 and 21 Sb4-units gain less energy by adding a further tetramer than particles with other sizes.
The magic numbers 36,52 and 84 can be explained by a model 22 , which concerns the packing of Sb4-tetrahedra on the faces of a Sb20-core (Fig.18). If one, two or four Sb4-molecules are put on each face of the Sb20-unit, packing shells are closed because the addition of a further molecule then is unfavourable from geometrical points of view. Consequently, intensity drops are observed in the mass spectra at the corresponding atom numbers.
8. COULOMB-EXPLOSION OF MULTIPLY CHARGED CLUSTERS
Multiply charged microclusters are not stable below critical sizes, because the repulsive Coulomb energy between the excess charges exceeds the binding energy of the particle 34 • The peaks of multifold charged clusters therefore
2* 3* 4* appear only above critical numbers of atoms, n , n , n , for twofold, three-fold and fourfold ionization, respectively.
Fig.19a shows a spectrum of Pbn with the threshold n2* = 32. If the ionizati on energy i s reduced from 70 eV to 35 eV, the probabi 1 i ty for doub 1 e i oni za ti on is considerably reduced. Then, the spectrum is free from Pb~+-peaks(Fig.19b). Coulomb explosion and critical numbers have also been observed with ionic clu-
2* ' 2* sters (Fig.20:Nai, n = 20) and van der Waals clusters (Fig.21: Xe,n =52). The observed threshold for Xen(n2*= 52) has recently been confirmed by computer Simulation studies 35
Two-, three- or fourfold charged clusters with masses m appear in the spectra at m/2, m/3 and m/4, respectively 36 • Fig.22 shows a size distribution of co2-clusters with the corresponding steps at the critical sizes: n2* = 44, n3*.. 108, n4*= 216. They are observed if the ionizing energy Ei is high enough for multifold ionization and disappear below the ionization thresfolds.
9. FRAGMENTAliON OF CLUSTERS BY ELECTRON IMPACT IONIZATION
It is a known fact in mass spectrometry that molecules can break into fragments after photo- or electron impact ionization. The fragmented molecular species can easily be identified in the mass spectra. In the case of microclusters however, the fragmentation is difficult to handle. If a cluster of n atoms splits off an ion of n~3 atoms for example, this fragment cannot be distinguished from a nonfragmented cluster of n-3 atoms. Variation of the energy of the ionizing electrons however, can give insight into the amount of fragmentation events.
A plot of the relative intensities (normalized to I10 ) for Pbnatdifferent
1149
Ei shows that the spectrum changes only in the very small size range for n ~ 8 (Fig.23). Therefore the steps at n=7,10 and 13, showing high stabilityofthese
clus:~r:~ed:a::ts:::~~: :~t:n~~m::; only the peaks for Sb~, Sb;, Sb:, Sb;, Sb~ and Sb; show strong variations with Ei 38 . The mostfrequent fragments are Sb; and Sb; as well as the ionized building unit Sb:. If the ionizing energy is
low enough peaks from the particles Sb:i_ 1, Sb:i-2 and Sb:i-3 have disappeared and the pure tetramer sequence is to be seen 38.
Size distribution of co2-clusters have been measured at Ei= 35 eV and 150 eV (Fig.24). Applying high Stagnation pressure large clusters are generated. At 35 eV the spectrum in the low size range 10 ~ n ~ 50 is free from cluster peaks, at 150 eV however this mass range shows high amount of fragment peaks.
10. INFLUENCE OF CONDENSATION PARAMETERS ON THE MASS DISTRIBUTIONS
All clusters produced by the method of inert gas condensation are condensed in a Helium athmosphere. No mass spectra were obtained when H2,N2,Ne,Kr or Ar were applied or when He was mixed with a small amount of these gases. In all these cases the clusters grow to large particles which are not detected by the mass spectrometer. This result is gained from the rate measurement at a crystal monitor in the neutral cluster beam as a function of the gas pressure 39 .
I f the oven tempera ture i s i ncreased ( Fi g 0 25 and ReL 29) or the He-temperature is lowered, the mean cluster size gets higher, both explained by the higher Supersaturation of the vapour,
With the supersonic expansion the size distributions are controlled by variation of pressure and temperature of the gas before the expansion (Fig.26 ). If lower temperature and higher gas density are achieved at the exit of the nozzle the mean cluster size gets bigger.
If the stagnation pressure is furthermore increased, condensation by successive nucleation proceeds to condensation by coagulation of the clusters. Both processes have typical size distributions,exponential or bell shaped curves, respectively. Fig.26 shows the transition between these two processes, the experimental curves tagether with a theoretical fit 41
The cluster sizes can even more be increased. Particles up to high masses
have been detected: Pbn, nmax ~ 400, 84000 amu; (Nai)n' nmax ~ 380, 56000 amu;
(CuCl)n' nmax ~ 2170, 215000 amu; (C02)n, nmax ~ 4000, 176000 amu.
ACKNOWLEDGEMENT
The author wishes to express his gratitude to his co-workers 0. Echt, M. Knapp, J. Mühlbach, P. Pfau, R. Pflaum, A. Reyes Flotte and E. Recknagel,
who enabled him to assemble the material for this review. This work was supported in part by the Deutsche Forschungsgemeinschaft.
1150
02 I
Fi~. 1 Cluster source for inert ~as concensation.
ON oven, c condensation cha~ber, o1 o2 collimate>rs, V valve for vacu~ connection,
G gas inlet, TU thermocouple
,....------ dritt space j" seporatoon space l
• dritt chomber TPJ
• TP2
source chomber
Fig. 2 Electronic time of flight mass spectrometer for
micre>clusters: (inert gas condensation source ):
0 oven, C condensation zone, G gasinlet, TP turho
molecular pumps, EO electron optics, IO ion optics,
TM film thickness monitor, DO B N apertures, SV
va l ve , P copper plate, CH channeltron
1151
1152
~ ~--~--------------~ .!:!! 10 =-~
50
1l
n 15 17
16
10
200
<0
18
19
17
JOO
Sb0
I Inert gos cOf"'denso.tonl
BO
100
Time of ftight I ~se' I
20
164 Time of flight I ~sec I
10
2!() lOO Time of flight I ~sec I
Figs. 3-6 Hass spectra of lead-, antirnony-, indium,
and bisrnuth-clusters
.... 0
Nof
Fig. 7 Mass spectra of sodium halide clusters
NanCln_1
( Inert gos condensohon I
Time of flight [lJsec)
Fi g. 8 Mass spectiurn o f NaCl-c l uster s
1153
d 211
~
~ m
>.o ;;; ~0 ;;; ·;;; c .. ~0
.=o
~ 0
0 0
r.o
] ~8 : l12
I 'iii ~ :S8
211
50
6/S
~
~~
11;'U
I 13
I 1 1S
I
12 I I
CunBrm I Inert gas condensatian I
18 21 21. 27
Cu0 Cim [ hwrt go.s condtosahon )
·~~~ 60 eo 100 140 160 180
Time of flight [~sec J
..... Agn Cln lW!; 21
23 51< - AgnCln-1 " 19lQr22 1q i
Ii 15
12 13 I! 1: Ii 16 11f>
Ii !1
11 Ii IJJ 1:
60 70 80 so 100 110 Time of flight ll!secl
Figs. 9-11 Mass spectra of CuBr-, CuCl-, and AgCl-clusters
1154
C'l 0
....I
2
2 ~~~LM~~--~~--~~~--~--L-~_L~~L-~ 7
Cluster size n
Fig. 12 Mass spectra of various Van der Waals clusters
1155
2! 1-c 2~ ~ "' § 46
~ ~·o •
>-!::Jl \2 ~ ~ 21.
16
eo
j I
1156
.. ~ ~ 4
"' ~ "-
.._~
~3 u '-'ü 8
!2
2-f
J- f
4- f
5-F
20 3(1
Qj
c c d
.J:.
~
"' ..... c :l 0 u <>
- C> C> >-N ..... "' c Qj ..... c
;::; (CFzClzln
...,.. '-'
u
~ "-u '-'
[CHF3ln Po= 500 mbar
Eo=40eV
u I
N
200
13
10
20
I
400
P. = 1400 mbar T.=300 K E,=40 eV
lO
SF6 in Xe
20 I
p0 = 200mbar
T0 = 200 K
10() 11)
TIME OF FUGHT (usecl
Figs. 13-15 ~ass spectra of CF2cl2-, CHF3- anc
SF6-clusters
l I 9 .,
'-.L...
zo )0
IC2HJn p0 = 700 mbar E; = 40 eV
~M 40 50 60
23
30
60
110
1.=246K
p. = 2000 mbar E, : 30eV
40 50
90 100 TK OF flljHT [usec[
Cluster size n
1157
19
,...
Xen P, :300 mbar
T, :175K
·;:;
~ o~L..Il~:JJ.JL!J!JUUUW~lli!!!ll~.~~~~~ -"--------1
Fig. 17 Mass spectrum of Xenon-clusters
Fig. 18 Model for tetrahedron packing: a "shell" is full if one, two, or four sb4-clusters are bound to the faces of a Sb20-core
1158
~ ~
-~ ~
1:' ·w; o C:N
~ ..s.~
,.. -;;; c: ~ c:
S!
~
N
ll
lQ l•l
l4Q
ll) lS
1•21
, I
Pbn E1 a 70tV
1S
Time of !light ( lJSec I
Pbn f 1• ls.tV
21 JQ
20
Fig. 19
Mass spectrum of Pb-clusters
a) electron ionizing
energy E1=70eV b) E1=35eV
-r t J 1 li '--''-- \k 1'-"l~v
u ... "'
"' ...
60 IOD 120 "0
II
IQ
1·11
T1me of fhght I lJseC I
20 Fig. 20
Mass spectrum of Nai
clusters; E1=70eV,
fragmentation after
electron bombardment + effects that (Nanrn_1)
L _ ____,.,__:=~;:..::::....:::::.....n---\lii;----------;oa--::::-:---::-----;;;::-;::-T.:';;:-; and (Na I l 2 partic les 100 120 Time of !light I lJSecl n n- 2 are detected
E, • 50t\l'
Time of flight (jJsec)
Fig. 21
Mass spectrum of Xe -
clusters; E1=SOeV. + The intensities of Xe 1
and xe,++ have been
electronically recuced.
1159
1160
..c:: .... -- 0 .... 0 -o
>.
1/)
c QJ ..... c
N
p0 = 4 500 mbar
10 = 225 K
E, = 280 eV
Specific cluster size n/z
Fig. 22 Critical sizes for Coulomb-explosion for
two-, three-, and fourfold charged co2-clusters
0 .----c:
q . .
, , , '
Ei [eVJ
,::,. =70 0 =35 V =25 D :15
5 10 15 n
Cluster size [atoms per duster)
Fi~. 23 Pbn: Plot of relative
intensities ( norrealized
to Pb10) for different
electron ionizing energies
150eV
Pbn Toven= 1225 K
Pbn Toven= 1315 K
lOO ~
Time of flighl Iu sec I
((02) 0
T0 = 225 K
p" • 1000 "'bar
Time of flight
Fig 24. co2-clusters:
variation of the
electron ionizing
energy
Fig. 25 Inert gas concensation
of lead clusters:
variation of the
oven temperature
Fig. 26
'!> 6 2 700 mbar
r= 2e 1000 mbar
t= 62
r, = ns K
U., = JSeV
Size dis~rihutions of co2-
clusters at different stag
nation pressures, fitted
with a nucleation approach,
< being the integration
parameter.
1161
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1162
(7) M. Leleuter and P. Joyes, Radiat. Eff. ~. 105 (1973); P. Joyes, J.Phys. Chem.Solids 32, 1269 (1971); F. Honda, G.M. Lancaster, Y. Fukuda and J.W. Rabalais, J.Chem.Phys. 69, 4931 (1978); J.P. Taylor and J.W. Rabalais, Surf.Sci. 74, 229 (1978); F. Honda, Y. Fukuda and J.W. Rabalais, J.Chem. Phys. 70, 4834 (1979); F.M. Devienne and M. Teisseire, 7th Int. Symp. on Molecular Beams, Riva del Garda, Italy, 1979, p.116; V.E. Krohn Jr., J.Appl.Phys. 33, 3523 (1962); W.L. Brown, L.J. Lanzerotti, J.M. Poate and W.M. Augustyniak, Phys.Rev.Lett. 40, 1027 (1978); K. Wittmaack, Phys. Lett. 69A, 322 (1979); T.M. Barlak, J.R. Wyatt, R.J. Colton, J.J. DeCorpo and J.E. Campana, J.Am.Chem.Soc. 104, 1212 (1982); J.E. Campana, T.M. Barlak, R.J. Colton, J.J. DeCorpo, J.R. Wyatt and B.I. Dunlap, Phys.Rev. Lett. 47, 1046 (1981); T.M. Barlak, J.E. Campana, R.J. Colton, J.J. DeCorpo and J.R. Wyatt, J.Phys.Chem., Dec.10, 3840 (1981); M. Szymonski, H. Overeijnder and A.E. DeVries, Radiat. Eff. 36, 189 (1978); J. Richards and J.C. Kelly, Radiat. Eff. ~. 185 (1973); H.T. Jonkman and J. Michl, Springer Ser. Chem. Phys. ~. 292 (1979)
(8) N. FUrstenau, F. Hillenkamp and R. Nitsche, Int.J.Mass Spectr.Ion Phys. 1!• 35 (1979); J. Berkowitz and W.A. Chupka, J.Chem.Phys. 40, 2735 (1964)
(9) P. Sudreaud, C. Colliex and J. van de Walle, J.Physique 40, 207 (1979) (10) D.L. Turner and D.C. Conway, J.Chem.Phys. Zl• 1899 (1979)
W.C. Lin and D.C. Conway, Abstracts of the 165th ACS National Meeting, Dallas, Texas, April 9-13 (1973)
(11) K. Gingerich,in Current Topics in Material Science, Vol .6, North-Holland Publishing Company, edited by E. Kaldis
(12) E. Dörnenburg, H. Hintenberger and J. Franzen, Z.Naturforsch. Al6, 532 (1961); H. Hintenberger, J. Franzen and K.D. Schuy, Z.Naturforsch. AlB, 1236 (1963)
(13) H.J. Kaiser, E. Heinicke, H. Baumann and K. Bethge, Z.Phys. 243,46 (1971) (14) S.H. Linn and C.Y. Ng, J. Chem. Phys. 22· 4921 (1981); M. Armbruster,
H. Haberland and H.G. Schindler, Phys. Rev. Lett. 47, 323 (1981); C.E. Klots and R.N. Compton, J. Chem. Phys. 69, 1644 (1978; C.E. Klots and R.N. Compton, J. Chem. Phys. 69, 163ö (1978); A. van Deursen and J. Reuss, Int. J. Mass Spectr. Ion Phys. ~. 109 (1977); J. Gspann and H. Vollmar, J.Low Temp.Phys. 45, 343 (1981); Y. Ono, S.H. Linn, H.F. Prest, M.E. Gress and C.Y. Ng, J.Chem.Phys. 73, 2523 (1980) A. Yokozeki and G.D. Stein, J.Appl.Phys. 49, 2224 (1978)
(15) K. Sattler in Current Topics in Material Science, North-Holland Publ.
Comp., edited by E. Kaldis, tobe published
1163
(16) K. Sattler, J. Mühlbach, E. Recknagel and A. Reyes Flotte, JoPhys. E13, 673 (1980)
(17) K. Sattler, J. Mühlbach and E. Recknagel, Phys.Rev.Lett. 45, 821 (1980) (18) 0. Echt, K. Sattlerand E. Recknagel, Phys.Rev.Lett. 47, 1121 (1982) (19) A. Reyes Flotte, 0. Echt, K. Sattlerand E. Recknagel,
J.Appl.Phys. 53, 1317 (1982) (20) J. Mühlbach, P. Pfau, K. Sattler and E. Recknagel, Z.Phys., in press (21) J. Mühlbach, K. Sattler, P. Pfau and E. Recknagel,
Phys.Lett. 87A, 415 (1982) (22) K. Sattler, J. Mühlbach, P. Pfau and E. Recknagel, Phys.Lett. 87A,
418 (1982) (23) P. Pfau, K. Sattler, J. Münlbach, R. Pflaum and E. Recknagel,
submitted for publication (24) M.R. Hoare, P. Pal and P.P. Wegener, J.Colloid Interface Sei.
75, 126 (1980) (25) K. Sattler, J. Mühlbach, P. Pfau, R. Pflaum, E. Recknagel and T.P.
Martin, submitted for publication (26) J.E. Campana, T.M. Barlak, R.J. Colton, J.J. DeCorpo, J.R. Wyatt and
B.I. Dunlap, Phys.Rev.Lett. 47, 1046 (1981) (27) W. Werfelmeier, Z.Phys. 107, 322 (1937) (28) J.G. Allpress and J.V. Sanders, Austral.J.Phys. 23, 23 (1970) (29) M.R. Hoare, Adv.Chem.Phys. 40, 49 (1979) (30) T.A. Milne and F.T. Greene, J.Chem.Phys. 47, 4095 (1967) (31) D.R. Worsnop, S.J. Buelow and D.R. Herschbach, tobe published (32) H. Haberland, private communication (33) J. Mühlbach, P. Pfau, E. Recknagel and K. Sattler, Surf.Sci. 106, 18
(1981) (34) K. Sattler, J. Mühlbach, 0. Echt, P. Pfau and E. Recknagel, Phys.Rev.
Lett. 47, 160 (1981) (35) J.G. Gay and B.J. Berne, submitted for publication (36) 0. Echt, K. Sattlerand E. Recknagel, Phys.Lett., in press (37) P. Pfau, K. Sattler, J. Mühlbach, R. Pflaum and E. Recknagel,
submitted for publication (38) P. Pfau, K. Sattler, J. Mühlbach and E. Recknagel, Phys.Lett., in press (39) P. Pfau, K. Sattler, J. Mühlbach, R. Pflaum and E. Recknagel, J.Phys.F,
Metal Physics, in press (40) J. Mühlbach, E. Recknagel and K. Sattler, Surf.Sci. 106, 188 (1981) (41) J.M. Soler, N. Garcia, 0. Echt, K. Sattlerand E. Recknagel, submitted
for publication
1164
EXPERIMENTAL STUDI.I!:S OF WATER-AEROSOL
EXPLOSIV~ VAPORIZATION
L.K. Chistyakova and V.A. Pogodaev
The Institute of Atmospheric Optics Siberian Branch, USSR Academy of Sciences, Tomsk, U S S R
The explosive vaporization of water aerosol due to the formation and destruction of metastable states of water occurs if the partielas are shock heated. The use of pulsed laser radiation ie one of the most promising ways for euch heating. In this case heating rates are determined by the pulse leading edge slope, ~ulse energy and can vary in the wide range up to """ 101 degrees/ s.
The study of aerosol explosive vaporization is of interest for a number of problems including determination of thermodynamic characteristics of a matter in a metastable state in technological processes, as well as determination of energy characteristics of radiation propagating through turbid media etc.
A large number of papers can be found in the literature which deal with the experimental studies of water droplets explosion in the field of pulsed laeer radiation of various spectral ranges, energetic and temporal parameterB of the puleee [1-4] •
Table I. The Energy and Tima-Dependent Parameters of the Lasers Used in the Experiments
Wave-J/cm2
Pulse Pulse Heat rate length .r.; Pulse rise repeti- degree/s ' "um duration,s time,s tion rate
Hz
0.69 0.1 + 103 10-3 5 10-6 single 3 6 10-9 103-106
2.36 0.5 40 5 10-9 400 108-109 10.6 0.1 + 50 3 10-7 5 10-B single 10 -10
1165
It has been found in the experiments that several characteristic types (regimes) of explosion can be separated out according to the value and rate of energy contribution to a particle which depend on the laser pulse energy, matter absorptivity and particle sizes.
The first regime occurs at heating rates up to 10• degrees/s. It is caused by the matter shattering when thermodynamic parameterB of an absolute instability phase limi t are reached (pressure"" 220 bar and temperature 578°K [3]). The characteristic features of this explosion are the following: shattering of the initial particle, essential asymmetry of the volume occupied with the explosion products, multistage character of the process (Fig.1).
The second regime occurs when the heating rate rises to 10 9 degrees/s, e.g., due to illumination with the pulse of a CO? laser radiation that results in significant modification of the dynamics of the process with energy consumption being the same. The explosion
. -·· ···· 0
b'ig. 1.
1166
.32 64 96 128 760
t ·10 6 • s
Sequence of photographs of water droplet explosion (first regime).
Integral photograph of water droplet explosion (second regime)
is accompanied with a shock wave and break down in the air; the angular distribution of the explosion products disperaal has central symmetry, the explosion products are confined in a sphere with the radius of 10 initial radii of the particle (Figo 2).
The theoretical modele for describing the first type of explosion are based on the use of the statistical theory of homogeneous nucleation and enable one to estimate only the energy thresholds of the total particle shattering, however, they do not provide information on the other parameters of the process [5-9].
The second regime of drop explosion can be sufficiently well described with the equations of the gaseous dynamics and the appropriate consideration of the matter state thermodynamic transitions. In this case a model is used wherein the matter expansion is considered as hydrodynamic flow [10]. One can obtain, using the above model, the estimates of the rates of surface motion and the speed of a shock wave accompanying the explosion, as well as the coefficients of radiation attenuation at explosion.
The experimental data on the explosion energy thresholds for the above regimes are presented in Fig. 3. A straight line for Ä = o.69]Am was obtained from
Fig. 3.
102 EKn, .J·cm-2 o• .. ••.. .. ! . : .
• .Y . ., ., ... G W .(~
/f • - }. ~0,69)1m I • -.A =2,3~m
• - ?. • 10,6)/m
. • 41 . .. ······
Water droplet explosion energy thresholds
1167
[5]. Satisfaetory agreement is observed between the theoretieal and experimental results. A straight line for i\ = 2 • .36 }" m is an approximation of experimental data. In this ease the theory and experiment differ signifieantly. The two groups of points for A = 10.6~m eharaeterize the explosion thresholds.
Figure 4 illustrates the averaged angular diagrams of partieles dispersion. The latter are formed at the initial drop explosion in the field of laser radiation pulsea at different wavelengths. The radiation propagates from left to right. The diagrams are eonstrueted with a 10° angular step, the polar radius value is proportional to a number of partieles ejeeted during the explosion along the given direetion. The type of the dispersion diagrams depends on the drop matter absorptivity energy pumping speed and energy distribution over the drop volume. Pulse repetition rate also plays an important role in foeusing the dispersion diagram. . The deerease of a number of partieles ejeeted along the direetion of the pulse propagation is observed for the laser Operating at ~ = 2.,36 jJ. m that ean be explained by the effeet of subsequent pulses on the explosion produets.
The information on the transformations in time of the volume oeeupied with the produets of the initial partiele explosion is quite neeessary for many problems. In rlef.(11]one ean find detailed investigations on the given problern for "- = 0.69 JA m.
1168
270.
90°
Fig. 4. Averaged angular diagrams of partieles dispersion
Fig. 5. Time-dependent variation of the volume occupied with the droplet explosion products ( Ro"" 10"" cm; Ä = 10.6 }' m)
Figure 5 illustrates the time-dependent Variation of the volume occupied with the producte of the co2 laser pulse induced explosion of the drop of the 10·1 cm initial radius. The numbers above the curves indicate the time intervale of 4•10·' s; 5•10"' s; 3•10.6 e. A characteristic feature of the drop explosion in the field of a CO laser is the quick formation of the diagram conto6r (in the illustrated case 5·10-1 s) and the formation of analmostideal sphere by the moment of the diagram formation (3•10- 1 s). The final radius of the volume occupied with the explosion products did not exceed 9R0 in the experiments (Fig.6).
~
~r 3 30 40 E, Icm-Z
Fig. 6. The ratio of the final radius of the volume occupied with the explosion products to the droplet radius depending on the laser pulse power
1169
The processing of the photographs of the initial particle explosion showed that the particle radius lt
Ä = 0.69 and 2.36 fo rn is mainly within (5-10) •10- cm. A dispersion composition of the explosion products
determines the sizes of a final volume and depends on the laser pulse power and the rate of energy pumping into a drop volume. The sizes of explosion products decrease with the increase of energy contribution rate therefore for the CO laser energy densities used in the experiment one c6uld not determine the dispersion composition, but nevertheless, one can say that it does not exceed (3-4)•10- 4 cm.
Thus the character of explosive vaporization of liquid aerosols and the rates of the process development depend significantly on spectral energy and timedapendent pararneters of laser radiation that should be taken into account when solving different problerne connected with this effect.
References
1. V.E.Zuev, A.V.Kuzikovskii, S.S.Khm~levtsov, V.A.Pogodaev, L.K.Chistyakova, Thermal action of optical radiation on small-size water drops, Dokl.AN SSSR 205:1067 (1972).
2. V.V.Kostin, V.A.Pogodaev, L.K.Chistyakova, The explosion of a water drop irradiated with a series of optical radiation pulses, ZhETF. 66:1970 (1974).
3. V.A.Pogodaev, A.E.Rozhdestvenskii, S.S.Khmelevtsov, L.K.Chistyakova, Thermal explosion of water particles under the action of powerful laser radiation, Quantum Electronics, 4:157 (1977).
4. A.A.Zemlyanov, Kuzikovskii A.V. and L.K.Chistyakova, Water drop explosion in the field of C02 laser radiation, in:"Investigation of Complicated Heat Exchange", Novosibirsk (1978).
5. A.V.Kuzikovskii, The dynarnics of a spherical particle in a powerful optical field, Izv.VUZov SSSR, Fizika. 5:89 (1970).
6. A.V.Korotin, L.P.Semenov and P.N.Svirkunov, Liquid drop explosion at strong overheating, in:"Atmospheric Optics. Truns.Inst.Experimental Meteorology", .Moscow, Gidrometeoizdat, 11:24 (1975).
1. A.P.Prishivalko, Evaporation and explosion of water drops at inhomogeneous inner heat release, ~uantum Electronics, 6:1452 (1979).
8. N.V.Bukzdorf, V.A.Pogodaev and L.K.Chistyakova. On connection between inhomogeneities of an inter.nal optical field of an irradiated drop and its explosion, Quantum Electronics, 2:1062 (1975).
1170
9. A.P.Prishivalko, L.G.Astafieva, Energy distribution inside light-scattering particles, Minsk (1974) (Preprint7Institute of physics, Acad.Sci. BSSR).
10. A.A.Zemlyanov and A.V.Kuzikovskii, Modeling description of a gas-dynamic regime of the water drop explosion in a powerful pulsed light field, Quantum Electronics, 7:1523 (1980).
11. V.A.Pogodaev, Rozhdestvenskii A.E. and L.K.Chistyakova, Transformation of hydrometeors into the mist at their explosion with an intense laser pulse, Izv. VUZov SSSR, Fizika, 3:34 (1980).
1171
LASER PROBING OF CLUSTER FORMATION AND DISSOCIATION IN MOLECULAR
BEAMS
G. Delacretaz, J.-D. Ganiere, P. Melinon, R. Monot, R. Rechsteiner, L. Wöste, H. van den Bergh* and J.M. Zellweger*
Institut de Physique Experimenta~e Swiss Federal Institute of Technology CH-1015 Lausanne, Switzerland
* Institut de Chimie-Physique Swiss Federal Institute of Technology CH-1015 Lausanne, Switzerland
BEAM PHOTON INTERACTIONS IN SF6
An SF6 molecular beam is irradiated with a cw C02 laser. The multiple effects of the infrared radiation are analyzed with several mass spectrometric techniques. Detailed information is obtained on the extent of the collisional region of the adiabatic gas expanslon. Infrared vibrational predissociation spectra of the van der Waals clusters are obtained.
Figure 1 shows a mass spectrum of a pure SF6 beam. Clusters (SF6)n are detected with n as high as 100. Cluster formation can be nearly completely inhibited by irradiating the monomer in the collisional region with a few kW cm- 2 • In the absence of laser radiation, clusters are formed by :
SF6 + SF6 +=t (SF6) 2* and (SF6) 2* + M--+ (SF6) 2 + M + E
When the laser lS tuned to the absorption band of the 32sF6 monomer we have :
SF6 + hv
* SF6 + SF6 (SF6)2** +
--+SF6*
+= (SF6)2**
M --+ ( SF 6) 2 + M + E
T . ( ) ** ( ) * he more exclted SF6 2 decomposes faster than SF6 2 and lS
1173
thus more difficult to stabilize. Hence vibrational excitation of the SF6 monomer can reduce the cluster concentration. A secend mechanism has been proposed by Teenniesand co-workers1. Here excitation of the monomer :
SF6 + h\1 -+ SF6*
is followed by near resonant V-V energy transfer to the clusters.
SF6* + (SF6)m -+ (sF6); + SF6
The excited clusters may then brake up
Fig. 1
-+ (SF6) + (SF6) + E m-n n
> I-Ci) z w 1-z H
w > ~ ...J w c::
20 I
mass-
Relative intensities of SF5+ (SF6)n_ 1 with n up to rv 100. P0 = 2 bar, T0 = - 48 °C, nozzle diameter 140 ).Im. Pure SF6.
with 1 < n ~ m-1, or be stabilized by collision. Both these mechanisms of cluster destruction depend on collisional energy transfer and thus can only take place in the collisional region of the gas expansion. We may employ this effect to demonstrate the extent of the collisional r egion as is shown in figure 2a) . Here the effective reduction of the SF6 SF5+ s i gnal , mainly due to laser excitation of the 32sF6 monomer, is plotted at different points of the gas expansion. A large laser induced reduction in cluster concentration is observed in the first 2 mm downstream from the nozzle. This is the zone where the pressure in the beam is high enough for collisional energy tr~nsfer to be effective. Further downstream, collisions no langer play a significant role, and the small amount
1174
of signal reduction is due to vibrational predissociation of bot dimers. Irradiation of the SF6 beam at the 10 P 36 line of the C02 laser causes excitation of the 34sF6 monomer rather than the 32sF6 monomer. The attenuation of the SF6 SF5+ signal in this case is strenger than may be expected from the ratio of the natural abundances (~ 20:1), indicating fast scrambling of vibrational excitation.
Figure 2b) also shows the effective reduction in the SF6SF5+ signal, but now for irradiation at the 10 p 26 line of the co2 laser at 10,65 ~. At this wavelength we directly excite t he 32sF6 dimers without excitation of SF6 monomers. In the collisional region of the beam this dimer excitation may be followed by the vibrational predissociation and/or intermolecular energy transfer. Both effects can lead to an effective reduction of the dimer concentration . The important difference between these data and those of figure 2a) is that the effective attenuation in the SF6 sF5+ signal caused by the co2 laser is relatively strong in the collision free region of the beam. This is due to the strenger vibrational predissociation of the van der Waals clusters in the collision free zone, i.e.
Fig. 2
h\J
SF6SF5+
100%
X
X
\ b)
50
+-~-+--~+-~-+--~+-~-+--._mm
0 5 10
Attenuation of the SF6 SF5+ signal measured for laser irradiation at different distances downstream from the nozzle. a) 500 W cm-2 at 10P 18, P0 = 2 bar, T0 =- 50 oc, pure SF6 · b) as (a) but for 500 W cm- 2 at 10 P 26. Data are not corrected for beam geometry changes caused by the laser.
1175
Although it is not known how exactly E is partitioned between internaland kinetic energy in this case2, the translational energy is probably sufficient to cause recoil of the fragments out of the beam. The wavelength dependence of this phenomenon yields the IR vibrational predissociation spectrum3.
Figure 3 shows the IR vibrational predissociation spectrum of the SF6Ar van der Waals dimer, measured at different points in the expansion of an Ar/SF6 mixture. It is interesting to note the narrowing and red shift of the spectrum as the measurements progress farther and farther downstream in the expansion. Several mechanisms can cause such a narrowing4 and the red shift for the colder SF6Ar complexes is expected.
Figure 4 shows the IR vibrational predissociation spectrum of SF6(Ar)m with m = 2, 3, 5 and 9. The strong spectral feature which develops as m increases coincides exactly with the maximum of the IR spectrum of SF6 in a solid Ar matrix5. This demonstrates that as far as the low resolution IR sp.ectrum of SF6 in an Ar matrix is concerned only the very nearest neighbours play a role.
Figure 5 shows the TOF spectra of the neutral molecules measured in a pure SF6 beam, withandwithout laser irradiation of the monemers in the collisional region. The energy deposited by the laser beam has 3 effects. (1) The mostprobable velocity of the beam along the beam axis is increased by about 20%. (2) The
Fig. 3
1176
100% 0
0
930 940 950
...J < z <!> iii lll :::E
100% cm-1
IR vibrational predissociation spectrum of SF6Ar at different distances d downstream from the nozzle. P0 = 1.5 bar, T0 =-10 °C, 1% SF6 in Ar. D d = 0 mm, e d = 0.4 mm, t:> d = 0.8 mm, ·0 d = 4 mm, • d = 4 mm,
T0 =- 57 °C.
Fig. 4
Fig. 5
100% Q 0 1\. ll I I I \ z I I 0
~ f\i ...J
:::J ~ 'i <( z :, ~ z UJ . ., C) I- .'' \\ iii ~ :' I~ 1/l ::;[ :I I \ ::;[ <( ' t UJ al
0
IR vibrational predissociation spectrum of SF6(Ar)m with m = 2 ( 0 ) , m = 3 ( T ) , m = 5 ( e ) , and m = 9 ( 0 ) • P0 = 1 bar, T0 =- 93 oc, 1 % SF6 in Ar. The vertical line shows absorption maximum of SF6 in a solid Ar matrix. d = 4 mm.
(\ I I
a , I ~b I I
I I I I
I I I
I I
I I I
\ \
I \ I \, ,
____ _) ' :::.....
0 2 3 4 5 6 7 8 9 t [ms]
TOF spectra of SF6 monemers in pure SF6 with (a) and without (b) laser irradiation of 6 kW cm-2 at 10 P 16 line. P 0 = 2 bar, T 0 = - 48 °C , pure SF 6 •
1177
Fig. 6
4 5 7 9 t[ms]
TOF spectra of SF6 (a) and (SF6) 2 (b),in pure SF6 without laser. P0 = 2 bar, T0 =-50 °C.
velocity spread or "local temperature" of the beam (FWHM) is increased by about 65%, and (3) the intensity of the beam measured near the beam axis is decreased by ~ 80%. The latter is probably due to a significant increase in particle velocity perpendicular to the beam axis caused by V-T,R energy transfer following laser excitation. This increase in off-axis velocity tends to spread the beam, thus decreasing the intensity near the axis. Similar but smaller effects have been observed by Bernstein and co-workers6.
Figure 6 shows an interesting feature of the TOF spectra measured in a pure SF6 beam without laser. It appears that the dimers are both faster and colder than the monomers. One possible explanation for this phenomenon is that the dimers are formed mainly from the colder and hence faster monomers.
TWO-PHOTON-IONIZATION SPECTROSCOPY
The electronic, vibrational and rotational structure of clusters of a certain size is most favourably measured by means of TwoPhoton-Ionization (TPI) : A tunable laser (hv 1) electronically excites particles from the ground state into a particular rovibronic level of the excited state. From there the excited particles get ionized with a second light source (hv2 ) whose energy is insufficient to directly ionize ground state particles. So far the method has successfully been applied by using continuous lasers and quadrupole mass spectrometry7. The resolution of these cw experiments is excellent. Severe limitations, however, are given by
1178
the small tuning range and low power level of cw laser systems : While the excitation step (hv1) can already be saturated at power levels of a few hundred milliwatts, the ionization process (hv2 ) requires laser powers > 104 Watt in order to get all excited particles ionized8. Forthis reason a pulsed laser and a time-of-flight mass spectrometer were arranged together with a supersonic molecular beam, where experiments could be performed in a spectral tuning range reaching from 4 microns to 190 nm at an unlimited mass range. The unfavourable duty cycle of the experiment of about 10-4 is highly compensated by higher laser powers and more favourable beam geometries. Furthermore., the character of the experiment allows time resolved spectroscopy of excited electronic states, and the observation of metastable ions.
EXPERIMENTAL
The laser system consists of a commercial Nd YAG laser oscillator and amplifier (Quanta Ray), a harmonic generator and harmonic separator, where the fundamental (A = 1064 nm), 2nd harmonic (A = 532 nm), 3rd harmonic (A = 355 nm) and 4th harmonic (A = 266 nm) can be selected and used for pumping the dye laser (see Fig. 7). The tunable dye laser output is either directly applied to the experiment, or it is frequency doubled, frequency mixed or Raman shifted. Simultaneously any Nd YAG-harmonic can collinearly be added at an adjustable delay for performing time resolved two-photon-ionization spectroscopy.
The las~r is irradiated at a right angle 7 cm downstream the molecular beam axis (see Fig. 7). The time-of-flight mass spectro-
IARIAIIll DUAY
IIIIIFT ll!I;IH
Fig. 7 Set up of the TPI-experiment
1179
meter is arranged perpendicularly to both axes, avoiding Doppler broadening of the mass peaks. The spectrometer is optimized to a first order spatial focus9. The ions are accelerated to 2. 8 keVenergies. The geometry minimizes the velocity dispersion in the interaction area. The mass resclution obtained this way reaches 1300 (FWHM). During routine measurements, however, this value is greatly reduced to about 100 due to the 100 nsec. window of the electronic data acquisition system.
The experiments were carried out on alkali cluster-beams. The oven system consists of a thermoelectrically heated jacket, which independently allows to heat the oven cartridge and nozzle . Temperatures of 1000 °C can be reached. The unit is placed in a water cooled, 3 fold tantalum radiation shield. The cluster beams were monitored by direct photoionization using the 3rd harmonic (A = 355 nm) and 4th harmonic (A = 266 nm) of the Nd-YAG laser as light sources. Irradiating a sodium molecular beam at an oven pressure of ~ 200 Torr and a nozzle diameter of 0.3 mm with laser light at A = 266 nm, mass spectra of Na-Na21 were obtained (see Fig. 8a). The sharp mass peaks of the aggregates are accompanied by broad signals of metastable ions. These signals, however, are strongly wavelength dependent. Fig . 8b shows an example, when the beam is irradiated at A = 355 nm, where practically no metastable peaks become apparent. The phenomenon indicates the occurence of strong fragmentation processes.
... .. , .. ... ... ... bl A• JS5..., ... ... ... ... 01 A• 266..., .. .., ... ... ... .., ... ...
.. Q
...
0 10 20 30 40 1•"""1 0 10 20 30 I•OOCJ
Fig . 8 Time-of-flight mass spectrum of a sodium cluster beam. a) Photoionizing at A = 266 nm; b) Photoionizing at A = 355 nm.
1180
RESULTS AND DISGUSSION
The TPI-spectra were recorded by setting a TOF-time window at the mass of interest, while the exciting laser hv 1 was tuned. Figure 9 a and b show results, that were obtained for the system Na2 A +X, by using a DCM-laser for the electronic excitation ( 1. 82 eV ~ hv2 ~ 2. 03 eV), and the thi rd harmonic Nd YAG- output for the ionization step (hv2 = 3.9 eV). The well res olved vibrational and rotational sequences (Fig. 9 a,b) indicate a strong collisional cooling. Values for Fig. 9 a,b compute to Tvib.= 75 ± 10 K and Trot. = 25 ± 8 K. In order to be sure of a clean 2-photon process, the laser power of hv 1 and hv2 had to be sufficiently
13. 0" 3'. o·
al 610 620 630 640 650 660 670 (nrn]
bl 637.6 642.2 [nm]
Fi g. 9 : Two-Photon-Ionizati on spectrum of the Na2 A +X band. (a ) vibrational sequence between 610.0 and 667.5 nm; (b) r otational sequence of the 9' + O" transition.
1181
attenuated, to operate in a linear power dependency. Typical peak powers,where still a high sensitivity was obtained, but no saturation or multiphoton processes yet occured, were around 10 kWatt/cm2 •
Two-photon-ionization spectra of Na3 were taken by using Coumarin 500, R 590, Kiton-Red and DCM lasers for the excitation process, ranging from 690-480 nm (see Fig. 10 a-d), while the second harmonic output of the Nd YAG laser or the dye laser itself was used to ionize the excited particles. So far transitions of Na3 in gas phase had only been found between 610 and 680 nm10 (see also Fig. 10d). Xa- calculations and matrix isolation spectra, however, gave evidence for more transitions in the shorter wavelength domain11,12. With the higher sensitivity of the pulsed arrangement these transitions became observable as well.
The time-of-flight technique allowed simultaneaus recording of the complete mass spectra, while the laser was scanning. This permitted a deeper insight into fragmentation processes : Initially, the oven conditions were set such, that predominantly only Na and Na2 were formed. Providing appropriate power conditions, no signal was obtained, as long as the two-photon-energy E = hv1 +
hv2 did not exceed the ionization energies of Na and Na2 (Eion = 5.14 eV for Na and 4.9 f or Na2 13. However, E was always kept well
a) CIOO
c)
Fig. 10 Two-photon-ionization spectra of Na3
1182
above the ionization energie of Na3 (Eion = 3.97 eV for Na3 13 ). Then the oven conditions were changed such, that Na3 was formed as well. Consequently - at proper excitation wavelengths - a clean Na3 mass peak occured (see Fig. 11a). Slight changes of the excitation wavelength, however, could provoke the occurence of Na2+ and Na+ fragments (see Fig. 11 b-c). Also strong metastable peaks could occur (see Fig. 11d).
The most plausible processes for the occurence of t hese fragments are :
al
Cl
Fig . 11 a-d
69
23 46 69
M [amu]
A1-589.7nm
M
bl
dl
+ e
+ e
A1=598.7nm
23 46 69 M
23 46 69 M
Mas~ spectra of Na3 TPI-processes at different excltatlon wavelengths.
1183
An cnergy analysis shows, however, that none of these processes is energetically below the 2-photon-energy14. No explanation can be given so far. Negative ion detection and an energy analysis of arriving metastable mass peakswill be necessary. Possibly the Na3 particles in the beam are not as much cooled down, as has been found for Na2; possibly they even occur in metastable states. Possibly also, accelerated photoelectrons in the ionization area are responsible for the observed fragmentation, or uncontrolled multiphoton processes may occur.
The tendency of the aggregates to break apart while being excited or ionized still seems to increase with growing particle size. We are reluctant, therefore, to attribute those spectra, which we recorded on the mass peak 92, exclusively to the Na4 molecule. An interpretation requires a systematic deconvolution of all fragments at all wavelengths, as shown in the 2-dimensional optical mass spectrum in figure 12.
Fig. 12
1184
700 [nm]
+ Na
2-dimensional mass spectrum of the alkali clusters.
ACKNOWLEDGMENTS
We are grateful to professor G. Stein for many helpful and stimulating discussions during the course of this work, and to the Fonds National Suisse for financial support.
HEFERENCES
1. T. Ellenbroek, J.-P. Toennies, J. Wannerand M. Wilde, J. Chem. Phys. 75, 3414 (1981).
2. M.F. Vernon, D.J. Kranovitch, H.S. Kwok, J.M. Lisy, Y.R. Shen and Y.T. Lee, J. Chem. Phys. 77, 47 (1982).
3. J. Geraedts, S. Stolte and J. Reuss, Z. Phys. A.- Atomsand Nuclei 304, 167 (1982).
4. P. Melinon, R. Monot, J.-M. Zellwegerand H. van den Bergh, to be published.
5. R.V. Ambartsumian, Yu. A. Gorokhov, G.N. Makarov, A.A. Puretzky and N.P. Furzikov, VICOLS Laser Spectroscopy, Conf., B.P. Stoicheff, A.R.W. McKeller and T. Oka, eds., p. 439 (1981).
6. D.R. Coulter, F.R. Grabiner, L.M. Casson, G.W. Flynn and R.B. Bernstein, J. Chem. Phys. 73, 281 (1980).
7. A. Herrmann, S. Leutwyler, E. Schumacher and L. Wöste, Helv. Chim. Acta E1, 453 (1978).
8. A. Herrmann, S. Leutwyler, E. Schumacher and L. Wöste, Chem. Phys. Letters 52, 418 (1977).
9. W.C. Wiley and C~. McLaren, Rev. Sc. Instr. 26, 1150 (1955). 10. A. Herrmann, M. Hofmann, S. Leutwyler, E. Schumacher and
L. Wöste, Chem. Phys. Letters 62, 216 (1979). 11. E. Scholl, Dissertation, Bern 1980. 12. M. Hofmann, S. Leutwyler and W. Schulze, Chem. Phys. 40, 145
(1979). 13. A. Hermann, E. Schumacher and L. Wöste, J. Chem. Phys. 68, 2327
(1978). 14. G. Delacretaz, J.D. Ganiere, R. Monot and L. Wöste, Appl. Phys.
B 29, 55 (1982).
1185
FREE MOLECULE DRAG ON HELIUM CLUSTERS
ABSTRACT
Jürgen Gspann
Institut für Kernverfahrenstechnik der Universität und des Kernforschungszentrums Karlsruhe Federal Republic of Germany
The observation of free rnolecule drag coefficients below 2 for helium clusters of either isotope is taken to indicate quantum fluid inviscity of the helium rnicrodroplets.
INTRODUCTION
'The drag on free clusters of rnolecules or atorns which rnove with respect to their environroent is usually deterrnined by free rnolecule interaction, due to the srnallness of the clusters. For clusters with less than sorne rnillion atorns, the longranging attractive van der Waals forces enlarge this drag considerably in cornparison tothat on a non-attractive hard body of the sarne size. 1 ' 2
Moreover, clusters are often rather volatile and rnay be partially vaporized by irnpinging particles. The recoil rnornentum of the evaporating material rnay then further increase the effective drag.
However, if the irnpinging particles are sufficiently energetic, they rnay also penetrate the whole cluster. The rnornentum flow transferred to the cluster is then srnaller than the intercepted one, and the drag on the cluster is correspondingly reduced. Usually, this will require rather highly superthermal energies, as rnay occur with accelerated clusters, in plasrnas, or in space environrnents. An exarnple is given by cluster bearns used as internal targets in storage rings for high energy particle physics. 3
In the case of helium clusters of either isotope, however, a rernarkable drag reduction is observed also in thermal flows of for-
1187
eign gases.~ In the following, the phenomenon will be discussed in terms of projectile penetration as a consequence of the quantum fluid properties of the clusters. In particular, the strong rarefaction of the gas of the socalled quasiparticles within the helium clusters is thought to be the reason for the low internal drag on the penetrating projectiles.
EXPERIMENTAL
As in the former studies, .the effective drag coefficients of the clusters are derived from measurements of the deflection of cluster beams by crossjets. 1 The helium cluster beams are generated by a converging-diverging nozzle at 4.2 K in the case of ~He cluster beams, or 3.2 K in the case of 3He cluster beams, respectively. Carbondioxide and xenon crossjets are used whose flowfields are obtained from stagnation pressure measurements. Beam and jet axes intersect at right angles in the experiments discussed here.
The speed ratio of the helium cluster beams is in the range of 40 to 60 while that of the crossjets is about an order of magnitude smaller. Hence, the speed ratio of the relative motion of the crossjet atoms or molecules with respect to the clusters is mainly determined by the crossjet speed ratio. As this is considerably larger than one, however, the free molecule drag coefficients of the clusters should be 2 if all the crossjet momentum flow which is intercepted by a cluster would be transferred to it.
The experimental results are shown as points in figure 1, with empty dots for xenon crossjets while filled and half-filled dots refer to carbondioxide jets deflecting ~He and 3He cluster beams, respectively.~ Obviously, all the measured drag coefficients are smaller than 2 at the given conditions.
The variation of the mean cluster sizes is obtained by changing the nozzle feed pressure. A special kind of time-of-flight mass spectrometer serves to determine the cluster sizes. 5 The full width at half maximum of the distribution of the cluster sizes is usually about as large as the mean size6 so that in a logarithmic size scale as in figure 1 the distributions would show up as rather narrow peaks. Consequently, in comparison with the investigated range of mean sizes, the distribution of the cluster sizes at given nozzle conditions can be neglected in the following considerations.
Control experiments in the same setup using nitrogen cluster beams with mean sizes of 103 to 105 molecules per cluster yield practically size independent drag coefficients of nearly 3 for carbondioxide, and about 2.5 for xenon crossjets.~ Neonclusters as well are not found to show drag coefficients below 2.
1188
2 0
u c Q)
1.5 u --Q) 0 u
1 Ol 111 ... "0
o.~o._4 ____ 1o._5 ___ 1_Jo._s=----1_J0_7 ___ 1_.oe
mean duster size N
Fig. 1. Drag coefficients of helium clusters as a function of the cluster size for carbondioxide (filled, half-filled) and xenon (empty dots) crossjets. Theoretical curves are fitted at the respective lowermost cluster size.
THEORY
Constant Atomic Drag Coefficients
Taking the incomplete transfer of the intercepted momentum flow as an indication of projectile penetration, the curves drawn in Fig. 1 are calculated on the assumption of a constant drag coefficient for the penetrating atoms or m~lecules inside the liquid helium clusters. Any effects of or at the cluster surface are neglected.
If c is the constant atomic drag coefficient referring to a suitably defined atomic cross section f, the relative velocity v of the atomic projectiles of mass m decreases along its path of penetration of length s according to
mdv/dt = mv dv/ds = -pv2c f/2 (1)
where p is the helium density. Integration yields for the ratio of the transferred to the initial projectile momentum
~p/p = 1 - exp(-pcfs/2m) 0
(2)
If all the relative momentum flow intercepted by the cluster cross section TIR2 is transferred, i.e. if all the projectiles get stuck,
1189
the drag coefficient of the cluster is 2. Hence, with penetrating projectiles, the drag coefficient is given by
(L'lp/p ) 2'ITbdb 0
( 3)
where the momentum transfer depends on the length of the respective straight trajectory for the impact parameter b:
(4)
Replacing the cluster radius R by the number of atoms of mass m He per cluster
N = (4/3)'ITR 3 p/m He
one gets finally
with
A (cf/m) (3p 2m /4'IT) 113 He
(5)
(6)
(7)
In Fig. 1, the curves represent Eq. (6) fitted to the experimental points at the respective lowermost cluster size by suitably choosing A. The curves are seen to reproduce the observed size dependence of the cluster drag coefficients at least approximately. The dashed curve results from transposing the curve fitted to the black 4 He points to 3 He by changing only the helium densities according to the ratio of the values at 0 K, 0.145/0.0823, and the helium atomic masses. Within the model, a higher transparency yielding lower drag coefficients CD is predicted for the 3He clusters, in accordance with Observation. This is remarkable as superfluidity should be a feature of 4 He but not of 3He clusters.
