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THERMAL rXNDUCTTVITY OF DENSE FLUIDS
A thesis presented for the Degreo of Doctor.of Philosophy
in the Faculty of Engineering University of London
by
Rafael Kandiyoti, B.S.
Department of Chemical Engineering and Chemical Technology,
Imperial College of Science and Technology,
London S.W.7.
Augus
ABSTRACT
The transient hot wire cell technique has been used
to build an apparatus for measuring the thermal conductivity
of liquids, at pressures up to 7000 atmospheres, in the
tomperature range 25.100°C. Measurements have been made
cin toluene with an estimated accuracy of 2 to 5% over the
wessure range. The analysis of the data has been developed
by solving the heat conduction equation assuming variable
p:nysical properties for the test fluid. In the solution of
this nonlinear equation, the Kudryashev.ZheMhor trans-
formations have been used.
The theory of Horrocks and McLaughlin, on the thermal
conductivity of simple liquids has been extended to chain
molecules and the model compared with data on the normal
paraffin homologous series.
The Chapman-Enskog theory, on the transport coefficients
for binary mixtures of dense systems has been combined
with the Lebowitz radial distribution functions derived in the
Percus-Yevick approximation. The kinetic, collisional and
distortional contributions have been factorized and the model
compared with real systems.
Acknowledgements
It is a pleasure to thank Professor A. R. Ubbelolule,
C.B.E., F.R.S., for his interest in this work.
I wish to thank Dr. E. McLaughlin, for his
sivervision, help and advice.
I would also like to thank Mr. Russell Harris,
M.L.. John Oakley, Mr. Len Tyley and the staff" o;; tho
DorWsnontal workshop for their constant help and
cooperation.
Last but not least I wish to thank Miss. C. Thurlw-
for her typing and help.