Atomic Drag Calculations
On the other hand, the assumption of constant drag coefficients for the atomic projectiles is not well founded in view of.the wide range of velocities passed through between impact and stopping. One
1190
can try to calculate the deceleration of the projecti~es in more detail, however, solving Eq. (1) for a velocity dependent atomic drag coefficient as given by the known sphere drag relations. Figure 2 shows examples of such deceleration calculations for the case of xenon atoms moving in liquid ~He.
For the drag coefficient, the full set of equations for continuum and rarefied flows proposed by Henderson7 is used. This set of equations, which are not repeated here, consists of one equation for all of the subsonic flow regimes, a second one for the supersonie flow regimes at Machnurobers larger than 1.75, and a linear interpolation for the intervening region. In the limits, the relations simplify to equations derived from theory, as Stokes, Epstein, or Stalder and Zuricks equations, while describing empirical data otherwise.
The drag coefficients are given as functions of the Reynolds and Mach nurobers, of the speed ratio, and of the ratio of the temperatures of the sphere and the flow. In the present case, this latter ratio is chosen to be one, assuming no heat exchange between the projectiles and the surrounding helium. The sphere radius is chosen tobe 3.1 R corresponding to 0.84 times the zero of the 12,6-LennardJones potential for Xe-He. 8 The factor 0.84 follows the usual hard sphere assumptions for helium atoms in liquid ~He. 9
The speed ratio is chosen to coincide with the Mach nurober because the kinetic energy of the atoms in liquid ~He due to their zero point motion9 corresponds to a speed practically coinciding with the speed of (first) sound. That this is so although the ratio of the specific heats in liquid helium is known to be one 10 is taken to be due to a particular speed distribution in liquid helium.
The xenon deceleration is first calculated for a liquid ~He temperature of 2.5 K, viz. well above the temperature of 2.17 K of the A-transition to the superfluid phase. The corresponding viscosity of 36 micropoise leads to a Knudsen nurober of about 0.4 and a Reynolds nurober at sonic speed of 5.5, justifying thus the need for using the complete set of drag relations. As shown in Fig. 2, a xenon atom of 400 m/s initial speed is found to come to a rest already at about 50 R depth of penetration, which is much less than the 450 R diameter of a ~He cluster of about 106 atoms.
Penetration is obtained, however, if the viscosity is assumed to be zero. Moreover, the calculated deceleration is then rather similar to that giving the curve fit to the empty dots in Fig. 1, corresponding to a constant drag coefficient of 0.66, as shown by the dashed curve in Fig. 2. Hence, this result is consistent with the picture of the drag on a helium cluster resulting from the momentum transfer of the projectiles to an inviscid droplet.
1191
Fig. 2.
400
1/) ...... E "'0 Q) Q) a.
'experiment '
1/) 200 E 0 iii Q)
>< -36~-JP
0 0 100 200 300 400 500
penetration depth <Al
Calculated deceleration of a xenon atom in liquid ~He as a function of the depth of penetration for normalfluid and vanishing viscosity. The dashed curve corresponds to the curve fitting empty dots in Fig. 1 .
Conclusion
If this explanation holds1 inviscity means that the Knudsen nurober vanishes also for atomic projectiles. The helium fluid moves as a structureless continuum even at these very small dimensions. (A similar conclusion can be drawn from the study of vortices in superfluid helium) • The helium clusters must be actually so cold that with both isotopes the helium atoms do not move any more independently but as a coherent quantum state. Viscosity then results only from the residual thermal excitations which form a gas of independent quasiparticles. 11 Within the small cluster volume1 however1 there are only very few of them left at sufficiently low temperaturesl and the viscosity vanishes as a result of the quasiparticle rarefaction.
REFERENCES
1 . J. Gspann and H. Vollmar 1 Drag Enhancement of Very Small Particles by the Intermolecular Forces1 in: "Rarefied Gas Dynamics" 1 R. Compargue 1 ed. 1 Commissariat a !'Energie Atomiquel Paris (1979) 1 p. 1193.
1192
2. J. Gspann, Small-Particle Drag as a Function of the Speed Ratio, in: "Rarefied Gas Dynamics", S.S. Fisher, ed., Progress in Astronautics and Aeronautics 74:959 (1981).
3. V. Bartenev, A. Kuznetsov, B. Morozov, V. Nikitin, Y. Pilipenko, V. Popov, L. Zolin, R. Carrigan, J. Klen, E. Malamud, B. Strauss, D. Sutter, R. Yamada, R. Cool, s. Olsen, K. Goulianos, I.-H. Chiang, D. Grass, A. Melissinos, Cryopumped, Condensed Hydrogen Jet Target for the National Accelerator Labaratory Main Accelerator, Adv. Cryog. Eng. 18:460 (1973) ~ J. Gspann and H. Poth, Internal Cluster Beam Target for Antineutron Production in LEAR, Kernforschungszentrum Karlsruhe, KfK 3198 (1981).
4. J. Gspann and H. Vollmar, Momentum Transfer to Helium-3 and Helium-4 Mieredroplets in Heavy Atom Collisions, J. Physique 39:C6-330 (1978).
5. J. Gspann and H. Vollmar, Metastahle Excitations of Large Clusters of 3He, ~He, or Ne atoms, J. Chem. Phys. 73:1657 (1980).
6. J. Gspann, Electronic and Atomic Impacts on Large Clusters, in: "Physics of Electronic and Atomic Collisions," S. Datz, ed., North-Holland Publ., Amsterdam (1982).
7. C. B. Henderson, Drag Coefficients of Sp,heres in Continuum and Rarefied Flows, AIAA-J. 14:707 (1976).
8. C. H. Chen, P.E. Siska, and Y.T. Lee, Intermolecular Potentials from Crossed Beam Differential Elastic Scattering Measurements. VIII. He+Ne, He+Ar, He+Kr, and He+Xe, J. Chem. Phys. 59:601 (1973).
9. G. V. Chester, Introductory Lectures on Liquid Helium Four and Three, in: "The Helium Liquids," J.G.M. Armitage, I.E. Farquhar, eds., Academic Press, London (1975).
10. L. D. Landau and E.M. Lifshitz, "Fluid Mechanics," Pergarnon Press, London (1958), p. 519.
11. I. M. Khalatnikov, "Introduction to the Theory of Superfluidi ty," Benjamin, New York (1965).
1193
VIBRATIONAL RELAXATION KINETICS IN A TWD-PHASE GA&-CLUSTER SYSTEM
ABSTRACT
A.A.Vostrikov, S.G.Mironov, and A.K.Rebrov
Institute of Thermophysics Siberian Branch of the USSR Academy of Seiences Novosibirsk, 630090, USSR
The goal of this paper is to investigate the relaxation kinetics of vibrationally excited molecules (VEM) under the conditions of a supersonic expansion accompanied by C02 and N20 condensation. The catalyst effect of clusters (Van-der-Waals molecules) on the rate of a vibrational relaxation has been established. The vibrational relaxation rate Re of molecules in bound state and the VEM relaxation probability P per a collision with a cluster have been determined as functions of a mean molecule number in a cluster (cluster size) N. The values of Re and P efficiently increase with N. They are independe~t of a phase cluster state and decrease with the cluster temperature when N=const. When N>IOO, P tends to unity. For N~const Rc<N20) and P<N20>>RC<C02) and PCC02). The VEM desactivation is accompanied by molecule evaporation from cluster surfaces.
INTRODUCTION
Recently the problern of vibrational relaxation in clusters and the VEM transition to the state bound with clusters during absorptionattractswidespread attention. In some works (see, for example1), the deviation of temperature dependence of V-T relaxation of gases from the well-known Landau-Teller relationship over the range of low temperatures has been found. This can be explained by the appearance of a great number of metastable clusters in gas and relaxation acceleration. Obviously this fact should be taken into account in calculations and analysis of the process concerned with supersonic flows, since a supersonic expansion is often accompanied by condensation. During this process the cluster size may change from 2 to
1195
thousands and tens thousands of molecules. In the meantime, while clusters grow, the molecule binding energy and the absorption potential depth increase, and, consequently, the phase state of clusters changes2. A supersonic expansion is known to be characterized by a different degree of relaxation of translational and internal degrees of freedom of molecules. Molecules in jet are vibrationally excited. It is to be cleared up what happens to the vibrational excitation energy, when the clusters are formed or when transfering vibrational excitation to molecules in clusters. It is evident that the condensation rate, size, phase state of clusters and gasdynamical parameters of a jet are changed if clusters accelerate V-T relaxation.
In this paper the heterogeneaus relaxation of vibrationally excited C02 and N2 0 molecules is investigated using the particles of a proper condensate formed in the jet behind a s0nic nozzle. A molecular beam generator used in the experiments is shown schematically in Fig. 1. Here 1 is the three-sectional vacuum chamber provided by a differentiated pump system, 2 - the C02 Laser, 3 - the means for obtaining glow discharge with a constant magnetic field, 4 - the gasdynamical source, 5 - the ionized gauges for detection of intensity, density and composition of a molecular beam, bolometer, 6- the mechanical modulator of a beam, c~r-223-2-IR-photodetector with a dispersion filter, HKM-1-IR- spectrometer. VEM were obtained with a gas flow moving through a glow discharge. The formation and growth of clusters were competed in practice before the excitation region. Therefore, over the range under consideration N molecules could be probably excited both in a gas phase or in clusters. The change in the VEM number was detected by measuring the spontaneaus
1196
~
I ? '\~ll
I Fig. 1. Experimental set-up
radiation intensity I of C02 and N20 molecules from the jet in bands of 4.3 and 4.5 f.lrlt, respectively. The data on condensation and properties of (N20>W clusters used in this paper were obtained, as earlier for C0~' 3 , by the molecular beam method.
RESULTS AND DISCUSSION
Curves 1 in Fig. 2 and 3 show I vs gas Stagnation pressure Po and N, for N20 with the gas stagnation temperature To=330 K, and the sonic nozzle diameter d*=2.8 mm; and for C02 I vs To~ N with Po=4.3 104 Pa and d*=1.9 mm. The arrow "a" points to the place of transition from the developed condensation, when stable clusters are formed which reach the intensity gauge. The arrow "b" shows the transition from the Liquid clusters formation to the formation of solid ones2. As is seen from Figs. 2 and 3, at the expansion without condensation, the value of I depends on the number of molecules in the region under study, i.e. proportionally to Po and Tö\5 .The appearance of clusters in the jet results in a qualitative change in the character of I as a function of Po and To.
Now Let us analyze the jet processes which can affect a spontaneous radiation value. The value of I is Likely to be determined by the number of VEM formed while passing through the discharge in a gas phase Ng and clusters Nd. In turn, the change in Ng and Nd is defined by the following processes:
1. Radiactive desactivation rate, Rr. 2. V-T relaxation rate in a gaseous phase, Rg. 3. V-T relaxation rate under impact upon the cluster, Rim· 4. V-T relaxation rate in the cluster, Re.
Here it is necessary to take into account the rate of exchange by vibrational quanta between gas phase and cluster, Rgc, and reverse process, Rcg at condensation (evaporation) and resonance V-V exchange.
The kinetic equations describing the variations of Ng and Nd in time, as well as the equations describing the variations in temperature T and density p are as follows:
dNf,/dt=-RrNg-RgNg-RgcNg-RimNg+RcgNd
d~/dt=-ARrN~-RcgNd-RcNg+RgcNg
dD.T/ dt= ( 2c* E* /Sk) ( RgNg+ RimNg+RcNg), T =T oo+D.T
pT=const
(1)
(2)
Here the coefficient A=(n/6)13 (6N~3-12NV3+8/N states to the fact that the cluster desactivation by radiation proceeds only via molec~Les on cluster surfaces; D.T is the value of temperature
1197
10
8
2
2 4 6 8
I
/0
/tJf 2 P..R.
I I I I l
101 !O'N
Fig. 2. Experimental (1) and calculated (2-6) curves for the radiation intensity I vs Po and N.
7
6
5
4
3
•ltt I I
6 4 2 tri' 6 4 2 10 Ai
Fi g. 3. Experimental (1) and calculated (2- 6) curves for the radiation intensity I vs To and N.
1198
change of gas due to V-T relaxation of VEM, e* is the total concentration of VEM at all the vibrational Levels; E* is the averaged value of vibrational excitation; k is the Boltzmann constant, Too is the gas temperature without discharge. Coefficient 5/2 in (2) takes into account the contribution of translational and rotational degrees of freedom into gas heat capacity.
The system of equations (1), (2) is solved numerically. When solving it, the variables t-x~ x=Vt/d* are substituted, where V is the flow velocity.
The radiation intensity from the jet is related to Ng(x) and N~(x) by the following equations:
(3)
I (x 1 x"J=R N'!. (x' x"J+AR N* rx 1 x"J o ~ rg ~ re ~
Here Q is the elementary volume of the jet region formed by the jet cross-section, from where radiation is detected, and x 11~ xd are the Locations of the boundaries of this region; x~~ x!f are the Locations of discharge boundaries; f(x") is the function taking into account a cylindrical shape of the discharge column.
In calculations it is assumed that Rim=avmePg· Here Vme is the frequency of molecular-cluster collision in a jet; Pg is the probability of V-T relaxation in a gas phase per one collision; a=exp(Ee/Em)~60 is the coefficient taking into account the increase of the probability of V-T relaxation of VEM in collision with a cluster, due to acceleration of molecules in the field of the attractive cluster potential Ee; Em is the energy of molecular collision.
Curves 2-6 in figs. 2,3 illustrate the calculation of the radiation intensities by (3) for different values of Re- In Fig. 2 curves 2,3,4,5,6 correspond to Re=3~102 , 7.5~10~, 3~105 , 3~106 , 3~10 8 s- 1 respectively. In Fig. 3 curves 2,3,4,5,6 are obtained for Re=3A102 , 3x10~, 1.2k105 , 3~105, 3~107 s- 1 , respectively. The increase in I with Po (Fig. 2) is explained by that in the region of developed condensation the condensate fraction in a flow is constant, and the value of N grows3. With increasing N~ the cluster number and the total area of the cluster surface decrease. Starting with some pressure, the area decrease factor becomes the most important, and the relaxation rate of molecules excited in ~as phase decreases via clusters.
As is seen from Figs. 2,3 within the framework of the VEM relaxation model, it is possible to describe the experimental dependences I(N), having constructed the Re vs N plot by the points
1199
of intersection of the calculated curves with the experimental ones. Presented in Fig. 4 are the no vs N plots constructed in such a manner for N20, and in Fig. 5 - for C02. Here curves 1 and 2 were obtained with varying Po and To, respectively. In Fig. 5 the dashed line illustrates the value of the relaxationrate in liquid co24· It is evident that with increa.sing N the value of Re tends to the value of the relaxation rate within a macrovolume of liquid C02 •
In supersonic expansion accompanied by condensation, with no external source of vibrational excitation, the generation of VEM in clusters is due to absorption of VEM or a resonance excitation transfer in VEM collision with clusters. Therefore, for v-r relaxationrate in clusters we have Ro=[(I/RgcJ+(I/Rc)]-1 • The total V-T relaxation rate in a jet via clusters is probably as follows: R[=Ro+Rim since Rgc~vmc, for probability of the v-r relaxationvia clusters the following expression is valid:
For C02 and N20 we have aPg<<I, therefore the basic relaxation mechanism at the expansion with condensation is the energy dissipation of vibrational excitation 'in clusters. Figs. 4,5 show P vs N(Po) (curves 1) and vs N(T0 ) (curves 2).
In the discussion of the results it is to be stated that:
(4)
1. Catalytic effect of clusters on vibrational relaxation becomes prominent, even during the formation of condensation nuclea (Figs. 2,3). 2. The aggregate cluster state (arrow "b" in Fig. 4,5) does not significantly affect the vibrational relaxation rate. 3. With decreasing cluster temperature the vibrational relaxation rate in clusters decreases. Really, as is seen from Fig. 4,5, curve 2 is positioned Lower, than curve 1. For the Latter the Stagnation temperature and, consequently, the temperature at the intersection point of the expansion isentrope and phase equilibrium curve is always higher than for curve 2. 4. The VEM desactivation mechanism for C02 and N20 is the vibrational energy dissipation in clusters.
What happens to the energy of vibrational excitation in clusters? In 1977 from the experiments with the molecular-beam generator5 it was estabilished that the condensation of SFs was retarded when introducing it into C02 Laser radiation jet <Fig. 1). This effect was explained by a vibrational predissipation of clusters, N=4-5 in size. The lifetime of clusters excited in such a way was estimated to be not above than 10-6 6. A direct bolometric measurement of density of flux energy with excited C02 molecules in a glow discharge evidenced that the vibrational excitation energy didn't keep in clusters. While clusters moved in the jet (during about 10-4 s) it passed on evaporation of molecules from clusters7.
1200
0(}
trf 06
' "' Cl(<. 'Q..
04
02
10 2 4 6 2 4 6
Fig. 4. The vibrational relaxation rate Re in clusters and probability of relaxation in a jetvia clusters P vs N.
107
OB
/06
~." 06
oi 'Q..
105 04
02
10'
0
2 4 6 8 tri 2 4 N
Fig. 5. The vibrational relaxation rate Re in clusters and probability of relaxation in a jetvia clusters P vs N.
1201
The character of Rc(N) dependences (Fig. 4,5) is in agreement with the results obtained in8, where it has been shown that in a dense gas, Liquid and solid H2 the vibrational relaxation time is defined by an intermolecular distance and temperature. When passing from small clusters to great ones, the molecule binding energy rapidly increases2,7 and the intermolecular distance in clusters decreases what Leads to increasing Re.
The VEM radiation desactivation from the bound state in clusters is of particular importance. From the radiation band displacement it would be possible todeterminethe binding energy of molecules in clusters. As it follows from the calculation for C02, due to a high vibrational relaxation in clusters the fraction of VEM radiation from clusters has a rather sharp maximum (about 10% from total radiation flux) in region of small N~2o~.
REFERENCES
1. P.F.Zittel, C.B.Moore, Vibrational relaxation in HBr and HCL from 144 K to 584 K, J. Chem. Phys. 59:6636 <1973).
2. A.A.Vostrikov, S.G.Mironov, B.E.Semyachkin, Molecular-beam investigation of nonequilibrium processes, Calculation of heat transfer in energy-chemical processes 86, Institute of Thermophysics, Novosibirsk (1981) - in Russian.
3. N.V.Gaisky, Yu.S.Kusner, A.K.Rebrov, B.E.Semyachkin, P.A.Skovorodko and A.A.Vostrikov, The scaling Law for homogeneaus condensation in C02 free jets, in: Progress in Astronautics and Aeronautics, J.L.Potter, ed., AIAA, N.Y. 51(II):1103 (1977).
4. C.Manzanares, C.E.Ewing, Vibrational relaxation of small molecules in the Liquid phase: liquid nitrogen doped with C02, CD4, N20, J. Chem. Phys. 69:1418 <1978).
5. I.M.Beterov, Yu.V.Brzhazovsky, V.P.Chebotayev, A.K.Rebrov, B.E.Semyachkin, A.A.Vostrikov, Influence of C02 Laser radiation on SFs condensation and nuclear beam intensity, in: Book of abstracts of the 7th International Symposium on Molecular Beams, Italy, Trente 298 (1979).
6. I.M.Beterov, Yu.V.Brzhazovsky, A.A.Vostrikov, N.V.Gaisky, B.E.Semyachkin, Investigation of SF 6 condensation in a free jet at the presence of Laser radiation. Quantum elektronies 7:2443 (1980) - In Russian.
7. S.G.Mironov, A.K.Rebrov, B.E.Semyachkin and A.A.Vostrikov, MoLecular clusters: formation in free expansion and with vibrational pumping: cluster-surface interaction, Surface Science 106:212 (1981).
8. M.Chateau, C.Delalande, R.Frey, C.H.Cale, F.Pradere, Vibrational population relaxation of compressed H2 fluid in 15-110 K range, J. Chem. Phys. 71:4799 <1980).
9. A.A.Vostrikov, S.G.Mironov, B.E.Semyachkin, Vibrational relaxation kinetics of co2 with clusters in a hypersonic jet, Journal ot Technical Physics 53:81 (1983) - in Russian.
1202
XVIII. GAS-PARTICLE FLOWS
LONG-RANGE ATTRACTION IN THE COLLISIONS OF FREE~OLECULAR
AND TRANSITION REGIME AEROSOL PARTICLES
William H. Marlow
Department of Energy and Environment Brookhaven National Laboratory Upton, New York 11973
INTRODUCTION
Rigorously speaking, an aerosol* is a nonequilibrium condition of matter. Given any nontrivial initial or boundary conditions, an aerosol ultimately evolves to a state of lower internal energy due to collisions among the particles and their thermodynamically irreversible coagulation. Estimation of these collision rates is the question to which the subjects discussed in this paper make their contributions. In particular, a description of the realistic long-range interaction force and its effects upon the collisions between particles in all ranges of particle Knudsen number, Kn (Kn = 1/a, 1 = mean free path of sustaining gas, a = particle radius), is given.
Reversibility in Molecular Collieions and Nucleation
In a gas at equilibrium, the molecular collisions occur under the influences of long-range attractive and short-range hard-core repulsive interactions1 which result in negligible sticking or reaction. This interaction energy is often expressed in the 6-12 Lennard-Jones potential form1
(1)
where E 0 and r 0 are depth and position of the minima and R is the molecular separation (for point molecules). Values of E 0 and r 0 characteristic of each pair of colliding molecules may be determined to give compositional specificity to the collisions. In contrast, post-critical clusters by definition are stable and
1205
undergo collisions that should be irreversible due to the depth of their attractive potentials. Since no equilibrium thermodynamic principles exist upon which to base the calculation of the size distribution of a coagulating aerosol, appropriate kinetic equations must be used. The range of their necessary applications extends from the critical cluster size (-ü.5 nm radius) up.
A common approach to computing the interaction energy of aerosols for use in estimating collision rates is to sum all of the pairwise molecular interactions between the bodies. Assuming that all the steric factors due to molecular asymmetries average out and that the bodies never approach sufficiently closely for the repulsive hard core to have an effect, the total long-range interaction energy based upon the l/R6 dipole-dipole or London-van der Waals interaction is
t.EH J v2
r 6 0
(2)
where i is the molecular density of body i, dVi is a volume element of i Vi is the volume, R12 is the separation of dV1 and dV2, and 2E0r0~ is the force constant of the elementary intermolecular interaction written to correspond to eq. (1). For spheres, explicit integration of eq. (2) gives
where R is the distance between centers of the spheres, ri is the radius of sphere i, r12 r1+rz, and A12 is the Hamaker constant defined as
(4)
The significance of eq. (3) is thftt it indicates the range of this energy (i.e., the region where ~E 8(R) ~ kT) is comparable with the spheres' dimensions despite the much shorter range of the elementary interaction, eq. (1). Mahanty and Ninham2 and Langbein3 both include accounts of the method of pairwise summation.
Gas Phase Particle Collisions
As alluded to above, description of post-critical cluster and particle growth must incorporate their binary collisions, a process that includes particles of all ranges of Knudsen numbers. If only Brownian diffusion takes place (as opposed to turbulence, external field gradients, etc.) the growth process is described by equations4 such as
1206
where n! is the number density of particles of size class ! and Kij is the coagulation rate density between particles of size classes i and j.
(5)
For particles of Kn < 0.25, the collision rate is normally derived4 by computing the steady-state flux of one spherical particle to the other according to the (continuum) diffusion equation which includes the Interaction potential of the particle.
In the free-molecular regime, Kn > 10, particle collision rates may be treated in the same manne~ as molecular collisions in the gas phase. As a resu1t, the particles are assumed to have a Maxwellian distribution of velocities. Their collision rates are conventionally computed by examining their impact parameters when an Interaction potential between the particles that is singular at contact such as eq. (3) is included. Since interaction potentials normally vanish at large Separations, in the strictly freemolecular case this boundary condition allows the impact parameter to be determined uniquely from the initial conditions. Using this approach, Lushnikov and Sutugin5 have shown that the freemolecular collision rate is
CX>
-g2 Kij
0 f (b*)2 g3 dg 2Kij e (6) 0
where 0 2
(21rkT/llij) 1/2
Kij = 2n1n2 ru (7)
is the collision rate without an interaction force, llij = (mimj)/(mi+mj) is the reduced mass of the interacting particles, and o* is related to a parameter determing the capture cross-section and must be numerically determined for each initial condition. Unlike the continuum case where particle Separations comparable with their radii place the particles in the hydrodynamic zone, free-molecular flow regime particles separated by distances somewhat under a gas molecular mean free path attract each other with energies comparable to the thermal energy with no modification of their trajectories due to molecular collisions. Thus, the long-range force clear1y is expected to be important in free-molecular regime collisions.
With the description of gas-phase collision processes and their compositional dependences as background, the following discussion will review recent results which facilitate the
1207
computation of free-molecular and transition regime col~ision rates between aerosol particles attracting each other via realistic Interaction potentials. The modern theory of this long-range Interaction will be introduced in place of the pair-summation approach of eq. (2) to faithfully portray the nature of the longrange Interaction with detail that the older method is incapable of. Finally, results of computations using the modern theory for the interactions between particles will be given and it will be used in the collision rate results. Pertinent experimental data will be compared with the compositionally dependent collision rate results and conclusions will be drawn for general questions of aerosol dynamics.
TRANSITION REGIME COLLISION RATE DENSITIES FOR ATTRACTIVE, SINGULAR CONTACT POTENTIALS
The transition regime, 0.25 < Kn < 10, is a difficult region for kinetic theory6 even without the inclusion of particle Interaction potentials. For a pair of particles approaching each other, one anticipates that at small surface-to-surface Separations s (e.g., 1 < s < 100 nm) the particles should follow freemolecular traject~rie;. However, Figure 1 (discussed below) shows that for separations where s < i, the dominance of the thermal energy, kT >> ~, cannot be a;sumed in general. As a result, computation of the collision rate in this free path zone by use of the customary impact parameter approach is not Straightforward since the initial conditions on the trajectories are not at infinity but are at finite particle separations where the Interaction potentials are non-vanishing.
A modified form of the free-molecular collision rate has recently been derived7 in a manner that both facilitiates generalizing Fuchs' Interpolation and provides a completely analytical expression, unlike expressions such as eq. (6) resulting from the impact parameter approach. The derivation first for the freemolecular collision rate between spheres is carried out in the center of mass system by relying upon an effective-potential picture of the binary encounter.
If H is the nondimensionalized Hamiltonian and 6E(r) is the nondimensionalized Interaction potential, then
(8)
with
V(r)eff (9)
1208
- 1 nm --- 100 nm
- t o-17r:....,-_,__._,__,_~,____..____.___._...........", 0 . 1 1. 0 c ,,,
Figure 1.
8
6
4
I 2
~ "' 0 z "' -2
-4
- 6
-8
k I 2 aiJ) 3 4 5 6 7 8 9 10 II
-r,.rz~ CENTER-TC-CENTER -SEPARATION
Figure 2.
Figure 1. Lifshitz-van der Waals interaction potential energies for identical spheres plotted as a function of s/r, the ratio of surface separation to particle radius, for water and tetradecane.19
Figure 2. Energy as a function of separation in the center-ofmass system. Veff is the effective potential (eq. (9) in text) for one angular momentum J and an example of the Lifshitz-van der Waals attractive potential. a (J) is the location of the maximum of Veff•
the dimensionless effective potential. The notation is v vY~/2kT so that the quantity corresponding to angular m;mentum is J2 = r2(;e2 + ;~2) where r is center-to-center separation of the spheres. Figure 2 is a graph of V(r)eff for one value of J. Note that the particles collide, for a particular Hamiltonian trajectory, if and only if the radial velocity ;r is negative and ;r2 > V(a)eff where V(a)eff is the maximum value of V(r)eff· Thus, computation of the flux of particles to the spheres of radii a(J) is sufficient to estimate the overall collision rate while obviating the necessity of computing ßE(r12) as is done, for example, in moment methods or for the collision rate via a Straightforward trajectory analysis.6 Since V(r)eff has a single maximum, it is readily determined by dV(r)eff/dr=O and the resulting expression implicitly defines a(J). With the Maxwellian distribution for the particles in the center-of-mass system given by
( 10)
1209
12
the flux of these particles to the surface a(J) is
F(a(J)) = 4lra2(J) J d~r ~r f(J,~r) vr<O
( 11)
Integration of F(a(J)) over all permissible values of ve and v~ then gives the total flux. After some changes of variables, the result is computed7 to be (per particle at infinity)
Ffm = -'!T Vav foo da _1_ [dE(a) + ad2E(a)] a2 (12) 2kT da da2
r12
• exp 1- ~T H ~~o) + E{a)Jl In addition to being entirely analytical, these formulas have a clear division of dependences that facilitates their modification as the boundary conditions change. This particularly applies in eq. (12) in comparison with eq. (6). The earlier collision rate is a functional of the impact parameter which itself appears as an explicit function of velocity, the variable of integration. Thus, no geometrical constraints can be readily imposed upon this freemolecular rate as would appear necessary in going from the freemolecular to the continuum regime. In contrast, the rate in eq. (12) is expressed explicitly as an integral over a geometric quantity a in terms of which all phase space conditions are expressed. Since Ffm is just a sum over F(a(J)), whenever boundary conditions need be imposed upon the flux, the upper limit of integration in eq. (12) is changed from infinity to whatever the boundary is. In the case of Fuchs' interpolation,6,8 the assump~ion ~s 1 ~2at the free molecular region extends a distance ö = (Al + A2) beyond the surface. Ai is the mean persistence distance for Brownian motion (or mean free path) of particle i derived from its diffusion coefficient Di' At the distance r12 + ö and greater from the center of mass, the spheres move according to pure continuum flow. This r12 + ö distance is then the boundary on the free-molecular flux. Clearly, trajectories at this outer boundary which also would lead to collisions, but for which a > r12 + ö, must also be included in the free-molecular part of the flux and this is done by recognizing that a(J) in eq. (11) is now a constant (i.e., cr(J) = r12 + ö for all J such that a(J) > r12 + ö). The full free-molecular flux is the sum of these two terms when the conditions imposed by Fuchs' interpolation are met.
The actual collision rate which provides the semiempirical interpolation8 between free-molecular and continuum rates is derived7 by equating those rates on the spherical surface of radius r12 + ö. The final resultant expression is
1210
c fm K12 4w(rl2 + ö) D12 'l' 12 'l' 12
2
[ 4(r12
r12 V av c
+ ö) Dl2 'l' 12 +
(13) 2 fm] r12 Vav 'l' 12
where D12 = D1 + D2 and vav =V8kT/'Irll12• The continuum correction according to these arguments.is
~~2 = [(rl2 + 0) ./"~ dr exp(E~;)/kT)]-1 r12+ö
From eq. (12) and the reasoning outlined above,
~~~ = ( -~ '( /r12+ö da a2[dE(a) +
2r 12kT/ da
r12 00
x exp /-(~,;} [ i ~~a) + E(a)JI + (r12 + 0)2
r12+o I
+ a d::~a)] expH~T) H ~a) + E(a)JI)
(14)
da[dE(a) ~
(15)
c fm The expressions 'l' 12 and 'l' 12 are the (multiplicative) corrections to the continuum and free-molecular collision rates in the absence of an interaction potential.
This computation of collision rates of particles interacting via attractive potentials which become singular as their separati6n vanishes is more general than just the Fuchs' model, since it is appropriate in any collision calculation where the final stage of approach of the interacting species is free molecular.
LIFSHITZ-VAN DER WAALS FORCES: THE UNIVERSAL INTERACTION OF CONDENSED MEDIA
Molecular Origins
Long-range intermolecular interaction potentiale between uncharged, ground-state molecules1 arise, in order of decreasing strength, from the interaction of their permanent multipoles, from
1211
the induced attraction between a permanent multipale and a polarizable molecule, and from the direct and induced interactions of their thermally and quantum mechanically fluctuating multipale moments (the "fluctuation interaction") whose lowest-order term is called the London-van der Waals interaction. The operational assumptions generally made are that the overall multipale moments for clusters of molecules and larger particles vanishes and that the fluctuation interaction potential has spherical symmetry. In general, the latter assumption cannot be made because of the tensorial character of the dielectric function (e.g., crystals or liquid crystals, particles with nonspherically symmetric structures).
The van der Waals interaction, whether for molecules9 or condensed media, results from both the direct and induced couplings of the interacting systems originating with their thermally and quantum mechanically fluctuating dipole moments. This is most simply described in terms of the interaction of a pair of hydrogen atoms, an example that readily generalizes to other atoms and molecules. The interaction Hamiltonian
e2 e2 e2 e2
HI =lg1 Q2 I - ~~ - Ell -122 - ~11 + IRl - Rzl (16)
serves as the perturbation for the Hamiltonian of the uncoupled atoms. Here, ~ and ~ are the position vectors, respectively, of the electron and proton on atom i and e is the elementary unit of charge. Carrying out the standard stationary quantum mechanical perturbation calculation and substituting appropriate wave functions gives a vanishing first-order contribution to the interaction energy. This is simply a manifestation of the fact that the atoms have no permanent moments so that terms of the form <oiH1 jo> vanish where jn> is the nth excited atomic state vectot. The lowest-order contribution to the ener!y is in second order and involves matrix elements of the form Hrln> summed over all excited sites. When the appropriate c ordinate system is chosen, the interaction energy becomes
t:.E ~ l<ojzlm>j2 l<ojzln>j2 (17)
m,n (E0 - Ern) + (E0 - En)
where Ei is the energy of the ith excited state and z is the coorlinate oi the electron relative to the proton in each atom. The ~1 - ~ -6 dependence arises from the product of ~1 - ~ -3 term which re characteristic of dipole radiation fields. Silce the induced dipole moments are therefore proportional to this term and the energy arises from the interaction of the induced dipole moments, the energy is proportional to their product. Note, however, the importance of the assumption that the interaction eq.
1212
(16) is electrostatic. Because of this, no currents are involved (in the sense of Maxwell's electromagnetic field equations) meaning that no time-dependence in the propagation of the electric fields arises. Consequently, no question of phase enters in the overlap of the matrix elements in eq. (17) relating the dipoledipole interaction to energy. This is called the nonretarded van der Waals interaction. When questions of propagation time and phase are taken into account in the retarded interaction, the attractive energy is weakened which has significance for the coagulation and growth rates of stable clusters and very small particles. In eq. (17), the dipole moments are those arising from transitions between the ground state and an excited state. These transition dipole moments are likewise related to the atoms' dynamic polarizabilities ai(w) (w is the frequenc~ cörresponding to the energy of the transition) and can be shown to give a very compact expression for the interaction energy of two point (molecular) dipoles
00
t.E = - 3~ J dga1 (is)az(is) 1rR6 0
( 18)
where w = iE; and R is taken as the distance of separation of the oscillators.
Incorporation of retardation in the interaction of two molecules2 requires solving Maxwell's equations in the presence of currents. The currents arise as a consequence of the fluctuations of the dipole moments of the interacting molecules treated as point dipoles in this discussion (the point-dipole assumption is not essential to the theory). This is a generalization to nonsteady fields of the picture of the nonretarded van der Waals Interaction discussed above. As a result of this variable perturbation of the molecular polarization P(t,t), a polarization current is induced and when the polarization currents on the interacting pair of molecules are summed gives the total current j(r,t) to be used in Maxwell' s equations. With "'
00
tk<r.t) = ö(t- ßk) J[dwak(w)eiwt E(ßk,w), (19a)
(19b)
In eqs. (19), ~ is the position coordinate of molecule k, t is time, r is the position coordinate, ak(w) is the molecular polarizabillty as a function of frequency , and E(ßk,w) is the electric field. Since E(~,w) = E(ßk,O) in eqs. (16) and (17), in the nonretarded case the only frequency dependence there is in the transition probabilities or polarizabilities. In this sense, the two molecules' polarizations fluctuate independently of each other
1213
with the static electric field serving only implicitly to couple the instantaneous polarizations arising from the fluctuations. However, eqs. (19) when used in Maxwell's equations are fundamentally different because the polarization current there is the only source of the electric fields. They serve to couple the dipoles to a degree which determines the strength of the interaction. If a particular dipolar fluctuation corresponds to a wavelength A >> R12• where lß1 - ßll = R12• then the electric field is essentially constant dur ng a fluctuation period A/c and the coup-
. ling should resemble the static case (e.g., eq. (18)). Conversely, if A << R12• then the electric field undergoes numerous cycles before it can couple to the other molecule which lowers the correlation of the fluctuations, and therefore their interaction energy. The solution to Maxwell's equations with the current of eq. (19) gives the energy2,10
(20)
(c is the speed of light) and this expression gives precise meaning to these observations on the effect of retardation. In the nonretarded limit where the dominant contributions to ak(i~) are such that A = 2~c/w >> R12• eq. (18) is recovered while in the fully retarded limit A << R12 or -->0 the interaction energy is proportional to Ri~· This dependence demonstrates the importance of separation in ffie van der Waals interaction where the actual form of the interaction law changes from Rl~ at short range to Rl~ at asymptotically long range. In the case of realistic molecular collisions among small molecules in the gas phase, the nonretarded form by itself is adequate. At molecular Separations where retardation becomes effective, the nonretarded interaction energy already is too weak to affect the molecular trajectories. Consequently, the Rl~ dependence in the long-range attractive part of eq. (1), the Lennard-Jones potential, has proved sufficient.
Integration of the pair-interaction potential in eq. (2) gives the classical formula, eq. (3), for the nonretarded interaction energy of two spheres. While this same procedure could be carried out with eq. (20) replacing eq. (1), there would be little benefit. The method of pair summation has an overriding deficiency in its representation of the long-range, or van der Waals, interaction energy of condensed media. That deficiency is that it fails to account for the fact that the electromagnetic properties of solide and liquide generally differ from those of their constituent atoms and molecules. Neither collective effects such as conductivity nor structural properties as may contribute to infrared spectra can be accounted for by noninteracting molecules. In
1214
the optical and higher frequency region, pair summation is of value, but its relative importance cannot be determined a priori. At the same time, the pair Summation approach does not account for the effect of temperature in driving the dipolar fluctuations, a deficiency that can never be addressed in the long-wavelength optical regions where it should be of greatest importance.
Condensed Media Interactions
In the London theory for molecules, the coupling of the fluctuating dipole moments of the two molecules lowers the interaction potential creating an attractive force. To construct a theory for condensed media, similar fluctuations characteristic of the media and not just their constituent atoms must be exploited. Lifshitzll was the first to address this problern of the "molecular" forces between solids by incorporating the spontaneously fluctuating electric fields of the medium (which arise in response to the thermal or blockbody background radiationl2) in Maxwell's equations for two interacting half spaces separated by a gap. He found an expression for the attractive force between the half spaces which included the effects of retardation. Various experimental confirmations of the theory have been made and it is currently considered well established (ref. 22 briefly reviews theory and experiment with extensive literature references). While Lifshitz' approach is suitable for the interacting half-space problem, other methods have proved more useful for other geometries. In all cases, however, € (w) is accorded the central role in parameterizing the condensed media interactions and therefore receives considerable attention on its own.l3,14
The van der Waals interaction energy of greatest interest for aerosol particle collisions is that of two spheres. Kiefer et ar.lS have derived highly accurate, readily calculable approximations to Langbein's3,16 expressions for the nonretarded interaction energy in this case. For some guidance on the importance of retardation, consider the exponential damping factor e-P (p = 2z~/c) in eq. (20). For p >> 1, or equivalently z >> A, a substantial weakening of the interaction takes place relative to the London energy in eq. (18). In the case of spheres separated by a distance of one diameter which is comparable to or Ionger than wavelengths (e.g., 20-2000 nm) of fluctuations contributing to the van der Waals energy, this weakening must be taken into account.
An ad hoc approach to the inclusion of retardation in the nonretarded interaction energy of two spheres is the incorporation of the factor
(21)
1215
By comparing eq. (18) with eq. (iO), Fz(~) can be identified as the spectral retardation factor for the point dipolar interaction. If this same factor is used in the nonretarded energy for two spheres, it should incorporate at least some of the most important effects of retardation. Thus, writing the energy17 as
00
t.E(z) = - ~ J g(~,z)Fz(~)d~ 8'!1' -00
(22)
the retardation factor Fz(~) gives a "zeroth-order" approximation to the retarded interaction. With Fz = 1, the nonretarded energy15 is regained and in the limit of vanishing radius, this expression should become exact.
Eq. (22) with minor variations has been used in the computation · of the Lifshitz-van der Waals potential for several materials. Figure 3 for lead spheres18 at 293°K and approximately 10000K where dielectric data were available are of interest for what they reveal about the thermal dependence of the energy (temperature appears explicitly in the form of ~E used in the reference). The attractive potential increases as temperature rises, but what is of particular interest is the fact that the relative rise in potential is greater for the 1 nm spheres than for the 10 nm spheres at equal surface Separations. For example, at 1 nm separation, the magnitude of the energy increases 35% for the 10
i "' '" 0:
'" :J: .. "' &> .. 0: e ~ '" z '" "' _J
" ~ 0:
'" 0 z ~
' N 1-
i "' ... :::;
1216
-·· -10 r1•1nm
f2•1nm
-IÖI7 ---- 293"K ---IOOO"K
-·· -10
_,. -10
-·· -10
-ol -10
-·· -10 /
SURFACE SEPARATION (nm)
10
: -u5 17
:3 0:
'" ~ -lÖte &> .. 0:
rt ~ "' z '"
-u5''
~ -IÖ 1•
" "' "' ... 0
z ~ ' N
'= :z: ~ :::;
r 1 ~"10nm r2•10nm
---- 293"K ---1000•K
I.
I
SURFACE SEPARATION (nm)
Figure 3. Lifshitz-van der Waals potentials for small lead spheres as a function of separation.18
nm pair but 85% for the 1 nm pair. The interplay of retardation and its wavelength dependence is responsible for this difference. While the precise percentages involved are dependent upon the particular form Fz(~), the effect itself is undoubtedly a real one. Figure 1 from reference 19 is a plot of attractive energies between identica1 pairs of water and tetradecane drops, this time at 298oK. The abscissa s/r scales the surface-to-surface separations by the particle radius. For nonretarded interactions calculated by either pair summation or Lifshitz theory, the potential ener·gy of spheres varies wi th this ratio. Consequently, the dashed and solid curves for each material wotild be coincident if retardation were absent. The energy splitting due to retardation also shows its composition-specific effects: the attractive energy of the 1 nm tetradecane spheres is greater than that of the water spheres of the same size while the reverse is true for the 100 nm spheres. This can be traced to the fact that water's van der Waals energy has a greater static and long-wavelength component than tetradecane which in turn is more active at the short wavelengths that become retarded more strongly as separation increases.
Computational and Experimental Results
Lifshitz' theory of the van der Waals potential was first app1ied to aerosols17 to estimate the enhancement of the collision rate density between water droplets relative to that rate between noninteracting hard spheres. The basic procedure of those calculations is first to compute a table of van der Waals energies as a function of separation for the spheres (examples of which appear in Figures 1 and 3). That table is employed in the construction of a cubic spline fit which is then used in eqs. (13), (14), and (15) to compute K12· To focus upon the relative importance of the long-range attraction, all collision rates are presented in 0 terms ofi the collisional enhancementcwhichf~s the ratio K1z/K12 where K12 is the value of K12 with~ 12 =~ 12 = 1 representing the hard-sphere case.
Table I based upon ref. 17 gives the enhancements of water droplet collision rates. As is expected, the importance of the interaction force decreases markedly with particle size and in most cases with increasing pressure. The exception in the latter case occurs for the 1 nm pair of particles which has an essentially constant collision enhancement over the pressure range examined.
In the transition regime, only limited experimental data is available on collision rates. Probably the best is that on the coagulation of 200 nm particles formed by the condensation of di(2-ethylhexy) sebacate (DEHS) upon AgCl nuclei in a He carrier gas.20 The data were taken over a range of pressures as a test of
1217
TABLE I. Collision rate enhancement for water droplets including (excluding) retardation factor from ref. 18.
Radii Pressures (atm) (nm) 0.1 1.0 10.0
1,1 2.44(2.46) 2.44(2.46) 2.37(2.38) 1,10 1.55(1.59) 1.54(1.57) 1.30(1.30) 1,100 1.09(1.12) 1.05(1.07) 1.01(1.01) 10,10 2.26(2.45) 1.94(2 .02) 1.21( 1.22) 10,100 1.35(1.50) 1.09(1.11) 1.02( 1.03) 100,100 1.31(1.42) 1.07(1.15) 1.07(1.17)
0 Fuchs' coefficient, K12• Figure 4 displays this data as frac-tional deviations from the unmodified Fuchs coagulation coefficient on the same graph as the values calculated here for 200 nm radius water droplets. What is noteworthy about this graph is the similarity of the pressure dependences. Since the water calculations are for a material that is considerably different from that on which the measurements were made, quantitative agreement should not occur. However, at low pressures both cases exhibit similar deviations from what is calculated by the Fuchs expression omitting particle interaction potentials. The water droplets' values are approximately twice the experimental values for the lowerpressure data reported with both being in essential agreement with the Fuchs' formula at the higher pressures. Since DEHS probably has a much smaller low frequency dielectric constant than water, a reduction should occur due to the lower attractive energy at these important, nonretarded frequencies.
The fact that even in the free-molecular regime, the enhancement decreases as particle size increases can be anticipated from Figure 1 where energy does not scale with s/r. This has consequences for the free-molecular "self-preserving" size distribution21 that constitutes an exact solution to eq. (5) (or its generalization to continuous size distributions) subject to several assumptions, one of which is homogeniety of the collision rate function ·Kij as a function of particle radius.
ACKNOWLEDGIENTS
This research was performed under the auspices of the United States Department of Energy under Contract No. DE-AC02-76CH00016.
1218
0.30
0.25
0.20
0.15
0 Q 0 ....
0.10 : I
N 0 rr:-
0.05
0
-0.05
0.05 0.1 0.5 1.0
PRESSURE (atm af Hel
Figure 4. Computed relative fractional enhancement of collision rate due to van der Waals forces for 200 nm water spheres (solid line); experimental points (circles) of the same quantity for similar-sized DEHS droplets by Wagner and Kerker (Table I of ref. 20). R12 is the ratio of collision rates including forces to that omitting the Interaction forces. Data and calculations for He carrier gas. The dissimilarity of HzO and DEHS dielectric susceptibilities precludes quantitative agreement. ·From ref. 17.
REFERENCES AND FOOTNOTES
* An aerosol is a gas-borne suspension of particulate matter. For the purposes of this paper, the particles will all be considered to be under 500 nm in radii and to undergo no chemical reactions. 1. Margenau, H. and Kestner, N.R., Theory of Intermolecular
Forces (second edition) (Pergamon Press-,-Oxford, 1971). 2. Mahanty, J. and Ninham, B.w., Dispersion Forces (Academic
Press, London, New York, San Francisco, 1976).
1219
3. Langbein, Dieter, Theory of van der Waals Attraction (Springer-Verlag, Berlin,lHeidelberg, New York, 1974).
4. Hidy, G.M. and Brock, J.R., The Dynamics of Aerocolloidal Systems (Pergamon Press, Oxford, 1970).
5. Lushnikov, A.A. and Sutugin, A.G., Kolloidnyi Zhurnal 36, 566 (1974). --
6. Brock, J.R., in Aerosol Microphysics, I: Particle Interaction, W.H. Marlow, ed. (Springer-Verlag, Berlin, Heidelberg, London, 1980), Chap. 2, p. 15.
7. Marlow, William H., J. Chem. Phys. 73, 6284 (1980). 8. Fuchs, N.A., The Mechan~ofAerosOls (Pergamon, Oxford,
1964) 0
9. Bohm, David, Quantum Theory (Prentice Hall, Englewood Cliffs, N.J., 1951) contains a lucid discussion of molecular van der Waals forces.
10. Casimir, H.B.G., and Polder, D., Phys. Rev. 73, 360 (1948). 11. Lifshitz, E.M., Sov. Phys. J E T P2, D(1956). 12. Callen, H.B. and Welton, T.~Phys~ Rev. 83, 34 (1951). 13. Hough, David B. and White, LeeR., Advances in Colloid and
Interface Science 14, 3 (1980). -- ---14. Parsegian, V.A., in-Physical Chemistry: Enriching Topics from
Colloid and Surface Science, ed. H. van Olphen, K.J. Myse~ (Theorex::LaJolla, 1975).
15. Kiefer, J.E., Parsegian, V.A., Weiss, G.H., J. Colloid Interface Sei. 67, 140 (1978). -
16. Langbein, D., J:-Phys. Chem. Solids 32, 1657 (1971). 17. Marlow, William H::-J. Chem. Phys. 73; 6288 (1980). 18. Marlow, William H., J. Cöf:foi~terface Sei. 87, 209 (1982). 19. Marlow, William H., Surface Sei. 106, 529-rl98I). 20. Wagner, Paul E. and Kerker, Milton:-J. Chem. Phys. 66, 638
(1977). - -- -- -
21. Lai, F.S._, Friedlander, S.K., Pich, J., and Hidy, G.M., J. Colloid Interface Sei. 39, 395 (1972).
22. Marlow, W.H., in Aer,Osor-Microphysics, I: Particle Interaction, ed. W.H. Marlow (Springer-Verlag, Berlin, Heidelberg, New York, 1980).
1220
NONEQUILIBRIUM STATISTICAL THEORY OF DISPERSED SYSTEMS
A.G.Bashkirov
Department of Heterogeneaus Media USSR Academy of Seiences Moscow 125040
A new approach to the theon,y of Brownian motion of a large particle in a viscous fluid was developed by the author in 1 • Its main virtue is the correct accounting of the fluid motion provoked by the moving particle.So the way to the construction of the N Partielee Brownian motion theory taking into consideration the hydrodynamical interaction between particles due to the fact that every particle moves in the fluid troubled by other particles is opened.This very interaction determines a specific behaviour of disperse system (se&. e.g. 2-4 and the bibliography cited there), so the theory presented here may be regarded as a first step to the construction of a statistical transport theory in such a system.
In the first section the Fokker-Planck equation for a distribution function of N large partieleB centers of mass will be derived from the Liouville equation for the assembly of all molecules,constituent the large particles and hydrodynamically nonequilibrium viscous liquid.
In the secend section a diffusion equation for large particles will be derived.
1. FOKKER-PLANCK EQUATION FORA BROWNIAN PARTICLES SYSTEM
with
( 1 )
The Hamiltonian of a system consisted of a pure suspended large Brownian particles is:
( ~ ~
7t= ~\p. +; L. 11,) + L. 1 & + _t_z"" I<)+ ' 1'11 1 l ~k l2"?z 2 e .et
+LVK-+ j[ P.Kt K;e d~.. Kt.lt.t 2e
liquid
where subscripts i ,j are used for Coordinates and momentum of molecules of the liquid, and subscripts K,l are used for
1221
coordinates and momentum of molecules constituent large particles,which are numbered by indexes K,L. The Hamiltonian includes the following interaction potentiale: Ui· is the potential of interaction between i-th and j-th mol~cules of the liquid, ~~ - ~tween k-th and l-th molecules of the K-th large par-t~cle, \!ti - between i-th mflecule of liquid and k-th molecule of the K-th particle, ~~ - between the k-th molecule of the K-th particle and the l-th molecule of L-th particle.It is necessary to note that potential ~: is to have an adequately deep negative part to ensure a bound state of molecules constituent a large particle. As far as nonequilibrium state of the system under consideration is connected with transfer of mornentum and energy, a quasiequilibrium distribution function for it is likely to be as follows:
(2) 4(f}=- C{1eX('{-jt:!~(x;l)[llfX)- iJ~t);o~?l where as usual .ß (i',t)=1/kT(j(,t), .ti(1) and p ('it) are an energy and more.entum densi ty in the system.
An essembly of N suspanded large partielas is described by centers 0f :nass coordinates and moruentum:
RK=- ~f~~ ) 7! =?ß~ _, where Nk is the number of molecules constituent the K-th particle.
The dynamical density is introduced as
h({f!, ~1)= ng-r~ -r ~kJtf(ll{~-.l:tZkJ. K J:.fJ. ,..., k.
It's average on the full phase space ~of the s~stem with a :;.onaquilibrium distribution function"P <{j{,i'f y , I~ 1 J , t) of all molecules of the system is N-particle distr~~~ion function of coordinates and momentum of all large partielas centers of :nass:
11iJf, i;J) = jlrn((~/ R:J(/ff):.<:n({/f./ ~J). A starting point uf a theory developed here is the Liouville
equation for a :1onequilibrium distribution function:
where :1:. is the Licuvi:Lle opera tor, that is the Poisson brackets wi th the Hamil tordan ( 1 ) o To deri ve a kinetic equa tion for f"' from eq. ('13) we introduce a protection operator :J> as
(4) PA =-.J{ft}jlr'n({iL ~J)A 1 ~{t)-=-/j,c/~ c
where A is an arbitrary function,;01. is the conditional quasi-equilibrium distribution function,which differs from (2) only by fixation of values of all the partielas centers of mass co-
1222
ordinates and momentum it .. P and by a normalizatiqp,tha~ is determined wi th the dynalhicaf densi ty as f,/r'n({fJc,!(1cJ)f/(f)=-t.
c It may be noted that a result of time dependence of ~ ,
the pro,ection operator P will also• depend on time. It is obvious tliat?=J', :/)(1:- .P )=0. With the use of the; operator ß> we can formulate a boundary condition for the Liouville equation (3) a t t=-oo in a form
<5> .l'ttJ[_ = f77'r-tJ]__ = f.t/tf)/N({~~ R;~ t)Jt-.-- j
that corresponds to an assumption that at initial time t= -~ there is no correlation between collective variables {~ 2K] and remaining variables, described by the quasiequilibri~ con-di tional fw1c tion Ac . We deri ve a kinetic equa tion for J wi th the help of the operator fiJ from the Liouville equation (3) wi th the boundary condition (5) in a lower (second) order on a small paeameter J' ={m~/M1 ,where M is a mass of a Brownian particle. Because this derivation is the same as the derivation of eq.(J.15) from 1 we present here the resulting equation
'()_fit/ P.._ '(} :fN -+ -t A rJ ~~~~] ) 0 (. R 1'1 ~ +?I~ iRA: + ( 'k + Ii; ~()=t ~ 7-r71' "E:ICIC'l (I~- üK}j -t-
'df'"~ > () II P. .... 1'1 rd/1'1) +k.T ~) +L ~· 'C; ·~(~-- v-")j + J.r --=4
1~ K/·(t:K} IC "" M 'tl;:; /
(6)
~A where FK =-W1/ilf' is archimedian.~ force affecting a particle wi th, the volume WK = !f.ya3 immersed in a liquid wi th a pres-sure gradient t1f1 .3
0
C,Kt =:/ jt~-t jtls jt~s' iiff: <'r(ZJ'r(~;tJ> -- 2: 1: IC t.
(7)
are coefficients of hydrodynamical interaction, ~K is a surface of the K-th particle.
This equation is a generalization of a Fokker-Planck equation for a particle ·in a inhomogeneaus liquid 1 .A similar equation•for a system of Brownian particles with potential (but not hydrodynamical) interaction between partielas and fluid was derived in ref. 6; but usually in its futher discussion coef-ficients are regarded as being of hydrodynamical nature (see e.g. ref. ' ). Here,unlike ref.S , we succeeded in more consistent derivation of this equation. In the case when the dissipative interactions in the system are absent (e.g. when the coefficient of liquid viscosity is equal to zero) the right hand
1223
side of this equation becomes equal to zero, so it takes the form of a Liouville~equation for a system of particle~ with interparticle forces FK in an external field of forces F; •
2. DIFl''USION EQUATION.
/ "'- (Jp - ..... .... - IV ,..... where -.. ... .....--~ ;«·1 ol/]tlf? ... o/~ol~/ ... / C(~~ f}= <1'7-..J C(R_,_J-/V(~ f) =- <.IJ.IM'?-- ~ - (ft Y.(i f 1> + kr~ , -' K- M - ~, // oPK
the barmeans the averaging i?Jl the QOordinat&s RI<(K=2,J, ••• N) veighted with the function C = j JP1 o •• IPNfand devided on C(R.f,t).
An equation for~ diffusion flow may be derived by multiplication of eq.(6) on Pl· and subsiquent integration on P~ ••. PNo Then in steady approx ·mation we receive
es){# =C(~t)rV-iJ)=- -/+IV{ [J.'Pfl(., Ck:l;! -<if~ +Ct.vflp J II Ii! _,.
where we have approximated ~z (~- Uz) '.::!.: ~tz (V- iJ} , in correspondence with ~ assumption on a negligible variation of diffusion velocity (V-~) on the distance 1 of hydr~dynamical~inter~~-~ tion. An average potential force is .<Ftr- = N<li'z>-=-NßiRz '0.,>/~.(; C-l(/(1 R..l) It is evident that for a sufficiently dilute system the influen~~ ce of this force reduce to a forbidding of_a mutual interpenetration lf particles. So in calcula tions of ....( F 12)'- w~ may approx~ma-te C (R1,K~,t)=-J0 (R,z)CC~,t) , where J.(ll,2)=[e;cp(-.,&t'(R,z)]/_{}_ lS an equilibrium pair correlation function. Irl reality the function l· ( R1§) may sufficiently differ from ;.,( R12 ), but in the calcula-}-0n f .( F 12'/'- i t is unimportant due to a shortrange nature of r . So we have ....,_ ,...
IV< f:>- =-;i/t~~ ?Je -f-'~(~1t) =-- nJ:r'w vC J n= !V/J2 z
On substitution of this equation into eq.( 8) we receive
1224
where J.'r(.i + nw}
~LL + N 'E..,z. is a diffusion coefficient
0 =-C_w __ ? - -~..t.t ...,. !V'l:.,.i!
is a thermodiffusion coefficient
is a barodiffusion coefficient
Calculations of the coefficients ~.ti and 'l;,z just on the base of eq.(7) are too hard problem, so we could use the corresponding results of the hydrodynamical theories, e.g. the work of Mazur 5 , who received C!;K-' in the form of power series of the ratio a/RIC~.o • But it has appeared that divergencies are rising from every member of such series on its averaging. So we are to deny such a naive perturbation theory and to take into account an effective influence of all partielas on each pair interaction. As it was shown in refs. 3 and 4 this influence leads to a screening of long-range ( ~ 1/RK~ ) hydrodynamical interaction. Taking into account this screening by the method of ref. 3 it is not difficult to receive
After substitution of this expressions into the formulae for 2> we o btain
( 9 ) 2J = ZJ __ t--:::+=n=w=-----==-0 .t + o.jne:s ' ... nw (i- ~/"'t=-s')
The numerator of !q. ( 9) containel a correction n W= ':f to 2>0 = J.T'!i!:::;. taking into account configurational hard sphere interactions tetween particles. It is more essential to coneider the correlation in the denominator due to the hydrodynamical interactions between particlee. Only the denominator correction retaines in the loweet a~roximation of n , so eq. (9) takes a form0 =Qt( 1-2,1.f'J' ). It ie notable that in the phenomenological theory of Batchelor 7 an analitical
1225
dependence of 2> on 'J was postulated and configurational interactions received a dominated influence, and as a result an equation 0 = .00 (1+1 ,45 '1 ) was derived, that corresponded to an increase of 2> wi th '1'
On our opinion an influence on the motion of selected particle from other partielas being in a haotic motion is to be not too different from the influence from fixed other particles, i.e. it is to suppress the diffusion. In the presence of the screening with a characteristic length,.\ =(61itt.n f'IZ the: corresponding decrease is to be proportional to R·
HEFERENCES 1. A.G.Bashkinov,Bnownian motion of a large particle in nonhomo
geneous fluid, Theor.Math.Phys. (USSR),49:940 (1982). 2. R.Zwanzig, Langevin approach to polymer dynamics in dilute so
lutions, Adv.Chem.Phys. 15:325 (1969)\ 3. K.Freed and M.Muthukumar, On the Stokes problern for a suspen
sion of spheres at finite concentration~, Journ.Chem.Phys. 68:2088 (1978). .
4. s.Adelman,HYdrodynamic screening and viscous drag at finite concentrations, Journ.Chem.Phys. 68:49 (1978).
5. P.Mazur, On the motion and Brownian motion of n spheres in a viecous fluid, Physica 110A:128 (1982).
6. J.Deutch and I.Oppenheim, Molecular theory of Brownian motion for several particlee,Journ.Chem.Phys. 54-3547 (1971).
1. G.K.Batchelor,Brownian diffusion of partieleB with hydrodynamical interaction,Journ.Fl.Mech. 74:1 (1976).
1226
THE MECHANISM OF STRONG ELECTRIC FIELD EFFECT ON THE DISPERSED
MEDIA IN THE RAREFIED GAS
A.G. Gagarin, V.H. Vigdorchik, and Yu.N. Savchenko
Department of Mechanics of Heterogeneaus Media USSR Academy of Seiences
Systems in the form of a gas containing suspended dispersed particles suffering the effect of strong nonuniform electric fields rather often occur in nature and in human activities. The transfer of energy, mass, and impulses in such systems greatly depends on the characteristics of the electric field, viz., the degree of nonuniformity, field strength, volume charge content, etc. These factors influence, in particular, atmospheric and hence, weather conditions, electrization of aerospace vehicles, and some technological processes.
The above described processes, despite their abundance, still lack a rigorous theoretical description.
In the classical mechanics, there are two approaches to the discription of the flow of fluids and gases, that is, a phenomenological metbad and a molecular-kinetic method.
With the former approach, the Euler or Navier-Strokes flow equation is derived from the general laws of mechanics with the use of empiracal coefficients of viscosity and thermal conductivity (~ and A), These equations are well known and have been solved with high precison or not so high precision but quite sufficient to find experimentally the ~ and A values for gaseaus and liquid media.
The other approach to the fow of fluids and gases in classical mechanics involves a molecular-kinetic metbad that depends on the Boltzmann equation derived by intuition. This equation was then theoretically proved proceeding from the mechanics equations and general principles of statistics mentioned in the works of Bogoliubov, Born and Kirkwood. Boltzmann bimself proved that in the zero
1227
approximation bis equation can be used to derive the Euler equation and in the first approximation the Navier-Strokes equation can be obtained. So, the kinetic theory has made it possible to determine with a high degree of precision the fields of application of the empirical constants (J..l, A and the like) and not only to prove the aerohydrodynamics equations and to define these constants with a high degree of precision.
Boltzmann has proved that the hydrodynsmies equation is valid for gases rarefied to a limited degree. For gases rarefied beyond these limi ts, not only the equations undergo a change (Burnett' s approximation); but also change the boundary conditions to a certain degree; for example, a skating effect appears, etc. These Statements are rather accurately confirmed by a great number of experiments for the flow of monocomponent gases and for the determination of tqtal flow characteristics of gas mixtures. New methods, however, were required for the solution of the Boltzmann equation (1, 2) at least to determine the mechanisms of flow of separate gas mixture components. Moreover, huilding the theory of dense gas mixtures involved generalization of the Boltzmann equation (3). The thoery of ~otion of rarefied dispersed media in a gas not surrounded by external electric or magnetic fields has been based on these two generalizations (4).
Following the work four dispersed admixtures will be described using the model of solid spheres as for molecules. In this case the following system of aerodynamic equations is obtained:
In this system (which corresponds to Euler's equations) interactions between gas molecules and admixtures are taken into consideration already in the zero approximation. These interactions are
1228
expressed by coefficients of intercomponent interactions
~'l" ' ,6S't"
which are known functions of gas particles' and admixtures properties in ths theory.
Equations of motions ponent - gas particles and are the following:
~ d ll..t
JLJ dlJz +
for two-component media in which one comthe second - solid particles of admixtures
For the flow in vertically placed channel the distribution of velocity of admixtures are obtained
lL _ cf:>t + cpz { z -h2} _ <f>z !Ut- <PtJUz {.i _ dz. ki') i!. (!j}- 2(Ji~+JUz) 'I kpL f.JU1 +J-Lz) c1z. kh
where
The anlaysis shows that fields of gas and admixtures velocities strongly depend on parameter of interaction
On Figure 1, the distributions of velocities are shown for different values of kh. For !arge kh >>i difference between gas and admixtures velocities are small (curve 1). However for small values of kh this difference is significant (cutve 2 - for gas, curve 3- for admixtures, kh = 0.5). The presence of admixtures results in altering the profile of gas velocity (curve 1 and 4 correspond to presence and lack of admixtures).
u~2 / uo
Figure 1.
1229
We do think that the above described situation is likewise ideal for the development of the theoretical description of processes occuring in external electric fields. Equations of the NavierStrokes type (5,6) have been already written long ago in the field of electrogas-dynamics and such quantities as viscosity and mobility have been introduced. These equations contain a great nurober of empirical constants. It should be borne in mind, however, that the equations proper have not yet been closely studied. At present, we do not practically know any work concerned with the definition of empirical constants incorporated in the electrogas-dynamics equatlions. Such definitions will not be apparently reduced to simple experiments that were made in classical hydrodynamics. Of great importance is the relationship between such characteristics as volume charge density, field strength, viscosity of charged and neutral components, mobility of charged particles, etc. It would be most important under such conditions to use gas-kinetic methods for specifying more accurately the constants incorporated in the electrogas-dynamics equations and not only the equations proper and the fields of their application. The creation of kinetic theory of charged particles, however, presents insuperable difficulties for the time being due to a great distance of interaction forces resulting in the divergency of collision integrals in the kinetic equations. That is why, Landau, Thomas and others, had to interrupt the integration limits at the Debye radius length which cannot be well grounded, as has been proved by Bogoliubov (7). New ideas have been suggested by Vlasov (8) in the theory of motion of charged particles. He believed it possible to neglect the diverging collision integrals and to take into account the Coulomb interaction as the so-called collective interaction. Vlasiv suggested a no-correlation principle which is not sufficiently grounded. So, the above-mentioned assumptions should be considered as very loose.
As can be understood from the above discussion, to describe the behaviour of dispersed gas systems in electric fields involves great mathematical diffuclties. So, the basic method of investigation of such phenomena is still an experiment.
This paper is concerned with the experimental study of the motion of dispersed particles in corona discharge which is a typical example of a strong nonuniform electric field with a volume charge. The experiment has shown that the main factor determining the motion of dispersed media in corona discharge is the electric wall. Described below is mechanism of its origin. Negative ions of gas are set up during a negative unipolar corona discharge near the coronaforming electrode. These ions flow under the action of the electric field towards the opposite-polarity electrode and in their motion come into collision with neutral molecules of gas imparting to them a kinetic energy. The rate of collisions depends on the density of gas and velocity of ions varying with the gas temperature and the field strength. The result of this interaction is an ordered motion of the gas mass, that is, an electric wind whose velocity is rather high. The effect of the electric wind on the motion of dispersed
1230
particles through a corona discharge area can be illustrated by the destruction of an incident jet of sperical glass particles, 1 x lQ-3 to 1.25 x lo-3 cm in diameter, in a needle-plate set of electrodes (Figure 2). As can be easily understood, an area free of particles is formed at rather high voltages between the electrodes. A similar effect has been obtained with a continuous jet of dispersed gas.
U•O
J
U· 25.v 1~ --~ I 1'/J ' I ' ,
.. ') , ""~
: I b L • 2,5
I U· 40.v ~J
e ~, L • 12,5
Figure 2.
f
From the comparison of experimental and calculated paths it was found that the particles move five to seven times faster than they would have done under the action of the Coulomb force alone at a real arnount of charge of the particle. The result of comparison also shows that their motion primarily depends on the jet flow of electric wind (Figure 2b).
1231
If we add a ring electrode to the same set of electrodes and place this electrode at the samepotential as the plate (Figure 3), all the negative ions will recombine across the ring electrode; this shows that no current flows from the plate electrode. Hence, the same jet of spheres placed between a ring and a plate will be in a area free from the volume charge of ions in the absence of an electric field.
a b
Figure 3
The mean mass velocity, however, acquired by the gas in the gap between the needle and the ring electrode is sufficiently high to ensure practically the same paths for the spheres as those observed in the former case. This fact confirms the suggestion that the mechanical effect of the electrodynamic jet on the dispersed particles is much greater than the Coulomb forces.
Thus, the experiments have prooved that the behavior of dispersed media suspended in a poorly ionized gas surrounded by a strong non~uniform electric field primarily depends on the aeromechanical factor, the electric wind.
The work was carried out under the direction and with the assistance of Academician Struminsky, V.V.
1232
REFERENCES
1. V.V. Struminsky, On a one method of the Bolyzmann kinetic equa~ tion solution, Diklady AS USSR, v. 158, (1964), N2, p. 248, In Russian.
2. V.V. Struminsky, On the methods of the kinetic equation system solution, Doklady AS USSR, v. 237, (19 77), N3, p. 533, In Russian.
3. V.V. Struminsky, V.I. Kurochkin, About the kinetic theory of the dense gases, Diklady AS USSR, v. 257, (1981), N7, p. 60, In Russian.
4. V.V. Struminsky, General theory of fine dispersed media. In "Mechanics of the multicomponent media in the technological processes." "Nauka," Moscow, (1981), p. 102, In Russian.
5. I.B. Rubashov, Yu.S. Bortnikov, Electrogasdynamics, M •. Atomizdat, (1971), In Russian.
6. I.R. Meloher, Electrohydrodynamics. Magnitnaia hydrodynamica, N2, (1974), p.3, In Russian
7. N.N. Bogoliubov, Problems of the dynamics theory in the statistikal physics, M. Gostechizdat, (1946), In Russian.
8. A.A. Vlasov, Many particles theory, GITTL, M-L, (1950), In Russian.
1233
GENERATION OF HIGH-SPEED AEROSOL BEAMS
B Y LA V AL NOZZLES
ABSTRACT
* W .J. Hiller and J. Hägele
Max-Planck-Institut für Strömungsforschung, *Institut für Physikalische
Chemie der Universität, 3400 Göttingen, FRG
High-speed aerosol beams of small aperture angles may be obtained by accelerating the aerosols in a Laval-nozzle flow. The mean radius of the spherical particles was about 0.2 fJ. m; as carrier gas air and He have been used. The results are compared with those obtained for a free jet flow out of a spherical orifice.
INTRODUCTION
In many applications aerosols serve as Carriers for single ato~ or molecules (e.g. for radioactive reaction recoils produced in a cyclotron ). Here, the transportation of the charged aerosols which are suspended in a gas takes place by laminar flow through small capillaries. The main disadvantage of these systems is their comparatively large inherent delay time, so that for short-lived nuclei the transport efficiency soon becomes very small.
The efficiency could be raised considerably if one would succeed in producing a high-speed collimated beam of aerosols. To generate such beams one could think of using Laval nozzles. Compared with a simple orifice -which has been employed in some recent experiments by other authors - one should expect smaller beam aperture angles, as the acceleration perpendicular to the laval-nozzle axis is much smaller, and higher particle velocities as the velocity gradient along the axis is not so steep. Forthis purpose, a set of Laval nozzles with different aperture angles downstream the nozzle throat has been investigated experimentally. For the sake of comparison a simple circular orifice has been studied too. Both, the throat as the orifice diameter amount to 1 mm.
1235
01r, He
aerosal _ generoter
flawmeter /
rest stete pressure
I
vacuum ~ha reference beom 1
nozzle
/ pVK pressure 1ns1de chomber
--:-~___.--
scattered light
probe - volume = (005mm)3
------>...,. to vocuum ----?" pump
FabrY: Perot - Interferometer 1- --~
I L ___ ~
photo muttiplier
Fig. 1 Scheme of Experimental Setup
EXPERIMENT AL ARRANGEMENT
A schematic view of the experimental setup is shown in figure 1. The aerosols are generated by spontaneaus condensation of NaCl-vapour (Fig. 2).
Then they Jre mixed with additional carrier gas and diluted to """107 particles per cm • The shape of the aerosols is nearly spherical, their mean radius amounts to .-- 0.2 fJ. m. This suspension then enters the Laval nozzle (or spherical orifice) from where it discharges into a vacuum chamber. The rest state pressure p0 of the carrier gas is p0 = 1b at 20°C, the lowest pressure in the vacuum chamber amounts to 0.44 mb for air and to 1.4 mb for He as carrier gas. All Laval nozzles have for their subsonic region the same shape, their supersonic part is formed by a straight cone with aperture angles changing in steps of 2.5° from 5° to 27.5°. Measi.Jrement of particle veloci ty outside the nozzle along the beam axis is performed by a Laser-Doppler-Velocimeter using a Fabry-Perot-Interofermeter for determining the frequency of the. light scattered by the aerosols. For this measurements a slightly focussed Argon+ laser beam enters the chamber through the nozzle coaxially to its axis. As observation of the scattered light takes place orthogonally to the illuminating beam the particle velocity is easily calculated (Fig. 3). The broadening of the scattered light is mainly due to the particle size distribution. For the calculation of the particle velocity it was assumed that thepeak of the scattered light intensity corresponds to the peak of the particle size distribution. The width of the aerosol beam becomes visible by the sc:;atterred light of a laser beam crossing the jet perpendicular to its axis, whe-
1236
10
5
H[%]
T. 0 c1=835°C
Qoren=20 cm3/s. N2
n =5 1Cl1cml
Crucible
L':lli::::::iit::i--i-- NaCl
Fig. 2 Partide Distribution and Scheme of Aerosol Generator
reas th2 joundary and part of the structure of the carrier gas jet is made visible ' by adding either to the gas or to the vacuum chamber J2-molecules which are excited to fluorescence by the 514.5 nm line of an Argon+ laser. Then, observation takes place by a microscope through a top window of the vaccum chamber.
RESUL TS AND DISCUSSION
As the velocity distribution along the axis of a free jeternerging from a circular orifice is quite well known (up to the Mach disk) we first will compare the measured particle velocities with the calculated ones. The main assumption for our calculations which are analogaus to those of Dahneke and Cheng 4 and which are discussed in detail in 5 are:
1.
2.
mp/mG« 1 Where mp and mG is the mass per unit volume of the particles and of the carrier gas resp. At rest state this ratio is smaller than 0.1 % so reaction from the particles on the flow are neglectabie.
~GI ~P « 1 Where 9G and <>p is the density of the carrier gas and of the particle medium resp. As our particles consist of solid NaCl this condition is fulfilled and the relevant force acting on the particle is the drag described by Stokes law as the particles are of spherical shape. The dragformula given by Stokes law has tobe corrected due to the non neglectable
1237
u = A. · !::" 11 = (285 m/s)
f ree spectrol ronge
{).v------t
Fig. 3 Typical Plot of Intensity I of Scattered Light versus Frequency v l:l v Doppler frequency shift u Partide velodty .A W ave length of Laser light
value of the particle Reynolds number Rep and to the Knudsen number K0 (the latter being in the range from 0.1 to 20). So, the equation ot particle motion along the axis becomes:
~T = k (v - u)
1~ 3 where k = ~L • c ( 1 + TI Rep)
and v u
d c
C = 1 + 2 Kn (1.25 + 0.44 exp (-0.54/Kn))
gas velocity along jet axis particle ve1oci ty along jet axis dynainic viscosity particle diameter slip correction factor
Fig. 4 shows a comparison between the measured and caJculated particle velocity for a He and an air free jet at a pressure ratio p0 /pVK = 100. (The "He" jet contains 10 96 N2 by volume coming from the aerosol generator). In the experiment the position of the Mach disk relative to the caJculated value is a little bit shifted downstream, but the slope of the velocity decay is in good agreement. Off course, the simple assumption that the velocity behind the Mach disk remains constant does not hold in practice.
For the case of the Laval nozzles the velocity along the axis has been calculated by a one-dimensional method neglecting boundary layer effects.
1238
VCM/S)
~ / ' Vth
7 I
0 I I
1/ VK
; Uth
/x" I'( . Ir! ~· . 500
I II He- N2 - NoCt, p 01p vK: 100
ß --- XCMI'1) 5
VCM/S) --/'Vth
I I / Uth
ö -o u 0 .. 500
VK
1/70- X
;\ 0 ~
V \ ~ }J \ /; ö
.).or- No Cl, p0 lp K: 100
____-/ XCMM)
s
Fig. 4 Partide Velocity along the Axis in a Free Jet Expanding from an Orifice. Nozzle exit at x = 5 mm
vth calculated carrier gas velocity uth calculated particle velocity XX, 0 measured particle veloci ty XX (orifice), 0 (converging nozzle)
1239
1240
750 throat nozzle exit
(/)
' E
250
1500
(/)
' E :::>
1000
throat
I I I I I
-----v,h
carrier-gas: air P. I P. = 50 0 0 VK
• •
10 20
nozzle exit
" "
normal shock
30
carrier-gas: He-N2
Po /pvK = 200
. . " X X X
LO X [mm)
v,h
Uth
50'~~~--~--~----+---~--------~--~--~~ 10 20 30 1.0 x [ m m]
Fig 5 Partide Velocity along the Axis of a Laval Nozzle vth calculated carrier gas velocity uth calcula ted particle veloci ty XX measured particle velocity
Fig. 6 Visualization of Shock Waves inside a Laval Nozzle. An Argon+ laser beam crosses a glass nozzle perpendicular to the axis in the supersonic region and exdtes the J 2 tracer molecules to fluorescence. Behind the oblique shock wave wruch protrudes out of the nozzle orifice fluorescence is quenched off. Flow medium He+ J 2, flow direction from left to right, pressure ratio ~ 200.
Fig. 5 displays the results for a Laval nozzle with an aperture angle et. = 5° and a nozzle exit diameter d = 4 mm. For the case of air as carrier gas the measured particle velodty at the nozzle end is only 3 % lower than the calculated one whlch we thlnk is mainly due to neglecting in our calculations
viscosityeffects. The jet leaves the nozzle underexpanded and at distance of 22 mm downstream the exit a normal shock closes the supersonic region on the jet axis. Compared with the results in Fig. 4 the maximum velocity is about 50 % higher due to the smoother velodty gradients in the transonic region. For the He-N2 mixture the difference between the calculated and measured particles is <lue to the fact that the pressure ratio amounts to only 200, a value, too small, that the flow medium could follow isentropically the expansion prescribed by the nozzle contour. Here one expects flow separation induced by shock-waves inside the nozzle. By using the aforementioned J2-method they are easily visualized (Fig. 6). But also in this case despite tne shock waves the particle velodty at the nozzle ex i t is considerably hlgher than the maximum velodty for the simple orifice. The rnost important property of these shocks, however, is tha t they are addi tionally focussing the aerosol beam. Even though the aerosol beam apertureangle by using a Laval nozzle with cone shaped walls in the supersonic region is smaller than the one expected for a beam out of a simple orifice, it will have a finite value.
1241
Fig. 7a, b Aerosol Beams generated by an Orifice (a) and a Laval Nozzle (b). Aerosols made visible by scattered light of a laser beam crossing the jet perpendicular to its axis. The distance between adjacent traces of the multiply exposed photo is 1.25 mm. Flow medium (a) air (b) He-N2, pressure ratio (a) ,..., 500 (b) ~ 200.
If one tries to reduce this angle by using Laval nozzles with smaller aperture angles ae. g. a = 0, theri, as i t is easily seen the particle veloci ty at the "nozzle exit" will be reduced to critical velocity, and, if the jet leaves the exit underexpanded it will behave like a jet coming out of an orifice. So, by properly adjusting the oblique shock waves one not only gets better focused
1242
beams but also the gas jet does not collaps at such an early stage by a strong shock. In Fig. 7a,b the progress achieved by this technique is displayed.
Our present work is concerned with the coupling of aersol beams to a high vacuum system.
REFERENCE
1 Macfarlane, R.D., Griffioen, R.D., Nucl. lnst. Meth. 24, 461 (1963)
2 W.J. Hiller, W.-D. Schmidt-Ott, Nucl. Inst. Meth. 139, 331-333 (1976)
3 Hägele, J., Untersuchungen an rotationssymmetrischen Überschallfreistrahlen mittels laserinduzierter J2-Fluorescenz, ?_, 1979 MPI für Strömungsforschung Göttmgen
4 Dahneke, B.E., Cheng, Y .S., J. Aerosol Sc • .!Q, 257 (1979)
5 Hägele, J., Experimentelle Untersuchungen an Aerosolstrahlen aus Mikrodüsen, Göttingen 1981.
1243
IIXETIC MODBL or A GAS SUSPENSION
Yu.Lunkin and V .Mymrin
A.r.Ioffe PhJsical-Technical Institute
Leninsrad, U S S R
ror obtainins a closed srstea of equations in which the
interaction ot phases in their siaultaneous aoTeaent is ac•
counted for froa a certain unitrins point of Tiew it ls aecess&rJ to eaplor the theorr describins-the aoTeaent of a sas and partielas contained in lt on a aicroscopic leTal /1/.
ror the sake of simpliticatlon a sas suspenslon will be
considered as a alxture ot one species ot sas aolecules and aonodisparse solid particles. Let us introduce in a conTenti
onal aanner the velocitr distribution of sas aolecules \(f.~l)
!be equatlon describins the chanse ot this tunctlon is
-r>:f .-(~ +V. {r-t f. ;v )~ = ~l + -'lr (1.1)
~~ =-//(!~;- f4f)tf/(c/y·"!{ (1.2)
The coll1s1on lntesr.al -~ , deflnlns tbe chanre ot ~ due to collisioaa of aoleculeswitb tbe solid partlcle sur
tace will be represented in-the expliclt fora on tbe basis
ot the tollowin~ assuaptions: 1) tbe flow around suspension
partielas ls consldered to be a tree.aolecular one, 2} tbe
process ot tbe lateractlon betweea aolecules and the particle
1245
is deseribed bJ a dittusion retleetion aodel, ') tha randoa aotion ot solid partielas is neslectad. Tban
~~ ~~tr{!-rt~/)f t!t'~- -.f.·l<o -~ "'3,-r.. (1.,)
-_f (,. _(j~ I} -J'.(fzJ' ( :;..; f e -~~-;;.,er{~"//] f·h" f·l<v · ,j l _ -~
h~~· I~ - nuber densi t:r ot suspension partielas 1 / .". V --'? ~~- aean velocitJ ot partielas at a point beins considered1
~ --loeal vector to the surtace ot the partiale, 1.;- teaperature ot the surtace ot the partlcle.
Tbe third assuaption peraits to usa ln considerins the solid partlcle pbase onlr sasodfnaaic equations tor lower order .oaents ot the distribution·tuqction whieh are obtained bJ the velocitr a~rasins of the eauation tor the particle distribution functlon
- ·I' - el .--_d_) ~ ~ :: ( ~ + ~ · ifP + ;; a;p 5;, ;: ~~ (1.4) tak1ns into account the relationships
J~Y~ dV~ ~-./ Mpi{ ~,~~/Tw I' Zlz# .- J ~ . .. (1.5)
J ~ ;r 1t "'-V = - n,r(jfo + ~ t;);,,~ "ullw Bere ~ ~".- ••••• ot 110lecules aad partielas respectlYelJ't C,- heat capacitJ per unit voluae ·of the particle substanca.
Still, for obtain1ne clostns·ralations in a carrrins sas one sbould solve the kinetic equation (1.1) whose diaensional
fora is ~/ E ~{ = f-l -f IX'~ c1.6)
Bere ~=- /1. - hudsen nuaber, o/: (/%; • parueter definins the 1ntluence ot a sas suspension, Y' • voluae percentase of particles, (/ - aean relative velocitr.
In the case beins considered % > i, tf ~ 1., % :;t 0 ,
!he last relation aar var:r ln a rather wide ransa. !heretor, the equation (1.6) contain tvo independent paraaeters and, senerallJ speakins, one cannot assert that the etfect ot particle intluence upon sas aolecules is a kind of disturbanc.:e.
1246
The solution of the equation (1.6) is sousht as a series
in terms of powers of the small pa.raaeter €
~ c to + t !.., + .. . (1.7) As it was shown in (1)~
~~ ( /.} "'- 0 ( e} for anr ol" :l, if tbe zero order term of the expansion (1.7) is a Maxwellian
tunction ~
tsh{.e.v':r) exP/-.:;rfV:.0j C1.8l
low on the basis of the estiaations .. de it is not di
ficult to obtain an equation for the distribution tunction in
a first approxiaation
:Jf Jo-~ (f.) ~ ft (i,~)+ 1/1 ai,J +.;, (ft) (1.Sf> Solvins the equation (1.13) by the aodified Chapaan
lnskos aethod /2/ we find the expression for ~ and knowlns the explicit form of f, we can set the expressions for tJae ..... _,. stress tensor p aud the heat flow vector f enterins into the Navier-stokes equation for the carrrlns sas
....... I ~ ... - . ~J P = ;., J:, ee dV = -~"* (vi:i- /, 1
1 = f~ufj, a2~~V =- ;{* (P7- i;) .( 't -:i
~ -z:[ -s 1. -( Vr -~aj ~c)l I+ &r:.-zt_ l+t _) Ke +(I~ -rk-.lj) r .eJlfS] (1.10) // h ,~,~;/ ,
f'.t .n6.)(z) L f -1
;!:A {l+ !!l!r: ... (. ~ ~)'"[ c~(,;z._3+~iJ+ (ri'.,s-.,.i/"iaf~]7 11 P lrit.f) _h _rt,z;(,, =:J
./l. ""' ( - --+)'l T 7 ., z-' ~ .- _" v ::- .z;;;- W- w, , O:::: '/ 'fW" 1 C ::- V- tl
Bere~~ - coefficients ot heat conduction and viscositr of
a pure ps, Z: and { - coefficients of expandion ~ { fo} in unreducible tensors /1/.
The results presented in this section are obtained
1247
usins the assupt1on ot aonodispersitJ ot.the sas .uspension but this.assuaption is not a priacipal oDe. In a slaitar -.nner oae .., obtaia the traasport coefficlents of the ,as suspeaslon in vhlch the disperse pbax consists of seTeral fractions of different parttele size.
~he approach deaoastrated aboTe aar be seneralized to include the case of tbe carrrins phase belns an lonized sas /3/. Under thesa condltions the aerosol partielas acqulra 1
as is shovn br experiaents /4/, a considerable nesatiTa charse. Bance, one has reasons to sussest that eTen the presence ot a relatiTelr small quantltJ of partielas can maricedlJ influence the tränsport processes.
fhe explicit ton ot the collision inte,gral. tor cbarpd coaponeats ot the sas phasa vith charse-carrJins particles;r& (~ .c o/is found b7 introducins asswaptlons analosous to those aentloned aboTe. In addition lt is assuaed that a) elastic reflection of electrons and ions froa tha particle surface is absent b) dlstance between aerosol partielas markedlJ exceeds the DebJe screenins radius.
And in thls case, the intesral is solTad in a conTantional aannar - bJ expandins tbe solutlon in a power series ot Sonin polinomials.
It one retains in each axpaasion aentioned onl1 one tera1 one obtains the tollowins expressions for the coetticients of electronic heat conduction le* and electrical con-
ducti Ti t1 ~ * f t -
J; .. J ff-!2r. 1/_ m, )-." [ P.\ Y:)t~{:;c)t,.J,_,-3.2S'"~ 2!,A,.,y ~z~ /Iee' t-tr•tf.
[ l.sf:fiJ + 2: ~Al' ( .Q~ )] /Et,tt 11e f
P. ( :;o/ = e :Jt!(.z~ 3- .2• 1 + 1. s-:;c - 13)
F; (;;c) =- e:;c( ~"+ 3~3-f .3.-tS"~zJ
A" ( ~) = "" .szr.r 1-ltJ.!l;tz; -fl szf! (3) 1248
(1.11)
wbere Äe • correspondins coefficients coaputed for
ionized sas wi tbout particles added, 2 .-2e2 ·- ~le :x- = ;:: A = ./ f- (t/-:~ele) ___ _ ~tp te · 1 2e'L --f?f"/1e ez ( ) 1.12
. ./L ~ :/ + (-2 o:l; }·z.r;' /i-1 rzze 2,_) 2. i! e 2 r ( 7 ~ ~ le
Let us illustrate tbe tbeory developed by considerin« the
disperse aixture tlov in a shoca layer formins at tbe surface of blunded bodies durins supersonic flisbt /6/.
For a carrying gas on shock wave we'll use the RankineBugoniot condition. Tbe gradient ot a velocity eo~onent tan
sert to tbe body surtaee we set trom tbe ealeulation ot pure sas flow around blunt bodies. ror the body surfaee tbe attaeb
aent and impermeab111ty eonditions are tisual, surfaee temperature is taken to be constant. Parttele parameters behind tbe
departed sboek wave are taken to be equal to those in the on
coains flow. The typical calculation results are.presented in tbe figure. Partteleparameters are denoted byp, ealeulated
values for pure gas are presented by dot-and-dasb lines. The increase of gas temperature are pressure in the pre
senee of partielas is due to tbe proeesses of aoaentua and
energJ exebange between phases. Wote tbe marked deerease ot tbe shock layer thiekness in a dusted gas. In tbe exaaple
treated above values ot heat tlow towards a body surfaee dependin!.. on sas heat eondaction ~ ;J." v;: , . triction coetti
eient ·~{Tl vt and shock wave departure t'1R-- C R ... blunt ... ness radiues) are as tollows: 0.0294, 0.0268,_0.251. ror pure sas tbe eorrespondins values are 0.0131 1 0.0180 and 0.301,
respactively (#~: 2, i~"'()= 11830, lf = 10-6, rr = 10~). Tbus the results presented testitynto the marked intlu
ence of solid partielas eontained in the flow upon the dis
tribution of sasdynaaic tunetions and the heat exehange.
1249
T .-------~---~. -V
1.0
0.5
0.25 0
o.o 0.15 n
Fi g. 1.
1250
BEl' ERDCES 1. Yu.Lun'kin, V.Mrmriil. Application of the kinetic theorr
for obtaiaiag the closed srstea of eqQatioas of g.aa,sus~ pension dy.amics. Pis'ma v Zhur. Tekh. Fiz. 19791 5, K 3-lD Russian.
2. V.Matsuk, V.Brkov. Extension of the Chapaaa - Enskos aethod to reactins gas ~xture calculation.,Zhur.vrch. mat.i mat. fiz. 1978, jä, I 1. - In Russian.
3. Yu.Lun'kin, V.Mrmri~ Influence of aerosol particles apon,transport processes in an 1on1zed gas. Zhur. tekh.
fiz., 1979. 49, I 4. - In Russiaa. 4. "Reolosia saspezij•, Mir, Moscow, 1975. 5. Chieleski R.M., rerziger J.R. Transport propertiea of a
nonequili'briua partlallr 1onized gas. Phys. rlulds, 19671
v.1o. • 2. 6. Golovachov Yu. P., Lun1kin Yu.P., MJmrin.v.r., Schäldt A.A.
Supersonic mot1on of 'bodies in dustr gas. Acta Astronautlca, 1980, v.?, K 3·
1251
XIX. GAS MIXTURES
KINETIC PHENOMENA IN THE RAREFIED GAS MIXTURES
FLOWING THROUGH CHANNE!ß
V .M. Zhdanov
Moscow Engineering Physics Institute Moscow, 115409, USSR
1. Introduction
Some pbysical effects (pressure effect in diffusion, separation of a gas mixture etc) arise in rarefied gas mixture flowing through a capillar,y or porous media under the partial pressure and temperature gradients. To analyse these effects th~ expressions for the cross-section averaged velocities and heat fluxes of the mixture components in the capillar,y must be derived. These expressions can be easily obtained in the free-molecular or Knudsen regime when Kn = }../R >> f ( }.. is the mean free path, R - characterictic radius of a channel). In the transi tion region ( Kn. #V ~ ) and even in the continuum limi t ( l<n.. - 0 ) accurate analysis of these phenomena is usually based on the application of the linearized Boltzmann equation.
According to the principles of nonequilibrium thermodynamics in the discontinuous systems /1/ we can write ~he relations between the cross-section averaged fluxes and the gradients of thermodynamical quantities for the binary gas mixture as
(1.1) (h)
(um)
== - f r-t dy~.ldz
r-f. dp ldl
1255
where ( h ) is the reduced heat flux, < u«) is o<. spe-cies drift veloci ty, (u,.,) = ; ';j ... (u..,.'j is the mean molar drift ve loci ty of the mixture , IJ c~. = n.. ... In. i s o~.. species molar concentration, p is pfessure and T is temperature.
The main purpose of the kinetic analysis is to obtain the expressions for the coefficients 1\ iK •
The number of the coefficients being calculated is sufficiently reduced because the Onsager reciprocal relations for the cross phenomena ( A;./( :::: A/(;. ) are verified. In a general case the coefficients A,K are the complicated functions of an effecti ve Knudsen number, the species molar concentrations; molecular masses and effective collision diameters.
In this paper we discuss the methods and the resul ts of the kinetic coefficients All( calculations for binary gas mixture flowing through the capillary. The results obtained are used for the interpretation of some physical effects arising in a gas mixture contained in two vessels connected with the capillaries or a porous slab when the partial pressure and the temperature gradients are present.
2. Knudsen regime ( l<n ~> 1.. )
Let us consider the technique of the calculation of A i~e in the free-molecular limi t for a gas mixture flow i~ a long cylindrical capillary. It is convenient to use a well-known expression for the number of gas molecules dn.A crossing a gas surface dA in unit time along the lines making the angle ~ with the normal to the surface and having velocities between tr and 1.r + d'lT wi thin the solid angle d w /2/
(2e 1) d nA = rt (J317i )311 e.xp (- J1F 1 ) 1T 3 CJJS 'f dc.o dv dA
Here n.. is the number gas densi ty, ß = rn. 12.1 T , rn is the mass of a molecule, 8 is the Boltzmann constant.
For the energy flux transfered by the same molecules we have
1256
The integration (2.1) and (2.2) over the velocities gives the total particle and energy fluxes through surface dA wi thin the solid angle dw
(2.~) dNA = (n.I;;H&TI27im) 111 CtJs'fdwdA
d fA :::: 21T rlNA
Let dA be the surface element of the long cylindrical tube cross section.For the small values of densi ty and temperature gradients we can take two sections at distance z and have approximately
n./Tt111 ~ rt T'11 + zd(n Tt11 )1Jz
n. I T I 311 ~ J1, T lf1. + z d ( /1, T 31:t) I d z
Following a well-known procedure for calculating the number of molecules crossing dA after their last impact wi th the wall surface element that is at distance z from given section /2/ we obtain the expressions for the total ~ species particle and energy fluxes through the capillary of radius R •
where .;~e .. is the fraction of molecules diffusely reflected by the wall (accommodation coefficient).
Using the defini tion of ~- and the relation p .. = n ... & T we can wri te the expressions of the fluxes per a surface unit of the capillary cross section in the form
1257
where I -l.ta a. = ( 8 R 13) ( 2 r T) j (-" = ( 2 - df!..c ) I ;;e..c
Using (2.5) for the expressions of the total reduce heat flux (h.) = I:((9E.c>- (SIJ.)p. (u..,)), of (u.,.,) and (u~,)- (u,) in the case of a binary gas mixture ( o1. = 1 ,2) we obtain
(2.6)
A "'"' = o. T( ~ + ~) n m.~ rn f./z
1 .t
AH = c. T ( (~, ~ n: m.i'-,1 + m~~~l
A1PI :: I\,..~, = o. T ( ~ -n m."z
i
1\ 'rn = 1\ ,..~ = - 1 f 1\ "'"'
1\t,=A,t =- fpl\tr."
)
(1. ) lfa mJ.
Note that in the analysis of the diffusion transport of a gas mixture in the capillar,y it is useful to define ~ species flux in the system of reference, where < u,.,) = 0 • In this case J1 ::: G't - r; ~t = rtl(,!la ( (u,) - ( U.r)) • Introduca the formal defini tions of' the diffusion coefficient 11:2 , the pressure-diffusion factor ~; and the termal-diffusion factor ~; in the free-molecular (Knudsen) flow of the bina
ry gas mixture. Then we have
Here
(2.8) I(
2)u =
1258
where
K o{T =-
( 1ft 1 ) t/t 1 1/1 , m b , = m t 111 ~ 1 + n1..1 b1 ~ ,_
Unlike the diffusion transport in the opposite limi t region ( l<n. << 1. ) the Knudsen diffusion coef-ficient does not depend on pressure,is proportional to the capillar,y radius and depends on accommodation coefficients de~ e It is essential that the pressurediffusion and tbe~al-diffusion factors in the Knudsen region are connected by the fo~ a simple relation
tc i K o(T :: - i O...f •
3, Intermediate Knudsen numbers
The calculation of the coefficients A,K for all Knudsen numbers for the channels of arbitrary sbape (parallel plates, cylindrical tube · etc) is usually based on the use of tbe linearized Boltzmann equation, For the isothermal flow of a gas mixture this problem was solved in /3, 4/ on the basis of the kinetic equation with BGK~odel collision ter.m in the Hamel-Oguchi form, A variational method /5/ (or the BubnovGalerkin metbod /3, 4/) was applied to solve the appropriate system of tbe integral equations for tbe species drift velocities. Wben tbe constant trial function for the velocities was taken tbe accurate values of the coefficients AiK in tbe free-molecular limi t ( l<rt ::>> 1 ) were obtained, In order to obtain tbe suitable results in an another limit ( ~n. - 0 ) tbe quadratic trial function must be used,
Unfortunately tbe BGK-model for tbe gas mixture does not allow to take into consideration the tbermal-diffusion effects. Nor does it guarantee an adequate accuracy in tbe calculation of tbe diffusion slip velocity (or tbe pressure-diffusion factor) even at the small Knudsen numbers, Using the models based on the equivalence of moments of the N-th order modeled collision operator and the moments of the full collision operator calculated with the N-th order approximation to the distribution function is more efficient. Suchmodels (for example, the Mc Cormack
1259
model /6/) represent proper~ the behaviour of a gas mixture in the continuum or hydrodynamical region. The kinetic coefficients for the isothermal flow of the gas mixture at small Knudsen numbers were calculated in ter.ms of model /6/ in/?/.
Consider some results fo» the isothermal flow of sas mixture at small Knudsen numbers. In this limit { Kn. << t ) the expressions for the average molar veloci ty of gas mixture (u.m) and for the difference of the average species velocities <.u.,)- <.c..c.",) in the cylindrical capillary may be written as /?/
Here u~s - u~s corresponds to the difference of the asymptotic (defined far from the wall) values of the species velocities. For this difference we can write the well-known expression /8/
In (3.1) - (3.2) 2 is the viscosity of the gas mixture 1 :Du is the binar,y diffusion coefficient, 0(8
is the pressure-diffUsion factor in the viscous fldw of the gas mixture /8/, 3 is the viscous s.lip coefficient and 5 11 is diffusion slip factor. It is noted that pressure-diffusion factor ~~ is different from ~; • This difference was expla1ned in /9~ ?/ by taking into account the contribution of the Knudsen l~ers when flux averaging upon the cross-section of the channels is made. Comparing (3.1) with (1.1) we find
1260
The condition /\1m= Amt results in ;;.,. ::: - Öu i.e. pressure - diffusion factor is equal to the diffusion slip factor (with the opposite sign).
The viscous slip and diffUsion slip coefficients may be calculated for a simple geometr.y problem when the mixture occupies a semi-infinite expance bounded by a flat plate. In this case the flow of gas along the plate is caused by the presence of concentration gradient and far from the plate the gas is maintained at constant velocity gradient normal to the plate. In these condi tions coefficients J and f5 1 were calculated earlier using the linearized kine~ic equation both with modeled and full Boltzmann collision operator. The results of the diffusion slip coefficient calculation by different methods has been summarized in references /7, 10/. The most suitable expressions for ~ and 6'.u are likely to be the results obtained by applying a variational technique to the linearized kinetic equation with the full Boltzmann collision Operator /11/. In /10/ the same result for ot1 was obtained by using the approximate method in
which the distribution function of the molecules incident upon the wall is supposed to have the same form as asymptotic function far from the wall but including unkDown constant defined from the condition of the tangential component momentum flux conservation. In /13/ cc f was calculated instead of 6'u. for the gas mixture flow between two parallel plates,with the moment equation system coressponding to the wellknown Grad1 s 13 moments approximation in passing to the continuum limit (the flow region far the wall) being used. Averaging the equations upon the cross section of the channel we obtain expression for ;;;.,.. .. - Su. similar to the data in /10, 11/ Neglecting small thermal-diffusion correction terms we can represent the expression for Q(f' (in the case dt1 ...- ;;;e.~,-= 1 ) as /13/
where ~; is the Knudsen pressure - diffusion factor (2.7) and ~~ is the first approximation to the pressure-diffusion factor in a viscous flow /8/
1261
2 = f 1 + Pa
Here ?a and f& -partial viscous coefficients /8/. Taking into account the ther.mal-diffUsion additive correction we must replace coe.;J1 by C-<ft J,~ 'look Ref /8/) and c"'; ]1 by f•'; ] 1 - J' , where ö is the correction to the diffusion slip factor calculated in /14~. General expression for a,. = - 6'11 for the case o., ':f: t is given in /10, 15/.