A paper based on Chapters 8 and 9 of this work
has been accepted for publication by Molecular Physics•,
CONTENTS
AB8TRACT
Acknowledgements
Contents Pafle
CHAPTER 1. INTRODIETION 1
CE PTER 2. THEORY OF THE TRANSIENT HOT WIRE CELL 7
2.1 Introduction 7
2.2 The General Equations of Heat Transfer 8
2.3 Solution of the Heat Conduction Equation for the Hot 0.20 Cell 11
2.4 Temperature Dependent Physical Properties 13
2.5 The Case of a Time Dependent Heat Source 17
2.6 Effect of the External Boundary 20
2.7 End Effects 22
2.8 Convection 24
2.9 Heat Transfer by Radiation 26
Ch27.21Ett 3. APPARATUS, PROCEDURE AND DATA HANDLING 31
3.1 Intwoduction 31
3.2 The Thermal Conductivity Cell 32
3.3 The Cell Casing and Bellows 36
PZ.ge
:1.4 The Electronic System 1" ,7
3.5 The Voltmeter Integration Period 142,
3.6 The Working Equation
3.7 The Pressure System r; r!
3.8 The Temperature Control System
3.9 Procedure 59
3.10 Processing of Data 61
4. DATA AND DISCUSSION 69
4.1 Introduction 69
4.2 Toluene Properties 69
4.3 The Wire
4.4 Toluene Thermal Contuctivities
4.5 Sample Calculation • 7
4.6 Sources of Error 31
cli7./..Y7R 5. THE HEAT TRANSFER EQUATION KITH VARIABLE PHYSICAL PROPERTIES
5.1 Introduction C5
5.2 The Infinite Slab 85
5.3 Solution of Equations (15),(17) and (18) 5.4 The Finite Difference Approximation
5.5 Results and Conclusions
CRAFTER 6. THE HARMONIC OSCILLATOR MODEL FOR LIQUIDS
Pie
OF SPHERICAL AND CHAIN MOLECULES 97
6.1 The Basic Equation 97
6.2 The Equation of the Cell Lattice Model 58
6.3 The Smearing Approximation 00 ./
6.4 The Harmonic Oscillator Approximation 100
6.5 Discussion of the Theory 101 6.6 The Cell Lattice Model for Pure Polymer
Solutions 105
6.7 The Heat Capacity 108
CHAPTER 7. THE HARMONIC OSCILLATOR MODEL OF THERMAL CONDECTIVITI 109
7.1 The Harmonic Oscillator Model of Thermal Conductivitios of Simple Liquids 109
7.2 Application to Chain Molecules 111
7.3 Comparison with Experiment 111
CHAPTER 8. TRANSPORT COatICIENTS OF PURE HARD SPHERE FLUIDS 115
8.1 Introduction 115
8.2 The Heat Flux Vector and the Pressure Tensor 116
8.3 Thermal Conductivity and Viscosity at Low Pressures
8.4. Thermal Conductivity and Viscosity for Dense Gases
8.5 The Equations of Longuet-Higgins and Pople
Lam
CHAPTER 9. TRANSPORT COEFFICIENTS FOR DENSE BINARY HARD SPHERE MIXTURES
9.1 Introduction
9.2 Transport Coefficients for a Binary Dilute Gas Mixture
9.3 Equilibrium Properties of Dense Hard Sphere Fluids +. 130
9.4 Reduction of Equations (6) and (20)
9.5 The Collisional Approximation to Thermal Conductivity of Dense Mixtures 137
9.6 The Collisional and Kinetic Contributions in Equations (39) and (49)
138
9.7 Comparison with Experiment
1444
CHAPTER 10. THERMAL DiFrubION IN DENSE HARD SPHERE FLUIDS 149
10.1 Introduction 149
10.2 The Theoretical Expressions 149
10.3 Comparison with Experiment 154
APPENDIX 1.
A Evaluation of TceDP/Dt 157
B Comparison of g vs. Kt. 159 APPENDIX 2 160
A Derivation of Equation 37 of Chapter 2 160
B Derivation of Equations (19) and (20) 169
APPENDIX 3
Programs 171
BIBLIOGRAPHY
125
125
126
1 CHAPTER 1
INTRODUCTION
The thermal conductivity of liquids is a subject of
interest both for the scientist and the engineer. Data is
required for engineering design purposes, and for testing
the validity of statistical mechanical models of dense fluids.
At present, thero is need in science and technology for more
data as well as for more accurate data.
In the measurement of the thermal conductivity coefficient,
there still seems to be problems that have not been solved or
avoided satisfactorily. So far none of the various techniques
utilized for measurements of thermal conductivity can be claimed
to have completely isolated the conductive contribution to heat
transfer from the radiative and convective contributions. Also
in measurements on mixtures, there is the complicating effect of
thermal diffusion.
On the other hand, theoretical predictions of thermal conductivity
are far from perfect; agreement to 15% is considered reasonable.
This does not mean however that the present levels of accuracy
in thermal conductivity measurements are adequate so far as
theory is concerned. A great deal of information can be obtained
from comparing calculated and experimental values of rates of.plariag,
2
of the conductivity with temperature and density; there
available data is in many cases deficient.
A. great deal of information on liquid structure can also
be obtained by extending the data range to high pressures,
which affords the evaluation of structural models over wide
ranges of density. In particular the change of sign from
negative to positive, of the temperature coefficient of thermal
conductivity, at high densities is an important index of the
success or failure of thermal conductivity theories.
The transient hot wire cell chosen here for thermal
conductivity measurements at high prossure has several
advantages over other techniques of measurement. Mille no
attempt will be made hero to compare in detail charachteristics
of the various techniques, as relative merits have boon
discussed extensively by Ziobland (Ref. 1) and Pittman (Ref. 2),
two advantages of the transient hot wire cell which have not
been discussed before must be mentioned here. The first is the
adaptability of the hot wire cell to high pressure techniques.