To illustrate the dependence of ;(P. on the molecular properties and the gas-surface interaction law it is convenient to consider the gas mixture with slight differences between species molecular masses effective collision diameters and accommodation coef-ficients (.Am I J..m. <.< t , ~r; I~ f> << I. , A ~ 12. ;Je << 1 ) • Using also the hard spheres inter.molecular force law we obtain /13/
d..f = (t.i4 1d +0,!303tle) ~~ + (0.0618-
-0.6653 i3e) ~ + (!.9322- 0. 033.9~) i~
In /15/ the method for calculation of 1\ '" for a nonisothermal flow of a gas mixture through the channel based on the averaging of the moment's equation system corresponding to the 20~oment approximation was developed. Such approximation takes into account the Burnett terms and enables to calculate all coefficients of the matrix (1.1) 1 which include the Knudsen corrections for the kinetic coefficients to the second order in the Knudsen number. For the gas mixture flow between two parallel plates the coeffici- · ents Aik were calculated in /15/ and for the flow in the cylindrical tube were given in the report /16/. The expressions · for A iK in their general form are ver,y cumbersome. For the particular case given above of the mixture with t:.m. 12.m. .t...<. 1 and Ao /2,s << 1 coef-ficients Amm. 1 /\m1 = A"m and /\ 19 are the same as the values for a one-camponent gas. For the gas flow in the cylindrical capillar,y we have (~= i) /16/
1262
.t-
1\ mrn = ~ 2 I ( i + 4 c", K n + 2. K n .t )
(3.7) Am1 = 1\~m =-~AT (1.- f.S04Kn)
/\~ 1 = ?- T 1 ( i - t. 0 t 6 Kn )
where K rt = 2 lpß 111 R -the Knudsen number, I.= ( 1S/lt) I~ Im. is the heat conducti vi ty of the gas, Cm = 1 ,0073 and Ar = 1,125 are viscous and thermal slip factors, which coincide with the results calculated by the variational technique in /17/. The calculation of the coefficients 1\. u , 1\. im and 1\ ~.., for the same mixture and for the hard sphere intermolecular force law gives /16/
Au= (Tip~1~1 )[:bu]z. ( i- 0. 652-Kn..)
Ä 1m = (T/p)[1)u]1 [(1.27'-t -i.Z35Kn) ~::;
- ( o. 5 !3 8 - t . i 7 t K n. ) -pf ]
Ai, = - T [.l)H ].l [ ( 0. 890 - i. tt1 l<n.) ~: +
+ ( 0. 33 9 - 0. 309 K n ) 2_[ ]
4. Some kinetic effects
Equations (1.1) together with the above expressions for /\il( can be basic for the analysis of some kinetic effects taking place in the gas mixture contained in two vessels connected with the capillaries or a porous slab. If the initial conditions are chosen so as the values of pressure, concentration and temperature in the vessels at the instant t :0 are different the mass and the heat fluxes through the capillaries will result and the thermodynamical parameters in the vessels will start to change. The equilibrium state will set in when the mass fluxes
1263
under pressure, concentration or temperature gradients will balance each other.
First consider the isothermal flow of the gas mixture. If the values of pressures in the vessels are equal but the concentrations are different at t = 0 the distinction of the species diffusion velocities gives rise to the difference of pressure between two volumes ("diffusion pressure effect")o The value of Af.. (t.) depends on the diffusion and hydrodynamical fluxes ratio and increases to a maximum when the total particle flow vanishes and thus the · diffusion flux is: compensated by the hydrodinamical flux. In the Knudsen region ( ~n. ::::>> i ) the rela.Xation times for pressures and concentrations in the vessels are of the same order of magnitude therefore the pressure difference con2es to maximum and then decreases rapidly to zero. The expression for Ap(t) is derived in this limit from the molecular balance of each component in the volumes /18/ takin~ into account the e~ressions for the fluxes (2.5). In an another limi t ( K n. <.<. 1. ) the values of the concentration at the ends of the capillary change slowly compared with pressure /18/ and the quasi-stationar,y maximum pressure difference sets in when the total particle flow vanishes. The condi tion (u..,..,) = 0 leads in this case to the relation
~ 1\ d (4.1) d z :: - 1\,..,t p cl~ ~i "'"'
An integration (4.1) over the capillar.y length L re sul ts in A P. = p ( L ) - p ( o ). In case the dependence of 1\ ""ltJ and /\.,.. 1 on concentration can be neglected for tlf I p <.< 1.. we find
For the gas mixture with Am. 12rn ~.<. i and Aoi2.S<< 1 the application of (3.?) and (3.8) leads to the result ( Kt?. << i )
1264
It follows from (4.3) that diffusion-pressure effect can also take place in the case of the gas mixture with equal species masses of the molecules but with different effective collision diameters of the molecules 5 1 and öJ. (for example in the case of N1 - C1 H" gas mixture) observed in the experiments.
The effect in a sense opposite to the previous one is the concentration difference arising at the ends of capillar,y if constant pressure difference between the vessels is maintained (gas mixture separation effect). Given t:.p we can find A~t if the diffusion flow on the capillary exit GJ..[f- ~~.(L)]- &1 y~,. (L) is supposed to be equal to zero anu the expressions for Go~. = n."~. ( u"') are used. In the Knudsen region we obtain
(4.4)
where o<.; (2.8) is defined by the value 'J1 ( L) • In an anotlier limi t ( K n << f ) for the case of the mixture wi th A rn I 2. rn <.< 1 and A s I Z ö << i we have
Finally consider some effects when the constant temperature difference between the volumes is maintained. Ä well-known effect in this case is the thermo~olecular pressure difference /1/. For the binary gas mixture the stationary state is reached when the mean molar velocity of the gas mixture and the diffusion velocities approach zero. Then using (1.1) we find
In the Knudsen region the substitution of the values A,~ (2.6) in (4.6) leads to the relation
(4.7)
1265
similar to the case of one-component gas. In an other limi t ( l<n. << 1 ) and for the gas mixture wi th Am 12m. << 1 and A ö /2 6 << 1 we have
(4.8) 1- 1. SOKn 1. + '-t. 03t<n
The presence of the stationar.y temperature gradient causes also the gas mixture separation effect. We can find ll_fJ1 in this case from the condition (ut)- <u.tj = 0 taking simultaneously into account the connection between Af and A T (4.6). Then
It is of particular interest that t:. ~1 = 0 in the Knudsen region but ll '/1 * 0 in transient region. In particularly for K.n « t
( t-) .t:.T ll~t :::. - o(T + z o(f' ~i YJ. T
In the case of the gas mixture with A.o /2.ö << 1
A ~1 ~ ( 0. f 53 z~ -0. f 76 ;: ) ~I. ~1
Am 12m<< 1 and
.t:.T T
Note that in the stationar,y state the usuall separation effect through thermal-diffusion is compensated by the opposite effect through pressure-diffusion in the capillar,y. As a result the total separation effect reduced and in the Knudsen limit goes to zero.
References
1. De Groot s. Mazur P. Non-equilibrium thermodynamics, M, Mir, 1964, 456 P•
2. Kennard E.H. Kinetic-theory of gases,Mc Graw-Hill, 1938.
3· Shendalman L.H. Low-speed transport of gas mixtures in long cylindrical tubes according to the BGK modal, J. Chem. PbJs., 51:2483 (1969).
4. Pochuev N.D., Seleznev V.D. 1 Suetin P.E. Flow of binary gas mixture at tbe arb1trary tangentional momentum accomodation, Zh. Prikl. Mekh. i. Tekhn. Fiz., 5:37 (1974). In Russian
1266
5. Cercignani C., Pagani C.D. Variational approach to bQundary-value problems in kinetic theor.y, Phys. Fluids, 9:116? (1966).
6. McCor.mack F.J. Construction of linearized kinetic models for gaseous mixtures and molecular gases, Phys. Fluids, 16:2095 (19?3).
?. Zhdanov v.M., Smirnova R.v. Diffusion slip and barodiffusion of gas mixture in plane and cylindrical channels, Zh. Prikl. Mekh. i Tekhn. Fiz., 5:103 (19?8) - In Russian.
8. Zhdanov V.M., Kagan Yu. M., Sazykin A.A. Effect of momentum viscous transfer on the diffusion in a gas mixture, Zh. Eksp. Teor. Fiz., 42:856 (1962) -In Russian.
9. Breton J.P. The diffusion equation in discontinuous systems. Physica, 50:365 (19?0).
10. Lang H.,Loyalka S.K.Diffusion slip velocity: theory and experiment, z. Naturforsche, 2?-a: 130? (19?2).
11. Loyalka s.K., Velocity slip coefficient and the diffusion slip uelocity for a multicomponent gas mixture, Pbys. Fluids, 14:2599 (19?1).
12. Loyalka s.K. Approximate method in the kinetic theory, Pbys. Fluids, 14:2291 (1971).
13. Zhdanov v.M., Baro- and thermodiffusion of gas mixture in the capillary, Zh. Prikl. Mekh. Tekhn. Fiz., 2:48 (1982) - In Russian.
14. Zhdanov V.M. On the theory of slip at the gas mixture boundary, Zh. Tekn. Fiz., 37:192 (1967) -In Russian.
15. Zhdanov V.M., Zaznoba V.A. Nonisothermal flow of gas mixture in channel at intermediate Knudsen numbers, Prikl. Matem. i Mekh., 45:1063
. (1981) - In Russian. 16. Zhdanov V.M., Zaznoba V.A. Nonisothermal flow of
gas mixture through cylindrical channel at intermediate Knudsen numbers, 13-th Intern. Symp. on rarefied gas dynamics, USSR, Nobosibirsk, Book of abstracts, 2:523 (1982)
1?. Loyalka s.K. The slip problemforasimple gas, z. Naturforsch., 26-a:964 (1971).
18. Waldmann L., Schmi tt K.H. Uber das bei der Gasdiffusion auffretende druckgafälle1 z. Naturforsch.,
16:1343 (1961).
1267
ON THE DISCRETE BOLTZMANN EQUATION FOR BINARY GAS MIXTURES
Nicola Bellorno and Luciano M. de Socio
Politecnico di Torino Corso Duca degli Abruzzi 24 - 10129 Torino (Italy)
INTRODUCTION
As reviewed and developed by Cabannes 1 , the original idea of replacing the full nonlinear Boltzmann Equation by a Discrete Velocity Model (D. V. M.) is due to Maxwell and was pursued by Broadwell 2
who proposed a simple two-velocity rnodel, by Cabannes hirnself 3 ' 4 ' 5
for the wave propagation and Couette flow and by Gatignol 6 • 7 , who developed various D. V. M.'s with application to the shock structure and to the formulation of the boundary conditions.
Various contributions have been given to the proof of the global existence of the initial value problern. The subject was introduced by Tartar 8 for the one-dimensional case and by Cabannes 9 for the fourteen velocity rnodel. Kaniel and Shinbrot10 and Illner11 proved the global existence for the initial value problern with assurnptions on the size of the initial data in two and several dimensions respectively.
This work deals with the problern of the mathematical rnodelling and analysis of binary gas rnixtures, which were not previously investigated. An eight velocity rnodel is proposed for the mixture and the proof of the existence of the solution to the initial value problern is also considered.
AN EIGHT VELOCITY DISCRETE BOLTZMANN EQUATION FOR BINARY GAS MIXTURES
In this chapter a plane D. V. M. is esthablished for a gas systern constituted of light "host" particles and heavy "test" par-
1269
tieles. Let m and ~· N and M, be the moleeular masses and the number densities of the two speeies in the ~as, respeetively. Only binary eollisions of elastie spheres will be eonsidered. An eight D. V. M. is developed aeeording to the following veloeity diseretization for the first and the seeond gas
-+ -+ -+ -+ -+ -+ -+ -+ -+ u {u1 ei, U = ej u = -ei, u4 -ej}
2 ' 3 -+ -+ -+ -+ -+ -+ -+ -+ -+ V = {V ]..Iei, v2 ]..lej, v3 - ]..Iei, v4 ]..lej}
1 -+ -+
where ]..1 = m/m and i and j are the unit veetors of a reetangular eo-ordinate f~ame. This ehoiee is eonsistent with the equations of momentum and energy eonservation in binary eollisions.
(1)
(2)
The evolution equation for the eight number densities Ni (i = 1, ••• ,4), Mi (i = 1, •.. ,4) shall eontain two eollisional operators, the first one eorresponding to eneounters between partieles of the same speeies and having the,,. same form as in the four D. V. M. for a single speeies • The seeond operator eorresponds to eollisions of partieles of different speeies. The basie set of equations is then
-+ ClN. /Clt + U •VN
i "' i LH ~ A .. (NN.Q,
l.J k - N.N.) + LH ~ E .. (N M.Q,
l.J k - N ,M.) (3)
l. l. J l. J
-+ L*H L *.Q,k ClM. /Clt + V •'ilM = ~ A ij (~M.Q, - M.M.) + ~ E .. (~N.Q,- M.N) (4) l. i "' i l. J Jl. l. J
H. h . . . where A. . l.S t e trans1.t1.on probab1.l1.ty eorrespond1.ng to an eneounter l.J * -+ • * -+ of the eoutle (u., U.) leav1.ng the eoouple (u , U.Q,) for the light
gas, and A*~. is tfie aJalogous transition proba~ility for eollisions between tesEJpartieles. On the other hand, the transition probabilities E refer to eollisions between test and host partieles: E~~
-+ ± -+ -+ l.J eorresponds to the eneounter (U., v.) leaving (U, V.Q,) and, analo-gously ' E*h to the eneounter {uj 'J v) ,leavinlC\. VQ,).
The transition probabilities ean be related to the eorresponding eollision probabilities a, a* and e, e*, respeetiveiy, as follows
k.Q, 2 (J{
k.Q, A .. a ..
l.J l.J
H (1+]..l)<J12e
k.Q, E .. e ..
l.J l.J
k.Q, ( 1 +]..12_) ~ E .. 0 12e l.J
1270
*k.Q, A ..
l.J
*U E ..
l.J
k.Q, *k.Q, e .. ' E
ij l.J
*k.Q, 2a2 e a .. l.J
(5)
(6)
(7)
where o1, o2 and o12 are the cross sectional areas of the host, test particles and of the pair. Relations (6) and (7) correspond to headon collisions and to ninety degree collisions, respectively. Collisions between particles moving in the same direction and in the same verse are not considered here.
As far as the transition probabilities 'a' are concerned for a
four velocity model:
k a .. ~J
On the other band, the meaningful collisions between pairs of different species, show probabilities 'e' and 'e*' corresponding to
24 *24 42 *42 13 *13 e13 = e 31 = e13 = e 31 = e24 = e 42
*14 - 41 e 41 - e14
32 - *32 e23 - e 23
1/4
1/2
23 - *23 e32 - e 32 = 1
+ +
(8)
(9)
( 1 0)
For instance,+the+head-on collisions (U 1, v3) can provide three
couples, namely: (U3 , V1) with ~robabili~y !+corresponding to reflection, and the two couples (U2 , V4) and (U4, v2), each with probabitlty 1/4, corresponding to ninety-degree scattering. Also, a colli-
+ + sion at ninety
+ + degrees for the couple (u1, v4) yields the only pair
(U4' V 1).
form Following these considerations eqs.(3, 4) can be put into a new
+ aN./at + U.•'VN.
~ ~ ~
+ H 1+]1)0 d(M.N. + 12 ~ ~ +2
+ (1+]1 2 )!co {M (N 12 i i+1
!M. 3N. 1 + !M. N. - 2M. 2N.)}+ ~+ ~+ ~+1 ~+3 ~+ ~
+ N. 3 ) - N. (M. + M. ) } ( 11) ~+ ~ ~+1 ~+3
+ dM./dt + V •'VM = c]lO (M M - M.M. 2) +
~ i i . 2 i+1 i+3 ~ ~+
+ !(1+]l)o12c{N.M. 2 + iN. 3M. 1 + !N. 1M. 3- 2N. 2M.)}+ ~ ~+ ~+ ~+ ~+ ~+ ~+ ~
1271
+ (1+)1 2 )!co {N.(M. + N ) - M (N +N )} (12) 12 ~ ~+1 i+3 i i+1 i+3
where, for n integer, if (i+n) > 4 then: N. = N. , and M. = M. 4 (i = 1 ,2,3,4). ~+n ~+n-4 ~+n ~+n-
Once the solutions N(x, y, t) and M(x, y, t) of eqs.(11, 12) are known for given initial and boundary conditions, then the relevant macroscopic quantities can be recovered. In particular the local nurober density is
n = n(x, y, t) = N + M = L. ~
N. + L. M. ~ ~ ~
(13)
the mass velocity is
+ + + + + W = W(x, y, t) Wx i.+ Wy j=(c/n){[(N1-N3)+ )l(M1- M3)) i+
+ + [(N2 - N4) + )l(M2 - M4))j} (14)
and the stress tensor ~ is given by
\ + + + + ~ = ~(x, y, t) = ~.{mN.(U. - W i)(U. - W i) + -- ~~~X~ X
+ + + ~ + m M.(V. - W j)(V. - W j)}
p ~ ~ y ~ y (15)
Moreover, the pressure p and the temperature T can be related to the local densities as follows
p = p(x, y, t) = knT (mc 2 /3){(N- (N 1- N3) 2 /N-
- (N2- N4) 2 /N +)12M- (M1 - M3) 2 /M- (M2- M4) 2 /M)} (16)
from which
c 2 = (3 k/m T(N + M))/{(N + )1 2M- (N 1 - N3) 2 /N -
- (N - N ) 2 /N- (M - M ) 2 /M- (M2 - M4) 2 /M)} 2 4 1 3
and, in equilibrium conditions
c 2 = (3 k/m T)(N + M)/(N +)1 2M)
1272
( 17)
(18)
EXISTENCE FOR THE INITIAL VALUE PROBLEM
The proposed model is now considered from the point of view of the existence of solutions for the Cauchy problem. The guidelines of the proof given in ref./10/ are followed here when dealing with the proof of the existence for the initial value problern as described by the D. V. M. for bounded time intervals.
Equations (11, 12) can equivalently be written, at given initial conditions, as follows
+ df./dt + w •'Vf
~ i i
f. (x, y; t=O) ~
L (Gij f f . k n k.Q. k .Q, j, ,x,
\j!.(x, y) ~
+ +
H L .. f.f.) (19) ~J ~ J
(20)
where~ i = 1, .• ,4 fi =Ni, wi Ui; i = 5, .. ,8 fi = M(i-4), ~i = V(i-4) and where the transition probabilities have been compacted into the terms G, which refer to the transitions with positive contributions to the fluxes, gains, and into the terms L, which refer to transitions with negative contributions to the fluxes, lasses.
A direct look at eqs.(3-12) and their comparision with eqs.(19, 20) indicate the following properties of the transition probabilities: a) Collisions between partiales of the same species
b)
k,.Q.,i,j 1 ' .• '4
k,.Q.,i,j = 5, .. ,8
ij u k.Q, Gk.Q,= Aij = Lij
ij *(k-4)(.Q,-4) Gk.Q,= A (i-4)(j-4)
k.Q, L .. ~J
Collisions between partiales of different species
k,i 1 ' .. '4 j,.Q, 5, .• ,8 ij k(.Q.-4) k.Q,
Gu Ei(j-4) = L .. ~J
k,i 5, •• ,8 j,.Q, 1 ' .• '4 ij *(k-4H u
= = GH = E (i-4)j = Lij
(21)
(22)
(23)
(24)
Moreover, for both kinds of collisions, the additional properties hold:
k = .Q,; i j G L 0 (25)
ij H Gk.Q, L ..
~J (26)
i,.Q, 1 ' .. '8 ' ij ij L. k{ (Gk" - 'L ) J' )(, k.Q.
(27)
1273
The above equations supply some pertinent properties of the collisional operators. Moreover, eqs.(11, 12), according to the transformation proposed by Kaniel and Shinbrot10 , can be written, under the same hypotheses of their paper, in the following form
L j,k,R.
f~i)(O, z) = ~.(z) ]. "' ]. "' (i) + ) f. = f.(t, z + w.t ]. ]. "' ].
z = {x, y}
"'
(28)
(29)
(30)
Let one consider the space A of the continuously differentiable functions f., with f.: D•I +~,Dis domain of the variables x, y and I is th~ domain ~f the variable t and let A be equipped with the norm
max (x,y) E D
lf .I ].
(i 1, .• ,8)
where { = {f.}. Note that if f.e Athen f.E c1 • t · d f" · · 1 b 1 od•r The fol ow1ng e J.nJ.tJ.on can now e state :
(31)
Definition (Kaniel and ShinbPot): A function set ~ = {f1, ••• ,fs} is a "mi"ld solution" of eqs. (19, 20) if f e A and satisfies eqs. (28,29).
The following result can then be recalled:
i) The local mild solution of e~s.(19, 20) exists, foP every ~., in a sufficiently bounded intePVaZ I = [0, t*], with t* invePseZy ~ pPopoPtional to II ~~~ .
ii) If any miZd solution is monotone decpeasing in no~:
t2 ~ t1 => 11~211 ~ 11~111 then the mild solution can be gZobaZly extended.
Therefore a "mild solution" exists, in terms of differentiable functions, in a sufficiently bounded time-interval according to the above-mentioned results applied to the eight D. V. M ••
The exposed method can be used to prove the global existence of the "mild solution" in the space A for the Cauchy problem, only if sufficient conditions can be found in order to state that the mild solution is monotone decreasing in norm as in ii). These conditions have not yet been found for the proposed model 12 • If, on the other hand, the space A is replaced by the space of the continuous
1274
functions (namely without any differentiability requirements) and eqs.(19, 20) are written in the form
f.(t, z + ~ t) = ~ (z) ~ 'V i i'V
+ f Q.(s, z + ~.s, f(s)) ds I~ rv ~ rv
(32)
where Qi(~, fi) is the proper collisional operator on the r.h.s. of eq.(19), then the proof,which has been recently supplied by Illner11
and which is also based on the concept of mild solution, can be directly applied, in order to prove the global existence for the Cauchy problem, under suitably bounded initial conditions in the space of the continuous functions fi: I+ Cb( R 2 ). Here Cb is the set of the real-valued bounded continuous functions on R2 •
Illner's method is, in fact, very general and the proof is given without any limitation either on the number of the selected velocities or on the structure of the collisional operator and on the space distribution of the initial data, as far as they are bounded and the collisional operator is "self consistent", where this term is here used to state that the mass, momentum and energy equations are satisfied.
As a conclusion,one should observe that the proposed methodology for the construction of mathematical D.V.M.'s can be applied to generalized models with an extended number of velocities. The increase of the number of velocities does not alter the procedure for proving the local or the global existence in the initial value problem.
Acknowledgement. This work supported in part by the M.P.I •. The authors express appreciation to professors H. Cabannes and R. Illner for the helpful suggestions.
REFERENCES
1. H. Cabannes, The Discrete Boltzmann Equation, Theory and Application, College of Engng. Rept., U. of California, Berkeley, (1980).
2. J. E. Broadwell, Shock Structure in a Simple Discrete Velocity Gas, Phys. Fluids, 7: 1243 (1964).
3. H. Cabannes, Propagation des ondes de choc dans un gaz a 14 vitesses, C. R. Acad. Sc. Paris, 279A: 761 (1974).
4. H. Cabannes, Etude de la propagation des ondes dans un gaz a 14 vitesses, J. de Mecanique, 14:705 (1975).
5. H. Cabannes, Couette Flow for a Gas with a Discrete Velocity Distribution, J. Fluid Mech., 76: 273 (1976).
1275
6. R. Gatignol, Kinetic Theory for a Discrete Velocity Gas and Application to the Shock Structure, Phys. Fluids, 18: 153 (1975).
7. R. Gatignol, Kinetic Theory Boundary Conditions for Discrete Velocity Gases, Phys. Fluids, 20: 2022 (1976).
8. L. Tartar, Existence globale~ pour un systeme hyperbolique semilineaire de la theorie cinetique des gaz, Seminaire GoulaouicSchwartz, Ecole Polytecnique,(Oct. 1975).
9. H. Cabannes, Solution globale du problerne de Cauchy in theorie Cinetique discrete, J. Mecanique, 17: 1 (1978).
10. S. Kaniel and M. Shinbrot, The Boltzmann Equation: Some Discrete Velocity Models, J. Mecanique, 19: 581 (1980).
11. R. Illner, Global Existence Results for Discrete Velocity Models of the Boltzmann Equation in Several Dimensions, Preprint n. 26 Fach. Math. U. Kaiserslautern, (1981).
12. R. Illner, Private communication.
1276
PECULIARITIES AND APPLICABILITY CONDITIONS OF MACROSCOPIC DESCRIPTION OF DISPARATE MOLECULAR MASSES MIXTURE MOTION
0. G. Buzykin, V. S. Galkin, and N. K. Makashev
MPhTI, Dolgoprudny, USSR
INTRODUCTION
The essential feature of gas mixture flows with disparate particle masses is that at the hydrodynamic regime (Knudsen number K-.0 ), translational temperatures of components may differ by their own value due to a retarded energy exchange between light and heavy particles. Such flows are described by multi-temperature hydrodynamics (MTH). Formally, the MTH description is possible due to simplification of collision integrals for particles of different kinds (cross-collision integrals) by an expansion in E. =(rn/M)112 , where m , M are characteristic masses of light and heavy particles.
NumeroU!I references are devoted to the derivation of MTH and multi-velocity hydrodynamics (MVH) equations and to the study, by their means, of uHrasound propagation, shock waves structure, etc. The paper scope does not permit their review, and we refer to some of them, where is a bibliography for mixtures of m onol. 2 and polyatomic3-6 gases. General equations of the MTH and MVH are given by the generalized Chapman-Enscog method6- 8 (G-method), which is developed for a structureless particle mixture 7 and for those of polyatomic gases with chemical reactions and with MTH effects6. 8. From these generat equations, simpler ones for E.- mixtures at arbitrary ratios of concentrations and collision cross-sections may be obtained by expanding in e« 1 . However, when constructing the G-method in the above papers, a detailed analysis of conditions, where the hydrodynamic description (HD) is applicable for mixture flows in generat and the MTH (or MTH and MVH) description in particular, was not carried out. Suchanalysis is given below for a binary mixture of structureless particles, and the estimates underlying the G-method are refined. In addition, some examples of solutions for different translational temperatures are presented.
1277
EXPANSION OF CROSS-COLLISION BOLTZMANN INTEGRALS
Let us expand the cross-coßision integrals 1 J (f, F) • J CfT -"f F) crdk: d~ 11 ,
J(F,f) = S<F'f- fj)udkch~'"' where ü= cr('i\ ,g·i<) , in e. under moregenerat as-sumptions accounting for "hypersonic" regime for a heavy component when U.m»c11,.- (2.kT,./M)112.. Parameters of heavy and light components are denoted by great and small symbols, respectively, or by subscripts M , m . It is assumed that component velocities u.l'l- u..., -c"' ... -l2.kT.,./tn)112. , component temperatures ~-_T11 ..! and_th~s ~- ~c ,:e.c;o. The charact~ristic valu~s are marked with an aster.sk, l: ... ~11-u.,.. , c • ~m-u11 , C0=~.num. The expans10n results m J ( t., f~) == = J'o'(f.,f") + J'.'<t.-..tl'>) -t- J'a.'(f.,..,1\.)-t-... where J'o'(f,F}= HfCTo)- f(co)] "llodk'
J (1) (t ' f) " 0
J(R.l(j,F) "2.e.2.J ~[f(T0)(~·k)kj'1°]dk + i)c.j - -
+ ~<J r { 21 [ f (To)- ftcol} ~2.~o + 2. ~f~Tol ktkt.k,kdo + ~f<'tol aol)o k1. kJ.l dk ' "<1'1 J uC.vCJ uCtuCh • u Ct CL J
J'G)(F,f) = 0
J'''<F.fJ= -!~.Jtcc>ctr , L
J'a.'<F,f) = e2 aa.F r f<cHc·k>dfkj ac,ac; J
When expanding J (f ,F) the molecularvelocities are taken in um- reference system, while expanding J(f, t) il"- system is used. Note here that J 101(f, F) differs from the common Lorentz Operatorin t'tol ;1. Je0 J • The expression for .. J'',(F,f) is transformed to J'11(F,f) = -(R~1/~,..] dFjClc , R~1 .. SJ111(F,f)M~ 11 •
·d.~ . Thus 1 , this term cancels with one of the summands in the left part of the equ~tion for F . According to the estimate used in the G-method 7 , antisymmetris parts of the linearized cross-collision integrals are small compared to symmetric ones and should be taken into account in the next approximation in K . At e.«. 1 , the antisymmetric part of J 111 (F,f) is of the order of the symmetric part. Butitdoesnot influence the algorithm of the G-method 7 because of the formal nature of the above procedure. In fact, both summands J'11(F,f) cancel with respective summands of the left part of the equation for F . Therefore the system of nondimensional Boltzmann equations for a binary mixture and the equations for il,.. ,illll can be written as follows:
= [8/(1+l5)]J(F,F) + [e./(1+&)]J(11(F,f)+ •••
1278
= [(!t/(1+ö>]J<f,f) + [8/(1+a)][J'",<f,F)+e~J''1 <f,F)+ ... ]
(1.3) K S :r. ll" u.,L +Ka P.,q = 1 [ R'1> 1 R<!o> ] 1 t:. "'?" Dt a ~j w "'' + e '"'· + · • • "" ·t-t·a c;;.
K c:.ll. 11111 l.l. 111;. + K o Pmq = 8 ( R '"' 2. R'2l 1 8 A (1.4) ""m qm Dt O~j 1 +8 mi + €. mi -+ •.. ,._ 1+8 Cm,.
In Eq. (1.1)- (1.4) the notations Dc(./Dt =d/dt + u~~~.ra!öxj , l. = i!11 - ilm•
K = (n,. b~ ll , n = n 111 +n 11 ---n,., 8 = (nMfn 111),., ~ =('b!/b~), Sol=(u,Jc .. vre used, where L is minimum scale of finite change of flow parameters, b~ is characteristic
heavy particle cross-section. Here andin section 2 it is assumed that S 11 - Sm.E.- 1 ; Mach
number Ma.-u.,.;a.,.~ 1, a..,.-cM,.((1+5)/(6+E.2.)]~12. hence Sm' 0(1),
SM40{(1+5)''lLlS+E.R.)- 1/!I.}; ,...E(e. 2 ,1) ,inthemoleculargasmixture p-1 and for a gas with a fine dispersed admixture p ~ ( eS11 , f.) •
CONDITIONS OF HD APPLICABILITY
The necessary condition for the HD applicability is infinitesimal left parts of
Eq. (1.1), {1.2) compared to right ones. Then from (1.1) the inequality is found
(2.1)
Accounting for Eq. {2.1), from Eq. (1.4) at 8»E , one obtains Rm- A/c111,.-K(1+8)/
/t'> < 1 and at 8-' e from Eq. (1.3) it follows A/cm,. ... ma:.lC! ( KS~, K ) «.
< (E.+ 8)ma:x (SM, 1).6 1 . Thus under conditions of the HD applicability IJ./c 111,.« 1
independently of p , e , 5 .
Let us determine the Iimits of the HD applicability from the condition
K«. K,. = mi.n(KM,Km), where KH, Km are Knudsen numbers at which F and t, respectively, differ by their own values from the maxwellian distributions. The value
of K 11 is given by Eq. {2.1). To determine Km the corrections to the maxwellian
fo [T111 ,c!] should be obtained from Eq. (1.2). As a result we find Km= [Ö{('>+
+h2>r''/(1+~). The comparison of estimates for KM and Km gives the Iimit of the HD applicability K,.
{2.2) K * = K 11 ~ ( e + 8 )/ ma:x ( S 11 , 1 }
K,. = Km~ [(1 + 3eh!B] 112
for a molecular component mixture ( ~-1) and
{2.3)
K" = K"' , ma.~ (5 11 , 1} > A
for a gas with a fine dispersed admixture ( e< <.I"'< e.).
' B ~ I'> 1/2.
1279
The MTH may occur if in the main approximation in K and e the maxwellians with different temperatures are solutions of Eq. (1.1 ), (1.2). It requires the operators responsible for temperature equilibration J'a.1(F,f) and Jll-l(f,F) tobe small compared to J(F,F) and J(f.f) , respectively. The analysis of the resulting equations showes that this requirement is met at t.« 8 « ~/1!. 2 • Also, it is necessary that the relation f T111 - T.., 1/T" .., 1 m ust be met in hydrodynamic scale (i. e. characteristic flow time is of the order of the temperature relaxation time). Thus, we find that the MTH region is defined by Eq. (2.2) or Eq. (2.3) and the conditions following from the equations for T111 , TM
(2.4) Ku 4 K « K.. , K** = E/S.., := E M~ [((he.t.)/(1+.,)] 411 •
Here the flow velocity is assumed tobe between the Iimits SM"' 0{1) and Ma.= 0( 1).
In the kinetic description of gas flows with a fine dispersed admixture, it is usually supposed that ~"' qm , i. e. 8, e2 . As a consequence from above such flows should be described by one-temperature gasdynamics. From Eq. (1.3), (1.4) it is evident that the relative difference of the component velocities A/u..-1 at K_, K « K* and e. «~ « r.>/E.. ~ , while A/u..« 1 at K« K-. Also, it can be shown that the MVH and MTH equations have amore and uniform accuracy than the common HD. An important conclusion from Eq. (2.2), (2.4) isthat MTH is possible only for e. - mixtures.
MULTI-TEMPERATURE PROCESSES IN CHEMICALLY REACTING GAS MIXTURES
In an exotherrnie reaction of gases with disparate molecular masses, the most of energy released in an elementary act is taken away by light particle. Owing to a retarded energy exchange between light and heavy particles, the MTH condition may occur. Proper gasdynamic equations are obtained formally with'in the G-method 8 and more detailed, using a nonelastic collision model, where the direction of relative velocity of partief es produced in the reaction is independentofthat of particles before a collision 5, 6. 9. Particular calculations of the spatially-hom ogeneous reaction A .. + Ar• .... A,.+ Aj in a diluent gas of molecules At. at m ...... m.~-m., rnl""rnj -m"- M » m are made on a base of the rigid-sphere molecule model and the reacting rigid-sphere m olecule m odel used to describe elastic and inelastic collisions, respectively.
Usually the HD with partial temperatures for each component is, however, · excessive. A less detailed description is needed, where various translational temperatures are substituted by Tm. for light components and T t1 for heavy ones . The Fig. 1 gives examples of calculations for partial temperatures under conditions close to the reaction Hz.+ f: - Hf'+ H in nitrogen6 • The simplification consists in neglecting the internal molecular degrees of freedom which is considered in defining the exotherrnie effect of the reaction.
The conditions of the HD applicability at MTH effects are w .. , «.(m/M )"II. , m.ft1' w,.i. where w4 i. is the reaction probability. The second relationship is met if the reaction time is less than that of equilibratiori oflight and heavy component temperatures. The first one is satisfied if the reaction time is much more than the maxwel-
1280
3
2
1
Fig. 1. Initial volume concentrations (A\10 :(A,..l_;fA\lo=-0.3:0.~~0.4 initial tempera· ture T0 = 300 K.
lization time of distribution functions of heavy components when the macroscopic description can be used. For the above reaction it is not satisfied, which results in a substantial difference in temperatures of components with comparable particle masses. According to section 1, such case is inconsistent with the HD. lt means that the spatially hom ogeneous reaction of H2. and F with slight dilution of chemical reagents should be calculated at the kinetic Ievel of the description. The data in Fig. 2 for partial temperatures (left) and for the two-temperature approach (right) relate to a speculative reactian 5, 9 satisfying both MTH applicability conditions.
F or spatially inhomogeneaus flows the mean componen t velocities are different. Under the HD applicability the mean velocity difference is Atj = IL'it- ilsl«cm,.· The corrections inserted by E,i to the exchange and reaction terms in the partial temperature equations5,6 are of the order of (f.tj/cm .. )2 and may be omited in the main approximation. lnelastic collisions contribute into the partial momentum equations the exchange terms
where the former5,6 nomenclature is used.
1281
Fig. 2
GASDYNAMICS OF A STRONG EVAPORATION OF BINARY e. -MIXTURE
The disparity in masses of evaporated particles brings some distinguishing features into the strong evaporation. At first, if the masses of incident particles and forming the surface ones are disparate, the energy accomodation coefficient of light particles «m can greatly differ from a unity. The known solutions for the Knudsen layer over the surface with a strong evaporation are found assuming ocm" 1 . A necessary generalization is simplified because we need the study of the flow of any component in the Knudsen layer only when this flow in the main approximation corresponds to the one-component gas evaporation. Such solution is obtained by the moment approach assuming the distribution functions r; and f; of the particles leaving and inciden t to a wall to be f.: a fotv + f ~ = Ut\ev( h.w/T.)&/2-e-:x:p(-hw~ 2.) + n,. { h.,./r.)'6/ 2. • ·ex.P(·h.~~2). f; = n..,(h..,;mstllexp(·h..,[<~."-u_)t+~!.•li!.J},h.«"tn/2k1.where f • ., and f ~ are distribution functions of the molecules evaporated and reflected by the surface , T,. depends on the accomodation coeffitient 0( , y - axis is normal to the wall, a. is evaporation factor, n.., , T 00 , u.., are parameters on the outer boundary of the Knudsen layer. The analytic relationships are obtained
T. a 2.vr.'S + ~-t[1-(1-a.)(1-ot.)] h.t/2. ..!!! l!lv • , S =u.., - , Tw .., YFSl~.5+S") -t- '-t.(1-(1-a.)(1-«.)}
(4.1) n. ... tn • ., ii N.., .. a.(a.1. 1 + 2.-ff. S)-1 -8_ 1,._, ?t 1 =exp{-Sa.)-m(1-erfS),
~2. = (1+ 0.5 S2.)e-xp(-S2.)- 0.5-/li' 5(2.5 + Sa.)(1-erfS) ,
which are in good agreement with precise ones ·except for the maximum evaporating mass flux.
1282
At second, the flow out of the Knudsen layer is essentially noneqilibrium for the strong e - mixture evaporation as the component temperatures on the external boundary of the Knudsen layer differ by a finite value (for an estimation Eq. (4.1) may be used considering that even at u."'-1.1." the value Sm.,..«S..,ao ), and the mean velocity of the heavy component U:" at the wall may be farless than that ofa light one. It results in equilibration of component temperatures and mean velocities out of the Knudsen layer. The flow properties in this region mainly depends on the parameter Bo• Pevl'l/ Pevm , where Pev is saturated vapor pressure, as the mixture mean mass Velocity on the outer boundary of the nonequilibrium region depends on ao . At a gteat deviation of the mixture state from the equilibrium, this velocity is near the so und speed at the wall timperature tle - (2 kTw/M)•It [(1 +Öe>/CBe + e.l.>r't Here 8e=(n,./nm)e-8o if80 )11 ,and 8e-80 (et.+68 ) 1/1t at 80 c.1 .Thesubscript "e" denotes the values outside the nonequilibrium region.
The analysis similar to that in section 1 shows that all these phenomena over an evaporating surface may be described by MTH - MVH for the binary mixture e 111« 80 « e-t. , when the evaporation of M - component is strong and that of a light one is weak (out of this in terval the nonequilibrium region is substan tially kinetic). Proper equations coincide in the main approximation with those obtained earlier 1 despite that for 80 « 1 the velocity u.r1e» (2.kTw/M)11t. For the maxwellian molecules the equations are
•
3 k (T . )' , - II! I. "', .. , T E T nMu. M~Tp,..(u.)\1 -gSt"flmn,..ek...,mM<Tm-•M)• •
~ knrn.l.l.(Tm>~ + Pmlu.)~ + (q,m~)~- ~ kT"'CnmV111 )~ = -E ,
q.m~ = -).(Tm>~+ ~ Pm V111 , ~ = [5 ktT"/(2r.m)][ CD~1~ +2 ~: <D;~i~
They include the additional tenns due to a great heat conduction of light specie. Consequently, the solution depends on larger number of governing parameters compared with strong one-component gas evaporation. That is the third feature of - mixture strong evaporation. The Eq. (4.2) are reduced to the first order equations for q: q..,IJh1:.p;;.~, N '"nl'1/nMe , p = PmiPme , 't' = T.,./Te and ~ = T,../Te on the finite intervalof
aQ
rz. = (588 58 /Te) S<q,. ... 'J/~)dy lj
,.
The singularity at '1::. 0 may be eliminated by means of expansion of the solution in 'l . The Fig. 3 shows the integral curves which make possible to get the solutions for
arbitrary wall location '1. .. "lmQ~ or Knudsen layer temperature jump which is equal (-8--r)/ T at 11 = '111\Q.x •
1283
p,-r, .S N P2 ~g q. 't"2. - 1
0.9 2.6 - 2.
- 3
0.7 -4
~g '2 -5
- 2
Fig. 3. The parameters with subscripts "1" and "2" correspond to Be .. 0.1 Se=- 0. 3 and 6e = 0. 1 ; Se= 0.8 respectively.
REFERENCES
1284
1. V. S. Galkin, On derivation of two-temperature gasdynamic equations by means of modified Chapman-Enskog method, Izv. AN SSSR. Meh. Zhydk. i Gaza. 1:145 (1981)- In Russian.
2. R. G. Huck and E. A. Johnson, Physical properties of double-sound modes in disparate-mass gas mixtures, in: "Rarefied Gas Dynamics" Pt. 1, ed. by S. S. Fisher, Vol. 74, Progress in Astronautics and Aeronautics, AIAA, New York (1981).
3. A. I. Osipov and E. V. Stupochenko, Disturbance of maxwell distribution in presence of chemical reactions. Reacting monocomponent system in heavy gasthermal bath, Teor. i Exp. Himija 6:753 (1970) -In Russian.
4. S. N. Lebedev, V. B. Leonas, Yu. G. Malama and A. I. Osipov, Application of Mon te-Carlo method in chemical kinetics,fu: "Com bustion and Explosion", "Nauka" (1972)- In Russian.
5. 0. G. Busykin and N. K. Makashev, Multitempersture property of translational degrees of freedom of m olecules due to exotherm ic reactions, in: "Rarefied GasDynamics", eh. 1, Trudy VI Vsesoyuznoy Konf., IT SO AN SSSR, Novosibirsk (1980)- In Russian.
6. 0. G. Buzykin and N. K. Makashev, The exotherrnie gas phase reactions as a reason of appearance of multitemperature flows of polyatiomc gases, Zh. Prikl. Meh. i Tehn. Fiziki 1 : 87 (1981) - In Russian.
7. V. S. Galkin, M. N. Kogan and N. K. Makashev, Generalized ChapmanEnskog method. II. Multivelocity multitemperature gas mixture equation, Uchenye Zap. TsAGI 6 (1): 15 (1975) - In Russian.
8. V. A. Matsuk and V. A. Rykov, On Chapman-Enskogmethod forchemicaUy reacting gas mixture with internal degrees of freedom, Zh. Vychisl. Matern. i Matern. Fiziki 18: 1230 (1978) - In Russian.
9. 0. G. Buzykin and N. K. Makashev, Multitempersture processes in chemically reacting gas mixtures, Uchenye Zap. TsAGI 12(4): 75 (1981)- In Russian.
NUMERICAL SOLUTION OF THE BOLTZMANN K:nmriC EQUATION
F OR THE BINARY GAS MIXTURE
A.A. Raines
The Leningrad State University, USSR
The process of energy and momentum exchange in binary space-homogeneous gas mixture with different molecular masses and collision cross sections is investigated on the basis of the Boltzmann kinetic equation.
The solution is realized by a regular numerical method with the use of a conservative algorithm, which provides an exact conservation of partial densities, total energy and total momentum of mixture.
An example is given for a computation of methaneargen mixture with real molecular rnasses ratios (rn, I m2 = 0.401) and collis ion cross sections ( d, I d 2 = 1.121). The effect of intermolecular interaction law on relaxation process is studied for the model of hard spheres with a variable diameter. The evolution of distribution functions, the time dependence of energy flux between the components as well as temperatures and entropies of every component and the mixture as a whole has been obtained.
Let us present the system of equations for the binary gas mixture as follows:
+OO l!'ii g[
(1) g{1 "~J J j((f'-fJ)~11 j(q,; K)!Sin8d8dPdü+ -oo o o
1285
Here
r-
..... 2!1[' qr
+ 2~ J J J (( f 2,- f1 F2 ) 6121Cq,2 • R)ISineded<J>dü 2 -oo o 0
~-6tl.ii
g~2 = ~qrJ J r ( F: (- F2 F) ß22 1(<\22 K)ISin eded~dü + -ora o o
-~-- 2fL Cj[
+ 2k J J J ((f~- F2 f,) ~ 21 l(q 2; K)l Sin8d8d(j)dÜ 1
-oo o o
f1 =f/1,Ü1), f=f1(t,ü), f2=F2(t,ü2), F=F2 (t,ü)
q,,= u-u1. G-22" ü-ü2, q,2"Cl21" u2-u,
K (Cos EJ, Sin 8 Cos()), Sin 8Sin~)
6l· ( L,j = 1, 2)- total collision cross sections. For 4hard spheres with a variable diameter the collisi-on cross sections are:
r- -w 6-.= <; .. [m C2 /{2(2-w) KTo}]
LJ lJ r r
where rnr- reduced mass; C r - relative veloc i ty; W = 2/( ~ -1), q - the index of inter-
molecular interaction law.
For hard spheres we have ~ = OQ ( W = 0) and so b~=g[d~j (i.,j=1,2) where di.j=di;dj, di.,dj-
being effective diameters of molecules.
As examples of molecules with inverse power law we used 7 = 9 and ~ = 5 ( W = ~ and W = t )
1286
System (1) is solved by Euler method:
f, ( t + ~ t ) " f1 ( t) + 6 t [ I 11 + I 121
F2 ( t + 6 t ) = F 2 ( t ) + ö t [ I 22 + I 2 , 1
To construct a conservative scheme at every step At a correction of distribution functions is introduced:
~ ( t ) = f 1 ( t ) ( 1 + p ( ü ,) )
,..... F2 ( t) = F 2 ( t) ( 1 + G ( Ü 2))
Q( Üa) = IJo + ß1u2 + IJ2v2 + os W2 +&4 ( u;. + v; + w~) Coefficients ai. and Oi. are determined in such a way as to exclude any numerical error in conservation laws.
The above correction method for a binary gas mixture generalizes that one proposed in 1 • A partial correction was also studied when only the conservation of partial densities was guaranteed.
At the stage of relaxation it is required to guarantee the exact performance of finite difference analogues of conservation laws, namely, conservation of partial density of every component, total energy and total momentum of the mixture. In the problem on isotro~ic relaxation the unknown coeffic ients are 0 0 ,
a4 t I.Jo ' ß4 ; in the problem on gas injection - ao t
a, t a4 t llo t 01 t tJ .... Owing to the :ract that the number of unknown coefficients exceeds the number of equations, we mak:e the following. Let fln , 6 I , ll. E - be errors introduced at the relaxation stage to density, momentum flux and energy respectively. The total error in energy computation is ll. E = t:J. E1 + ll. E 2 Suppose A E1 -~- SD Lrfi- E2 -Then E _ SD · AE E _ A E and instead
l:J. 1 - 1 + SD ' l:J. 2 - 1 + SD of one equation for ll. E we arri ve to two equations for t:J. E 1 and t:J. E 2 • Similarly, the total error in momentum computation disintegrates into two parts.
As a result the orig'inal system of 6 (or 4) equations with 6 (or 4) unknown values disintegrates into two systems of 3 (or 2) equations.
1287
To verify the accuracy of calculations a test problern was used, i.e. a computation of ho:nogeneous isotro~ic relaxation in hard spheres gas, given in 2,3. The initial conditions were
- 1 (3 Ü2) F2(1u 2 l,t=o):::(T2o)lh exp -2~
nro ( ) n 20 "' 1, T,0 " 0.0583 0.5 , T2o: 1.9417 ( 1.5)
The second problem that of a heavy gas (argon) tnJection into a light gas (methane) is considered as a space homogeneaus one. This problem is solved in cylindrical coordinates in velocity space. Computations are made for some initial ratios of densities and temperatures. The process of gas mixing is studied both at the level of macroscopic values and at the level of the distribution function for the components. The initial conditions are:
F (u v t = o) = 1 exp (-l. ( u2- Ux)2 +V~ ) 2 2, 2 • (T ) % . 2
2o T 2o
where
V,=YV 2 +W1z', V2 =Vv~+w~·, nn10 =1, 1,0=0.5, T20=1.5 1 20
The results obtained in the problem of space-isotropic relaxation are given in Figs 1-4. The mean relative statistical deviation of our results from that obtained in 2 anywhere not exceeds 3% when the data from S have a 5% statistical error in low and high velocity domain.
The resul~s ootained in the problem of space-nonisotropic relaxation are given in Fig. 5.
1288
F ig. 1.
Fig. 2.
f -2. i'~
0.6 -1
O.lf \ \
0.2
"" 0 .., 2 ,-I fLI
Methane distribution function at t = 4<I where Cl - mean time between subsequent collisions of the first gas when mixture is in equilibrium.
T 2/l
\ \
\ \
f,5 \
\ \
\
' '' ......... '-..,
0 2 4 6 8 w t
Methane and argon temperature nonconservative scheme
- x-x partially conservative scheme ---- completely conservative scheme.
1289
Fig. s.
r - 2 1.-u. 0,7
0,5
Q3
Qf
Fig. 4.
1290
Deviation from Maxwellian runction. initial Maxwellian function
- ---- nonconservative scheme ---- completely conservative scheme
0,5 f,O t5 2,0 2,5
Metqane distribution function att=2~,6~ 100 random samples
4 random samples l\XXXXX
T
t,O
0
F ig. 5.
--- -- - - ----- - ----------------I .
2
2 (j 8 10
Change in longitudinal and transversal temperatures of methane.
Figs 6 and 7 show the effect of intermolecular interaction on relaxation process with the use of the model of hard spheres with a variable diameter (VHS model)4.
0,5
0 2 'I 12 ~~ (tj l
Fig. 6. 6T2 =T 2 - 1 ( T2 - argon temperature) 7~00- hard spheres
----- ~·9- inverse power law ----- 7"5
1291
Qj
Q.}
Qf
F ig. 7.
", ... ~ ............. ,, ~''-"' ~'
~' ~,
6,5 1,0 1,5 2,0 2,5 3,5 ll{!
Methanedistribution function at t=2q ~:oo - hard spheres
---- - ~:9 ---- ~. 5 - inverse power law.
Fig. 8 gives the comparison of solutions for classic inverse power law and for the VHS model with 7 = 9 in the one component isotropic relaxation problem. The initial condition is:
f(U,t=o)= ~ exp(-3u 2 )+-q.exp(-u2 )
F ig. 8.
f(UJ
0 Q5 2,0 2,5
classic inverse power law ---- VHS model
The calculated data for 7 = 5 are qui te similar to that given in Fig. 8.
1292
General Conclusions
1. The method of direct solution of the Boltzmann equation provides a sufficiently accurate computation of both the macroscopic parameters and the distribution functions of gas mixture.
2. The efficiency of the algorithm considerably increases when the conservation laws are exactly s atisfied (Figs 2, 3).
s. The conservative method permits real reduction of a number of random samples at the computation of collision integrals preserving a satisfactory accuracy, thus considerably reducing the time of computation (Fig. 4).
Heferences
1. Aristov v.v., Tcheremissine F.G. The conservative splitting method for the solution of Boltzmann equation, USSR Comp. Math. and Math. Phys., 20:191 (1~80) - in Russian.
2. Rykov V.A.,Chukanova T.I. The solution of Boltzmann kinetic equations for the binary gas mixture relaxation, Numerical methods in the theory of rarefied gases, Moscow Comp. Center of the USSR Academy of Sciences, 36 (1969) - in Russian.
3. Denisik S.A., Lebedev S.N., Malama Yu.G. On a control of a non-linear scheme of Monte-carlo method, USSR Comp. Math. and Math. Phys., 11:783 (1~71) - in Russian.
4. Bird G.A. Monte-carlo simulation in an engineering context, Rarefied Gas Dynamics. Proc. of the 12th Intern. Sympos., 1:239 (1981).
1293
XX. SPECIES ISOTOPE SEPARATION
GAS OR ISOTOPE SEPARATION BY INJECTION
INTO LIGHT GAS FLOW
S.F.Chekmarev
Institute of Thermophysics Siberian Branch of the USSR Academy of Seiences Novosibirsk, 630090, USSR
INTRODUCTION
When a mixture jet enters a secondary light gas flow, a spatial separating of the species takes place, which is due to the difference of molecular mass of the species and that of molecular size as well. Such a flow can be used for gas or isotope mixture Separation. The problern was previously studied in R~~s.1-9. Herewe report the results of theoretical investigation of the problem. The approximate moment method described in Ref.5 is used. Conceptions underlying the method are similar to those of Ref.1, but in cantrast with the latter we use another <moment) form of equations, which allows to calculate spatial distributions of the species directly.
STATEMENT OF THE PROBLEM
To have a large scale spatial Separation of the species, the mixture :jet is, roughly speaking, to be divided into jets of the only species. For that, colliding with molecules of the light gas flow the species molecules must be deflected slightly in each collision, a number of the collisions being large enough. In addition, the feeding mixture jet is tobe collimated, so that the thermal velocity of the molecules should be small enough as compared with the overall velocity of the jet flow.
According to those general conditions discussed in Refs.1,5 we specify the problern as follows. A hypersonic jet mixture produced in a slit nozzle enters a light gas flow. For simplicity, the mixture is regarded as binary one <i=1,2). Molecular mass of the light gas in assumed tobe small as compared with that of the species
. 1297
being separated (m 0 <<mi, here the upper index, 0, refers to the Light gas, and the subscript,i, to the species). Then, the species densities are taken being small as compared to the Light gas flow one <ni<<n°), so that the collisions occuring between the species molecules are neglected by comparison with those of the species molecules with molecules of the Light gas flow. Under two Last conditions species molecules will scatter into small angles. At Last, it is supposed that the mixture jet entering the Light gas flow does not disturb it.The above conditions taken as a whole make it possible to simplify the problern and obtain a reasonable practical solution.
We shall consider the simplest separating element in which one blade is used to divide the feeding mixture flow into enriched and depleted ones. The flow pattern is schematically shown in Fig. 1.
Let is define characteristic quantities of the separating process. The cut may be written as
Here, ci is mole fraction of i-species in the feeding flow, and Gi is the partial cut of i-species, defined as
(1)
(2)
where Si is the total flow of i-species, and Sip is the part of that being in the product flow. The separation effect is defined as
(3)
It is our objective here to determine a rel~tionship between E and 8, which allows to estimate an amount of the separative work.
y --------
'fr vic X
Fig. 1. Mixturejet entering a uniform Lightgas flow.
1298
APPROXIMATE MOMENT METHOD
To calculate the quantites desired we have previously to find spatial distributions of the species flow densities.
With an interaction between the species molecules being neglected, a motion of either species may be considered as independent of the other. Therefore, first we consider a hypersonic jet of an only species entering a Light gas flow.
To obtain a solution of the problem, especially an analythical solution, we need to use certain simplifying assumptions. The principal one will be that a motion of a species may be represented as a composition of two processes, namely, a motion along an average trajectory of the jet and jet spreading with respect to it.
With using the system of the first ten moment equations, based on the Linear Boltzmann equation, it is possible to derive an equation for the average trajectory of the jet, and to obtain solutions for the typical cases of the jet spreading~-6. The results needed in the paperwill be given, in brief, below. The main condition used to close the moment system is that a species moves at "hypersonic velocity", i.e. at thermal velocity being small as compared with overall velocity of the species flow. The condition is provided by the hypersonic velocity of the feeding jet at the nozzle exit and, then, by small-angle scattering of species molecules due to which thermal velocity of the species molecules increases slowly.
Equation for Average Trajectory of the Jet
If we integrate the momentum equation over the jet width and omit the terms of the order of 1/M associated with stress tensor components, we obtain
(4)
Here t is the time, ri is the radius-vector of the average trajectory of i-species jet, and ~i=mim0/(mi~0) is the reduced mass. 1/Ti is the mean frequency of i-species molecule collision with molecules of the Light gas flow, determined with effective cross-section Q< 1 >= =f(I-cose)dQ. A similar equation was used in Ref. 2.
Jet Spreading: Entering a Transverse Hypersonic Light Gas Flow
Let the Stagnation temperatures of the Light gas flow and the species jet flow be comparable. Then, since mi>>m0 is assumed, a mass centre of the system of species and Lightgas molecules colliding is nearly the same that of the species molecule. It is
1299
known, that in a centre mass system molecules scatter isotropically, if they collide as hard spheres. So, if we treat the collisions as hard-sphere ones, the species molecules will be scattered by the Light gas symmetrically relating to the average jet trajectory. Therefore, jet spreading with respect to its average trajectory, until a deviation of the jet issmall enough, will be described by formulae similar to those of the jetentering a quescent Light gas5. Specifically, species flow densities, gi, which are of interest here, follow the normal distribution Law
s · 1 x-x · 2 g - _1-_ exp [- -(-<-....:k.) ] r Ra . 2 a .
(5)
X'& X'&
where the standart deviation in the x-direction, axi, is given by
(6)
Here k is the Boltzmann constant, t is the elapsed time, and T0 is the Lightgas temperature. Another notations correspond to Fig. 1.
INVESTIGATION OF SPECIFIC CASES OF SEPARATION
Mixture Jet Entering a Uniform Light Gas Flow
The flow pattern is shown in Fig. 1. The following conditions are assumed:
- molecules collide Like hard spheres; - Q1=Q2, that is the case of "pure inertial separation", which
is typical for separation of isotopical species in ground states; - the Light gas flows at hypersonic velocity; in this case
maximum separation effect is achieved; - at the nozzle exit V1=v2=vc; - the stagnation temperatures of the feeding jet and the Light
gas flow are comparable; then, for mi>>m0 we have Vi<<u0~ so that the collision frequency takes the form J/Ti=n°Qi(u 0 t-xi).
Under these conditions one can easily obtain a solution of (4). With the elapsed time excluded, it can be written in the form Xi= =xi(y) 3 from which it is possible to find a spatial divergence of the average trajectories. If deflection angles of the trajectories are small enough, the divergence is represented by a difference ßx=x1-x2. For ßm<<mi (it is assumed that ßm=m2-m1>0) we have approximately ßx=-(3xi/3mi)yßm. Thus we obtain
- u 0 ]..! - U 0 ]..! - ßm ßx=[(--y-I)exp(--y)+J]----o (7) Vc m Vc m m
1300
Here x and y are expressed in units of mean free path of species molecules: x=n°Qx and y=n°Qy. Besides, mean values of m and ~ are used, which are m=(m1+m2J/2 and ~=mm0/(m+m0 ).
The expression (6) may be written as
- ff ~i 3 kTo Vc Xi 2 1p Uo -y)~f-. a .=-- [ I+ -2 --oo-2 (I+ <>-=- ) ] ( - !"" x~ 3 m . m u u y Vc
~
Now, with the expressions for ~ and axi determined, we can find characteristic separation quatities (1)-(3). Taking into account (5), we obtain the partial cuts as follows
Here the following notations have been used: blade position parameter
(8)
( 1 Q)
jet trajectory divergence factor
jet speading difference factor
a=(a /a )b-1 X1 X2
The subscript b indicates that all the values correspond to a certain position of the blade.
(11)
(12)
Excluding ~1 from (3) and (9), we obtain two functions, which can be written in general form as E=E(81~a~~1) and 82=82(81~a~~1J. Using these functions and the expression for the cut, 8 <1>, it is not difficult to calculate the relationship between E and 8.
This follows from (3) and (9) that the separation results from two effects. The first is a separation due to a divergence of the species jet average trajectories, it is characterized by the factor ~1 (11). The secend is a separation because of a difference of the species jet spreading, it is characterized by the factor a <12>. Note that in the case of "pure inertial Separation" according to (8)
Taking into account that a>O and ~1>0, one can easily see that the above effects cooperate on the right of the blade and counteract on the other side.
(13)
1301
For cut, 8, being not closed to 0 or 1, E is of the order of Eo, where Eo is the separation effect corresponding to the blade position in the midway between the species jet average trajectories <~1=0>, here approximately e~1/2. So, it is reasonable to use value Eo for preliminary estimations. If ~1<<!, we obtain
Eo~[4(2+a)/v'TI]~1 (14)
For ~m<<m the relation(13) gives a<<l. Therefore, the factor ~1 <11) i s of prime impotance. The importance of such a factor was previously remarked in Ref. 3. Expression for ~ 1 is easily found with using (7) and (8). One can see that ~1 increases as the "nozzle exit - blade" distance increases.
If the "nozzle exit - blade" distance is large enough, a divergence of species jet average trajectories will be represented by difference ~y=y 2-y1, which can be found from. the solution of (4) as ~x has been done (7). Corresponding distributions of species flow densities follow the normal law (5), where y and oyi appear instead of x and Oxi~ respectively. Expressions for ayi will be similar tothat for oxi (6), except for the coefficient before the first term in the square brackets.
As it is seen from (8), miximum of the separation effect will be achieved if T0 =0. Under this condition, at large "nozzle exit -blade" distance the separation effect appears to be "freezing" with the value E=Emax4 . If T0 j0, it does not reach Emax because of remixing process of the species separated at small distances.
It is usefull to determine a maximum value of Eo. Analysis of the above solution and the similar one for Vc>>u 0 , described in Refs. 5 and 6, shows that Eomax can be written as
Eomax~Egas diff A /(m+mo )/m0 ' <15)
where m=(m1+m2 )/2. The factor A is a function of M0 and Vc/u 0 •
Unfortunately, we did not succeed in determining of the function in a general form. It seems to be more expedient to use the MonteCarlo simulation for that purpose.
According to the above mentioned, the factor A increases with M0 and gets its maximum value at M0 ==(T0 =0). Under this condition, with using the Monte-Carlo simulation results we obtain A~3 for u0 =vc4,7 and A~1,3 for u0 =5vc (B.L.Paklin). Note, by the way, that the simulation results exhibit a good correlation with each other in accordance with the relation (15). For instance, in the case of 23ey23 SUF6 entering the flow of H2 4 we have A=3, and in that of C2+N2 entering the flow of He?, AZ3,13.
So, taken into account the order of A, one can see from (15)
1302
-V. 10
~ ~~::::: -------R--- --
=1:-b ---- __ .._X
Fig. 2. Mixture jet entering a Light gas expanding fLow.
that the Separation effects achieved by given method are more high as compared with the other gas dynamics methods1D,especiaLLy for a Large vaLue of m/m0 •
Mixture Jet Entering a Hypersonic Light Gas Expanding FLow
Pattern of the fLow under consideration is shown in Fig. 2. The expanding hypersonic fLow can be produced in a nozzLe, or behind the nozzLe by exhausting of highLy underexpanded jet. At smaLL distances, approximateLy up to the point of mixture jet returning (R-point in Fig. 2), the separation process is simiLar tothat under the jet entering the uniform Light gas fLow. The separation effect achieved is just a LittLe higher than in the Latter case. But behind R-point the separating process foLLows another way. Because of djvergence of the streamLines and cooLing of the Light gas, there is no remixing of species separated up to the R-point. Moreover, the Separation effect sLightly increases ~x(Yo-1 )/2, where y 0 is the ratio of specific heats of the Light gas. MoredetaiLs can be found in Refs. 5 and 6.
Using of SeLective Excitation of Species MoLecuLes
Being excited, atoms or moLecuLes have the other vaLue of their collision cross-section. It was shown in Ref. 9, and can be $een from (4)-(6), that the reduced cross-section difference, ~Q/Q, would contribute to the separation effect as much as the reduced mass difference, ~/m. According to the estimations, given in Ref. 9, under reasonable conditions the selective ex~itation of atom resonance Levels by Laser radiation is Likely to yield ~QeffiQ up to 0.4. Thus, for instance, the value of Eowax, which for "pure inertial separation" is given by C15), increases by the factor of ~0.4m/~m.
CONCLUSION
The expression for Eo C15), tagether with the relations (9), permits to estimate an amount of the separative work associated with
1303
the given method of separation. The value of Eo, corresponding to the separation effect at the cut ezJ/2, shows that the method appears to be promising, especially for m/m0 being large. In practice it is more appropriate to use an expandin~ light gas flow, because, first, in such a flow there is no remixing of species as in a uniform flow, and, second, there is no problern of its performance. At last, we believe that it is worth to pay attention to the possibilities, which the changing of collision cross-section of species molecules being selectively excited by Laser radiation provides.
RE FE RENCES
1. J.Gspann, Mass Separation in Molecular Beams by Crossed Free Jets, in: "Rarefied Gas Dynamics. Proceedings of the 9th International Symposium", v. 2, M.Becker and M.Fiebig, ed., DFVLR- Press, Porz- Wahn <1974).
2. J.B.Anderson, Low Energy Particle Range, Journal Chem. Phys., 63:1504 (1975).
3. J.Gspann and H.Vollmar, Mass Depended Molecular Beam Focusing by Cross-Jet Deflection, in: Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, p. 1, J.C.Potter, ed., AIAA, New York (1976).
4. R.J.Gallagher and J.B.Anderson, Isotope Separation in CrossedJets Systems, in: Rarefied Gas Dynamics, v. 1, R.Campargue, ed., CEA, Paris (1979).
5. S.F.Chekmarev, "Theory of Gas Mixture and Isotope Separation by Their Injection into a Light Gas Flow", Preprint 47-79, Inst. of Thermophysics, Novosibirsk, USSR <1979) - in Russian.
6. S.F.Chekmarev, Theory of Inertial Gas Mixture and Isotope Separation by Their Injection into a Light Gas Flow, in: "Proceedings of the 6th All-Union Conference on Rarefied Gas Dynamics, v. 2, Inst. of Thermophysics, Novosibirsk (1980) - in Russian.
7. Yu.S.Kusner, B.L.Paklin, and V.G.Prikhodko, Air Separation at Injection into a Light Gas Flow, in: "Proceedings of the 6th All-Union Conference on Rarefied Gas Dynamics", v. 2, Inst. of Thermophysics, Novosibirsk (1980) - in Russian.
8. A.V.Bulgakov, Yu.S.Kusner, V.G.Prikhodko, and A.K.Rebrov, Separation of Gas Mixture Components in Interacting Flows, in: Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, v. 1, S.S.Fisher, ed., AIAA, New York (1981).
9. V.G.Dudnikov, S.F.Chekmarev, On Separation of Particles Selectively Excited in Crossed Jets, Pisma v Zhurnal Tekhnicheskoi Fiziki, 9:1174 (1981) - in Russian.
10. R.Campargue, J.B.Anderson, J.B.Fenn, B.B.Hamel, E.P.Muntz, and J.R.White, On Aerodynamic Separation Methods, in: Nuclear Energy Maturity", P.Zaleski, ed., Pergarnon Press, Oxford (1975).
1304
IIOLECULAR DD'lUSION THROUGH A PINE-PORED PILTER VERSUS
RESONANTE IR-RADIATION INTENSITY
V.A.Kr8vchenko, A.N.Orlov, 8nd Yu.N.Petrov
P.N.Lebedev Physic81 Institute of the USSR Ac8demy of Seiences, Leninsky prospect, 53 II7924, Moscow, USSR
INTRODUCTION
Under l8ser radi8tion 8 molecular interaction with the surf8ce is changed th8t m8y result in sh8rp vari8-tion of 8 nature of diffusion 8nd moleeular 8dsorption in a fine-pored medium. A prineiple possibility to eontrol the moleeular flow by laser radi8tion through a fine-pored nonabsorption filterwas shown in /I-4/. It was reve8led that electronsor vibrationally excited molecules undergo the gre8ter barriers when diffused tbrough the fine-pored structured then the unexcited ones. But a meohanism of this phenomenon, however, was not clarified. In /I,2/ decreasing of the molecular flow of boron trichloride, being exeited by co?-laser radiation through BaCl filter with the pore diameter ... 40 f'I'Yl was explained by increasing of sorpt ion of the vibr8tionally excited molecules. To explain the first experiments on molecular bromine diffusion, excited by argon l8ser radiation,othrough 8 fine-pored quartz filter (pore diameter 80 A), it was advanced a supposition on formation of the stable chemical compounds with bromine atoms, formed 8S a result of dissociation of the electron excited bromine molecules in the pores /3/. However, the following bromine investig8tions /5/ showed, that a ohemical mechanism of keeping the excited bromine molecules in the pores is not decisive,a main process - is physical sorption /6/. On the basis of the experiments on vibrationally excited toluene molecules traversing a fine-pored mica, it was also 8Ssumed that together with sorption a membrane effect for the exciting molecules takes place /4/, i.e. under l8ser r8diati-
1305
on a decreasing of the resonance molecular flow through a fine-pored structure is caused not via their accumulation inside the pores {filtration process), but via a change of velocity of molecular diffusion in the resonance radiation field which is caused by decreasing of diffusion coefficient. The latter is decreased because of adsorption potential depth changed due to polarized action of resonance radiation upon the molecule /5,7/.
A difference in behaviour of the resonance {with induced dipole moment) and nonresonance molecules in the fine pores has a same nature as the difference of adsorbent interaction between dipole and nondipole molecules. An interesting property of molecules with large dipole moment lies in its mutual surface orientability. If the pores are regular and strictly orientated, these molecules can be orientated with respect to the whole sample, that was observed in the experi~ ments on orientational polarizability of C02, CCl4, H20, c7H8 , CH3I, C5H~N, c3H6o, CH3CN in a fine-pored mica and polymer film /8/.
EXPERIMENT
The diffused gas flow experiments controlled by IR-radiation are easily conducted with fine-pored mica, non-absorbed up to 6-7~Wl and having good thermochemical properties. We have used the mus~ovite mica samples K20 )( 3Al2o3 x 6sio2 x 2H2o = (All 2,r. LAlSij} ~oi0x {OH)2K• A technoiogy of fabricAtion ofthe ~50 I diameter pores was the following: the mica was bombarded by protons with the energy of IO Mev, avalanche density Ioi5cm-2 followed by its etching. The dimensions of the channels developed were defined with respect to a mercury ~~egram of the sample, standart KCl solution conductivity and electron microscopy methods. The pores are of rectelinear geometrical cross-section forme {rhomb~, the deviations of the channel parallelism are <~ 5 , that refers to a lack of mica amortization for such a technique of filter fabrication. A thickness of the sample was 40 )A-m, diameter - I cm.
We have investigated the change of the porous crystalline mica permeability for toluene molecules under resonance CO-laser radiation. Gas flow traversed the filter was recorded by a quadrapole KM-2 mass spectlometer. Gas pressure at the input of filterwas
Io- torr, the gas flows through filter were up to IO mg/hour {Knudsen flows).
The most intensive radiation spectrum lines of
1306
~ I.02 ~
~ r.oo ~ 0
~ 0.98
0 ~ 0.96 ~
0.94
0.92
~ ~ ·~ ~ ~ ~ ~ M 0.90
(wt/cm2) Fig.I. The relativ. e toluene flow changes veraus
radiation intensity: a - flow 3 mg/h, b - 0.5 mg/h, c - O.I mg/h
h ~ t e continuous Co-laser are in the range of I750-I880cm. In switching on laser radiation the toluene transmission through a filter is sharply .decreased. Diffus-ed flow of toluene molecules through the pores is al-so decreased.In the controi experiments decreasing of nonresonance gases (N2,SF6) transmitted through the filter under radiation was not revealed. The effect of laser influence is unchangeable very long (it is even constant after transmission of toluene of hundrede of milligrams).
A good ~resonance property of toluene makes it possible to observe laser effect under the relatively low radiation powere. Thermal influence of laser radiation upon those powere• transmiasion was not observed (filter heating leads to increasing of filter permeability for all the gases investigated).
The relative changes of the gas flowsunder reaonance laser radiation veraus radiation intensity ia of special interest /9/. We have studied these dependences for toluene with the same porous mica varying the intensities from O.I to IO w/om2. A presence of the strictly expresaed resonances on the curves of these dependences (see Fig.I) in intereating as well. With smooth changing of radiation intensity the periodic variations of laser influence upon the resonance molecular flow is observed. But there are the radiation intensity regions at which the moleoular flow through a fine-pored filter becomes larger then when it is free of radiation. Note, that such a flow increasing
1307
'IJD.d.er laser radiation is well distinguished from the thermal influence. The characteristic times of flow establishment in switching on and -off laser radiation -I sec. To avoid thermal influence the flows were measured in switching on -laser radiation for a short time (I~20 sec), a thermal effect here was small. A value of the laser effect observed very much depends on a degree of filling of the filter aperture with radiation. In, condu!Jting a great number of measurements we have to diaphragm a beam so, that not to illiminate the filter edges in the places of glueing, therefore the 1isted values of flow measurements are relative ($ee Pig.I). With a total filling aperture the flow varia• tions in switching on laser can be reached 20~25% under the same values ot radiation intensities and gas flows.
A detail consideration of the transition processes in switching on and -off radiation allows to clarify a number of specific peculiarities. In the fields of laser radiation intensities which involve decreasing of the resonance molecular flow under irradiation (see l!'ig.I) at long radiation influence after qu·ick decreasing of flow it is slowly increased (thermal effect), and after switching off radiation - it slowly drops (see Pig2b). It is interesting to observe in .another fields of intensities the appearing of the sharp peaks in switching on and -off radiation (see Pigs.2, c,d,e). The peak time is ,.. I sec.
1308
Pig.2. A scheme of resonanee radiation influence
a- upon molecular flow through a filter
b- the molecules are not orientated on the surface
from c to d ~ the orientational effect increases.
THEORY AND DISCUSSIONS
The observed effects can be understood if to proceed from a view of polarized effect of electromagnetic radiation upon the resonance molecules, diffused in the porea. A change of the adsorption potential -AU for the molecule, polarized by laser radiation is realized due to interaction with the same molecules on the surface, as well as with a dipole image induced in the surface material. It can greatly contribute in a change of U in the case of free pure surface. The adsorption potential growth decreases the surface diffusion coefficient D:
fnn -d..U/KT ( 0.. = 0.06 - 0.45)/I{V,
A general expression for the interaction potential between two molecules, caused by polarized interaction of laser radiation was obtained in I 7 /:
f:-+ ...,. .... ...,. ... _:1 ..,. ... ...,. ... ~d 1 ,d2)-3(d1 ,n)(d2n~X(E0 ,d1)(E0 ,d2 )
u12= t- 2 j X 411 R
[
(JJ ...... ] e i <,f- R-k.R > ------........ ----+ c.c:. < 81 + i rl > < s2- i 't2 >
(I)
where 8 = "'! - We - is the laser frequency tuning out of resonant I and 2 molecule line centers • .. ~ - are the homogeneaus line widths, dj - are the matrix elements of operators of the dipo
le transition moments, :rk - is the wave vector, ... .. -+
- is the molecular radius vector, R=r2-r1 , n=-~/lR.I. Change of the adsorption potential 6. U , caused
by interaction of the polarized molecule an8 its mirror image, induced in the material of wall can be easily obtained from the above formula
..L -- '-. d4E2cos2 » ( 1 +Cos2 ~ ) ( 3.t 82. + ~I l2. ) t.Uo- d.. ":ll. 2 (\2. 2. ~2. 2.
I6R()ll ( «':t + 'lt ) ( o2 + 't2 )
(2)
... -+ if E is perpendicular to the surface. When E is paral-lel, the first cosine is changed by eine. Here R - is the distance between molecule and surface, 9 - 1i the
1309
angle between E and d; 8:r, Sa , 'IT. and 1~ - are the molecular line widths and tunings, respectively, in the gas phase (index I) and for the images (index 2), ~2 -is the refractive index out of surface.
AU is always ~ 0, i.e. the adsorption potential is depresied under resonance field. A strong dependence of AU0 on molecular orientation is important. When A U}kT
the orientated interaction of resonance laser radiation on the adsorbed molecules become noticeable.
Consider the interaction of polarized molecule with the adsorbed ones. If to account, that change of the adsorbed potential AU can be approximated by a sum of the pair interactions, then on going from summing to integration for the arbitrary orientated molecules on the surfacJl, when the coefficient of the surface fllling ~I, for E~of the surface and homogeneaus line broadening we obtain: 8d4E2e2 J..2.cos2 ~ ~ 8't82. +1t Y~ ) AUJ.= - 311" 1i2< A2. +I6R~>3 2 < 8l + tl· >< s: + rl > <J>
here 8x' 82 ' r'I and Y2 - are the tunings and molecular line widths respectively in the gas phase (I) and on the surface (2).
At R0 commensurabled with the molecular sizes it should be considered also the molecular interaction with the molecules of nearest environment, calculated by summing of the pair interactions ~ UJ.• This sum depends mainly. on the molecular location and qrientation~
At an· accurate resonance ( l'i > b"2. or Y1 > OI ) the projections of the induced molecular dipole moments on the surface are simultaneously directed along and against the field. Therefore, with an arbitrary orientation ,.2L the diffused '!POlecuJ,ar moment out of the gas phase AUJ.. = 0. When ö2l-t2, 81:~tt the projections of the molecular dipoles induced on th~ surface and gas phase will be parallel with 8r and ö2. having the same signs, and antiparallel in case of different signs. If the molecule is at R0 distance over the adsorbed molecules, then it is given by the equation (I), but if it is over the center of a square with diagonal 2a in the angles of which the adsorbed molecules are placed A U1 will be given as follows:
(a2-R~)d2 AU~ = 4 • A (4)
( 8 2+ R~ )5/2
wher-e A :{exp[i(<lt)R-kR]/Cclx+ift)(Sa-lJi)+c:.c}CS.,drJ(E0,J2)/4.ha. In this case while averaged the potentiale (I) and_i4)
1310
according to the different molecular coordinatea, AU~ ia not va~lahed.
When ~ is directe~along the aurface and large R0 one can write ( 8t. = ~2 = 0):
~ P d4E2.cos2 9
With R0 comparable with the molecular aizea, tated ~erm ahould be added to thia potential
' 32d2(b2 - a 2) AUu = - (4R2 + a2+ b2)5/2
0
• A
an orien-
(6)
if the molecule ia over the center of a and b aide reetangle (along the field) in the anglea of which the adaorbed moleculea are placed.
The diffuaed moleculea will be depoaited in the definite placea of the surface and orientated, if the potentiale A U-1. and •Ut >r kT. Value AU/kT will charecter~ze a degree of molecular orientation along the field E. It the adaorbed mo}eculea are orientated, the absolute values AU and AU will increaae, that reaulta in greater orientability of the depoaited molecules. Finally, a degree of molecular orientation will depend on filling of the surface with AU( 9 ) and AU' ( 9 ) , A U0 ( ~ ) aa well as adsorption potential free of field - U ( Ci) ) •
Thua, a polarized action of laaer radiation upon the resonance molecules decreasea their diftuaion coefficient in the case of alight filled surface. At aufficient number of molecules on the aurface their orientation under the resonance field may become noticeable. Molecular orientation will reault in increasing of the diffusion coefficient. Note, that the definite orientations can domain veraus a degree of the surface filling.
The effect of the diffused flow decreasing observed in the experimenta under laaer radiation is explained by deposition of the adsorption potential. At some values of radiation intensity the molecular orientation on the pore surface takea place, that weakens this effect. Filter heating under long radiation caused to molecular deorientation, i.e. slow decreasing of the diffused flow (aee Pig.2e). Under chaiotic molecules a thermal effect will be reversed (see Pig.2) The firstpeak can be explained by thermal molecular evaporation from the sample surface (Fig.2·C, d, e), the second and third ones - by the transition processes
1311
of molecular orientation. The peak's duration is deteP. mined by the time of molecular accumulation on surface and that of molecular diftusion through the sample.
In conclusion we note that the found molecular property of self-orientation on the surface under resonance radiation i' general and will be exhibited in a wide range of waves and for a large group of the finepored materiale.
REPERENCES
I. Gochelashvili K.S., Karlov N.V., Karpov N.A., Mdinaradze N.I., Petrov Yu.N., Prokhorov A.M. Laser isotope Separation under filtration diffusion. Pis• ma v ZHTP,2 I6,72I(I976)- In Russian.
2. Gochelashviii K.s., Karlov N.V., Petrov Yu.R. Laser isotope separation under filtration gas ditfusion. Trudy FIAN, II4, I68 (I979)-In Russian.
3. Karlov N.V., Meshkovsky I.K., Petrov R.P., Petrov Yu.N.,Prokhorov A.M. Laser control with molecular screen permeability. Pis'ma v ZHETP,30,48 (I979)In Russian.
4. Kravchenko V.A., Lotkova E.N., Meshkovsky I.K., Petrov Yu.N. Control by molecular transmission through a porous crystal with IR radiation. Pis•ma v ZHTP,7,I9,II97(I98I)- In Russian.
5. Orlov A.N., Petrov R.P., Petrov Yu.N. Laser radiation influence upon the molecular sorption in finepored filter. ZHTF (I982)-In Russian (tobe publ.)
6. Petrov Yu.N. Laser influence on gaa diffuaion in fine-pored filtere. Proc. of Intern.Conf.Laser-80, 852 {I980).
1. Karlov N.V., Orlov A.N., Petrov Yu.N., Prokhorov A. Interaction between molecules in the field of resonance electromagnetic radiation. Pis'ma v ZHTP, 8,7,426 (I982)-In Russian.
8. Kravchenko V.A., Pe~rov Yu.N., Kuznetsov V.I., Alexandrescu R, Komanichi N., Mikhailescu I., Morzhan. Discovering of orientational polarizability in the fine-pored membranes at adsorption of dipole molecules. Pisfma v ZHTF, 8,I4,848 (I982)-In Russian.
9. Kravchenko V.A., Petrov Yu.N.Discovering of the resonance IR-radiation molecular diffusion through a fine-pored filter. Pis'ma v ZHTP.(I982) (tobe publ.)-In Russian.
IO.Frank O.Goodmatl. Interaction Potentials of gas atome with cubic latticee on the 6-I2 pairwiee model.Phye.Rev.,I 64,No 3, III3 (I967).
1312
ON LIMITING SITUATIONS OF
GAS DYNAMIC SEPARATION
Yu.S.Kusner, B.L.Paklin, and A.K.Rebrov
Institute of Thermophysics Siberian Branch of the USSR Academy of Seiences Novosibirsk, 630090, USSR
The gasdynamic separation of very heavy molecules with dif~ fering mass is of a theoretical and practical interest. The present paper treats the spatial species separation taking place during injection of a gas mixture into Light ~as. The separation of a mixture of molecules of masses 349 and 352 a.u.m. which were injected into hydrogen uniform flow (Fig. 1) was studied1 by Monte Carlo method, for a case when the separated mixture was injected perpendicularly to the hydrogen flow at a velocity V equal to flow velocity U, and the temperatures of all gases were zero. It ~as assumed that molecules in the mixture did not interact and collided only with molecules of the Light gas. With these assumptions the problern is essentially simplified andeasily solvedby Monte Carlo test particl~ method.
The same scheme of gasdynamic separationwas investigated in the present paper. The purpose is the determination of conditions under which a separation factor ß (for the Lighter mass relatively initial composition of a mixture) has its maximum value at a given cut 8 (the condition for producing maximum separation work oU for a given 8). Since it is known1,2 that for nonzero temperatures of gases the separation factor for a given 8 decreases with increasing temperatures, it may be expected that the obtained dependencies ß(8) will be Limiting in this sence.
From the analysis of equations describing the variation of velocity of a heavy molecule in the hard sphere model (elastic colLisions) for the above separation scheme it follows that value K=V/U is the only parameter (for the predetermined molecular masses). The importance of this parameterwas indicated by Chekmarev3. Value A=(noa)- 1 , where a is the cross-section of collision of mo-
1313
Fig. 1. Scheme of gasdynamic separation of gas mixtures with molecular masses m1 and m2 by their injection into Light gas flow with molecular mass mo.
Lecules of mass m1 and m2 with H2 molecules and no is the number density of H2 molecules, serves as a unit Lengtr. Fig. 2 shows dependencies E(G) CE is the enrichment factor, E;ß-1) for K=1 at x=300 and for K=4 at x=1200. Function E(G) is the same in both cases: E (G)~ln8 and value of E is much greater than the free-molecular value Ef.m.=vm2/m1-I.
E
\ • • • • • 0.20 • •
• • • •
• • • - ---1 •
• 010
-tn7r Fig. 2. Enrichment factor versus 8 for different K: 1 - K=1 at plane x=300, 2 - K=4 at plane x=1200.
1314
Fig. 3. E(G ) at plane x =1200 and different values of K versus the theoretical dependency E(G )=Eoo(G)(l-exp(-K)) which is represented by solid Lines.
The value of ß(G) obtained for e=O.S and K=1 is in a good agreement with the value of separation factor a (defined for "tails" a=ß2 for 8=0.5) received by Gallagher and Anderson1. Functional dependency E-lnG (Fig. 2) qualitatively corresponds to that obtained by Kusner4 for small e for the initial stage of the separation process at the first collisions when the heavy gas velocity Vy~v (E(G)=-4Ej'.m.lnG). The present calculations give much higher coefficient of proportionality, for K=4 it is approximately 8 times as Large. In the case of conservation of dependency E(G)-lnG for all separation stages the optimum cut value Gopt. obtained from the condition of the separation work maximum value for small 86U-E28( 1--e) : G0pt~e-2 retains its value for all separation stages.
The separation factor E is freezing at increasing x because of molecules of the mixture at increasing time Lose their initial velocity and acquire the velocity of Light gas. Therefore one may regard values E(G) as Limiting if plane x=const in which the Separation of species is defined is far enough downstream.
Dependencies E(G) for the plane x=1200 and different K are shown in Fig. 3. Values E(G ) practically are not changed for K exceeding 4. The influence of the parameter K is well described by relation
E(G )=Eoo (G)(I-exp (-K))
represented in Fig. 3 by solid Lines. Within the Limits of inevitable statistical scatter the results obtained by Monte Carlo method and
1315
the above dependency are in satisfactory agreement. The Limiting enrichment factor value Eoo(G) is attained when K-oo. For K=4 the enrichment factor is 0.98 E00 (8) which implies a sufficiently fast convergence of results when increasing K.
The behavior of quantitative characteristics of separation process when approaching to Limiting situations is essentially defined by temperatures of the mixture to be separated and carrier gas. The method of calculation permit to divide the influence of temperatures. First Let us consider the influence of the temperature of species being separated upon enrichment factor when the temperature of carrier gas is zero and after that Let us investigate the separation of heavy molecules beam at uniform velocity under injection into Light gas with nonzero temperature.
Suppose that the distribution function of velocities of molecules in the mixture is Maxwellian with mean molecular velocity V. Except parameter K it is naturally to introduce a new parameter Sin=V/v2kTinlm1 which is the molecular velocity ratio for one of the molecular mass of the mixture to be separated, where k is the Boltzmann constant, Tin is the temperature of mixture in the point of injection. As well as in the above case dependencies E(G) are considered for sufficiently distant plane x=const for the definite value of K. The experimental investigation of the separation for high values of K may be realised for relatively small values of molecular masses of mixture to be separated. That is why at this step of investigation the mixture of gases with molecular masses 28 and 32 a.u.m. was used. The dependency of enrichment factor for e=O.S and K=10 versus Sin for carrier gas - helium is shown in Fig. 4. The dashed line is E in the case of zero temperatures of gases for e=O.S and K=10. The results are attributed for a plane
E
1 0
1 2 3
Fig. 4. Separation of air by injection into helium at plane x=150 for K=10, 8=0.5 and different values of Sin·
1316
x=150. For another values of 8 behavior of E versus Sin is analogous and for all 8 is well decribed by formula
d8, K)=Eo (8., K) {1-exp ( -SinJ)
where Eo(G.,K) is the enrichment factor versus 8 for ideal case of zero gas temperatures for definite value of K. When Sin-ooE(G.,K) is asymptotically close to its Limit Eo(G.,K).
Another case- the temperature of Light gas is not zero. Suppose that all molecules of entering mixture travel at the same velocity V, it denotes that Sin=oo. Program of calculation for this case become more complicated due to including of calculating of collision partner velocity for a test molecule. Suppose that the velocity distribution function of Lightgas is Maxwellian. In this consideration parameter S=U/v2kT/m 0 where T is the temperature of Light gas, mo is the mass of its molecule is introduced. It is found that for small values S-I the Separation factor for the definite value 8=const has maximum downstream, then it falls at increasing x due to great chaos of collisions. For Large values of S the dependency of separation factor at increasing x is more smooth. Maximums E(8) for the mixture with molecular masses 349 and 352 a.u.m. entering hydrogen for K=0.5 and different values of S are shown in Fig. 5. The results obtained for masses 349 and 352 a.u.m. are described by dependency
E(8)=E 00 (G)(l-exp(-K)}(I-exp(-S)}.
Curves in Fig. 5 are constructed in proposing of ~auss distribution function of Coordinates y in intersecting plane x=const by trajectories of test molecules 5,6.
As it seen from the above expression E is changed in the
E
0. 1
Fig. 5. The influence of the carrier gas speed ratio S upon the separation factor: 1- S=4 at x=200, 2- S=2 at x=100, 3- S=1 at x=25
1317
same way by changing K and S for every e. Therefore the dependency E(G) is universal and it follows from the condition of maximum separation work that the sameoptimal value of cutwill be for any situation in this scheme.
Common regularities obtained in this simple scheme of Separation allow us to carry out estimations of the effect of gasdynamic separation. We may confirm on the base of results obtained that sufficiently Large value of the parameter K is the main condition for getting of maximum separation effect. But it need not forget the role of thermal parameters Sin and S for every concrete gasdynamic separation device.
Authors wish to thank Dr. M.S.Ivanov for useful discussions.
REFERENCES
1. Gallagher R.J. and Anderson J.B., Isotopeseparation in crossedjet systems, in: Proceedings of Rarefied Gas Dynamics, XI-th Int. Symp., 1:629, Paris (1979).
2. Kusner Yu.S., Paklin B.L., Prikhodko V.G., Separation of air by injection into Lightgas flow, in: "Proceedings of the 6-th ALL-Union Conference on RarefiedGas Dynamics", 2:176, Novosibirsk <1980) - in Russian.
3. Chekmarev S.F., Gas or isotope separation by injection into Lightgas flow, in: Present Book.
4. Kusner Yu.S., About the theory of gasdynamic separation, Journal Doklady AN~, 295:2 (1981)- in Russian.
5. Chekmarev S.F., The theory of gas mixtures and isotopes separation by their injection into Light gas flow, Preprint 47-79 of Inst. of Thermophysics, Sib. Br. of the USSR Academy of Sciences, Novosibirsk, (1979) - in Russian.
6. Anderson J.B., Low energy particle range, ..:!._. Chem. Phys. 63:1504 (1975).
1318
A STUDY OF REVERSE LEAKS*
E.P. Muntz, Shao-Sheng Qian,+ R. Duenas, D. Benjamin and A. Shoostarian
Department of Aerospace Engineering University of Southern California Los Angel es, Cal ifornia 90089-1454
INTRODUCTION
The penetration of a background gas into a higher pressure gas source, what we call a "reverse leak", is of interest in a number of appl ications. Protection of the cold optical components of infrared telescopes is one example (Ref. 1). Another is the escape to the high pressure region of small amounts of toxic materials through microscopic holes in a pressure barrier, for example tritium escaping the interior of a fusion reactor (Ref. 2). A third example of background penetration into a source region is the Jet Membrane isotope separation scheme (Ref. 3).
BACKGROUND
There is an extensive Iiterature of studies concerning background gas penetration of under expanded jets. lt is from this work that a general perspective can be obtained on the background gas penetration of a source. Campargue in France,4,5 Rebrov and his co-workers in the Soviet Union6,7,8 and Muntz and his co-workers in the United States9-13 have studied various aspects of the invasion of a background gas into a plume of gas expanding into a low pressure region. The details of this work can be found in the references. For our purposes here, it is worthwhile to review certain physical concepts that are useful for helping to understand this flow phenomena.
There are two important dimensions associated with the penetration of background species into a gas plume. ln Fig. (1) these dimensions are illustrated. Consider the general case of a gas at a source pressure Ps expanding
·'· Supported by Utah State University
+Visiting Scholar, Department of Engineering Physics, Tsinghua University, Peking.
1319
into a low pressure background gas at pressure PB· One of the important dimensions is the distance from the source, along the plume centerline, at which the background concentration has been reduced to e-1 of its value far from the source. This distance is called rp. The other distance is the length of travel of a typical jet molecule in the background before it has experienced a coll ision with a backgrounJ molecule. This distance is called rA. ln both cases these distances can be non-dimensionalized with the source exit diameter, D, used as a characteristic length to give Rp and RA. For the classic case of a sufficiently high pressure in the source and a sufficiently low pressure in the background, simple expressions can be developed (Muntz, Hamel and Maguire,9 or Zarvin and Sharafutdinov7), for Rp and RA. These simple ex~ressions have been val idated by more rigorous analysis and by experiment.7,10, 1
The flow regime of the interaction between a gas plume and its background can be described by a so-called plume Knudsen number where Knp = RA/Rp· For Knp<O.l, the plume background flow is in the continuum region with relatively thin shock waves and mixing layers. For Knp>l the background gas can more or less freely penetrate the plume up to a distance Rp from the source. For conditions where Rp<l the background gas begins to penetrate into the source itself. Other things bei~g the same, the distance Rp is controlled by the pressure Ps or Rp = Vrelns~BjD/C 8 , where Vrel ~s a typical relative coll ision speed between background ana jet molecules, Cs is the mean background speed directed at the
BACKGROUND
R = r/0
1.0
0 R
Rp
Fig. 1. Nomenclature for background penetration of a source gas.
1320
source, ~Bj is the backgro~nd-jet molecule coll ision cross-section and n5 the source number density (n 5 = p5 /kT).
Flow conditions where Rp>l have been extensively investigated by Rebrov et al.,6 Zarvin and Sharafutdiriov,7 Deglow and Muntzl2,13 and by Brook et a1.14 However, it is the region where Rp<l that is of interest in the case of the purging protection of infrared telescope optics. This flow regime has not yet been studied in any detail.
A flow regime map in the parameter space log Knp, log (p 5 /Ps) is shown in Fig. (2). On it, the parameter region of interest to this study is indicated as the cross-hatched region. The molecular flow region with Rp<<l is not of interest because it permits essentially free penetration of the source. The curves for Knp>l can be constructed from the relationship
( - 2 Knp Vj/C 8)(mj/m8) (p 5 /P 8)/R p
2.0
Il 1.0
- 2.0
0
Rp<1
Knp > l
SC AlTERING
~
Rp > 1 Knp>1
PLUME INVASION
TRANSITION REGION
1 2-: Knp?:O . l . Rp >l
CONTINUUM REGION
Knp«1 . Rp>l
log (P5 /P 8 )
SOUACE PENETRATION PARAMETER SPACE MAP
Fig. 2. Parameter space map of background penetration of source gases.
1321
EXPERIMENT
We have completed a series of experiments on the characteristics of background gas penetration into a source region. These were done in the apparatus that is shown in Fig. (3). ln all cases the area of the sampl ing probe was less than 4% of the area of the orifice or tube used in the experiment. The relative background gas number fluxes extracted by the sampling probes for an orifice and length to diameter ratio tubes of 1 and 2 are shown in Figs. (4,5,6). The solid 1 ines shown in the figures are a theoretical prediction of the background number density based on the MHM analysis9 but using free-molecule source gas number density fields derived from Patterson.l5 The predictions asymptote to a constant value in the source because the mean flow velocity of the source gas goes to zero here. This is a physically correct result but the "prediction" is of course based on "ad hoc" assumptions about the relevant collision dynamics.
For the conditions of the experiments the background gas entering the source region can only leave by reverse diffusion back through the source. Thus, the number density of the background in the source region (away from the vicinity of the orifice) is a result of the balance struck between the "reverse leak" into the source region and the diffusion of backgroundout of the source in order to satisfy mass continuity.
1322
pressure heod of MKS Boratran Pressure Meter
to diffusion pump
to moss spectrometer ._____
SOURCE
bockground gos
BACKGROUND
PROBE
""'" ,"-1~P -I ' to diffusion
pump
Fig. 3. Schematic of ex~erimental apparatus.
N2 Source A Background
D = 1.5 cm Or1fice
Predictions ore solid lines
Points ore experiment
0-6 -5 -4 -3 -2 -1 0 1
SOURCE ORIFICE r, DISTANCE FROM
2 3 4 5 6 BACKGROUND
ORIFICE (cm)
7 8
Fi g. 4. Sampl ing probe relative mass fl ow for Argon background penetrati on i nto a nitrogen source gas through an o rifice .
~ ~ 0 .7
LL. ~ 0.6 w_j CDLL 0 w 0 .5 O::CD
N2 Source A Background D=1.5cm
L/D = 1.0
~ ~ 0.4 ~~ ::::i l:) 0.31---...--!~!.---o._Z
15.0
~ i2 o, 2t;r--&~r--o--&--c.pa-"" (/)~
+ + + +
~ 0.1
0 - 7
20.0
- 6 -5 -4
SOURCE
-3 -2 -1 0 I
Tube Entronce
Predic tions ore solid lines
Pomts ore experiment
I 2 Tube Exit
3 4 5 6 7 8
BACKGROUND
Fi g . 5. Sampling probe r e l ative mass f l ow for Ar gonbackground pene trating a nitrogen source through a L/0 = 1 tube.
1323
By adopting this model it is possible t o make estimates o f the size of t he "reverse leak". Because many of the results a re for Knudsen numbers where f ree molecule flow is not a good a pproximation we adopted the foll ow ing procedure . lt was assumed that the form of the molecular velocity di st ributi on function of the background in the source region wa s the same as that of t he source gas . The discharge coefficient (mass flow divided by the free-molec u le mass fl ow) f o r background gases through an orifice and the t wo tubes were measured as a function of Knudsen number . With this info rma t ion and the known source gas Knudsen number in any exper iment, as weil as t he measured background number density Ievel in the source region, an esti mate could be made of the background
N~ I I I !!. Source ~! w 0.9r- A Background X
(!) + 6 0: D = 1.5 cm
X
..... ::3 0.8 X 6 -<l L/D = 2.0 X +
~ ~ 0.7- X +6
~ X+
LL g 0.6 X + 6 -w LL + 6 CD 0.5r- X -0 w 0: CD X 6 Points ore experiment Cl. 0 0.4 P5 = 10.0fLI1gx
+ (!)0: X 6
+ zCl. 0.3 X X X X -(!) + 6 erz 15.0 + ::!::::::i 0.2 + + 6 <lCl. + + + 6 Cf);:!:
0.1 20.0 66
<l 6 6 6 6 (/)
0_4 I I I I I I
-3 -2 - 1 0 1 2 3 4 5 6 7 8
SOURCE I I BACKGROUND Tube Tube Entronce Exil
Fi g . 6. Sampl ing pro be re la tive mass f low fo r Argon background penetrating a ni t rogen source th rough a L/D = 2 t ube .
gas mas s flow from the source in to the backgrou nd . Since thi s had t o equal t he mas s flow of backg round in to t he sou rce reg ion , an es timate of t he si ze o f the "reve r se leak" i s availabl e.
The "re verse leak" va l ues obtained as described a bove, ra tioed to the reverse leak if there were no source gas pressure (and the reverse leak was in free molecul a r fl ow) are shown in Fig. (7) f or orifice and an L/D = 2 t ube . Note t ha t i n all cases the Knud sen number of the o ri f ice o r tu be ba sed on its d iameter and t he bac kg round pressure were grea t er t han one.
1324
Orifice, 0=0.635 cm + Orifice, 0 = 1.5 cm t::. L/0=2, O=t.5cm
PREOICTION FOR ORIFICE
Fig. ]. Reverse leak mass flow compared to the free flow mass flow through an orifice or short tube.
CONCLUSIONS
The re are significant "reverse leaks" even at source Knudsen numbers of 0 . 2.
The size of the "reverse leak" varies exponentially with source Knudsen number, even though the flow regime of the source gas is changing .
There is only a slight difference between an orifice and L/D = 2 tube when the "reverse leak" i s scaled to the free molecule flow reverse leak into the source reg ion.
REFERENCES
1. A. Guttman, R.D. Furber, E.P. Muntz, "Protection of Satellite Infrared Experiment (SIRE) Cryogenic Infrared Optics in Shuttle Orbiter", Optics in Adverse Environment Proc. Soc. Photo-Optical Instrumentation Engineers, vol . 26, 1980, pp. 174-185.
2 . J. Brook, priva t e communication, 1980.
3. J . W. Brook, V.S . Cal ia, E. P. Muntz, B. B. Hamel, P.B. Scott, T. L. Deglow, Aerodynamic Separation of Gas Mixtures and Isotopes of Sulfur and Uranium by the Jet Membrane Process, AIAA J. of Energy, 1981.