The physical size and limiting dimensions of the hot wire coil
are such that the ono designed for this work was acommodated in
a conventional 1.5' ID, 6n OD pressure vessel with a range of
up to 7000 atmospheres. Co•.axi.al cylinder and particularly flat plate .
techniques require rather larger pressure vessels. This
complicates problems related to construction and sealing.
The second advantage of the technique is its suitability
for measurements on mixtures. Approach to the state in the
presence of a temperature gradient leads to the setting up of
concentration gradients because of thermal diffusion. It is to
be expected that the short duration of the experiments with the
transient hot wire cell (20 - 30 seconds) will go a long way to
prevent partial separation from affecting the measured thermal
conductivity values.
In this work, the apparatus designed and constructed for
high pressure measurements will be described and data on toluene
for pressures up to 6250 atmospheres over a temperature range
of 30 to 90°C presented. The analysis of the data is based
essentially on the work of Horrocks and McLaughlin (Ref. 5) and
Pittman (Ref. 2). Here however, in analyzing the data, effects
relating to the temperature dependance of the thermal conductivity
coefficient, and the change in power supplied to the system ale
treated as part of the mathematical statement describing the
system. This implied solving a non-linear partial differential
equation. The method used for this particular application was
also examined as a. tool for solving partial, differential
equations with temperature dependent physical properties.
While on the one hand an apparatus has been designed,
constructed and tested for measuring thermal conductivities of
pure and mixed liquids, and certain improvements made in the
analysis of the data, attention has also been paid to the
theory of tran.;:pert properties '4
The hot wire technique can be used for measuring the
thermal conductivity of non-polar, non-conducting liquids.
(The method has also been extended to gases; that however is
beyond the scope of this work.) The most important single group
among those is the normal paraffin homologous series. Here the
Horrocks and McLaughlin model (Ref.. 3) for the thermal
conductivity of simple liquids has been extended to liquids
composed of chain molecules by using Prigogize7s cell model
for pure polymers. (Ref. 4). As will bo seen, the thermal
conductivity, in this approximation, is a function only of the
molecular force constants and the density. Hence calculation
of 7, as A function of pressure follows from data on high
5
pressure densities for the homologous series. Results for
atmospheric pressure calculationswill be presented here, but not
those for high pressure, as data is not available for comparison.
It must be said however that the temperature coefficient of
thermal conductivity does not go through a sign inversion for
calculations up to 10,000 atmospheres.
As mentioned before, the fact that the hot wire method is a
transient one makes it particularly useful for measurement on
liquid mixtures. This is because for a mixture in a non-uniform
temperature field, there exists a velocity of diffusion in the
direction of the t(averature gradient. The mass flux is then
given by
J. = D n 12
xl D a T T 8 r a.
where x, is the mole fraction of component 1
r is the position variable
D12
is the mutual diffusion coefficient.
and D is the thermal diffusion coefficient.
The potential availability of a technique for satisfactory
thermal conductivity measurements on liquid mixtures raises the
question of the state of theories on the transport properties of
dense fluid mixtures. Tho most general theory to date has been
6
that of Chapman and Enskog (Ref. 6). The derived equations
have been generalized to dense fluid mixtures by Thorne (Ref.6a).
The work is based on the rigid sphere interaction potential and
as such, can be combined with the radial distribution functions
of Lebowitz (Ref. 7) derived by using the Perms Yevick
approximation (Ref. 8).
This analysis was first used on mutual diffusion in
binary mixtures by McLaughlin (Ref. 9). Here it is extended to thermal conductivity, viscosity and theimal
diffusion. While the treatment, on the whole, does broadly
reproduce charachteristics of liquid mixtures observed
experimentally, better quantitative agreement must wait for the
treatment to be applied to more realistic intermolecular
interaction potentials.
The Lebowitz radial distribution functions were also used
in recalculating results from the theory of Longuet.Higgins,
Pople and Valleau (Ref. 10), for isotopic mixtures. While for the
mutual diffusion, thermal conductivity and viscosity coefficients,
equations resulting from the two theories are at least partly
related, for thermal diffusion there seems to be no correspondance.