4. R. Campargue, Compte Rendu, 268A, 1969.
1325
5. R. Carnpargue, J. Chern. Phys., 52, 1975, 1969.
6. A. Rebrov, S. Chekrnarev and R. Sharafutdinov, J. Applied Mech. and Tech. Phys. 1, 136, 1971.
7. A.E. Zarvin, R.G. Sharafutdinov, J. Applied Mech. and Tech. Phys. 4, 11, 1976.
8. A. Rebrov, in Rarefied Gas Dynarnics, L. Potter ed., Progress in Aeronautics and Astronautics, vol. 51, 11, 11, 1977.
9. E.P. Muntz, B.B. Harne] and B. Maguire, AIAA Journal, 8, 1651, 1970.
10. J.W. Brook and B.B. Harne], Phys. Fluids, 15, 1898, 1972.
11. J.W. Brook and B.B. Harne], E.P. Muntz, Phys. Fluids, 18, 517, 1975.
12. T. Deglow, Ph.D. Thesis, University of Southern Cal ifornia, June 1977.
13. T. Deglow and E.P. Muntz, J. Appl. Phys., 50, 2, 589, 1979.
14. J.W. Brook, V. Cal ia, E.P. Muntz, B.B. Harne], T. Deglow and P. Scott, AIAA J. of Ene rgy, 4. 5, 199, 1980.
15. G.N. Patterson, lntroduction to the Kinetic Theory of Gas Flows, Univ. of Toronto Press, 1978.
1326
INVESTIGATION OF NONEQUILIBRIUM EFFECTS IN
SEPARATION NOZZLES BY MONTE-CARLO SIMULATION
ABSTRACT
Wo Schwan, M. Fiebig, and N.K. Mitra
Ruhr-Universität Bochum, Institut für Thermo- und Fluiddynamik, FRG
A.U. Chatwani
DFVLR - Köln, FRG
A tracer Monte Carlo procedure is used to study the nonequilibrium effects between heavy isotopes and light carrier gas and the separation processes in a cylindrical separation nozzle. For trace concentrations of the isotopes in the carrier gas the transient radial and azimuthal velocity and temperature slip between the high mass disparity components are calculated for Knudsen numbers from free molecular to nearly continuum flow. The carrier gas Mach number is high subsonic (M ~ 0.9), the isotope Mach number is hypersonic (M ~ 10) and the mäss disparity 175. Even for carrier gas Knudsen nuMber 0.02 the radial isotope temperature becomes more than four times the carrier gas value and the radial diffusion velocity reaches values more than twice the thermal speed of the isotopes. The separation processes are functions of Knudsen, Mach number and deflection angle, while in the corresponding continuum theory only Knudsen number times deflection angle and Mach number occuro The transient enhancement of the isotope separation effect over its equilibrium value reaches a value of 2.2 for Kn = 0.02 and becomes zero with the free molecular limit. c
INTRODUCTION
The standard sepfration nozzle, developed at the Karlsruhe Nuclear ~~3earch Center , serves for the enrichment of the light isotope U F6 • Typically a gas-mixture of 5% UF6 and H2 as carrier gas
1327
is used at normal temperatures and pressures. A centrifugal force field is generated by the streamline deflection due to the fixed cylindrical outer riozzle wall, Fig. I. The centrifugal force field serves as the driving potential for the separation of the isotopes. It is proportional to the square of the velocity and inversely proportional to the deflection radius. Therefore high azimuthal velocity and a small deflection radius are ~ss~ntial. The light carrier gas (index c) in great molare excess (x /x << I) allows acceleration to high velocities (400 m/s) of the UF~ (lndex u) at moderate expansion ratios while a small radius of curvatures (Ra ~ IO ~m) is achieved by special production techniques.
The gas mixture fl?1~ at high subsonic speeds (Mach number M ~ I) speed ratio S = M (y/2) , while the heavy isotopes are in the high supersonic regime (M >> I) because of the low carrier gas concentration and high ratio M /m = I76. In spite of the high velocity the small channel width (~ ~uiO ~m) leads to a Reynolds number of approximately IOO and a Knudsen number of o.oi, Kn = const M/ReSc (Sc Schmidt number).
Nonequilibrium effects are expected at encountered Knudsen numbers because of the large mass disparity m >> m o An analysis of the relaxation times of the binary mixture ind~catescthat the relaxation time for the temperature equilibrium between the isotopes an2 the carrier gas is not small against the characteristic flowtime • In addition non-Maxwellian velocity distributions are expected for isotopes near the wall. Hence the applicability of the Navier-Stokes equations is doubtful and a direct solution of the Boltzmann equation, for example by Monte-Carlo simulation, is desirable.
The physics of the thermodynamic nonequilib~ium can be investigated by intensifying the nonequilibrium effects • This intensification can be achieved by using trace concentrations for the UF6 and large Knudsen numbers. For trace concentrations the effect of the UF6 on the carrier gas is neglegible. At subsonic speeds the carrier gas
Fig. I.
1328
Scheme of stand~3g curved separation nozzle for enrichment of U F6isotopes.
will remain in local thermodynamic equilibrium and the flow and density field will correspond to the streamline curvature. If the walls are moving with v = wr, an exact solution of the Boltzmann equation, valid fo2 all Knudsin numbers, exists, namely T = const; v = wr, n = n0 exp (S ); n number density. This equilibrium solution is assumed for the carrier gas. Starting with initial conditions for the isotopes T = Tc; v = v and n = ~ • n , deviations in temperature and velocity~etw~en tfie isot8pes ~nd tfie carrier gas will develop till the isotopes finally adjust to the equilibrium solution far downstream from the entrance.
The nonequilibrium between isotopes and carrier gas is generated because the heavy isotopes are moving essentially in straight lines at Kn ~ 1 when S >> 1, ~ << 1 and m /m >> 1, till the walland the in-
u • • U . h Ud • C b 'l'b • 1 termolecular coll1s1ons br1ng t e a JUStment to t 2e equ1 1 r1um so u-tion, T = T ; v = vc = wr; and n n exp (S ). u c u u uo u
MODEL AND METROD OF SOLUTION
The simulation procedure follows e~sentially the Tracer MC-method presented by the authors at the XII RGD •
Geometry
The diverging channel of the standard separation nozzle is modeled by a cylindrical simulation region of constant w!dth a and deflection angle ~ larger than the maximum impact angle ~ which depends on the ratio of the channel width to the deflection radius a/(a + R. ), F . 2 1n 1g. a.
+ ~ = arccos [1 - a/(a + R. )J 1n (I)
The simulation region is divided into uniform deflection elements in the azimuthal direction and a staggered grid in the radial direction with extremely small cell sizes near the deflection wall to achieve the needed resolution of the flowfield without increasing the computer time. In the entrance region, because of the nearly uniform density distribution of the isotopes, the extremely small cells do not contain enough molecules for meaningful collision computations. This necessitates a different grid size for collision computations compared with the flowfield computations. §taegering is achieved by the coordinate transforma~ion with r/a = (r) • For NG ~ 1, the uniformly divided cells in r represent staggered cells in the physical plane.
Model gas
The ternary mixture consists of the carrier gas of mass m = 2 and the isotopes of mass m = 350 with m1 = 300 and ~ = 400. The large mass difference ßm =u~ - m increases the separation effect and reduces the computer time require! for a given accuracy. The collision
1329
cross sections of the rigid hard spheres molecules cn : n • n I 3 r. 6) h 1 · · b ccH ancu u' F6~u : .J : represent t e actua 1nteract1on etween 2 a .
Initial and boundary cönditiöns
The initial conditions are shown in Fig. 2b. Temperature and isotope mole fraction ~ are assumed to be uniform, The Mach number at the channel midpointu(r/a = 0.5), based on the average mass of the mixture m = ~ m + ~ m , is M 5 = 0.9. The ratio of specific heats y = 5/3 for ail gompoHen~s. All(~em~eratures are normalized with the temperature at the channel entrance T = T • The azimuthal velocity profile represents a solid body rotatfon. c
v,~, = w (r + R. ) 't' 1n
(2)
For the number density, the equilibrium solution for constant mole fraction is chosen. Normalization by the number density at the midpoint of the channel entrance results in
n (r) 2 r + Rin 2 = exp <t M(0,5) [C0.5a+R.) - Il).
1n (3) n (r = 0.5 a)
The initial Knudsen number of the carrier gas based on the midpoint stagnation density is
Kn = [ 12 n n (r = 0. 5 a) a] -I c cc 0
(4)
The normalized stagnation density at the channel midpoint is
Fig. 2a. Cylindrical channel simulating the separation nozzle.
Fig. 2h.
1330
0.5 r·.lo
Initial dimensionless number density and azimuthal velocity profiles.
n (r 0
n (r
Oo5 a)
OoS a)
I y- 1 2~
= ( 1 + ---2--- M(Oo5) ) (5)
The isotopes are reflected diffusely and fully accommodated by the solid body rotation. The wall temperature is constant. The magnitude of the reflected velocity component in the azimuthal direction is chosen from a Maxwellian velocity distribution function, superimposed by the velocity of the moving wall. The reflected velocity component in the radial direction is chosen for example at the deflection wall from the distribution.function
F. l. -oo
r [-v. 2 ß. 2J d v.
(6)
b vir exp tY l. I.r with ß. = (m./2 k T ) 2 and v. ~ O(i stands for 1 and h, light
l. 1 w l.r and heavy isotope) The exit is represented by a hard vacuum so that all leaving molecules are losto At the entrance an upstream leaving molecule is reinjected with the initial velocity into the simulation region.
RESULTS AND DISCUSSION
~ The Tracer simulation is carried out for an isotope concentration x OoOS% and carrier gas Knudsen numbers Kn = 0.02, 0.1, Oo5 and f~ee molecular flow. The ratio of the channelcwidth to the deflection radius is a/(R. + a) = 1/3. The deflection angle was chosen such that the transient Miliancement effect of the separation and the asymphtotic equilibrium state are included for large Knudsen numbers. The simulation region is divided into ten (radial direction) by twenty cells. The grid constant for collision calculation is NG = 4 and for sampling NG = 14.
Average flow properties of the isotopes
The distribution of the isotopes in the simulation region at a time after the simulation has reached a steady state (the entering and leaving molecules are balanced) are shown in Figs. 3 for different Knudsen numbers. For Kn = 0.1 (Figo 3a) the flow is nearly free molecular. The molecules mo~e in straight trajectories corresponding to the entrance velocity until they strike the deflection wall. Not before ~ = n/4 the streamlines follow the curvature of the deflection wall. The isotopes are concentrated at the deflection wall in a range of Oo8 < r/a < 1.0. The inner part of the simulation region is empty. A staggered grid at the deflection wall is necessary. For Kn = 0.02, reduced migration of the heavy molecules towards the outer w~ll occurs by reason of carrier gas-isotope collisions (Fig. 3b)o
1331
Fig. 3. Distribution of isotope molecules in the simulation region for carrier gas Knudsen numbers 0.1 (a) and 0.02 (b). (Tracer simulation: M = 0.9, ~ = 0.0005, m /m = 175 ~ - m1 = 100, a/(R. + a) = IY3, ~ = 6 • a~ 18 x 20 ~ells, scaggered grid NG =1~ and 14, V= w(r + R. ), diffuse re-flection at moving wall) 1n
Nonequilibrium between isotopes and carrier gas
The velocity deviations between the isotopes and the carrier gas are shown for the azimuthal and radial velocity in Fig. 4 for the wall region (0.9 ~ r/a ~ 1.0). Each grid point represents the average radial or azimuthal velocity of the isotopes minus the corresponding carrier velocity at the midpoint of the cell. The isotope velocity, samples in cell (j), is
V • uqJ
E . V • uqJ E N •
UJ
(q represents direction r or 0, N sample size). (7)
As pointed out previously for Kn = 0.1, the movement of an isotope molecule from the entrance of th~ channel is practically undisturbed until the molecule strikes the wall. The velocity vector is divided into a radial and an azimuthal component. Neglecting the thermal velocity (S >> 1), the limits of the radial velocity v 1 . and of
h . hul 1 . "bl b rmax, 1m t e az1mut a ve oc1ty v~ . 1 . are poss1 e y · ."m1n, 1m
V 1" = w • R. cos ~+ rmax, 1m 1n +
v min,lim = w • Rin sin ~ ' where ~+ is given by (1).
(8)
(9)
In Fig. 4a the azimuthal velocity of the heavy species limps behind the carrier gas velocity until all the molecules have had a wall collision. The moving wall accelerates the slow molecules in direction of the equilibrium profile.
1332
05
~ llkT.Jm,l
Fig. 4.
25
T~.~.- Tc• 1,
12 5
Fig. 5.
10 v,u . vc, llkl,lm/11
05
Velocity slip between isotopes and carrier gas for the azimuthal (a) and radial (b) direction, Kn = 0.1.
c
25
120' Tu, · Tcr -,,-12 5
120'
/
Kinetic-temperature slip between isotopes and carrier gas for the azimuthal (a) and radial (b) direction, Kn = 0.1.
c
1333
The radial component of the isotope velocity is identical with the diffusion velocity because v = 0. This diffusion velocity reaches Mach numbers M ~ 5 and vanisfi~s at the deflection wall, Fig. 4b, The large diffusign velocity arises purely by geometrical considerations because the isotopes move nearly along rays until they hit the wall when they are reflected with the thermal radial velocity corresponding to the wall temperature. The deceleration near the wall gives rise to large increases of the kinetic temperature in the r direction. The carrier gas temperature in the deceleration region Fig. Sb is equal to the temperature of the mixture at the channel entrance T •
e
The temperature nonequilibrium in the azimuthal direction is shown in Fig. 5a. The velocity of the incoming molecules decreases geometrically until they have had a wall collision. The average azimuthal velocity after a wall collision corresponds to the velocity of the wall and is larger than the average azimuthal velocity of the incoming molecules, thus the velocity distribution function is broadened. The variance of the velocity distribution function is a measure of the kinetic temperature, so T dJ increases over the equilibri~m temperature. Whenn all molecules h~ve had a wall collision, ~ > ~ the nonequilibrium decreases very fast.
The calculated maximal slip effects between isotopes and carrier gas as a function of a Knudsen nurober are given in Fig. 6a and b. The maximum azimuthal velocity slip, because a constant given by geometrical relations for Kn + oo tends to zero for Kn + 0, The increasing number of collisions b~tween the carrier gas andcthe isotopes prevent the isotopes more and more slipping with regard to the azimuthal velocity of the carrier gas. In the radial direction the deceleration effect of the collisions during the diffusion of the isotopes through the stationary carrier gas diminishes the maximum velocity slip from its maximum at Kn + oo to the finite value of the continuum limit.
c
0.5
Yurmax -Ver
(2 kl /m l112 • c
Yue min -Vc~
(2kT.tm)12
-0.5
" V, v--V - - free molecular I im
~
"'· or, --····~T] 1 Ol--1----/1'--~-----1-+---i
~J 0.01 0.02 0.1
(o)
0.5 1.0 0.5 1.0 --oo Knc-
Fig. 6,
1334
Calculated maximum velocity (a) and temperature (b) slip between isotopes and carrier gas as a function of carrier gas Knudsen number.
For the kinetic temperature a decreasing Knudsen number means redistribution of energy among the molecules by collicions in different directions and to other components. The remperature slip in the radial direction between isotopes and carrier gas amounts to three times the carrier gas temperature even at Kn = Oo02, while the azimuthal temperature slip equals the carrier gasctemperatureo
Separation between isotopes and carrier gas
As the flowfield of the carrier gas is fixed in the simulation and the isotopes are concentrated at the deflection wall, carrier gas and isotopes are separated. A measurement for this separation is given by the mixture separation factor A • Expressed by the cut of the carrier gas cut 6 and the cut of themisotope mixture 6 it reads
e (T - e ) u A
m c u
e (1-e) u c
(I 0)
The 6 -profiles are constant along the deflection angle (equilibrium stat~ for the carrier gas), thus only the development of isotope cut 6 with increasing deflection angle is of interest, Fig. 7. The cut o~ a component i 6. for a radius r at the constant deflection angle is defined by ~
r e. = r ~ 0
n. v. ,~, dr ~ ~'I'
(II)
The equilibrium partial number density and the equilibrium velo-city profile v. = w (r + R. )
~ 2 ~n a ln n. mi vicp ~ = (r + R. )kT ar
~n
permit the calculation of the equilibrium value 6~ ~
exp [A. (r + R. ) 2J - exp [A. R. 2J ~ ~n ~ ~n e~
~ exp [A. (a + R. ) 2J - exp [A. R. 2J ~ ~n ~ ~n
(12)
(13)
with A. = m. w2/2 kT. Because the mass ratio of the mixture to the carrier~gas ß/m = 1.087 for ~ = 0.0005 the equilibrium value for the carrier gascis nearly equaY to the initial profile for the isotopes e~ ~ e n. oo.
~ u,'l'=
For Kn = 0.1 and cp > 90° the position of the isotope streamline does not chänge any more (the upper integration boundary r is fixed for a constant cut 6 ) the diffusion is complete, equilibrium has been reached. By comparison the 6 -profile for Kn = 0.02 and cp = 90° showes the reduced migration of tMe heavy molecul~s to the deflection wall r/a = I because of increased isotope carrier gas collisions which slow down the mixture separation. The diffusion is not yet complete, equilibrium has not been reached,
1335
Fig. 7.
0.9 1 ~
~-
Development of isotope cut radial gas Knudsen numbers Kn 0.1 and the equilibrium profil~s.
profiles 8 for carrier u 00 00
Knc = 0.02; Sc' Su are
Separation between the isotopes
The Knudsen number influences also the separation between the light (1) and heavy (h) isotopes. A measure of the Separation is the separation effect of the isotopes EA' defined by
el - eh
eh (1-el) (14)
It is shown in Fig. 8a as a functign of cut and deflection angle for Kn = 0.1 • The equilibrium value EA is calculated with the aid of equati5n (14). In the entrance region (small ~) the separation is induced by concentration near the deflection wall, see ~ = 45°. The insignificant enhancement over the equilibrium separation value at ~ =
t EA
1.0
Fig. 8.
1336
Iai
• • u · • il' • 10'
--t;:
1.0 e"
The separation effect EA versus tion angles, Kn = 0.1 ~a), Kn c c value)
I 1.s I I • I
E A I . II' I D 10' 1 0 II ' 1 0 1)5' I
1.0 I --IE:'
0 o~----:o.'=""s ---e-. -'u {b)
cut 8 for dif1erent deflec= 0.0~ (b); (EA equilibrium
60° and Kn = 0.1 refer to the fact that at relatively large Knudsen numbers th~ intensification of the separation by collisions between isotopes and carrier gas is not effective. These collisions hinder the diffusion of the light isotope more than the heavy isotope to reach the equilibrium state and cause a transient enhancement of the separation above the equilibrium value. For Kn = 0.1 and ~ larger than 90° the isotope Separation is complete ana the calculations are in good agreement with the equilibrium profile. This was already expected from Fig. 7. For Kn = 0.02 the transient enhancement is clearly visible, at ~ = 90° andc135°, Fig. 8b. The ratio of the separation effect to the equilibrium value EA/E~ ~ 2 at 6u = 0.25.
A summary of calculations for different Knudsen numbers is given in Fig. 9 for a const~nt cut 6 = 0.25. From calculations using the Chapman-Enskog-Theory (Kn ~~u1) it is known that the separation is only a function of ~/ReSc Ör ~ • Kn (Kn ~ M/ReSc) for a fixed speed ratio and nozzle geometry. At larg~ Kn c~he separation is wall induced. For free molecular flow the angle c~ gives the angle where all molecules have had at least one collision with the wall and the separation is started. The continuum solution is calculated by means of a finite difference m~thod which solves two independent Chapman-Enskog diffusion equations for a curved channel with the corresponding velocity profile v = wr. This description in terms of two equivalent binary systems neglects collisions between the light and heavy isotopes and increases the tendency to produce a transient enhancement, because isotope-carrier gas collisions are completely effective. The maximum separation effect for MG-simulation decreases with increasing Knudsen number and Kn ·~ is not a scaling parameter. c
Fig. 9.
lO t
2_
Kn. I
~ • 0.02 s.·10 4
.. 01 e •• o 25 -cootinuum two bmory o/IR1.•ol·11l
0 '\ diftUSIOn rq
I ~ \ \ \ \ \ "-' fh, \ ...... _ . ............ ' ..... 6/
j_J,.. I
I
~ / __ ",l
01 ~· ~r I'
1.0
0 0 OD2 0.06 DOS 01 0.12 014
Separation effect at a constant cut (6 = 0.25) normalized by the equilibrium value EA/s: as a fugction of scaling parameter Kn • ~ for different carrier gas Knudsen numbers. c
1337
CONCLUSIONS
From the tracer Monte Carlo calculations for UF6 isotopes and H2 carrier gas in a cylindrical separation nozzle with moving walls the following can be deduced.
For trace isotope concentrations, high subsonic carrier gas speed (M ~ 0.9) and hypersonic isotope speed (M ~ 10) and a Knudsen number Knc = 0.02 the transient radial temperatur~ slip is still four times th~ carrier gas temperature and the radial diffusion velocity is still more than twice the thermal UF6 speedo This implies that the Navier Stokes approximation at these near continuum Knudsen number is at least doubtful. The azimuthal slip velocity is of the same order as the diffusion velocity even though the moving wall boundary conditions suppresses the radial gradients of the azimuthal components. The azimuthal temperature slip becomes much smaller than the radial one at small Knudsen numbers because the azimuthal length scale for adjustment to equilibrium conditions becomes much !arger than the channel width.
For Knudsen numbers Kn > 1 the free molecular limit is practically reached. The slip eff~cts reach a maximum and the circumferential extent of the transient nonequili~rium region is independent of Kn and given by the geometric angle ~ (I), where all incoming moleeures have had at least one wall collision. In the continuum limit (Kn << 1) the only slip effect present is the radial diffusion velocit~ and a function of Kn • ~ and M as well. For intermediate Knudsen numbers 0.001 ~ Kn ~cl the slipceffects depend separately on
c Kn , M and ~. c c
The transient enhancement of the isotope separation effect over its equilibrium value due to isotope carrier gas collisions, which inhibit more the migration of the light than heavy isotope-components to the deflection wall, increases from zero for free molecular flow to a value of 2.2 for5Kn = 0.02. This is higher than the limiting value given by Rebrov aE the last symposium. A tracer continuum model which neglects additionally the collisions between heavy and light isotopes gives a maximum overshoot of 2.5. Whether this discrepancy can be attributed to the difference in the model or is due to nonequilibrium effects cannot yet be decided.
REFERENCES
1 Becker, E.W., Bley, P., Ehrfeld, U. and Ehrfeld, W., "The Separation Nozzle - An Aerodynamic Device for Large-Scale Enrichment of Uranium-235", Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 51, Part I, edited by J.L. Potter, AIAA, New York (1977), pp. 3-16.
1338
2chatwani, A.U., Fiebig, M. Mitra, N.K. and Ehrfeld, w., "Nonequilibrium Effects and Their Modeling in Separation Nozzle", Ratefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 74, Part I, edited by Sam s. Fisher, AIAA, New York (1981), pp. 517-540.
3chatwani, A.U., Fiebig, M., Mitra, N.K., Schwan, W., Bley, P., Ehrfeld, W. and Fritz, w., "Tracer Monte-Carlo Simulation for an Isotope Separation Nozzle", Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 74, Part I, edited by Sam s. Fisher, AIAA, New York (1981), pp. 541-557.
4 Berkhahn, W., Ehrfeld, w. and Krieg, G., "Berechnung der Uranisoto-penentmischung in der Trenndüse bei kleinen UF6-Molenbrüchen im Zusatzgas", KFK-Bericht 2351, Gesellschaft für Kernforschung, Karlsruhe (1976).
5Bulgakov, A.V., Kusner, Y.S., Prikhodko, V.G. and Rebrov, A.K., "Separation of Gas Mixture Components in Interacting Flows", Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, Vol. 74, Part I, edited by Sam s. Fisher, AIAA, New York (1981), pp. 607-616.
1339
SEPARATION OF BINARY GAS MIXTURES AT THEIR EFFUSION THROUGH A CAPILLARY AND A NUCLEAR FILTER INTO VACUUM
V.Selesnev,B.Porodnov,V.Akinshin, V.Surguchev and A.Tarin
USSR, Sverdlovsk Urals Polytechnical Institute
INTRODUC TI ON
Despite practical importance, the phenomenon of gas mixture separation at the effusion through regular and porous membrains into vacuum is not extensively studied both theoretical by and experimentally. Experimental investigations on regular channels [1,2] are confined to free molecular and intermediate flow regime. The number of investigated gas mixtures and the diaphragm materiale is very limited. In theoretical works [3-5] , as a rule, an assumption inadequate to experiment about entirely diffusive reflection of gas molecules by surface is used. Restrietions by Knudsen numbers ( Kn<<1 or Kn~71 ), by volume flow rate not exceeding the possibilities of diffusion extending gas to diaphragm, by the value of pressure difference on the channel (AP/P<<1 ) take place.
In the present werk experimental results by separation of gas mixtures He-Ar, He-D2 , Ha-D2 in the wide range of Knudsen mllllbers ( 0,02< Kn< 100 ) on a glass capillary and a nuclear filter have been made. Experimental results are interpreted on the base of the theory with arbitrary Knudsen numbers and accommodation coefficients, at any pressure difference and flow rates.
EXPERIMENTAL TECHNIQUE
Measurements were made by the stationary flow me-
1341
thod on the installation whose principal scheme is shown in Fig. 1. At effusion of a gas rnixture into vacuum, when one of the bloaking devices 9 is closed, constant pressure in the retort 1 was kept by moving the piston 10 with the help of the operating signal of the pressure detector 2. In the course of the experiment the composition of a gas in the retort 1 did not change, as a viscous flow regime was kept on the capillary 3, and the separation of rnixture on it could be neglected. The difference of movement velocities of components through a capillary or channels of the nuclear filter 8 lead to the shift of concentrationdC in the retort 4 in comparison with initial concentration. In the course '· of the rnixture effusion through the channel, a steady concentration distribution was fixed in the system. The value of the fixed concentration in the retort 4 was determined basing on the dependence of gas mixture flow rate through a channel on it composition. Due to high pressure detector 5 sensitivity (-10-~Pa/Hz) it was possible to measure the concentration shifts AC-10-.5" wi th the accuracy not le~s than 1%. A glass mel ted capillary wi th the length t'~r = (29, 62 +0,01) cm and the radius R~r= (5,05 ± 0,01)·10-2 cm and a nuclear filter from polyeth;ylenteretalate film. with the thicknesst'.s-= (1,0: 0,1)·10-3 cm, porosityfS = (9,2 ± 0,7) ·10-1, the channel radius R.fl = ( 1,3 ± 0,1 )· 10-G cm have been chosen as channels being investigated. In the construction of the filterholder there was an extending channel 7 with the radius RJ= (0,3036 ± 0,0006) cm and the length fe/ = (1,7 ± 0,1) cm.
2 5
3
Fig. 1. Schematic diagram of the experimental set-up.
1342
EXPERIMENTAL RESULTS AND DISGUSSION
Fig. 2 shows experimental values of the concentration shift as a function of inverse Knudsen nurober S in the logar'i tbmic scale.
Experimental data (1,2_,3) _were obtained in experiments on separation of the mixtures' He-Ar, He-D2 , H2-D2 on a capillary. The figure 4 corresponds to the experimental values AC , measured at Separation of the mixture He-Ar on the nuclear filter. Initial concentrations of mixtures in all cases are equal 50%. As is seen from Fig. 2, the concentration shift on the mixture He-D2 changes the sign in the intermediate regime ( points 2
. {
0 2
e J ~ I{
1o·'r-----==iij*=~~=-r-------"K----~---~
10"1 - ----+-----r----
s
to·sa.----....L..---_;ji----~----L---~ ... 3 -2 --( 0 { ~$
Fig. 2. Dependence of the concentration shift AC upon the inverse Knudsen nurober s •
1343
with a negative sign are marked by the line ), while in the free molecular flow regime (6<<1) has values differing 20 times from theoretical predictions using the model of entirely diffuse reflection ( line 5 ). Experimental data on separation on a capillary and a nuclear filter seemed to be shifted on the axis S by an order of the magnitude.
To explain the obtained experimental dependences L:lC (eS ) , consider the steady isothermic effusion of a binary gas mixture through a diaphragm. Gases penetrate through a working channel not in the same ratio in which they are before entering it. Due to this fact there appears the difference of concentrations ( in other words, the concentration shift )
( 1 )
where c.,o = n1o/(n.tf) +n20 )= n1o/n() is the molar concentration of a light gas mixture component ( index "0" denotes characteristics on tbe entrance into the diaphragm ) ; C'/'= J{/( 11+ 12 )=h.,Uff(n.,U1 +1?2 ~) is the part of flow of light component in total flow of gas mixture through diaphragm; n,. , Uc" ( ( = 1 , 2) are the numeriaal density and projection of the average molecule velocity on the axis of channel of i-th mixture component, respecti vely. The value ~17 is equal the molar concentration of a initial mixture, because the flow of gas mixtures is stationary.
According to t·he independence of the components flow upon the ~ coordinate (along the channel axis ) and defini tion <!/ and cfn we can get [61
C _ c;c:(u{o_ u;) ( 2 ) A - o u.o couo • c., f .,. 2 2
On the basis the non-equilibrium thermodynamics of diaphragm systems [7,8] convenient to write in the following manner;
( 3 )
( 4 )
where l.,pp, LpJ), L])p, L.]).JJ are the kinetics coefficients
1344
of flow gas mixture through diaphragm, f' is the pressure.
An effusion of mixture into vacuum can exclude the secend terms from right parts of eq. (3) and (4)[6]. Inserting eq. (3) and {4) ( for entrance cross-sectionof channel ) into eq. (2), get result:
1 o 1 {Po oJ oc.O U'Jjp 0 0 U~fl 1 ~1' I
tiC'= r1 2 ~ = c1 c2 L. ( poe.,o) • ( 5 ) I.J.pp PP , 1
For calculate of kinetic coefficients at arbitrary Knudsen numbers and incomplecte tangential momentum accommodation of molecules on the wall of channel the system of model Hamel equation 9 with diffuse-specular boundary conditions f9r the distribution function has been solved.In [10] analytical expressions of kinetic coefficients for the binary gas mixture movement in· a long cylindric channel with the radius ~ at arbitrary rarefication are given. In the free molecular flow regime these expressions lead to the formulaes of Knudsen flow and in the viscous regime ( Kn ... O ) they lead to the relations:
R2 JJ f c2 Lpp= 8 ? ; LpJ) =LJ)p = rl.. p; lJI:r> = ({C 2 P, < 6 >
where R is the radius of the .channel; 7 , eiJ are the viscosity and diffusion coefficients; o{ is the barodiffusion constant.
Inserting eq. (6) into (5), we get for the concentration shift in the viscous flow regime ( Kn-.0 ) :
( 7 )
Theoretical dependences AC' ( S ) .\. obtained by eq. ( 5) are shown in Fig. 2. The curve s 1. , jJ ., f!.7 are obtained for the mixtures He-Ar, He-D2 , H2-D2 respectively. Calculating .ar parts of diffuse molecule reflection were taken from experimental data on Poiseuille flow of one component gases in glass capillaries. For He,Ar,He,De they amounted to 0,920; 0,980; 0,968; 0,953 , respectively. As is shown in Fig. 2, calculated values A{- coincided with experimental data for all three pairs of gases in the wide range of Knudsen numbers. Maximun1 divergences (~10%) correspond to the viscous flow regime and
1345
are caused by insufficient accuracy of the Hamel model. As in the experiment, the theoretical curve AC ( S ) on the pair He-D2 intersects zero value at the intermediate Knudsen numbers. In the free molecular flow regime due to the incomplete tangential momentum accommodation the concentration shift for the pair He-D2 is AC'= 1, 7 4 ·10-2 instead of A~ = 8,5·10-'~ co-rresponding to the entire diffuse- reflection of molecules.
To explain different posi tions of the curves A~( S) on a single capillary and a nuclear filtre we shall consider the problern of steady gas mixture effusion in conditions when in. front of the diaphragm there is quite a narrow channel ( see Fig. 1 ) of the length P~ and the radi us . Rr1 •
Suppose that the nuclear filtre could be represented as a set of straight cylindric channels. The length of such channels is tf, the radius is /i.f. The area of the filtre transport.~ross-section is Si' the porosity is 8 = S~o/ 5,;,1 = S.f/(?rR;). Place the orig~n of coordinates H = 0 into the entrance cross-section of the fil- n treholder. In the conditions of experiment the part C1 of the steady flow Q.f the first component in the total flow is given, and it is necessary to find the concentrationCiin the volume in front of the filtreholder. Mixture flows in the filtreholder and in the filtre pores are described by eqs. (4) and (5). For a wide ( in comparison with the filtre pores ) entrance filtreholder channel at all pressures being realized in our experiments, the viscous flow regime is observed. Therefore, from eq. (3),(4) and (6) one can get:
Rj dP . ., Q d~f '12. U2= Tf"ot~ • ( ~ - o(. ) o(J ot~ = n~ S,/
2 .
u1= Rot.oiP+ <- .1._ ti. ) ~ ~<'1= 'J1 • 8 ? d~ c-., CJC ~ n1 Sol
( 8 )
( 9 )
The solution of the given equations leads to the following mixture concentration dependence upon the coordinate ~ of the channel filtreholder
~:- Cf(i)_ r~(~f~}7 (!/'-C'fo -exp nSot3J 1· ( 10 )
The concentration on the entrance into the filtre will be determined by
1346
c:-c.,t. = e..rp· (JIJd'} ' < 11 ) ~~n-C/1
where '1 = '11+ J1 ; JJ =nSc/~lfJ is the m7;xim}F. diffusion flow on the entrance channel, ..1 C = C, - C., 1s the conc7.rtr1tion shift being measured in the experiment, and ('1 - (t'~ is determined by eq. (5). Thus, it is easy to obtain the expression by which one can calculate the concentration shift in the experiments with diffusion resistances:
" o t.cJ. L..JIP f P,"r/1 e pr .. 7!7 ) . < 12 > ~C= C'1 - C, = ~~ 2 l (Poc.') · ..{ ./ol PP I ., -]IJJ
At sufficiently low pressure J<<Ja', e .._... 1. In this case the diffusion resistance of the filtreholder channel does not influence the results. This circumstance corresponds to the ini tial area of the curve fX in Fig.2. With pressure increase in the system the flow rate becomes of the same order as the maximum diffusion flow 1J, as a result of it, the distortion of the concentration fields which leads to the reduction of /::ir long before the appearance of the intermediate flow regime in the filter channels begins. The calculation of ~e by eq. (12) has coincided with the experimental values of the concentration shift with the precision of experimental data. I' proves the correctness of the consideration of the diffusion resistance of the entrance channel influence which is in front of the nuclear filtre.
HEFERENCES
1.
2.
J.A.W. Huggil, The flow of gases through capillaries, Proce Roy. Soc., 212A:123 (1952). D.E. Fain, W.K. Brown, Neon isotope separation by gaseous diffusion transport in regular geometries, Raref. Gas Dynamics, Proc. of X-th Intern. Sympos., J.L. Potter, ed., 1:65 (1976). R.D. Present, de Bethune A.J., Separation of a gas mixture flowing through a long tube at low preasure, J. Phys. Rev., 75:1050 (1949). V.D. Selesnev, P.E. Suetin and N.A. Smirnov, Separation of binary gas mixture in the whole ran~e of Knudsen numbers, Jurnal Tehn. Phiz., 45:1499 {1975). S.T. Hwang, K. Kammermeyer, WMembranes in Separations", A Wiley-Interscien. Public.,New York, (1980~.
1347
6. A.A. Tarin, V.D. Selesnev, and B.T. Porodnov, Separation of binary gas mixtures at their effusion through a capillary into vacuum, Dep. VINITI, N 3413-79 (1979).
7. c. de Groot, P. Masur, 1962,"Non-equillibrium Thermodynamics", Nort Holland, Amsterdam.
8. V.D. Selesnev, P.E. Suetin, and E.V. Kalinin, Isothermal diffusion of gas mixture or liquids in channels, Dep. VINITI, N 4169-77 (1977).
9. B.B. Ha.mel, Kinetic model for binary gas mixtures, J. Phys. Fluids, 8:418 (1965)o
10. N.D, Pochuev, V.D. Selesnev, and P.E, Suetin, Binary gas mixture flow at arbitrary tangential momentum accomn1odation, Jurnal Prikl. Meh. Tehn. Phiz.,5:37 (1974).
1348
XXI. IONIZED GASES
EFFECTS OF NONIDEALITY IN QUANTUM KINETIC THEORY
Werner Ebeling
Sektion Physik der Humboldt-Universität Berlin, DDR
NONIDEALITY PARAMETERS
The increasing interest to the nonideal plasmas is due to its importance for many problems of practical interest as thermonuclear fusion, 1lliD-generators, gasphase nuclear reactors, extreme pressure devices, highly doped or exci ted semiconductors, liquid metals etc.1•l. Our investigation deals with a component quantum plasma characterized by the interaction energy ~ which is assumed to be given by local pseudopotentials. The consequent quantum-statistical treatment avoids divergences which must, in classical theories, be overoome by arbitrary out-off prooedures at small distanoes. The essential length in this conneotion is the thermal de Broglie wave length of the eleotrons
( 1 )
The main task of this paper is the exaot aooount for the equilibrium correlations which influence strongly the tranapart properties. For example the structure factor is of primary importance in liquid metal theory. For this reason the equilibrium correlations are treated here exactly. Our point of view is, that in dense plasmas the equilibrium correlations are more impotant than dynamical screening effects. In order to give a quantitative definition of non-ideality in a plasma, we introduce the following dimensionslass parameters
1351
( 6 /2 - mean kinetio energy per degree of freedom)
1
r = ~,..t I t,ki'tl = el./de I d = (3/'+rn)3
( 2)
't=1 5 6 log (TIK]
Fig. 1 The region of nonideality in the densitytemperature plane for eleotrons r&~ 1 and for ions r,. ~ 1 •
1352
KINETIC EQUATIONS
The collision term in the kinetic equation for the distribution function fa (r,p,t) of charges belanging to the species a is given by the pair correla-tions. The equation for the pair function reads if neutral partielas are neglected
( 3)
Here the parenthesis stands for a functional depending on the conditional third order correlations. After replacing these quantities by their equilibrium values we get the binary collision approximation (BCA)
(4)
where ~ ab represents the potential of effective force in equilibrium
(5)
Further simplification of eq. (4) yields the effective potential approximation (EPA)
(6)
where h b is the equilibrium correlation function which a is assumed to be given. Finally we ~ropose the direct correlation approximation (DCA) 1. In this method the r.h. s. of eq. (3) is approximated by
1353
(7)
where ~ah(x) is the Ornstein-Zernike direct correlation function. Various approximations for C may be used, the simplest is to replace Cab by theab equilibrium direct correlation function. Eqs. (4-7) are still solved by the exact .equilibrium distribution. Using Fourier transforme the nonequilibrium_.solution is in EPA and in DCA easily obtained. We note that in EPA the exact equilibrium correlations are taken into account, however the twoparticle dynamics is approximated. Nevertheless we get for Maxwell distributions ~ the exact local equilibrium correlations. In BCA the solution to eqs. (3-4) is expressed by two particle propagators. In this way we get the Coulomb collision integral in EPA, DCA or BCA, which contains effects of the external electrical field, of retardation in time and of nonlocality in space. The various contributions may be represented in the following way 3,4 I (O) = 0(1)- usual collision term a Ia(1) = O(E)- effect of the external field
I (2) = O(d/dt) - effect of retardation in time a Ia(3) = O(d/dr)- effect of nonlocality We want to underline that the effects included in the contributions Ia(1), Ia< 2>, Ia(3) are of primary importance for nonideal plasmas, describing mainly nondissipative interaction effects. The influence of the nonideality on the dissipative effects is contained in Ia through the equilibrium correlations hab• in the quantum theory the eqs. given above
remain nearly unchanged if f and h are con-sidered to be the one-particle a ab Wigner function and the quantum correlation function, which is defined now by
1354
the diagonal elements of the two-particle density matrix .r in the following way
1,6 ()() = 9.-.( ... ,r,r+)C',r.f-x)-j
The requirement that hab remains correct ~ields in EPA the effective poter:tial (a+ b)
TRAl~SPORT EQUATIONS
From the kinetic equations the transport quantities may be obtained by the momentum method. Avoiding lengthy calculations we give here some results only for the electrical current and the heat current
(8)
The relaxation factor Q is civen by
The ther.modynamic quantities are expressed by integrals over h in the usual way and are connected by dpP. = ab ned p e• The nonideality appears in Q, p9 and
JA • The kinetic coefficients 1l , ll and ~ are the usuai ones, however the Coulomb logarithm of a nonideal plasma is modified by correlations and is in EPA given by
1355
N
Lie = !dkkycpie{k)o/ie(k)/1~rr~lf} 0
(1 0)
We see that in the given approximation (EPA) the whole problem is reduced to the two-particle cor.relation functio~ in equilibrium for which many results are available 4,6. In certatn approximation we may use e.g. the formula
L { C" [4>,..,- 'lpJ 1 flQt, r) =-lqb(r) ftP k T - (11) 8
N where S denotes the two-particle Slater sum and <P the statically screened potential. Using Taylor expansions for small distances, Wigner expansions for large distances and Padfi approximations for the intermediate region one gets
s- (r) = eJtp[ A-l~+12A'f-1 R 1 1 ca~;; 1- R2+ fZA l-.t Rlf
-i &,~, e"p [A-'fR- ~'], R = r/}t (12)
A = e., I S('f){ii('fJ(c3>+r~s>+~iiy 00 (13)
. s dJt·lr [1- e~p(-rrr/~) 1 0
We note that the contribution of bound states was included into a mass-action constant of Brillouin-Planck-
1356
Larkin type. Therefore the sum over discrete states contains only two terms of the Taylor expan..qion ( '13 and 15). In other words eq. (12) represents the Slater sum for free charges only. In comparison to the classical result the correlations at small d~stan-
-0,4
-0,6
T:104K
-0,8 log [ncm3 ] ~
20 21 22
Fig. 2Potential part of the chemical potential
ces are strongly decreased due to quantum uncertainty effects. The Coulomb divergencies disappear and one gets finite results for all transport quantities. By using these informations we are able to calculate the integrals over h appearing in our theory. The results are most ab conveniently represented by Pad~
1357
a.pproxima·tions which provide the correct behaviour at the low- and high-density border of the region of nonideality Y (Fig. 2). As an example we present the chemical potentual (interaction part)
(14)
The three temperature functions a0 (T), a1(T), a2(T) are chosen in such a way that the Debye law, the quantum corrections at low density, the Hartree-Fock contribution and the lattice energy at high density are represented in the righ.t way (Fig. 2). Similar curves may be drawn for Q, p pGt and p pOT which appear in the transport equa!ions. We note that the effects of Coulomb nonideality tend to weaken the transport proce~~3s like the electrical and heat corrents. In camparisie to the classical Spitzer result the conductivity e.e~ is strongly reduced due to the relaxation effect des2 cribed by the Q-function and the nonideal Coulombscattering described by the L-function. Additional short range forces however increase the conductance as well as the influence of ionic correlations. Finally we note that the electronic contributions to the tranapert processes are strongly influenced by the degree of ionization which can be calculated from the condition that the chemical potentiale of the plasma and the atoms are equall-.
REFERENCES
1. w. Ebeling, ICPIG-15, Invited Papers, Minsk 1981 2. w. Ebeling et al., "Transport Properties of Dense
Plasmas". Akademie-Verlag Berlin (in press) 3. G. Röpke et al., Ann. Phys. (Leipzig) 36:377(1979);
39:133(1982); Physica 101A:243 (1980) 4. Yu.L. Klimontovich, w. Ebeling, Soviet Physics
JETP 36:476(1976) 5. R. Balescu "Statistical Mechanics of Charged Par
tiales". London 1963 6. G. Kelbg et al., Ann •. Phys. (Leipzig) 14:310(1964);
19:186(1967); 21:235(1968); 22:1{1968); 383; 25:80(1970)
1. w. Ebeling et al., phys. stat. sol. 104:193(1981); Ann. Phys. (Leipzig), in press
1358
MOLECULAR MASS AND HEAT TRANSFER OF CHEMICAL EQUILIBRIUM MULTICOMPONENT PARTIALLY IONIZED GASES IN ELECTROMAGNETIC FIELD
A.F. Kolesnikov, I.A. Sockolova, and G.A. Tirsky
The Institute for Problems in Mechanics the USSR Academy of Sciences, Moscow, USSR Moscow-Physical-Technical Institute, USSR
INTRODUCTION
The derivation of the hydrodynamic set of equations and transport coefficients which describe diffusion and heat transfer in multicomponent partially ionized gas in the presence of an external magnetic field is considered by methods of the kinetic theory of gases. It is assumed that the collisions of partiales two-body and elastic, and that partielas do not possess the internal degrees of freedom. What is more, an external electromagnetic field does not influence upon the collisions of partiales, and there is also no effect of electrons interaction with plasma oscillations on transport coefficients. The conditions under the approximation in question is valid are know.n /1/.
In the present case the expressions for heat and diffusion fluxes and formulas for transport coefficients have been obtained by the Chapmen-Enskog and Grad methods /1,2/. The latter formulas are fairly eleborate for practical calculations. The aim of this work is to obtain mass and heat transfer equations, and also exact for.mulas for transport coefficients in a more simple form convenient for solution of hydrodynamic problems. The for.mula for the complex thermal conductivity A = )...J.. + i* 'A" derived in the work generalizes into the Curtiss-Muckenfuss formula in the event of heat transfer in a magnetic field. The exact Stefan-.Maxwell relations are obtained for diffusion fluxes in a magnetic field, which are necessary, in particular, to calculate effective transport coefficients in case of chemical
1359
reactions at equilibrium occurring in the flow. The work shows that it is more expedient to consider thermodiffusion ratios rather than thermodiffusion coefficients as the basic independent coefficients. Thermodiffusion ratios are expressed here directly in terms of collisions integrale and cyclotron frequencies.
As a whole, this work is the generalization of the authors' previous work /3/ with account taken of an electramagnetic field.
BASIC SET OF TRANSPORT EQUATIONS
The initial set of transport equations for diffusion and partial heat flows in N-component plasma in the presence of an external magnetic field takes the form 74/:
- .. " - \l ,J oo-dt: ~f!'' f, f7·~- W;d~)b11 i -p Ay· ~- (1)
II 1:! 'P--f,~ Av·~,
X· Vfn T~ = 3/ii [fr+l) m;Xr(A}i bx ;, + (2) ' rt 'I rtr+512) K T i' 'r
N ro - II l::! "P -+ ~ /\;; l{- + [; L 1\;i ~10 ·rr= t, ...• f- n r' u , J""' f=' , ,,, ~
d; = VX; + (Xi-Ci) v !n p + (C;fJin- rz,·e;) EJKT .... '
Here VJ is the diffusion rate, X; is the molar concent-ration, ~ is the mass concentration, m; is the mass, ~ is the charge, uJi is the cyclotron frequency of the j-component; P is the pressure, T is the temperature, k is the Boltzmann constant, q is the space charge, n is the partiale number density, t is the number of approximations in expansioi!.,,o~ t,he J}.istribution function in Sonine polynomials; E =E+ Vo,!H , where Eis the electric fie~d intensity,~B is the magnetic induction vector, ( G = 8'//t>IJ and Vo is the mean mass velocity. Vectors ~· are interreJated with reduced partial heat flows of component: f.;t -=-q.i/pi , where P· is the partial pressure. The vectors fJr> (p=2, ••• ,J-I)hre expressed in ter.ms of higher maments of the distribution function /3/.
The set of mass and heat transport equations (1) and (2) has been obtained in /4/ on the basis of the
1360
solution of the Boltzmann equations by the ChapmanEnskog method wi th regard to the arbi trary number of te~s in expansion of the perturbation function in Sonine polynomials. A similar set of equations can be derived by the Grad method /5/.
In the absence of an exter.nal magnetic field (or w, << 1) transport equations have been analyzed on the basis of the equations (1) and (2) in /3,6/ where, in particular, the Curtiss-Muckenfuss fo~ulas for the heat conductivity and Stefan-Maxwell relations for the diffusionfluxesin case of higher approximations for transport coefficients have been generalized. Besides, it has been show.n that it is expedient to introduce the~odiffusion ratios rather than multicomponent the~odiffusion coefficients as independent coefficients. In the presence of an magnetic field all results obtained in /3,6/ extend to the flow camponent parallel to the magnetic field and essential transport coefficients.
HEAT TRANSFER IN A MAGNETIC FIELD
Consider the set of equations (2) for the components of flow and gradient vectors no~l to the magnetic field. We can write eq. (2) in the fo~ :
" ,_, ,., ,., - J. "' 1'0 ...... J.. Z:Z: Av 'f;, = il:i v.ttnrtfrv -~ /\i, lt (3) J=f /l=f j=f II "I
where 'l. ~" _ 1\'f. + i* 38 rtr+IJ mirzi(/)i r. ·l1 r. ~ ~) I ''I - V 71 r(/"+5/2) I<T ay 'i" 'l* =- 7
The following property of coefficients remains valid for coefficients ~';" :
'A~ ='fi.P! t J.~ =o <4) 'i 'I" ) , .. , 'I Solving the set of equations (3) in respect to vectors fJ,. (p=1, ••• ,J-i) and summa.zing the reduced partial hea~ flows over all components the reguced heat flow in the direction no~l to the field ~ is obtained'
-l II .....,J. N _.J. n_ -= ~ o i =- z; Pi{:-, = (5) 7 r=t 1' ,/ z=1 '
=- ~ v.J.T +PE fr. V:J. ,.. j=t 'I "I
Here A is the complex the~l conducti vi ty of mixture, -K~ are the complex thermodiffusion ratios.
1361
For the complex coefficients following expressions:
we have the
-,l('j. =-I* I
0 Xs 0 .... -u
XrArs
0 1\2{ rs
0 x:s
1\/Z rs
-u Ars
0
I • •'
1\'0 - /f /2.
"i Ars Ars · · · · 1\20 21 ,.. 22
Ars Ars ... · 'i
1\ H~ !\ ,.,, A Hl. I'J 1'8 I'S ....
0
A'"" ,., /\211
I'S
,... """' /\,.$
( f:. Kr. = 0 M = '- f ) J=l J I
I "- I /\- ~-,~· Where is the deter.minant of the matrix 4
(6)
(7)
(r,p ~ 1 ). - ,.., The complex coefficients ~ and KT.t can be ex
pressed as the sum of the real and imaginary parts:
\ = ~.1. + i'" )/' kTd· = kf. -t- i""k~· (J·= i N) /\ , d d , ... , (8)
where the notation "~" notes transport coefficients in the normal direction to the field, and the notation "/\" in the transverse one. Then for the heat flow in a magnetic field ~ we can obtaine the known expression /1/
lj: Cj" + ij.J.. =-)/ V"T- ~J. VJ.T- )./'b ~tVT + (9)
II n -11 .L - .L A '"* :-r + p Z:: ( kr- V.. + kr. l{ + k~ h 11 ~- )
j=t ' "I ' 4 fl
Thermal conductivity and thermoduffusion ratios can be calculated from for.mulas (6) , (7) directly without preliminary determination of all other transport coefficients - diffusion, thermodiffusion and A'
1362
in contrast to the standard Chapman-Enskog procedure/1~ Note that formula (6) is the generalisation of
the Curtiss-Muckenfuss formula for the thermal conductivity in case of heat transfer in a magnetic field and with regard to the arbitrary number of approximations for transport coefficients in the Sonine polynomial expansion. Formulas (6) and (7) for the complex transport coefficients formally coincide with the formulas for transport coefficients in the absense of a magnetic field.
HIGER APPROXIMATIONS FOR STEFAN-MAXWELL RELATIONS
Turn to the derivation of the Stefan-Maxwell relations for diffusion flows in a magnetic field. For vectors normal to the magnetic field we get from ( 1 ) :
_j .L tl ,.. oo 11 J. II 1..:.1 op ~ .J. Q' =-r..",ol( vl( - z L 1\i" rl(" , 10) K=f K=l p:l 6
Wb.ere
1\-oo 1\ oo 0 ff m r JK -= 1 '''K - z rrfiJ· ( cl(wl( -~·I(~->
Eliminate the vectors f.c~ (p=1, ••• ,M) from ( 10) ~ing expressions obtained for them, then the vectors aj will be expressed in terms of diffusion flows and temperature gragient, that is Stefan-Maxwell relations will be obtained. Taking into account, that
# ,..,
KT· = -EL. I K=l P=l
.- ( tp_.K:--' Ii\ I
0 0 , , , dKS .. 0 0 1\0fO ,_ II A {P A IH ~ Xr Ars ... "rs ... "rs 17\ I · 0 •••••••••••
0 H { lt. H p A. HM Art, ... "rs .. · "rs
0 J\ O( 0/of J.s · ·. Ajs
1\ fO /\,_I{ A 1H f'l< r$ · · · I ' rs
...... A MO J\ Hf ,... HM
I \ rK rs · • • !\ rl
,.;...., ( :E fP,ol!. = 0 ) j=l '
the Stefan-Maxwell relations can be presented as
( 11 )
( 12)
~L II -40 - -J. ,_ c( = - I: ( A1·1( - lf.l( ) ~ - k~o v.1. tn r < 13 )
• 1(::( ' II
1363
The complex coefficients il,.i are symetrical ( ~- = ~-,· ) and take account of the corrections for the coerficients · }\~ owing to higher approximationsL ~ A
'J Introduca ip~- in the form ffy- = <.Pv·+i*t.'i.- wi th due regard for (8) , and the Stefan-Maxwell relations be-come:
~1. ,/ II ..;. -
d- = L {/f· C-WJ( (CK -0:1() + f/'.'1< f1)} b 1t ~ + ( 14) 'J t<=l 'd tl I
" X.~ -.1 -.l) J. g__ 1\ ... ln + z: ~ ; J. r~-r I- k7-r;.LUIT- Kr;· h" v T /(::( ,,~~. ~~(1)
Here ~~~= ~~(T) are the corrections taking into consideration the ~ffect of higher approximations on diffusion in a tranverse direction, h"~<tF)-= f~'fJI)= 1/(1-~f) where .A-;·= Cf~·tf)/.A;- are the correct~on factors for binary coeffic~pnts~diffusion. In the presence of a magnetic field /i" # f,·· , that is the anisotropy of diffusion coefficitnt3 in the longitudinal and tranverse direction to the field takes place if at least the second approximation is allow for. Stefan-Maxwell relations for diffusion flows in a magnetic field with regard to /3/ are written as
_7 "' I m " ~ - XXI< _, ;-11, ~- -: ~l~KT Cj·WI( (C..c- ~J') + ",K (f}} bx ~ ~:~tJfj:tf}(~- ~)+
~.·3!~ - J. -.1~ d 1/ p__ .J. J./L A _. !n ( ) + ~.L ,JVx -~.) -k~-'1 VIT- K~- '1 wT-K~·hJtV T 15 ~"Kf~ 'i"tp 'I
In formula (9) for the heat flow and in the Stefan-Maxwell relations (15), thermal conductivity, ther.modiffusion ratios and multicomponent correction factors for binary diffusion coefficients act as transport coefficients. Such presentation of relation between the flows and thermodynamic forces is most expedient for solving the specific problems.
NUMERICAL RESULTS OF CALCULATIONS FOR TRANSPORT COEFFICIENTS IN HIGH APPROXIMATIONS
The numeri~al calc~lations of a number of transport coefficients J."r;) , kr,-lfJ, flff) have been carried out for different approximations of expansion in Sonine polynomials. There convergence in 1 has also been studies. The calculation have been made in the region of
1364
pressure and temperatures (P=1.013 103N/m2 to 107N/m2, T=2000 to 20000 K) which embraces all the characteristic states of multicampnent gas mixtures - from the neutral state to the total ionization. Earlier the calculation of transport coefficients given in ' traditional form (that is the coefficients >-.'tf) , D;lf)and D;/f)) in higher approximations ; was conducted for number of particular gases, for example in /7/ and /8/. As a result of the performed extensive study of the higher approximations effect on the all effective transport coefficients in the transport equations for chemically equilibrium air we have constructed the chart (Fig.), which shows the necessary order in higher approximations when calculating the effective transport coefficients with errors ~ 5%.
The curve 1 shows the boundary on the left of which (region I) the thermal conductivity and viscosity coefficient may be calculated usi~ the first nonzero approximations {i.e. ~ (2),- }1{1)), while the correction ratios for all effective coefficients may be neglected ( i. e. /jj_ =0, I<Ti =0). On the righ t of the curve 1 ( region II;III,IY,Y) the thermal conductivity must be calculated in the third approximation (the second nonzero approximation). On the right of the curve 2 (region III, IY,Y) the viscosity coefficient must be calculated in second approximation t ; on the right of the curve 3 (region IY,Y) the correction coefficients beco.me essential and only on the ri~t of the curve 4 (regionY ) the thermodiffusion ratios J<rr. (1) must be calculate for the calculation effective coefficients. In this region the correction coefficients / 1i.· {1) and the thermodiffusion ratios k~ (f) may be calcula~ed in the second approximation (first nonzero approximation).
P. atm
10.0 .A (2,1 I .Al?>) TI }41!1 }l~)
~.o 5~ (i\= i f~j(1}=i
Krj{i\:50 Krj(4)::a iY
0.~ ~(~) Jl(2)
O.Oi 5000 100011
5y (\) Krj (2.)
Fig. 1
1365
The numerical results show that the corrections to the effective coefficients, such an effective Shmidt number Sj•H and Lewis number Le,;•H due to the valuce of f~ C;> amount to 12% and for the other effective coeffic~ents these corrections give contribution ~ 5%. When calculating the effective Prandtle number Ö~f of highly ionized air the thermal conductivity Ä (,) should be computed in third order approximation , and second approximation is needed for the viscosity coefficient so determinantes in the expression for the thermal conductivity and viscosity have the same order 2xN. The using of the lower ordere of approximations leads to an error in Prandtle up to 60%.
For neutral components of the partially ionized air the thermodiffusion ratios K~· (f) and corrections ~~~(,) are small (KTNI/~0,01, 'fi.J· ~ 0.004), andin the Stefan-=Maxwell relation these co~fficients must be only account for the such pairs in which at least one of the component is ionized particle. Thus the complete set of all the tranapert coefficients based on the higher approximations ( A (3), f'1(2), l<rz-(2) and Jz·(2)J is needed only for highly ionized air (.z1 -0.5) ~corresponding to region Y on Fig.
REFERENCES
1. Ferziger J.H.,Kaper H.G. Mathematical theo~ of tranapert processes in gases.-Amsterdam-London {1972)
2. Chapman s.,Cowling T.G. The mathematical theory of non-uniform gases.- Cambridge University Press (1952)
3. Kolesnikov A.F.,Tirsky G.A. Hydrodynamic equations or partially ionized multicamponent mixtures of gases with tranapert coefficients in higher-order approximations.- In: Molecular gasdynamics, by v.v. Struminsky. Nauka, Moscow (1982)- In Russian
4. Kolesnikov A.F. Transport equations for high temperature ionized mixtures of gases in an electromagnetic fields, Trudy Instituta Mechanicy MGU,by G.A. Tirsky, Moscow, 39:39 (1975) - In Russian
5. Zhdanov V.M.,Yumashev P.N. Diffusion and tranapert of heat in multicomponent completly ionized plasma, Journal AMTPh 4:24 (1980) - In Russian
6. Kolesnikov A.F., Tirsky G.A. Hydrodynamic equations and tranapert equations of ionized multicomponent two
temperature gas mixtures.-In: Models in continium medium mechanics, Novosibirsck (1978)- In Russian
1. Capitelly M.,Devoto R.s. Transport coefficients of high temperature nitrogen, Physics Fl.,11:1835(1973)
8. Sockelova I.A. Transport coefficients for air in temperature region from 3000 to 25000 K and pressure 0.1, 1,10,100 atm, Journal AMTPh, 2:80 (1973)-In Russian
1366
SPECTROSCOPIC STUDY OF A PLASMA FLOW ALONG THE STAGNATION
STREAMLINE OF A BLUNT BODY
M. Nishida and A. Nakajima
Department of Aeronautical Engineering Kyoto University Kyoto, Japan
INTRODUCTION
The dark space which is the region of reduced self-luminosity ahead of a shock in ionized gas flows is much interesting concerning a shock problern in pre-ionized gas flows. According to Grewal and Talbot, 1 the dark space results from the elevated electron temperature ahead of the shock. Namely, there exists a broad zone of elevated electron temperature ahead of the electron compression region, caused by the high thermal-conductivity in the electron gas. When the electron density remains unchanged in this region, the elevated electron temperature reduces the recombination rate. Since the self-luminosity of the flow is due to the recombination radiation, the dark space is observed in the region of the elevated electron temperature.
The self-luminosity is visible radiation which is caused by radiative transitions between bound electroni'c levels. For argon of the present interdst, visible spectral lines having strong intensity result from radiative transitions 5p+4s and 4p+4s so that the reduction in the atoms excited in electronic levels 5p and 4p are expected to make the dark space. Therefore, the behaviour of the atoms in these levels is to be investigated to reveal the dark space. In the present work, the atoms excited in state 4p[3/2] are treated.
The present work is concerned with measurements of the axial variation of excited-level population densities from the upstream of a detached shock caused by a blunt body to the downstream of it, and a comparison of these with theoretical predictions. The predictions were obtained by solving the rate equations for electronic
1367
levels wherein observed electron temperatures and densities were employed to determine collisional transition probabilities.
EXPERIMENTAL PROCEDURE
The experiments were carried out in a low-density plasma wind tunnel operated by a dc are-type discharge. A schematic diagram of the experiments is shown in Fig. 1. Argon was used as a test gas. A blunt body having a nose radius of 4 cm was placed at a location 8.5 cm from the nozzle exit. The body was electrically insulated from the tunnel wall so that the potential on the body was the floating potential. The surface temperature of the body was controlled by cooling the body with nitrogen gas, being kept within 407±5 K. Since the blunt body is connected with the noz zle as shown in Fig. 1, spectroscopic measurements were able to be made at any station between the nozzle exit and the body wall by varying the position of the nozzle relative to the optical axis.
Typical operating conditions are as follows: argon mass flow rate = 0.11 g/s, stagnation pressure = 1.8 kPa, test section pressure = 10 Pa, stagnation temperature = 3400 K. Flow parameters along the centerline were estimated from the impact-pressure measurements and measured stagnation conditions by using the isentropic flow r elationshi p . The stagnation temperature was calculated by using simple choked-flow relation. 2 Under previously cited flow conditions, a Mach nurober was 4.4 at the blunt body position. With
~trometer l-V A _ 0 Micro-Photomuttiplier Converter Amplifier Converter Computer
0-ring Light Guode Pope
' To Pumps
Coolant Gas
Fig. 1 . Schematic diagram of experiment
1368
the measured Mach number used, the atom density was estimated as 1.9x1o2l particles/m3 at the body position. From a comparison of the atom density with the observed electron density, the degree of ionization is seen to be much less than unity so that the over-all flow field may be considered to be governed by neutral atoms. Under the assumption of a viscous shock layer, shock layer equations were numerically solved to estimate the shock layer thickness. The result shows that the shock layer thickness is 0.67 cm.
In order to make radial survey of spectrum an optical fibre was used. A lens mounted in the light guide pipe with 2.5 mm x
2.5 mm cross section collects light parallel to its axis coming from the plasma and focuses it onto the end of the optical fibre. Light emitted from the other end of this optical fibre is refocused by two lenses onto the entrance slit of a spectrometer which is equipped with a photomultiplier tube. The optical system was calibrated for absolute intensity measurements by using a standard tungsten ribbon filament lamp. After an Abel inversion for radial data was made on a computer, the spatially resolved population densities were determined. The emission measurements were made on the line at 7635 A. These measurements were carried out at 20 stations between 7 cm and 0.2 cm upstream of the blunt body. The population densities of state 4p[3/2] are deduced from the data of 7635 A.
Electrostatic-probe measurements were also made along the flow centerline to determine the electron temperature and density. The measured values of the electron temperature and density are employed for predicting the population densities.
THEORETICAL PREDIGTION
The excited-level model employed in this work is the same as used in the earlier work. 3 The present model contains 20 levels, and the populations of the highly excited levels above 9s were assumed tobe in equilibrium with free electrons. From a comparison of the relaxation time of each electronic level with the characteristic flow time, 4 we can write, for level 4s,
d[n(2)/n]/dy = ~(2)/nu (l)
and for levels higher than level 4s,
~( j) = 0 (2)
where ~(j) is the net production rate of level j, n(2) the population density of level 4s (j=2), n the global density, u the flow velocity and y is taken in the direction of the flow. The expression for ~(j) is the same as given in Ref. 4.
1369
According to Drawin and Emard, 5 the collisional frequencies for atom-collisional transitions are negligible compared with those for electron-collisional transitions in the present conditions so that the atom-collisional transitions are ignored. Therefore, only the electron-collisional transitions are considered. For such transitions, Drawin's expression6 for excitation rate constants was employed.
Eqs. (l)-(2) can be solved to provide the population densities of all the levels. In the calculation, the electron temperature and density, flow velocity, and global density are needed. The electron temperature and density are obtained from the experimental data. The flow velocity and global density are deduced from the stagnation conditions and impact-pressure measurements.
RESULTS
· Figure 2 shows the distributions of the electron temperature and density measured along the f low centerline. The electron temperature is elevated in the region from y=3.75 cm to y=0.2 cm and reaches 2500 K at y=0.2 cm, where y is taken as a distance from the body wall. It should be noticed that the electron density is nearly constant in the region from y=3.75 cm to y=2.75 cm while the
::.:: ..
1--
3000.-.---.---~--~--,---,----r7,~ I
0 Te Preodic teod Shock Position-: I
o ne I
~H 2000
? ? b ? ~~hh~oHhHTf !
~ ~ o 9 "f9oy~29~~~too9?ttf~ 1000 qo
y.J
1020
1: ;:::
.. c
1019
Fig. 2 . Axial distribut ions of electron temperature and density.
1370
electron temperature is elevated in this region.
Figure 3 shows a comparison between the population density distributions of state 4p[3/2] for both cases when the body is placed in the flow and when it is removed. The result for the case when the body is placed, was deduced for two cases; one is the case when the absorption of Ari 7635 radiation by atoms in state 4s[3/2] 0 is taken into account in the Abel inversion, and other is the case when such absorption is not included in the Abel inversion. The radial profiles of absolute density of absorber (atoms in state 4s[3/2]0 ) and its temperature are required for absorption to be taken into account in the Abel inversion. The population densities of state 4s[3/2]0 on the flow centerline were determined by assuming an isentropic flow. Also, the radial profile at each station was approximated by that obtained by Limbaugh7 while it is assumed that the jet radii of the emitter (atoms in state 4p[3/2]) and absorber (atoms in state 4s[3/2]0 ) are approximately equal. It was also assumed that the absober temperature is constant in the radial direction. In addition the Doppler broadening was taken for the spectral line. Thus obtained population density distributions are
E u .....
1.4
!!. 1. 2 'Ä ·-c ::::: 1. 0
0.8
0.6
0.4
0.2
4p[3/2J
- ·<>-· absorption considered }on thestagnan(4s)=10181/m3aty=7cm tionstream
hne ofa -o- without absorption blunt body
--X-- without a blunt body
I
7
\ ~
~
'
6 5
predicted shock position
4 3 y,cm
2 0
Fig. 3. Axialdistributions of population density of state 4p[3/2 ].
1371
shown in Fig. 3, which are norma1ized by the population density at y=7 cm. In considering the absorption, the population density of state 4s[3/2f at y=7 cm was taken ta be 1018 partic1es/m3 and 1016
particles/m3 • The result far the latter case is nearly the same as the result obtained when the absorption is not considered. Even when the va~ue of the papu1ation density of 4s[3/2] 0 at y=7 cm is taken tobe 1018 particles/m3 , the popu1ation density is on1y approximate1y 1.25 times 1arger than that for the case without absorption. In Fig. 4 the va1ue of the population density obtained when the body is p1aced in the f1ow falls below that withaut the body in the zone fram y=3 cm to t=l.5 cm. In t his zone the radiation i ntensity is reduced and this zone i s the dark space for t he 7635 A radiation. It may be mentioned from the above resu1ts that the dark space exists in the region from y=3 cm ta y=1.5 cm. In this region the e1ectron density is near1y unchanged whi1e the e1ectron temperature is elevated, as shown in Fig. 2. Therefore the dark space is seen ta be assaciated with t he e1evation in the electran t emperature.
COMPARISON OF EXPERIMENT WITH PREDIGTION
The camparisan of the experiment with the predictian is shawn in Fig . 4, where g(j) is the statistical weight af state j. In the prediction the measured e1ectron temperatures and densities were used for estimating the co11isional trans ition pr obabi1ities . We
1015
M
E
~ c
1014 '"""
4pt3/2l
0 absorption considered n(4s) = 1018 11m3 at y: 7 cm
o without absorption
--- prediction
7 6 5 4 3 2 0 y,cm
Fig. 4. Camparisan of experi ment wit h pr ediction far state 4p[3/2] .
1372
tried to solve Eq. (l) for three different boundary conditions of n(2)/g(2) at y=7 cm such as 101 5 , 1016 and 10 17 particles/m3 , where g(2) is the statistical weight of level 4s. The reason for the choice of these values is that the 4s-level population densities observed in the earlier work4 fall among these values. The predicted result for the three values gave the very close result for the population densities of state 4p[3/2].
In Fig. 4, it is seen that the experimental population density decreases in the region, y=7 cm to y=2 cm, and then increases. The prediction shows the density distribution very similar to the experiment, but in the downstream of y=l.5 cm the predicted population density increases less rapidly than the experimental one. The decrease in the predicted density in the region, y=7 cm to y=4.5 cm, is due to the decrease in the electron density. The prediction shows the reduced density in the region, y=4.5 cm to y=2 cm, in spite of nearly constant electron density. In this region, the electron temperature is elevated. Hence, the reduction in the predicted population density in the region, y=4.5 cm to y=2 cm, results from the elevation in the electron temperature. From a comparison of the experiment with the prediction, it may be mentioned that the reduction in the ,experimental population density of state 4p[3/2] ahead of the shock is caused by the elevated electron temperature.
CONCLUDING REMARKS
The spectroscopic measurements of the dark space were made for the case when the blunt body is placed in the flow and for the case when it is withdrawn. In the former case the spatially resolved populatiQn densities of state 4p[3/2] observed ahead of the detached shock are reduced in comparison with those observed in the latter case. This phenomenon is undoubtly owing to the existence of the body, and so called the dark space. In the region of the reduction in the population density, the electron density is nearly constant while the electron temperature is elevated. The population density distribution predicted by using the observed electron temperatures and densities shows the reduced population density in the region where the electron temperature is elevated. A comparison of the experiment with the prediction shows that the reduction in the population densities of state 4p[3/2] is caused by the elevation in the electron temperature ahead of the shock. This elevation in the electron temperature results from high-thermal conduction of the electrons from the high temperature region behind the shock. As the electron temperature increases, the equilibrium population densities of upper levels will be reduced. This reduction propagates down to lower levels through collisional and radiative de-excitations. Thus the population densities of level 4p are reduced, which leads to the dark sapce. Therefore, the dark space
1373
can be observed only in the plasma where the upper level populations are in equilibrium with free electrons and where the transitions between upper levels are controlled by electron collisions.
HEFERENCES
l.
2.
3.
4.
5.
6.
7.
M. s. Grewal and L. Talbot, Shock-wave structure in a partially ionized gas, Journal of Fluid Mechanics, 16:573 (1963).
R. B. Fraser, F. Robben and L. Talbot, Flow properties of a partially ionized free jet expansion, The Physics of Fluids, 14: 2317 ( 1971) •
A. Kimura and M. Nishida, Spectroscopic study of a partially ionized freejet expansion flow, Transactions of the Japan Society for Aeronautical and Space Sciences, 22:162 (1979).
A. Kimura, K. Tanaka and M. Nishida, Optical absorption measurements of excited argon-atom densities in an ionized freejet expansion, in: "Rarefied Gas Dynamics," S. S. Fisher, ed., American Institute of Aeronautics and Astronautics, New York, (1981).
H. W. Drawin and F. Emard, Atom-atom excitation and ionization inshock waves of the noble gases, Physics Letters, 43A:333, (1973).
H. W. Drawin, Collisional and transport cross sections, EURCEA-FC-383, Association Euratom-C.E.A., Fontenay-aux-Roses, France (1967).
C. C. Limbaugh, Excited state distributions function of argon in a freely expanding arcjet plume, in: "Rarefied Gas Dynamics," American Institute of Aeronautics and Astronautics, New York, (1977).
1374
ON MODEL KINETIC OPERATORS AND CORRESPONDING LANGEVIN SOURCES FOR
A NON-EQUILIBRIUM PLASMA
I. Paiva-Veretennicoff * and V.V. Belyi * * Fakulteit van de Wetenschappen en Toegepaste Wetenschap
pen, Vrije Universiteit Brussel, Pleinlaan 2 1050 Brussel, Belgie
* Izmiran-USSR Academy of Science, Moscow Region 142 092 USSR
ABSTRACT
A model kinetic operator is proposed, describing at the level of the first Chapman-Enskog approximation, the relaxational and hydrodynamical behavior of a weakly coupled, n-component plasma, each component being near to a local equilibrium state (n temperatures, n velocities).General expressions are derived for the associated Langevin forces, appearing when long-range fluctuations are present in the system. They display new couplings between the hydrodynamical variables. As a result, new Landau and Lifshitz formulae are obtained ; some of them show a non-Markovian behavior, typical for plasmas.
The one component case is discussed briefly as well as the analogies and differences with the neutral gas of hard spheres. All the relaxation times are evaluated explicitly in terms of the parameters of the system.
INTRODUCTION
In nature (e.g. in the ionosphere) and in the laboratory (e.g. in fusion devices), plasmasoften exhibit long-Pange ftuctuationa, with a characteristic wavelength much longer than the Debye length. These fluctuations deeply modify experimentally relevant properties of the system, like form factors, correlation functions or transport coefficients.
1375
A kinetic theory of these fluctuations was first developed by Kadomtsev 1 • For the electron .gas an extensive review can be found in Gantsevich et al 2 • Here, we shall use the notations of Klimontovich 3 , who presented the theory for non-equilibrium multicomponent gases and plasmas.
It is well known, that the long range fluctuations in an open, weakly coupled system, can be described by a fluctuating kinetic equation of the Boltzmann type (for neutral particles) or Landau or Balescu-Lenard type (for plasmas). The deviations of the one-particle distribution function with respect to a (non-equilibrium) reference state is then determined by a linearized kinetic equation with an additional stochastic source term, called the "Langevin source" in analogy with the theory of the Brownian mo-
·tion. The intensity of these Langevin sources as well as all the experimental relevant properties of the system can be shown to depend crucially on the Unea.rized coZZision operator and its inverse 2 •'3 , which is linked with the so-called "memory functions".
However,in any explicit calculation e.g. of form factors or transport coefficients one introduces, sooner or later a truncation procedure, which preserves only the essential properties of the true collision operator, and leads to much more tractable expressions. The most widely used collision model is the Bhatnagar Gross and Krook (BGK) model 4 • An overview of extensions and generalizations of these collision models can be found in Cercignani 5 • We have no space here to refer to all the literature published in this field. Let us just point out that going from multicomponent Maxwellgases 6 to mixtures of polyatomic molecules 7
and to one-component plasmas in a large range of values of the plasma parameter 8 , the different analyses all consider the system to be near thermal equilibrium. They aim at the understanding of molecular dynamics simulation data of form factors and correlation functions.
In this paper we propose a model kinetic operator for a weakly coupled n-component plasma near the local equilibrium state, with an expression for all the characteristic frequencies. In cantrast with most models of multicomponent systems 9 there are no more adjustable parameters. Our model is shown to describe accurately (i.e. on the same level as the first Chapman-Enskog approximation) the evolution of the moments of the distribution function connected with temperature and velocity differences, as well as the species stress and heat fluxes. The consequences of this model on the intensities of the Langevin sources are discussed. Generalized fluctuation-dissipation theorems are derived. In a final section, the one-component case is discussed, and the analogy is mentioned
10 with the results for a Boltzmann gas of hard-spheres
1376
THE MODEL COLLISION OPERATOR
We start with the set of Landau-Vlasov kinetic equations governing the evolution of the one body distribution functions fa(qp;t) of a weakly coupled multicomponent Coulomb plasma of particles with charges ea and masses ma (a is the species label ; a = I ... n)•. When long-range fluctuations (with characteristic wavelengths larger than the Debye length AD) are taken into account, the distribution functions and the electrical field become fluctuating. The small-range fluctuations (A < AD) have been smoothed out and their dissipative contributions are taken into account through the collision integral Iab{fafb}. We shall not consider here the dissipation due to the long-range fluctuations, leading to the well-known "quasi linear" diffusion terms I 3• Indeed, if the like-particle collisions dominate the time evolution of fa, the system reaches rapidly a ZocaZ equiZibrium state, in the neighbourhood of which "turbulent collisions" are negligible 3 • Let ea measures the local temperature in energy units, na the local density of each species and Va the local velocity. These functions are smooth functions of space and time. The deviations with respect to this state, satisfy a set of coupled fluctuating linearized kinetic equations with additional "Langevin source" terms with known stochastic properties 3
a H + v . .,! öf + e E 0 t a oq a a
a ö f + e öE • a f 0 op a a Opa
- öi öf ya(qp;t) a a (I)
and the cross correlation functions of the Langevin forces are given by 3
Y a (ql Pl t1 )yb (ql ~ tl )
.{(ai <Pl>+ orb<~})ö bö(Pl-~>f cj;t> a a a
(2)
• The results presented here are also valid for the BalescuLenard kernel, differences will only appear in the values of the cut-off wave vectors.
1377
Here the jUnctions Ia(p) and Iab<PoP1) only appear for non-equilibrium systems and are linked with the action of the full collision operator on the reference state. They can be evaluated explicitly and present no difficulty 12 •
It is the Zinearized ooZZision operatoP öia(P), appearing in (2) and (3) that plays the central role not only in the determination of the intensity of the Langevin sources, but also in the determination of the form factors and transport properties of the systems 3 • 12 • Any theoretical approach leading to concrete results, involves sooner or later a trunoation pPOoedure, which amounts to modeZing this operator. We shall follow here the method of Sirevieh (ref. 13, see also 5, 14) but shall propose a more compact projection operator technique and Hilbert Sface formalism. Therefore we first introduce the state ket vector lh>, describing the state of the multi-component plasma, near local equilibrium :
I h> = r h1 • h2 • • • h 1 n (3)
where the components ha are defined through
cSf = f 0 • h a a a
and a Langevin source ket of lh> is determined by
A A A
(a .. 1
a lh> + ~lh> + vlh> - örlh> = IY> t
n) (4)
Ynl • The time evolution
(5)
where ~ is a diagonal operator, describing the free-flow and V is the non-diagonal linearized Vlasov operator, which results from the field terms (1) by using the Poisson equation explicitly. öl is the linearized collision operator.
We then introduce a scalar product in the standard way
<1Pix> ~ ...!_o f dp ~ X (p) 1jl (p) n a a a a=l a
(6)
which allows us to define matrix elements of all the operators appearing in (5) e.g. (7)
Due to the non-~quilibrium character of {~,a • 1 ••• n} the collision operator cSI is non herrmitian and most of the elegant, simplifying properties of the global equilibrium case 10 ' 12 arenot applicable.
1378
Defining öva- v- Va' the matrix elements of öl becomes
A
<ljllörlx>= -I: a
In view of the const~ctionA of a model for öi, we introduce two p~ojeation operato~s, H and N with the properties
(8)
and
H + N • Id (9)
Here Id is the identity operator. H is the projection operator onto the "hydzoodynamiaaZ subspaae", spanned by kets {lljl.>} corresponding to the polynomials of lowest order in the momen~um variable. The five (times n) first polynomials correspond to the collisional invariant& per species : density, momentum, kinetic energy. But the space will be extended to higher order polynomials. Th~ir
number and order depend on the physical processes one wishes t~ treat "exactly". They correspond to non-conserved quantities like pressure tensors, and heat fluxes per species. The projection operator N projects the state vector on the "non-hydrodynamical subspace".
with
We construct H in the following way
n 13 H• I: I:
a•1 i•1
al b <ljl. ljl.> •ö .• ö b 1 J 1J a
(10)
(11)
The kets I1Pf> have one non-zero component ~Pf(Ü). Per species, every jUnation 1Pf (i • 1 ••• 13) of the properly dedimensionalized velocity ü, corresponds to one of the thirteen first Grad polynomials.
1379
Introducing this dimensionless velocity,
u = (p- m V )(6 m )-112 a a a a a
(12)
We have for instance
(13)
r=1,2,3 (14)
a 1ff ( 2 tjls = 116 ua - 3) (15)
tjl6 • • • tl11o are of second degree in u and correspond to irreducible tensors of second rang.
a -1 (u2 - 5) (16) tjl10+r TO .U a,r a
The choice of this basis is obviously suggested by the derivation of the plasmadynamical equations in the framewerk of kinetic theory. All the quantities appearing there are obtained by taking the scalar product ( 6) of the state vector lh> with one of these polynomials. Indeed, the deviations per species with respect to their local equilibrium values are for instance
- the number density deviations
- the velocity density deviations 0
6 112 n° 6V • n° (~ <tjl8 11h> r • 1, 2, 3
a a,r a m8 r+
- the temperature deviations
We now introduce the truncation procedure
A A A A
H 61 Nlh> • H 61 Nlh> = 0
(N 61 Nlh>) = - V (Nih>) a a a
1380
(17)
(18)
(19)
(20)
(21)
By (20) we ignore all non diagonal elements : in accordance with the spirit of the Grad approach, we assume that we can approximate the evolution of the plasmadynamical variables (34-38) by a set of aZosed equations. This assumption can be relaxed, and leads, in the one-component case, to renormalized transport coefficients appearing in the intensities of renormalized Langevin forces • 1 4. For the non-equilibrium multi-component case, however, the explicitation turns out to be too cumbersome to be presented here.
Assumption (2!~ defines in a standard way, non-hydPodYnamiaaZ PeZaxation times va for the different components. The frequency va is chosen to correspond to the inverse of the longest non-hydrodynamical relaxation time, in our case :
V • a C22)
where the kets l$~4 > (a • 1 ••• n) are constructed with the GRAD polynomial of fourth degree in na
a 1 4 2 $ "' -- (u - 10 u + 15) 14 ,~ 120 a a
(23)
lt can indeed be shown that all the polynomials of higher tensorial character, even if they are of lower degree, like the flow of the
iresaure tensor yield shorter relaxation times·. Although the kets $~4 > arenot dipeatZy linked with any physical quantity, they ap
pear, via the orthogonalization procedure, when the flow of the heat flux is taken into account in the non-hydrodynamical subspace. The choice of (22) is in agreement with McCormack 14 and is consistent with the fourth order Maxwell molecule model of Boley and Yip 6
The explicit value~ of v4 - which are still (slowly) varying functions of space and time - are given in reference 12, in terms of all the local equilibrium parameters.
With assumptions (20), (21), our model kinetic operator becomes
• This will be presented of the 3rd Conference on Fluctuating Phenomena in Physical Systems, Vilnius, September 1982.
1381
A 13 (6IIh>) = - ~ h + a a a E v ~~ (u )<~~lh>
j=1 a J a J
n 13 + E E
a=1 i ,j=1 a al Al b bl ~. (u )<~. 6I ~.><~. h> 1 a 1 J J
(24)
It is a tedious tast to evaluate the ~atrix elements <~fl6il~j>. In the multi-fluid picture no part of the hydrodynamical basis correspond to collisional invariante, which means that, due to the unlike-particle collisions, quantities of the same tensorial character (like the ~~~+ 1 > and ~~~ 10>) are linked to each other. Moreover the non-equil1brium situat1on, leading to the non-hermiticity of the collision operator, also leads to couplings between kets of different tensoriaZ ch~cte~, like densities and velocities. However, most of the latter matrix elements can be neglected, if we admit a "weak anisotropy", i.e., that the macroscopic velocity differences between the species be much smaller than the thermal velocity of each species. We give the values of all nonvaniehing matrix elements in reference 12.
INTENSITIES OF LANGEVIN SOURCES AND GENERALIZED LANDAU AND LIFSHITZ FORMULAE
Using our model kinetic operator, the cross correlation functions of the Langevin forces given in (10) become
Ya <Ci1 P1 tdyb <Ci2 P2 t2 > .. + 6Cr1-r2 >Ht1 -t2 > 13
{ + 2 '\1 6 b 6(p1 -~ )f0 - 2 v E f~ (pt)tß (~ )~~(Ü~)~~<U:>6 ab a a a a i•1 a 1 1
(25)
To find the generalized Landau and Lifshitz formul•e, one has to consider the time-evolution of all the quantities appearing in e.g. (18). The source terms in these fZuctuating equations is proportional to <~fly>. Their correlation functions are easily obtained from (44). As an example, we give the intensity of the Langevin source in the momentum baZance equation, a simple and new result,
1382
• m n 9 V a a ac ac (26)
It is also easy to check that these Langevin sources do not contribute to the totaZ momentum and totaZ energy of the system, which are conserved quantities.
The intensity of the Langevin sources for the higher-order hydrodynamical variables (i,j ~ 6) display a non-Markovian behavior. This is due to the presence of the Vlassov term in the equation (5), which is responsible for the appearence of high frequency motions 16 (plasma modes). For the sake of simplicity, we shall show this in the one-component case. The generalization can be found in 12 •
THE ONE-COMPONENT PLASMA AND HARD SPHERE GAS 1 0
... In both cases (resp. "P" and "G") 61 is hermitian and diagonaL
in the hydPoaynamicaZ subspace. The non-vanishing matrix elements have the same strructuzoe
... <111. I öl 1111. > = - tll
1 1
2 • - 3 A1
e4 n2 li'L ml'2el'2
(L is the Coulomblogarithm)
i - 6 ••• 10 (27)
i • 11 13 (28)
(29)
For the hard sphere Boltzmann gas, we adopt the notations of Sommerfeld 11 The non-hydrodynamical relaxation frequency v is
(30)
which allows for the following very simple expression of the model kinetic operator
5 10 6IIh> = - i A1 lh> +! A1 I l111-><111.lh>- -31 A1 I 1111-><ljl.lh>
3 i•1 1 1 i•6 1 1
(31)
We insist on the fact that although the microscopic heat flux have
1383
disappeared in (31), this operator gives the "correct" heat conductivity (equivalent with the first Chapman-Enskog approximation). Substituting into (2) :
10 + i A1 f0 (pJ)tc' (pl) l: ljli (pJ)ljli (Pl)
i•6 (32)
If one takes the Fourier transform in time of the macroscopic equations, the deviations of the pressure tensor and the heat flux can be shown to be
ös (w) • öcr - Sne V öe r r 2(-iw + i A1)m r
where ÖWij and öcrr are the time Fourier-transformed Langevin sources in the plasmadynamical equations, divided r~sp. by the coefficients (-iw+Al) and (-iw + f A1 ). Using (32), their spectral functions are easily found -
and el
(öcr.öcr ) k • 2 ReA(w) --k ö. 1 r w, B 1r
where n(w) and A(w) are the frequency-dependent viscosity and heat conductivity of the system
n(w) ne • A1-iw
For a neutral gas, where only low-frequency phenomena can occur, (w <Al), the classical results are recovered. Fora plasma, this is only the case if local electroneutrality is assumed. These results agree completely-with the ones obtained by Rozmus and Turski 16 , who used the plasmadynamical projection technique.
1384
REFERENCES
1. B.
2. s.
3. Y.
4. P.
5. c.
6. c. 7. P. 8. R.
9. J.
10. I.
11. R.
12. v.
B. Kadomtsev, Zurn. Eksp. Teor. Fiz. 32, 943 (1957). English translation : Sov. Phys. JETP 5, 771 (1957). V. Gantsevich, V.L. Gurevich and R. Katilius, Rivista del Nuovo Cimento 2, n° 5 (1979), V. Klimontovich, Kinetic Theory of Non Ideal Gases and Non Ideal Plasmas, Ed. Nauka, Moscow 1975. English translation: Pergamon Press (1982). L. Bhatnagar, E.P. Gross and M. Krook, Phys. Rev. 94, 511 (1954). Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York (1969). D. Boley and S. Yip, Phys. Fluids 15, 8, 1424 (1972). c. Philippi and R. Brun, Physica 105A, 147 (1981). Cauble and J. J. Duderstadt, Phys. Rev. A 23, 6, 3182 (1981). M. Greene, Phys. Fluids~' 2022 (1973).
Paiva-Veretennicoff and V.V. Belyi, Model kinetic equations and fluctuating hydrodynamics in one-component plasmas and gases, submitted to Physica (1982). Balescu, Equilibrium and Nonequilibrium Statistical M~chanics, Ed. J. Wiley (1975). Russian translation. Mir Moscow (1978). V. Belyi and I. Paiva-Veretennicoff, Model Kinetic Equations Associated Langevin Sources, and their Role in Fluctuating Plasmadynamics. To be submitted to Journal of Plasma Physics (1982).
13. L. Sirovich, Phys. Fluids 9, 2323 (1966). 14. F. J. McCormack, Phys. Fluids 16, n° 12, 2095 (1973). 15. L. D. Landau, E.M. Lifshitz, Sov. Phys. JEPT 32, 618 (1957). 16. W. Rozmus, L.A. Turski, Phys. Lett. A71, 344 (1979). 17. A. Sommerfeld, "Thermodynamik und Statistik", Vorlesungen
Über theoretische Physik Band V, Wiesbaden (1952).
ACKNOWLEDGMENTS
One of us (V.V. Belyi) expresses his gratitude to Professors Prigogine and Balescu for their hospitality during his stay at the U.L.B •• We also thank Professors Yu.Klimontovich, R. Balescu and G. Severne for encouragements, discussions and critical reading of the manuscript. To Monica Weckx we express our appreciation for her careful typography.
1385
XXII. RELATED FJELDS
RAREFIED GAS DYNAMICS AS RELATED TO CONTROLLED THERMONUCLEAR FUSION
H. 0. Moser
Institut für Kernverfahrenstechnik, Kernforschungszentrum Karlsruhe, Postfach 3640, 7500 Karlsruhe, Federal Republic of Germany
ABSTRACT
Rarefied gas dynamics problems occur at the forefront of both magnetic and inertial confinement fusion research. Prominent examples are the control of power flux and impurity content in tokamak plasmas by means of divertors, the refuelling of plasmas by pellet injection and the protection of the first wall by a gas blanket. The large variety of problems can be reduced to a few basic ones among which the contact between a plasma and a neutral gas is the most important.
INTRODUCTION
In view of the close relationship between Rarefied Gas Dynamics and Plasma Physics it appears quite natural that similar relations also exist between RGD and Controlled Thermonuclear Fusion. Indeed, they are so numerous, that the scope of this article will be restricted to describing only a few important contributions of RGD to the solution of problems at the forefront of fusion research.
As generally known, the main feature of devices that shall be useful for fusion reactors is to maintain a plasma of nuclear fuel particles, e.g. a D-T mixture, long enough at sufficiently high density and temperature so that the energy released by fusion reactions exceeds the energy used to produce, contain and heat this plasma. In the case of magnetic fusion this is done with magnetic fields confining the charged particles, while inertial confinement fusion is based on the idea that in highly compressed matter the fusion
1389
reaction rate is fast enough to burn a significant fraction of the fuel before it can expand unter the action of the pressure force.
In MCF, many different magnetic field geometries are investigated1'2 . However, the greatest effort has been spent on a toroidal field geometry which is called Tokamak. In spite of the importance of other devices like magnetic mirrors, Stellarators, Elmo bumpy tori etc. the following discussion will be limited mainly to the tokamak an example of which is shown in fig. 1.
All of the gasdynamics on tokamaks dealt with in this paper is related to the particle loss and refuelling and to neutral beam heating. This is illustrated in fig. 2. Particle refuelling is done by a simple gas inlet through nozzles into the vacuum chamber, a method called gas puffing, or by injecting mm-sized pellets of frozen hydrogen. Furthermore, the loss of energetic particles from the plasma leads to an interaction with the wall of the vacuum chamber which is the cause of a release of wall atoms like Fe, C or Mo. These are called impurities and have to be kept far from the plasma because of the radiation loss they will induce. The divertors are devices serving to keep the plasma free from impurities by_ scraping off and pumping away the outermost layer of the plasma . Finally, neutral beam heating involves gas dynamics, too, when the originally accelerated 1ons are to be neutralized by means of gas cells or jet targets and when the pertinent gas load has to be pumped.
Devices for inertial confinement fusion are much different from those for MCF (fig. 3) 3 •~. A small pellet of the nuclear fuel is shot into a reaction chamber. When it has reached the center of the vessel a number of driver beams are fired into the pellet. The fuel
T R .. NSfORMU~ PRIMARV WINOINGS
PLASMA
VAC.UUM VESSEL
Fig. 1. Schematic of the Princeton Large Torus (PLT) (courtesy of H.P. Furth /1/).
1390
lmpurity release
Recycl~
Plasma loss
Divertor
Gas puffing } Reluelling
Pellet injection
Fig. 2. Typical processes involving gasdynamics in a tokamak.
is compressed to a very high density and heated to ignition temperature. Then, owing to the pressure build up by the fusion burn the pellet explodes. The energy released is distributed onto neutrons, target debris and radiation. The various devices for ICF experiments and reactor designs differ mainly by the drivers and by the methods used to protect the first wall of the reactor cavity from bombardment by the reaction products. Drivers used or proposed are lasers, electron beams and light or heavy ion beams. The concepts of first
I 1111 lOLAIC LAH11 t'VI- HACTO. IYSttM I ~LUT INJIC:TOR
... · .. · .. I : .--: ,u2otun1~:: ·; : • ~ t"uo•CI TUII '--;;UW;--fl_-l...
T:z lKtiACTION tt :~ cooecl ST.IAM ~ROTATIO •o•)
Fig. 3. Schematic of an ICF reactor (Conn e t al. /4/).
1391
wall protection range from gas atmospheres to liquid metal curtains surrounding the pellet and absorbing most of the energy.
The gas dynamics in ICF is found, of course, in the ablationdriven implosion of the pellet, but also in the pellet acceleration and transport and in the gas atmosphere protecting the first wall. The paper will be restricted to the latter item.
Many of these seemingly heterogeneaus issues can be grouped into four basic categories, namely - the contact between a plasma and a neutral gas - particle beams and flows - the interaction of particles with solid surfaces at kinetic
energies beyond 1 eV and - the momentum transfer by gas flows.
Hereinafter , examples belanging to the first two groups will be discussed.
CONTACT PLASMA-NEUTRAL GAS 5
Let us assume that in a plane geometry as depicted in fig. 4 , going to the right one comes into a plasma region characterized by the number density n and temperature T of the ions and electrons, while on the left side a region of neutral gas having a number den-
MUitCII gas rf11ion
boundory ~ion
ioni:r:olion rf11ion
i nlernol plosmo region
Fig. 4. Density and temperature of ions and neutrals at the boundary between a magnetically confined plasma and a neutral gas (courtesy of IAEA /5/).
1392
sity n and temperature T is found. A plasma ion travelling from right ~g left will penetra~e down to a certain depth in the neutral gas and then lose its charge, excess energy and momentum by collisions with the gas particles. In a similar way, a neutral particle going from left to right will end up as an ion. Both plasma and neutral gas will tend to fill also the space occupied by the other. Depending on the physical conditions like nurober density, existence of magnetic field or others, the particles may flow freely or only diffuse owing to collisions or to the confinement performed by the magnetic field. The equations describing the situation theoretically comprise always the gas dynamic equations for the balances of particles, momentum and energy together with further relations, depending on the given case, like equations of state, Ohm's law etc ..
Pellet Injection
A pellet of frozen hydrogen is shot into a hot plasma. Its surface is bombarded mostly by plasma electrons. The surface material is evaporated and a gas cloud surrounding the pellet is formed. It expands radially away from the pellet. Now incoming electrons must penetrate the neutral gas cloud where they lose energy before arr1ving at the pellet surface. Fig. 5, taken from a recent review by Milora6 , visualizes the situation.
The question of interest is how far a pellet of given mass and velocity can penetrate into a given plasma. To answer it the ablation rate, i.e. the rate of change of the pellet radius, drp/dt, must be calculated. According to the generat description given above the equations used to find the ablation rate are the gasdynamic equations for the balance of mass, momentum and energy of the cloud, supplemented by equations for the evaporation of the pellet material at the pellet surface and for the degradation of the energy flux carried by the plasma electrons when they pass through the shielding cloud . The ablation rate, drp/dt, derived in this model, in-
\ 2H" • 2t" ICOlLISIONA. P.ASMA EXPANSION'
-'"+l,:,:P- -ß
r1 - PELLET RADIU S (O.Ot eV/ MOLECULE)
r 1 - IONIZATION RADIUS (-36 eV/ ION)
0 - E XTERNAL PLASMA HEAT FLUX ( ne Ce / 4 • 2 kle) 0 1- ATTENUATEO PLASMA HEAT FLUX
Fig. 5. Model of pellet ablation in a magnetized plasma (courtesy of S.L. Milora /6/).
1393
creases with the plasma electron density and temperature and is the higher the smaller the pellet is. It is in reasonable agreement with experiments, particularly with respect to the penetration depth, as is shown in fig. 6.
For future plasmas with higher densities and temperatures the shielding mechanism may have to be refined by inclusion of other effects like the energy deposition by energetic ions and a-particles, the ionization of the cloud and its shaping under the influence of the magnetic field.
Gasdynamic end-stoppering in linear magnetic fusion devices
Linear magnetic fusion devices are characterized by a linear, cylindrical geometry of plasma and confinement system with primarily transverse confinement by a quasi-static magnetic field whose field lines close on themselves only outside the device. As a consequence, in an LMF device of finite length, the plasma is lost at the ends and effective end-stoppering methods have to be applied in order to maintain the attractiveness of LMF devices for nuclear fusion 7 •
Among several methods proposed there are the material end plugs which are relevant in the scope of this article. Asolidwall or a dense gas is placed at the ends of the device. The axially streaming plasma interacts with the plug material and is stopped or at least retarded. Ideally, at the dense-gas/hot-plasma interface, a rarefaction wave travelling into the plasma and a shock wave moving into the dense gas are formed. The shock wave leads to high pressures by which the plasma flow is reduced or even stopped. This is 2 also called thermal plugging. Power fluxes in the range of 1olO W/cm are required to reach the limit at which the plasma outflow velocity
- 6
P'LASMA ~AOIUS (cm)
2!1o 20 '~ tO
lp • t20 \ A
Br • 3.2i!.G
r., • l ,3&t0'3 cm.· 3
T, (O) I 760 .~
·~ 20 2~
PCNETFI:ATION OEPTM (cm:
Fig. 6. Experimental and calculated pellet ablation prof iles in ISX-A (courtesy of S.L. Milora /6/).
1394
becomes zero. For lower power fluxes a retarding effect remains on the plasma flow by the inertia of the cold gas blanket. Fig. 7 shows calculated results concerning the plasma outflow velocity as a function of power flux and gas density 8 •
Divertors
After the digression to LMF devices the gasdynamics aspects of a tokamak subsystem which is called divertor will be discussed. When explaining the nature and operation of a divertor it must first be mentioned that the edge of a tokamak plasma needs to be defined in order to avoid contact with the wall of the vacuum vessel. This is usually done by a solid limiter. The scrape-off performed by such a mechanical limiter can also be obtained by choosing a magnetic field configuration where the field lines outside a certain radius are diverted and go into a separate pumping chamber. The characteristic feature of the magnetic surfaces in the case of a divertor tokamak is that there is a last closed magnetic surface, called separatrix, which separates an inner volume of nested magnetic surf aces from an outer one where one or more field nullsexist (fig. 8). Plasma particles diffusing out of the separatrix are rapidly transported along
1.2·10°
\ \ heotwove
_, \ confinemMt 10 \ 9 \
cm \ \
102 \
u0 = l2 106cm/s \ \
·> 10 u,= 1.2·106
-------- - ------- ~\.:. ',, ~r~ '<>;.\
10' '?~ 1,2·107 \~
' \ 9,
~ 10 retordotion by cold gos inertia
10 10° 10" 107 v.<~ttlcm2 10° 10'0 1d' ~w
Fig. 7. Thermal plugging of an LMF device . Escape velocity contours of a D-T plasma as function of blanket gas density p1 , and absorbed power W (courtesy of B. Ahlborn /8/).
1395
1.6
Oß
0
Fig. 8. Simplified cross-section of ASDEX tokamak (ASDEX team /10/).
the field lines into the divertor chamber. This process defines the plasma edge. As further consequences of a magnetic limiter, impurity atoms coming from outside the plasma have a certain chance to be ionized within the scrape-off layer while escaping hot plasma particles can partly be absorbed under more controlled conditions in the divertor chamber. In the first process, an impurity ion which would otherwise penetrate into the plasma and contribute to the radiative energy loss is eliminated, in the second a hot plasma particle which would otherwise hit the limiter or the wall of the vacuum vessel and probably produce impurity atoms in the vicinity of the plasma can be neutralized at a greater distance from the plasma. The expected positive influence of a divertor on the impurity level in a plasma has been demonstrated by several presently operating tokamaks 9 .
Fig. 9 illustrates the various particle flows and shows in particular the crossing of the plasma entering the divertor with the
plasma
Fig. 9. Model of particle balance in divertor discharges ~n ASDEX tokamak (ASDEX Team /10/).
1396
neutral gas coming out. It has been measured that the flow of the scrape-off plasma leads to an increase in the neutral background pressure in the divertor chamber of more than two orders of magnitude above the main chamber level (fig. 10). This finding, obtained with negligible pumping in the divertor chamber, has been interpreted as indicating that the plasma flow into the divertor chamber acts as a diffusion pump, leading to a certain gas compression1~ The analogy to the foregoing section is evident. Furthermore, the neutral gas within the divertor chamber represents a remote gas blanket serving a dual purpose. First, it redistributes the power of the incoming plasma more uniformly on a greater surface than the narrow strip of intersection between the magnetic field lines and the neutralizer plate. Second, it is more distant from the plasma than a mechanical limiter or the wall of the main chamber. Consequently, impurity atoms released from any wall in the divertor chamber have but little chance to get into the plasma. Instead, they accumulate in the divertor chamber and, by their radiation, contribute much to the redistribution of plasma power.
The physics of divertor action, and in particular the gasdynamic aspects, are presently very actively worked on.
Gas blanket
From the remote gas blanket just mentioned it is but a small step to the gas blanket surrounding the hot plasma as a cool plasma mantle and acting as a buffer between plasma and wall. The space between a plasma edge as defined e.g. by means of a limiter and the wall of the vacuum vessel is not empty. Plasma particles diffusing out of the plasma and charge exchange neutrals strike the wall. Several processes like backscattering, sputtering and desorption
IC 1013
10 •10'' '10'~
:0 8 .§ 0.
6 N" • 5 ~ 15 4 • ..; 3
I i
/n.dl / i
i /
I
i i
/ ,' .',.'
~,I/ ,..~(/ 0 d1scharge
~·V
0~~--~------------~ 0 6
Fig. 10. Neutral hydrogen pressure in divertor, p0 , and plasma chamber, Pp• versus plasma density n . Line density of scrape-off plasma in divertor chambe~, /nedl, is shown for comparison (ASDEX Team /10/).
1397
give rise to a flow of hydrogen and impurity atoms back into the plasma and, for the anticipated heating powers and particle contents of fusion devices now under construction or in the conceptual design phase, can lead to an inacceptably high wall ablation. To control the particle and energy flow between plasma and wall was one of the main incentives leading to work on the gas blanket. Already in 1968, Lehnert 5 gave a theoretical analysis of the problern (fig. 4). He also reviewed work clone in this field up to 1979 11 .
The main question to answer is how the parameters of both plasma and cold gas have to be chosen in order to keep the configuration stable. For, if the density of the cold gas mantle is too high and the energy flux from the plasma too weak, then the cold gas will extinguish the plasma. On the other hand, in the case of a too streng energy flux and an insufficient gas density, the plasma will eat up the gas blanket. Conditions for and domains of stability have been found both theoretically and experimentally. However, it is still an open question whether such conditions could be created in present-day tokamak experiments. On the other hand, it is expected that the conditions can be fulfilled for full-scale reactors.
In this context, a further approach to control the particle and energy flux from the plasma to the wall has to be mentioned. It is based on the conversion of the energetic particle flux into radiation by means of the interaction with the cold gas mantle. If this cold gas mantle is sufficiently dense it can also stop the CX neutrals emanating from the plasma so that the wall receives mainly radiation. Transport codes used to calculate the energy and particle balances for those cases show that "photosphere"-like solutions do exist 12 (fig. 11). The stability of these configurations is a matter of concern in the same way as it is for the radiationless cold gas blanket.
15
PIMW)
10
s
0 10 20 30 LO rlcm)
so
Fig. 11. Radial distribution of power flow in different loss channels showing "photosphere"-like behaviour in a tokamak (Lackner and Neuhauser /12/).
1398
Gas protection
The last example in this chapter is taken from inertial confinement fusion.
When a fuel pellet explodes inside a reactor chamber the wall will be bombarded by radiation and particles, such as neutrons, aparticles and pellet debris. The protection of the wall against this energy load is one of the difficulties in designing !CF reactors.
The 10 to 14 MeV neutrons have a large mean free path compared with the first wall thickness and will pass through without depositing a significant amount of energy. In centrast to the neutrons, the target X-ray and debris energy will be absorbed in the first few microns of an unprotected wall, thus leading to an unacceptably high erosion rate 3 ' 13 by evaporation, spallation and sputtering. Besides the wetted-wall, fluid-wall or magnetic-protection concepts the protection by a gas atmosphere has been studied.
The optimum gas density depends on the requirements of wall protection as well as on the conditions set by the transmission of the driver beams and the fuel pellet.
The physical processes occurring when a fuel pellet ex~lodes in a gas atmosphere are depicted schematically in fig. 12 1 • The soft part of the X-ray spectrum can be absorbed and then reradiated by the gas atmosphere. Since this reradiation extends over a much Ionger time interval than the original pulse width, the thermal load pulse on the wall becomes appreciably flatter. If the gas density, typically ranging from 1 to 10 Torr at 0 °C, is high enough to absorb the soft X-rays, the debris ions are readily absorbed, too. This density range is well suited for the transmission of light ion beams which propagate in highly-conducting, magnetized plasma channels. For laser and heavy ion beams, however, difficulties with gasbreakdown and beam stability may arise 3 ' 14 •
At these densities a we11-known phenomenon is expected to happen which is called fireball. It has been studied extensively as
fiRST WAU .. Rt S"O~~~~'Sl
Fig . 12. Physical processes in a gas filled !CF reactor cavity (Hassanein et al. /13/).
1399
part of the nuclear weapons programme 15 . The condition for its occurrence is that the hot plasma is surrounded by a sufficiently dense gas. Then, at the interface between plasma and cold gas, plasma radiation is absorbed and reradiated at longer wavelengths by the gas atoms or molecules. The visible part of the reradiated spectrum gives rise to the fireball phenomenon.
The concept of the fireball is analogous to the already mentioned photosphere in tokamak discharges. Its theoretical evaluation seems to require a broader knowledge of the physical data of gases in the thermal energy and density ranges of 0.1 - 100 eV and
14 3 18 -3 • 3 3xl0 - xlO cm , respect~vely
PARTICLE BEAMS AND FLOWS
The examples belonging to this group have been selected from the formation of high-energy neutral beams as required in MCF and from a special gas protection scheme proposed in ICF.
In MCF, intense beams of mostly hydrogen or deuterium atoms in the energy range of several keV to several 100 keV are used or envisaged mainly for plasma heating purposes 16 ' 17 , but also for creating electrostatic potential barriers, suppressing instabilities or as pump beams 17 • Less intense hiyh-energy beams have been proposed as a means of plasma diagnostics 8• These high energy neutral beams are produced by electrostatic acceleration of ions and subsequent neutralization in a gas cell or a cross-jet.
Here, typical RGD problems arise as will be shown by the following example. In a neutral beam source based on positive ions a neutralizer gas cell follows immediately the accelerating structure (fig. 13). In order to obtain sufficient neutralization a minimum
Fig. 13. Schematic of neutral beam injector (courtesy of M.M. Menon /16/).
1400
target thickness, also called line density, /ndl, has tobe maintained in the gas cell. It depends on the ion energy and lies typically in the range of 3xl015 to lxl016 molecules per cm2 • The gas target is kept constant by a steady gas flow. Taking into account the geometrical dimensions of typical neutralizers (aperture 0.1 m2 ,
length 2 m) high throughputs on the order of 20 Torr 1/s are required. Obviously, high pumping speeds of the order of 106 1/s must be installed to keep the background pressure below about 10-s Torr which, in turn, is an important requirement since reionization losses of the neutralized beam must be minimized. Any measure to reduce the gas throughput at constant line density is thus highly desirable. Hemmerich and Deksnis 19 propose to cut the neutralizer in two halves and to operate them at different temperatures, e.g. the section near the ion source cooled to 77 K, the other heated to 700 K. Since, as they show, the temperature of the flowing gas accommodates rapidly to the wall temperature the gas is cooled in the first section and the throughput can be lowered while maintaining the line density. The small increase in conductance in the first section which also results from the temperature reduction can be compensated by the temperature increase in the second half (fig. 14). Recent measurements have shown a density increase in the LN2-cooled section of a tandem neutralizer by a factor of 3.3 as measured with a nude Bayard-Alpert type ionization gauge. The accommodation length la required to reach 90 % of the peak density is about equal to the minor duct width. 20
c (V<)
120
1
110 1\
100
I
90
80 :.....,.
10 ~ \II
I i I i II 10 ... 5 10·1
I II nK,»>k' 1
I ll 300K/
I III \II 300K
1
! 'I I 'I llli!i I, Ii I ~ J i li;;i
I I
II [V
I ~'I I V
# '11 ,_ """"" olo • ....... ~-~ l I i
I II -I
I/ · i II I• 17K14SIJK
I I ~
V
V
!-1 I
i
: II I II
~17K/~
I' 17K I
i I II
! I II
K
6P (Torr)
5 10"1
Fig. 14. Measured conductance of a tandem neutralizer (Hemmerich and Deksnis /19/).
1401
A last example shall be taken from ICF. As mentioned, the protection of the first wall of an ICF reactor cavity against X-rays and target debris is an important task. In the HIBALL study21 a combination of the wetted wall with the gas protection is proposed, ensuring that before a shot the cavity is at high vacuum to allow a good transmission of the heavy ion beams. Porous SiC tubes through which liquid Pb83Li17 is flowing surround the reaction volume and form a cylindrical wall with a 5 m radius (fig. 15). After a pellet explosion, the X-ray energy is absorbed within a thin sheath of the PbLi liquid which is thus heated and evaporated . The vapour flows towards the center of the cavity and intercepts the target debris. Absorbing the energy of the debris, the vapour is further heated and radiates part of the heat back to the wall where fresh PbLi is evaporated. When all the energy is degraded to only thermal motion of the vapour atoms, the hot gas is recondensed on the porous tubes the temperature of which is maintained by a forced flow of the PbLi liquid. The time history of the evaporation, condensation and the mass in cavity is shown in Fig. 16 .
CONCLUSION
Rarefied gas dynamics contribute to solve important problems in present-day nuclear fusion research. A wide field of applications is added to the classical topics of rarefied gas dynamics. This has been
Fig. 15. HIBALL reactor cavity (Badger et al. /21/).
1402
Mo •13000 g
MASS IN CAVITY
10~~--~----~----._--~~--_.~~
10-6
Tl ME (sec)
Fig. 16. Pb vapour recondensation after a pellet explosion in HIBALL (Badger et al. /21/).
substantiated by examples taken from both magnetic and inertial confinement fusion among which are the refuelling of plasmas by pellet injection, the control of power flux and impurity content in tokamaks by divertors and the protection of the first wall by a gas blanket.
Acknowledgment
The author is grateful to Drs. G. Haas, K. Lackner, F. Wagner, J. Hemmerich and H. U. Karow for providing useful informations and illustrations.
REFERENCES
1. H. P. Furth, 11The Tokamak", E. Teller, ed., Fusion, Vol. 1, Part A, Academic Press, 1981, and references cited therein. Sheffield, John, "Status of the Tokamak Program11 , Proc. of the IEEE 69:885 (1981) and references cited therein.
2. F. F. Chen, "Alternate Concepts in Magnetic Fusion", Physics Today, May 1979, pp. 36-42, and references cited therein.
1403
3. H. U. Karow, S. I. Abdel-Khalik, "Thermophysical Questions and Materials Problems in Inertial Confinement Fusion Reactors", High Temperatures - High Pressures 12:373 (1980) and references cited therein.
4. R. W. Conn et al., "Solase- A Conceptual Laser Fusion Reactor Design", Univ. of Wisconsin, Madison, UWFDM-220, 1977.
5. B. Lehnert, "Screening of a High-Density Plasma from Neutral Gas Penetration", Nuclear Fusion 8:73 (1968).
6. S .. L. Milora, "Review of Pellet Fueling", J. of Fusion Energy 1:15 (1981) and references cited therein.
7. W. E. Quinn, R. E. Siemon, "Linear Magnetic Fusion Systems", E. Teller ed., Fusion, Vol. 1, Part B, pp. 2-37, Academic Press, 1981.
8. T. Sinnott, B. Ahlborn, "Reduction of Particle End Losses from Linear Magnetic Fusion Devices", The Physics of Fluids 20:1956 (1977).
9. M. Keilhacker, U. Daybelge, ed., IAEA Technical Committee Meeting on Divertors and Impurity Control, Garehing 1981, and references cited therein.
10. The ASDEX Team, presented by M. Keilhacker, "Divertor Experiments in ASDEX", ibid., pp. 23-26.
11. B. Lehnert, "Main Features of Cold-Bianket Systems", Introductory Course on Plasma Physics for Thermonuclear Fusion, Joint Research Centre, Ispra, Italy, 1979.
12. K. Lackner, J. Neuhauser, "Simulation Code Predictions Concerning the Existence of a Radiating Boundary Layer", in Ref. 9., pp. 58-61.
13. A. M. Hassanein, T. J. McCarville, G. L. Kulcinski, "First Wall Evaporation in Inertial Confinement Fusion Reactors Utilizing Gas Protection", Univ. of Wisconsin, Madison, UWFDM-423, 1981.
14. D. L. Cook, "Technological Aspects of Particle Beam Fusion", Sandia Laboratories, Albuquerque, New Mexico, SAND 80-0466C, 1980.
15. Y. B. Zel'dovich, Y. P. Raizer, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena", Vol. 2, Academic Press, New York, 1967.
16. M. M. Menon, "Neutral Beam Heating Applications and Development", Proc. of the IEEE 69:1012 (1981).
17. T. C. Simonen, "Experimental Progress in Magnetic-Mirror Fusion Research", ibid., pp. 935-957.
18. D. E. Post, et al., "Technique for Measuring the a.-particle Distribution in Magnetically Confined Plasmas", Princeton Plasma Physics Laboratory, PPPL-1592, 1979.
19. J. L. Hemmerich, E. B. Deksnis, "Conductance, Thermal Accommodation and Beaming from Neutralizer Gas Cells", Proc. 9th Symp. on Engineering Problems of Fusion Research, Chicago, 1981, IEEE Pub. No. 81CH1715-2 NPS, 1:795, and J. Vac. Sei. Technol. 21:86 (1981).
20. J. L. Hemmerich, E. B. Deksnis, private communication. 21. B. Badger, et al., "HIBALL- A Conceptual Heavy Ion Beam Driven
Fusion Reactor Study", KfK 3202, 1981.
1404
VAOUUM BJEO!ORS WI!H APPREOIABLI OIEVII P.LOWS IB OBAIBELS A! LOW BEDOLDS BtJJilmRS
V .G. Jarinov
.Aviation Institute Xua,n
IKDODUO!IOll
In the vaat nwaber ot publioationa dedioated to e3ectora there are 0D17 a tew where the characteriatics ot an ejeotor as a funotion ot Reynolds nuabers are inveatigated. As shown b7 Xu1pin1, PlotDikoT and Tilloshin2 , Philatov a,4 , the influence ot i.eynolda D.lDlber mq be couiderable. Oalculation aethod ot obtaiDing ejeotor charaoteriatios,repreaented b7 Philatov, is oonstructed on the basis ot experillental data and so is ot empirical character.
EJBO~OB BQU.A!IOBS
In the theory ot jet device the ejeotion equationa are ot great illportance. Let the parameterB of active (aarked with one prime) and passive (aarked with two priaes) gasea in cross-seotion 0-0 (tig.1) and mixed gas in cross-seotion 3-3 be distributed uneven. Using the oonservation laws, let us write down subject to usual as.waption:
I II I II ~ NI ..,u_,_, ,.J.,. G., + G1 ::: G1 , E( + E" = c a , ..11 + ..J( - .J3 + Y".f! , ( 1)
G=mS ~J) cl~, E=mc"SP*{ii'CJ,(J)dF, ':1=SP*.f(J)dFj (2) (F) T* (,:) (F)
where are G - the rate of fiow; E - the total athalP71 :1 - the co:mplete aoaentwa; ~5 - the triction force
1405
Pig.1. Diagram of the ejector. I - jet boundary.
(; 2 lf+l ] f/2 of gas on mixing chamber walle; m= [K _\iN" .J._ ; K+tl R
1\- the adiabatio exponent; R - the gas oonstant; I I I
(-1\+l)iH" 1 1\-./ 2.)H ) '2 ~ lf-1 2.)/H" q,(~)= -2- .A,I-J\+1J , -t(J =(J +i/l-tr+l.../ - gaa dy-namio :tunctionsJ J = w Ia. - the velooi ty coefficient;
'( 21<' 0 \ 11:1.. q= 1\+.f "r,., , T - te:mperatu.»e; w1 th a star the drag parameters are markedJ w - the velocity. Instead of uneven tlows (passive, aotive and mixed) let ua introduoe into conaideration equivalent to them onedimensional oanonical tlows.
!rhe state of one-dillensional gas tlow is defined by three independent parametera, e.g. P 11 , T"and .A • !hus for finding of averaged paramatere of each canonioal tlow we shall have three equations w1 th three unknowns:
"' mP'*q.(J) -G G r-..* K+-1 "' "' J T *' , F ~ , c,. T = E- , ---zK Ga -z ( J ) = ':J. ( 3)
Bere the wavy line ( rv ) denotes averaged paraaetera of the oanonioal one-dilllensional tlow, ;;(.;) = ..1 + J-i , valuea G , E. , ~ are defined by formulae ( 2), Ci==
= ( ~~ Rrilr/z • Using the averaged parameters of the oanonical one-dimensional tlows, att~r re-arranging we shall have instead (1)
I - * (. "'t) rv l ~" f(<: 'l:.-,/rn§·'+~(n t 1)' ~;+6j~'i(,t~- , f'" ;;~, \ ( 4)
1406
1\' ~""2.-/ h :t(.i") (."~) (. I ) + .i..f. J,.,. = K?> ~ /f 1Q..- ./, fil tr'll.- I Vi" I + ~ J,
i ~?! 2 ,; .\ I 2. I .I I '
K "3 - v(ne-'+ r)(n+t) ....--1\-;-!,.1-~
~(K' 1\J\ • I'_ = d 6~'CJ(Jl') .S(I\' 1\1= ~ (!.:;-)R':-1(1\-f) ' /9(' I oll (."'') ' > •) 1\: lli
1\=~eu ve 64 P Cf, ~~ 1 z ':\tr-~0~, i' lk+l} - J
where are !il= /5q~/Po"*- the compression ratio, p - the
pressure, n=G''/G' - the ejection coetticient, fi= h'o*/h!'~
h- the specitic enthalpy, Ei4=B'Y'Po'*,~~=-'i~;*/~*,P*=P;'lf..11 *,
cl = F~"/F~ ~ F - the area ot cross-section, ::C =h~/h~"*,
f?J= 2'(?;(Y\a.+~T 1~3 Ld~- 1 - the reduoed length ot mixing
chamber,o.t:B"Y~'*• By derivation ot tormulae it is taken into account, that 'h~ ='h; , the torce ot triction c!p4?>=2- 1 ~ld?/G3 ~~ , where ~- triotion lose coetticient ot mixing chamber. ~he adiabatio exponent ot llixed tlow
c.' "" e:';-' K5=[K""(n+qi-)](n+ 1'\'C; ,
Cp- the heat capacity at p =const. Expressions (4) represent the ejector equations. !he appreciable teature ot supersonic ejectors is their work on tlow choking regimes. As Bhown by Vasiljev5 , either the tlow choking regimes are realised at the expence ot ori tical regimes, or the choking ot the mixing oh8Jilber is available. At mixing chamber choking regimes the calculation ot ejectors is made wi th the help ot ejection equations jointly with the condition by known values ot ES4', ESt, <5?J , .f3 • li'oJP cri tical regi:aes i t is necessary to oonsider the tlow at the ini tial seotion ot mixing chamber.
TBE l'.LOW A~ TBE INITIAL SECTIO:R Oli' MinBG CB.AJmER
Supposing the tlows at the initial section between oross-seotion 1-1 and choking oross-section 2-2 (tig.1) arenot mixed, heat exchange is absent. In this oase, when the parameters ot passive tlow are distributed uneven, we shall suppose as a tirst approximation that on cri tical regimes .1~ =1. Let us restriet ourselves wi th the most interesting tor the practice cases ""P~j{ ~ ?* < P~ax , "Pr!Qx - ratio ot pressures, at which the jet ot active gas tills the
1407
cross-section of the mixing chamber completely and n =0.
Using the conaervation laws, we shall get: GI/ G" GI I II II I I I II I II ,.,s,.
{= 2, f=Gg";E4 =E,_,E-~=Efl,; ~4 +!:4 =~t~.-+:t/1,+..,..12 , ( 5)
where the values G , E 1 '::J are calculated by formulae ( 2). Regarding the oondi ti on .1;: = 1 , we shall get instead of (5)
$( ~) = "':f') {I+ «{I - '1-~~1) r: "'') ,..., n I<'-~[ r..'' " .1'] i!(J:z_ = r:(J.,) -lr" k"V~ 2-z. '"/.,)+ z""2 '
1 r:! ~ -~ Dd-.f where 62. ==Pi.*/~'*, 6~ = Pi'YP."*, .f:t = 21C'"(K""+.i) ~-!. :.3 •
Let us find an expression for Pt* • Regarding that in thia oase 'ft• = 'P." and 3{• =1 ( choking takes place in the croas-aection 1-1), we shall have
.j
(6)
p4 _ Ei~1 Ji'(/) 'h'(J) = (/- :+-: l:t)K--1' l - 6~ 1(JJ) ,, 1 o.j (7)
Thus, on oritical regimes, when p* :::> P-1* 1 the oalculation of ejectors is made with the help of ejeotor equations (4) and equationa (6) .known values 6i,6/',~3 ..t3,G~,G'd',.f:~..; ifP~~-=rt, instead (6) the equation (7)' is used.
The equations include quantitiea ~ I 'fi14f I 'f:tll- II 'P."* II f."* 'P.4* 6.( r:: - , E) - 2. 6 - -4 s - !l.. r.5 -'f..t .- 2.- '\5.'* ) .j - 'f~' * ' :1.- - 'f>{" 41- ) 3-~ II '
whioh represent the ratio of complete preasure of canonioal flow at the end of either section to the complete preasure in the beginning of it. These quantitiea take aocount of the change of distribution of actual flow parameters in control cross-aection in comparison wi th the ideal aase. If the distribution of parameterB in corresponding cross-seotions, determined theoretically or on the basis of experimental data, is known, the quantities mentioned may be cal-culated. -
It ia obvioua, that the derived equations are correot also in the aase of weak rarefied gas fiow, when the degree of rarefaction of gasea is oomparatively not great and it is not neoessary to use the oonoeptions of kinetic theory for their flow disorip-tion.
1408
ABOUT THE OALOULATIOB OP SUPERSOBIO NOZZLES YISOOUS PLOW
Por the determination of parametera diatribution in the nozzle outlet seotion a nozzle visooua flow problea in the slender-channel approximation was aol ved. On the baeis of ini tial set of la:aiü.nar boundary-layer equations a sequence of linear parabolic equations with constant ooefficients was conatructed, the boundary problema for which have an analytical solution. With the aid of the sequence mentioned the aolution of the nozzle continuum flow boundary problem in the way of suocessive approximation·was made. Tagether wi th the theoretical analysis of nozzle flow experimental investigations were made. Static and total pressure fields in nozzles outlet cross-aection were measured. The nozzle divergent parts were oonical.. The inveatigations were made wi th air. Aa a result of oalculations and experiments a relationship was got
8; = 2[i- Jq,(JltJ.)'] +n-'K Re~:·'2'A(ft?,;)-,1, 5 - d (8) where 8; - the ratio of the boundary-layer thicknesa at the nozzle outlet cross-section to its radiua;Re~, ~ - aocordingly Reynolds and Kach numbers, determined from parameters of non-viscous one-dimensional flow at the nozzle outlet oross-section (' 11t~ <G ) 21 is the angle of the nozzle supersonio section. ' V ATTT s I r.:. I r:: Ii C: II C: .f ,r &UuE 6. , 'V~, Q4, QJ., '\>~ t-"J.,.A~
. When calculating ejectors it is not ·reasonable, from the practical point of view, in each case of determination to find the distribution of parameters at the nozzle outlet cross-section as the calculation in this oase will be very bulky. It is reasonable beforehand to find relationship between the boundarylayer thiokness at the nozzle outlet cross-section, e.g. in form (8), tagether with the flow rate ooeffioient and the determining parameters. As c&lculations and data of other authors show, one can acoept, beaides, the relationship between the Mach number and the transverse coordinate, linear in the boundary layer. Taking into consideration the above, we shall get instead of (3):
(_l?._)K'~I 9(..1:~~.1fi = u-8.')2 + ..!.._ ~ "'+/ rr..1:J)8; x IC' +I ...1,~ E(J:a) ~ 24 (K!...I)~ /'11~ J1~
1409
X /2+ (1\'-l)rrt,~:t' {(K'-1)/'1,~fl.+/}}' I I ,....
(9)
(...!....)"''-' q.(J4!L) r(Jl)('t =I+ 1\'!ft~~-r, .. 4&' '1-8:')! 1-B')'n K"'+~ 1fi(.'J') bßltrl.ll .'\.1 I l1 I J, (10)
1& / o, h (. ~ ( 1{'-.f fl.) - ) 1\•/ :i. J. I
w ere E J, = 1- K·t-1 ..1 ~r-~ , tz::(.A = !-K0 J, t.2.is the Veloci-
ty coefficient of non-viscous one-dimensional flow at the noszle outlet oross-section, the index "13" marks the isentropy core parameterB, f'1,9 and .J,& are connected by a known relationship ML=JLJ1~-~J~-~. B,y known
tr+l K'l-1 I -reometrical. dimensions of the nozzle and values 8], f' ~ - nozzle flow rate coeffioient) two equations
9), ( 10) inolude two unknown quanti ties - d,~ .5; • After solving this system the value s; will be tound by tormula 1 _ .<i r.· ,, ) [ (. ,..., 17 - ~
64 - \. Cf, oAfJI. CJ, ../, )j •
Since usually the supersonic nozzle works by moderate degree ot underexpansion, we shall suppose 6~ •1.
:ror calculation ot f1. in ( 6) one must know l (fig.1). The finding ot this value is oonneoted with oalculation ot an underexpanded supersonio jet, propagating in a parallel flow wi th variable pressure. Taking into aooount, that the solving ot such a problem would have oomplioated the caloUlation ot e3ectors to a great extent, we shal.l suppose .f2. =0• We shall note, besides, that the mixing chamber wall friction losses at the part between oross-sections 1-1 and 2-2 will be indireot caunted up by deter.minating of 6.," , 6~ •
By determination of f~ 1 appearing in (4), as a first approximation one may take advantage of known from the publioations relationsbips for pipes ~=~(~). The Reynolds number Re we shall determine. here b7 paramatere in oross-section 3-3.
Por determination ot<53 , appearing in (4), one may take advantage of known from the 11 terature results of diftusors investigations.
1410
1\ \
' ~ ', ,, ', " ~
20
10 > ....._ - -0 0,1
~ig.2. Relationships P*=~(~. - calculation, oexperiment.
II c- II l!'or values ö1 and 'O:t in the case of air ejec-tors there were received relationships:
6;' = 4 + 3,18 - 23.59 + 39,93 ~2,44 f~ee/' (l~Ret)'- (~fle;')'J (.&;f?e:')4
- o,ll'i ~ + o, 24 U- J;· J = /+ :f_ r;·R. ')" + a~ Re:') 1\:::f e,
,.. • I" " J"'" i + 0>24 (I- ...1}') , .).,4 ~ 1 .:5 ; (11)
('12)
n " the Reynolds number K€4 is determined here by parama-tere of the averaged passive flow in cross-section 1-1 (characteristic dimension is calculated by for.mula (4Ftlr) 112.., where Ft is the area ot passiTe flow cross-section):
( 13)
1411
!he relationship (11), (12), (13) were ~ound with the help o~ experiaental~y reoeived exter.Dal Oharaoteristics P*=.f(n) at oritioal regilllea o~ ejeotor work.
" Let ua pq attention to followi:ng. AcQepti:ng fVI., •1 ('the choki:ng in the oross-aeotion 1-1 at oritioal regimea) and the mentioned range o~ Variation of Re:' , we shall find o,oo':/45~M/'/cRetJ"11l:50,/. It is evident here, that at low Re7J1olda uumbers ~or the passive flow we have, obvioualy, transi.ent regimea of flow.
!BI OALCULATIOB OP BJBO!ORS AJTD COJIPABISOB WID BIPDIVEllTAL DATA
There were made oaloulations of ejectora and comparison with experiaental data o~ Philatov(fig.2); at the figure are given for comparison plota (dotted line), received by caloulation supposing 6/ , 5~ ,6;', f!St_', 6?1 =1 J .f2.,J! =0 (in this oase we have one-diaensional theory equationa of Vaailjev}. The comparison showa satisfactorily coincidenoe o~ caloulation resul ts wi th experilllental. data; at the saae time the corresponding derivations from experimental valuea, received by calculation according to the equation of one-dimensional theory, are great.
BE:PEREBCES
1. B.Y.Xu1pin, The in11uence o~ Reynolda number upon basic characteriatios o~ jet compreasor, !ransaotions o~ Xazan aviation insti tute, 37:72. (1957) - in Ruasian.
2. A.E.Plotnikov, A.B.Timoshin, Some questiona o~ multistage oompression of gae in vaawwa ejectors, Bulletin o~ Ioscow Universi~. Series: Kathematics and meohanics. 1: 66 ( 1969) - in Russian.
3. A.P.Philatov, Investigations of superaanie ejeotors at low numbers Re , !xudy !sAG!. 1365: 3 (1971) -in Rusaian.
4. A.P.Pbilatov, Experimental investigation of oooled gas eJectora at low numbers Re, !rudy fsAGI. 1365:19 (1971) -in Russian.
5. Yu.B.Vasiljev, fheory of supersonio gas ejeotora wi th cylindri oal. mi:rlng chamber, in: "V ane mashines and jet apparatu.s", Kashinostroenie~ Koscow, 2:171 (1967) -in Russian.
1412
SIMULATION OF THE PROCESS OF
THE COSMIC BODY FORMATION
A .G. Sutugin, A .J. Simonov, V .N. Korpuaov, S .K. Ayzatulin, and V .Y. Koatromin
Karpov Inati tute of Phyaical Chemiatry Moacow 107120, USSR
There ia a great intereat to the problem of modeling of coamic body formation and an evolution of the comet tail becauae of the approaching to Earth the Halley'a comet. Main proceaaea playing a general role in the comet tail formation are well known. There are the evaporation from the comet core surface, the gaa dynamic expanaion into vacuum and the deviation of the tail by the aunlight pressure. However,all modele of the comet tail formation described early (Kaimakov and Sharkov, 1979; Cowan and A'Hearn, 1979; Brin, 1980) suggest the entrainment of particles from frozen comet nuclei by the flux of an evaporated substance as a dominant mechanism of the dispersed tail component formation. On the other hand a condensation of evaporated matter can occur during the process of the adiabatic vapor expansion into vacuum. Such a condensation results in the formation of particles with sizes comparable with visible light wavelengths and providing thus the high comet brightness. As comet tails contain substances capable to react to each other (NH3, 002, H20, HCN) it is supposed that products of these reactions having a lower Saturation vapor pressure than initial substances will condense more rapidly. The fact of the vapor condensation during the comet tail formation is confirmed by the observation of tails of second and third types on Bredikhin's classification (Wurm, 1963) only for large comets. An explanation of this phenomenon lies as it will be show.n below in sufficiently high values of an integral of the time dependence of expanding matter density. There can be no secondary condensation
1413
with the large particle formation in the cases of insufficient mass flux. If this is the case the comet tail of first type is formed. Components of this tail are gases or gas and molecular clusters. As they have a low inertion the sunlight pressure deviate the tail on the counter sun direction. Tails of the second and third type include inertion particles and have a more complicated form.
The subject of this work was to model the dispersed phase formation during the expansion of the large reacting gas mass into vacuum. For a design of an experimental scheme we regard the comet tail formation as the following. The evaporation from the comet surface is initiated by the radiation heating. Whereas the evaporation products flow with the undersonic Velocity about the eva-poration surface, the flux of evaporated, reacted and condensed products crosses the Mach surface on the definite radial distance. Then the expansion transforme to the supersonic flow and molecular collisions cease. Thus while the comet tail can be extended on million kilometers the dispersed phase condensation can occur in a thin layer about the sound surface only. We will name the cloud of condensate particles in the interplanetary space as cosmosol (on the analogy with aerosol).
As a transport of large masses of evaporating materiale in an upper athmosphere is difficult, the model ling of the large-scale flux into vacuum can be provided by decreasing of the experiment duration. It was only necessary that the total duration of the particle formation procesa was longer than the time of the individual particle formation. The condenaed phase formation in superaanie nozzlea is known (Saltanov, 1973) to have characteriatic timea about 10-3 sec. To produce an intensive flux of the evaporating material we ImlSt use the volatile component diapergation.
We will auggeat that in our caae the particle formation does not depend on a thermodynamic barrier attainment due to the low volatility of chemical reaction products and to the high supercooling during an adiabatic expansion proceas. The free energy of the cluster formation is negative even for the dimer formation. The bound state formation process is retarded owing to the necessity of the third body presence. If this is the case we can divide the particle forma.ti.on process on two phasea. The first phase with a duration denoted en is characterized by a relatively slow eluatering
1414
of reaction products. When clusters become sufficiently large for the growth by binary (but no three-body) collisions formed the fast molecular phase depletion starte. Further the particle growth on the coagulation mechanism is described by the expression (Lushnikov, Sutugin, 1976)
(r) = A 6 '-'-1 sr K T* 1/ 2 ~ dr' (1) ex "'c 0 o urrrJ'·
where r - radial coordinate, K - some coefficient of bonded state formation kineticß, 9ex - characteristic expansion time (say, sound surface radius divided by sound velocity), ~ 0 = KoXo - characteristic time of collision, Xo - initial.molecule concentration, u(r) - velocity of particles, T - temperature of gas divided by initial one, coefficient A depends on a choice of 9 ex• For the coagulation process the growth of the average particle size g is described (Sutugin, Grimberg, 1975) by
(2)
It is evident from expressions (1) and (2) that the particle size is defined by an initial substance density and ita expanaion velocity. In the caae 8(r =OO)< Sn the dispersed condensate formation doean•t occur.
It was used the following experimental acheme: a geophysical rocket delivers to an upper atmoaphere the aource of two fast-evaporated liquide which were ejected with varioua masa aupply into an environmental space. Then there were determined 1) the fact of the cloud formation, 2) the Velocity of the cloud expansion, 3) the brightneaa of the cloud above the planet terminator and 4) the visual life time of the cloud.
The main component of the cloud muat be water (aa the main comet component), but we didn't uae it in theae experimenta because it has too low equilibrium vapor preasure and too high latent heat of condenaation. These factora eliminate the poasibility of occuring of aubatance fluxea with aufficient density for a condenaed phaae to be formed. The use of the high-power heat source was impossible becauae of a weight limitation. Hence, for the hignest substance flow we used as reacting liquide two pairs of liquified reagenta - ethyldichlorsilane and diethylamine and hydrogen chloride and ammonia. Both pairs of reagenta reacted yielding chlorides - of diethylamine and ammonia, respectively. In the first case it was formed also a ailicaorganic polymer. An evaporated material of comet may contain inclusions of solid (mineral)
1415
particles which are drag~d by a vapor stream. To simulate such particle behaviour we introduced in reagentsthesoot in the suspended form. It was made also to simulate the behaviour of more coarse particles than condensed ones in the artificial comet tail. These particles may be considered as ready-made ones whirled out of the comet surface core with an evaporated matter. We used three types of cosmosol generators in our experiments: 1) the generator which contained two balloons with rea
gents ejected through two parallel perforate tubes by the compressed nitrogen pressure (about 20 athm's);
2) the plunger generator which was ejected particles by an action of smokelese powder gases (pressure about 70 athm's) and
3) the fast-disintegrating generator which contained two concentrical vessels with reagents.
A cosmosol cloud was formed when the light source (Sun) lied in 8-110 below horizon. Cosmosol clouds were photographed by cameras type AFA-BA-21/c coated by lightfiltere SZS-21 on the photofilm type 22 (light sensitivity 1200 units of GOST). The routine procedure of film development was performed by the developer ASP-30 and the photometric treatment was made on the Sensitometrie device FSR-41. The relative value of the light flux in the field of camera view was defined by an integration of cloud brightnesses on cloud square.
Table 1 summarizes data on the rocket experiment conditions. In two cases we used the holding of the rocket head part encasing around the generator as an additional method of the influence on cloud format~on conditions.
Experimental results could be interpretated in such a manner; we can affect on the cloud configuration and the cloud expansion by using of various type generators and various ejection process organization. There were two main shapes of clouds with a spherical symmetry: the dense sphere and the spherical layer. In studying of these clouds it should be remernbered that we seen the external view of clouds on the film; we can't see the possible internal cavity in the cloud. From the other hand, if the cloud has the ring-shaped form on the film, it means that the spherical layer is thin and has a sufficient density in a tangential direction only. In all cases except the case of fast-evaporated :iquids (ammonia and hydrogen chloride) there were observed the slow expanded core formation. Fast-desinte-
1416
Tab
le
1.
Co
nd
itio
ns
and
resu
lts
of
exp
erim
ents
No.
and
R
eag
ents
and
V
isu
ai
Alt
itu
de,
Ave
rage
E
xp
ansi
on
R
ema.
rks
datu
m
gen
erat
or
life
tim
e,
km
bri
gh
tness
, v
elo
cit
y,
of
test
typ
es
min
. lg
B
m/s
1 D
EA+E
DCS
4
160
-1,3
45
0 E
nca
sin
g u
nre
-2
6.0
8.7
7
(S)
mov
ed,
sph
eri
-cal
lay
er
2 -"
-3
01
.09
.77
15
7 -2
,0
250
Sp
her
e
3 -"
-1
,5
165
-1,4
20
0 S
ph
ere,
co
re
02
.09
.77
fo
rma.
tion
4
HC
l+N
H3
0,5
16
1 -1
,6
500
Sp
her
e,
sph
eri
-0
6.0
9.7
7
(P)
cal
lay
er
5 D
EA+E
DCS
1
150
-1 ,o
60
0 S
ph
ere,
ra
dia
l 0
1.0
9.7
8
(FD
) st
ructu
re w
ith
a
core
6
-"-
1,5
15
0 -0
,6
500
-"-
01
.09
.78
7
_ .. _
1
150
-60
0 -"
-0
5.1
0.7
8
8 D
EA+E
DCS
3
155
-1,2
33
0 S
ph
ere
wit
h
29
.08
.79
(P
) co
res
No
tati
on
s:
DEA
-d
ieth
yla
min
e,
EDCS
-
eth
yld
ich
lors
ilan
e,
S -
gen
erat
or
of
sip
ho
n t
yp
e,
P -
gen
erat
or
of
plu
ng
er t
yp
e,
.....
FD
-g
ener
ato
r o
f fa
st-d
esi
nte
gra
tin
g t
yp
e •
~
.....
.....
grating generators were characterized by an appearance of a radial structure in clouds. Such a structure was obviously the result of the random character of the ge~erator destroyment. The spherical expansion was occurr~d for this generator also.
In accord~nce with theoretical results of Glenn (Glenn, 1969) the evaporated product expansion into vaouum must be ocourred with a sound velooity. Hence the velooi ty of particlea drag.ged must grow to the value of hundred meters per a second. The cloud front moved in our experiments with Velocity about 200-600 m{s. This value is in accordance with Glenn's predictions, but the mechanism of slow expanded core formation is unknown.
To eatimate the extent of a correspondence between the conditions of comet tail formation and those of our experiments we make following considerationa. Unfortunately, such factors as the presence of the atmosphere which slowed the expansion down and the absence of the long time influence of the sunlight pressure don't permit to simulate the tail shape. However,the particle formation process and expansion conditions can be simulated because the condensation takes place only in a zone near the discontinuity surface. This zone has a size significantly ernaller than both the comet tail dimension and the observed cosmosol cloud size.
It is evident that the difference in the brightness and the visual lifetime of clouds observed may be related with the vapor-to-condensate conversion degree and the condensate dispersity. The expansion Velocity of the spherical cloud front depends on both the particle size and the conservation of the uncondensed vapor phase which carries condensate particles out.
The model of the cosmosol cloud formation from the fast-disintegrating generator must have some special features. Two situations may be realized in such a case. Firstly, the drop-vapor cloud formed under the influence of ini tial impulse can possesses sufficient densi,ty to form a common discontinuity surface around itself. Secondly, the vapor density in such initial cloud may be too low and the discontinuity surface may form around each individual drop and/or each group of drops.
It should be noted that the cloud formed by a dispergation of ammonium and hydrogen chloride (i.e., more volatile liquide) had the lowest brightness and the shortest lifetime. It is evident that in this case it is
1418
the low conversion degree of reaction products into the dispersed condensate. Hence the particle drag by uncondensed vapor and their velocities are significant.
The lowest expansion Velocity and the most manifested layer structure of clouds were observed for cloud formed in experiment No.1. It is seemed that the unremoved encasing was acted as a condenser which increased the extent of the chemical conversion. The formation of the sharp boundary of the internal cavity may be explained by a rapid completion of reaction and condensation processes in the more dense system.
The increasing of total and differential flux (compare tests 8 and 1-3) leads to the increasing of the expansion Velocity and therefore to the decreasing of the cloud lifetime. We observed neither the brightness increasing nor the growth of the dense cloud part It was some coinsidence with Glenn's conclusions about the decreasing of the expansion velocity when the flux increased.
Fast-desintegrating generators given clouds with maxima of the expansion Velocity and the brightness. This appeared contradiction we explain by the formation of solid unevaporating partielas during the process of mixing and dispergation of liquids. If this is the case there must be formed unevaporating partielas consisting of final products of this reaction. The presence of such process is confirmed by the lower brightness of the cloud formed in experiment No.6 (encasing unremoved). The presence of such interaction of liquide in this experiment may occur when they impacted on the encasing. Then it should form. a condensate layer on the encasing and hence the extent of the matter conversion into partieleB should be decreased.
The special interest presents the mechanism of the cosmosol cloud core formation. The decay of such cores may be explained by a natural turbulent diffusion only. These cores were observed even for generators of an instantanous action when all particles must have initial velocities of order of hundreds m/s. This fact points to the presence of the hindrance mechanism for the particle motion. It was possible that particles were bindered by rarefaction waves reflected from the discontinuity surface. The other mechanism of the drag may also exist. If the chemical reaction on particles went on an unsymmetrical way, the particle temperature would
1419
be nonuniform. This nonuniformity should result in a angential creep of the vapor athmosphere of the particle with the resulting force acting on the particle. In the same manner this force may be induced by a nonsymmetry of the Stefan flow field around the evaporating ang reacting with environment particle. It seems to us that such mechanism can cause the particle fractioning in a space when the comet. tail is formed.
REFERENCES
Kaimakov E.A., Sharkov V.I., 1979, Physical and chemical processes in comets, Nauka, Leningrad, in Russian.
Cowan J.J., A'Hear.n M.F., 1979, Vaporization of comet nuclei: light curves and life times, Moon and Planets, 21:155.
Brin G.D., 19SO, Three modele of dust layers on cometary nuclei Astrophys.J., 237:265.
Wurm K., 19~3, The solar system IV, eds.B.M.Middlehurst and G.P.Kuiper, Univ.of Chicago Press, Chicago.
Saltanov G.A., 1973, Supereonic two-phase flows, Vyshayshaya shcola, Minsk, in Russian.
Lushnikov A.A., Sutugin A.G., 1976, Simulation of aerosol coagulation processes, Teor. Osnovy Khim. Tekh., 9:210 1 in Russian.
Sutugin A.G., Grimberg A.N., 1975, Vapor condensation at the free jet cooling, Teplofiz. Vysok. Temper., 13:787, in Russian.
Glenn L.A., 1969, Recondensation from a particle-vapor source flow into vacuum, AIAA J., 25:593.
1420
INDEX
Accommodation, at interface, 677 Accommodation coefficients, 621 Adsorption, of molecular gas, 653 Aerodynamics, 385ff, 401
low-density, 385 Aeronomy, 205 Aerosol
beams, 1235 particles, 1205 water, 1165
Analytical methods, 221ff Asymptotic methods, 39 Atomic hydrogen beams, 787
Boltzmann equation, 3, 39, 67, 75, 115, 123, 237' 253, 269' 277' 285, 303, 577, 1269, 1285
Boltzmann-Krook-Welander equation, 413
Boundary-layer flows, 185 Boundard phenomena, 83 Broadwell model, 3 Brownian motion, 1221 Burnett equations, 193, 237, 253
Carbon dioxide-nitrogen study, 749
Cesium vapor jet target, 777 Chapman-Enskog expansion, 237 Cluster(s), 1097, 1195
diagnostics, 1141 dissociation, 1173 experiments with, 1131ff helium, 1187 kinetic theory of, 1073ff van der Waa1s, 1141
Cluster formation, 1073, 1173 kinetics of, 1087
Cluster isomerism, 1105 Coincidence techniques, 725 Co11ision integral, 245, 697 Collision processes, 697ff Co11ision rates, 1205 Collision time, 717 Comet atmosphere, 503 Complex-shaped bodies, 487 Complicated media, 161 Condensation, 895, 1011ff, 1019,
1033' 1043' 105 3' 1063' 1087 1121
Continuum gas flows, 269 Controlled thermonuc~ear fusion,
1389 Cosmic body formation, 1413 Couette flow, 569 Creep velocity, 83
Dense gases, 107, 123 Diffuse reflection, 561 Diffusion through a fine-pored
filter, 1305 Dispersed media, 1227 Dispersed systems, 1221 Dissipation relations, 19
Electron-beam diagnostics, 825ff, 839
Electron beam fluorescence, 807, 833 Electron-electron co1lision, 705 E1ectron-molecule collision, 705 Evaporation-condensation problems,
363 Explosive vaporization, 1165
1421
External flows, 515ff
Flow boundary layer, 185 continuum, 269 hypersonic, 349, 421 low-Reynolds-number, 469, 477 low-speed, 477 plume, 983 slip, 523, 595 supersonic, 461 through channels, 1255 two-dimensional, 293
Flows near surfaces, 545ff Fluctuations
large-scale, 19 thermal, 99
Free jets, 815, 849ff, 865, 879, 887, 895, 903, 923, 931, 939, 951, 1011, 1053, 1131
Gas flows, two-dimensional, 293 Gas mixtures, 1255ff
chemically reacting, 199 flow with disparate particle
masses, 1277 relaxing, 199 two-temperature, 115 two-velocity, 115
Gas-particle flows, 1205ff Gas-surface interactions, 621ff,
645 Gas suspension, 1245 Gasdynamic separation, 1313 Group symmetry, 245
Heating effect, 965 High-energy molecules, 177 High-energy scattering, 717 Hydrodynamic equations, 161 Hypersonic cone drag, 453 Hypersonic flow, 349, 421, 669
Impact pressure measurements, 461 Inelastic collisions, 725 Interaction potentials, 717 Ionized gases, 135lff Ion-pair formation, 725 Irreversibility, 19
1422
Irreversible processes, 115 Isotope separation, 1297ff
Jet interference effects, 385 Jet-surface interactions, 965ff,
975
Kinetic equations, 371 Kinetic theory of gases, 3ff, 19,
39, 51, 83, 91, 107, 115, 123
Kinetic theory of turbulence, 51 Knudsen layer, 545ff, 577, 605
Langevin sources, 1375 Lava! nozzles, 1235 Lennard-Jones particles, 1097 Local interaction models, 431 Low-density aerodynamics, 385 Low-Reynolds-number flow, 469, 477 Low-speed flow, 439
t~xwell molecules, 75, 221, 237, 577
Metastahle neon atoms, 795 Molecular beams, 717, 733, 761ff,
799, 815, 879, 939, 1141, 1173
Molecular clusters, 1121 Molecule-surface interaction,
687 Moment methods, 245, 879 Monte-Garlo methods, 269, 277, 313,
333, 349, 357, 363, 371, 535, 1327
Monte-Garlo simulation, 313ff Mott-Smith method, 253
Navier-Stokes equations, 237, 253, 269, 535
Nonequilibrium condensation, 1011 Nonequilibrium effects, 199 Nonequilibrium expansions, 849ff Nonequilibrium processes, 205, 849,
895, 1043, 1063 Nonequilibrium statistical theory,
1221 Nonideal gases, 19 Nonideality, effects of, 1351 Nonlinear processes, 19
Nonlocal·hydrodynamic models, 229
Nucleation kinetics, 1073ff, 1113 Numerical methods, 261ff, 269,
277' 285
Onsager-Kasimir's reciprocity relations, 115
Onsager's principle, 67 Optical pumping, 795 Oxygen-iodine kinetic studies,
741
Partially ionized gases, 19, 1359 Particle flows, 1205ff Physical models, 133 Plasma flow, 1367 Plume flow model, 983 Plume impingement, 965, 983, 993
1001 Polarization flux, 83 Point-intense explosion, 261 Polyatomic molecules, 213 Porous media, 477
Radiation processes, 205 Raman effect, spontaneous, 799 Rate constants, 697 Rayleigh problem, 561 Reciprocity, 91 Relaxation, 133, 705, 1097
rotational, 911, 931 rotational-vibrational, 193,
887 temperature, 221 translational, 879 vibration, 213
Relaxation gasdynamics, 133ff, 145
Reverse leaks, 1319 Rotational excitation, 705 Rotational relaxation, 911, 931 Rotational-vibrational relaxa~.
tion, 193, 887
Satellites, 385 Scattering cross sections, 717 Separation
of binary gas mixtures, 1341 of isotopes, 1297ff of species, 1297ff
Separation nozzles, 132 Shock waves in rarefied gas, 253 Simply-shaped bodies, 495, 515 Slip flow, 523, 595 Species separation, 1297ff Specular reflection, 561 Statistical mechanics, 3ff Statistical modeling, 371 Stochastic theory, 99 Strang pairwise interactions, 19 Supersonic flows, 461 Surface scattering, 637, 661 Symmetry, 91
Temperature relaxation, 221 Thermal accommodation, 621 Time-of-flight diagnostics, 807,
815' 8 79 ' 9 39 Trailing edge region, 523 Translational relaxation, 879 Transport terms, 185 Turbulence, 51 Two-dimensional flow, 303
Vacuum ejectors, 1405 Van der Waals molecules, 1131 Vapor-liquid interaction, 1019 Vibration relaxation, 213, 1195 Viscosity cross sections, 733
Wall effects, 561, 585, 595 Water aerosol, 1165 Weak collective interactions, 19
1423