Furthermore neither one of the two theories gives satisfactory
results when compared with experiment. This is to be expected
as the thermal diffusion coefficient is particularly sensitive
to the form of the intermolecular interaction potential.
CHAPTER 2
Tha.2a_2Ltbaffsansient Hot Wire Cell
Introduction
The experimental technique is based on the measurement
of the voltage change across a thin wire carrying a current
and immersed in a test fluid. The wire is joined to thick
current leads, top and bottom, and at distances conveniently
removed from its ends, welded to two voltage taps.
In order to determine 'he thermal conductivity of
the test fluid from this system, the heat conduction
equation appropriate to the system will be obtained,and solved
for the temperature profiles in the thermal conductivity cell;
the temperature profiles will then be related to electrical
measurements. In addition, any approximations made in the
derivation of the heat transfer equation and the solution
will be examined.
7
2. The General Equations of Heat Transfer
The equation of thermal energy (Ref 1) in the absence
of radiative heat transfer is given by
DU = - (V q) - P(V • v) T : C7v) , (1) Dt
where the rate of gain of internal energy equals the
sum of three terms which are successively:
1) .tat input by conduction
2) reversible energy increase by compression and,
3) irreversible energy increase by viscous dissipation.
In order to reduce equation (1), to the desired form,
the following relationship, derived from the first and
second laws of thermodynamics is used:
DU r 8P DV DT P R. :2 1._ TE-ST3v P Dt CvliP P (2)
Here the operator D/Dt is defined by
a a a a (3) Dt at + vx .53-c+vy
Combining eq.(2) with the Equation of Continuity
Ra = p (v•v) (4)
Dt
equation (1) can be written in the form
8
DT ap p c (0•q)- T(--)
vDt MEM a T v (v-v)- (T :v) (5)
9
which in turn can be re0.uced by the substitution of
Newtons and Fourrierus Laws to
DT ap p c = V'(XVT) -T C7'v)+ aT x (6) v Dt
where is called the dissipation function (Ref 1).
Equation (6) can be further reduced by using
and
• cp v p f3 -c = cd2 T
-dp = - p ,BT dP + p a dT
where
a ,dV, B (av)
= 4. Vt—cily ' T- V aP T
to
DT DP p C = V. (XVT) + T a +% • (7)
P Dt Dt
(Ref. 2)
Equation (7) describes the behaviour of the experimental
system but is mathematically intractable. It is therefore
further reduced by using physical argument:..
As heat is supplied to the central wire, expansion
of the liquid around the heated section gives rise to
1.0
radially symmetric free convection. Hence the operator
D/Dt reduces to D 3 3 Dt 31 "z a z
where z is the axis along the length of the wire.
Furthermore due to the existonce of a heating section
below the bottom voltage tap, the rising liquid
surrounds the mid-section of the heating wire, still
retaining the temperature distribution that would have
existed in the absence of convection, until cold liquid
rising from below the heated section reaches the bottom
tap. Hence before cold liquid reaches the bottom tap
BT DT 3 T 0 and 5.e = z cat
Clearly, this simplification imposes a time limit on the
duration of tho experiment.
Using a set of assumptions fully discussed in a later
section of this chapter, the velocity of the fluid along
the z-axis has boon calculated (Ref. 3 and Ref. 4a). Using
these results and toluene properties
it has been shown in Appendix lk that the term
T a DP/Dt can be neglected. It has also been found
(Ref. 4b) that the viscous dissipation term X is negligible,
under the rolevoni; conditions. Equation (7) then reduces to
P C =
p at (x v T)
3. Solution of the Heat oilthie.4ationfo( z.11-0. Het Wire Cell.
In order to relate the temperature changes in the
cell to the thermal conductivity that is being measured,
it is necessary to solve the heat conduction equation.' LI the
first approximation the following is assumed:
a) Free convection and viscous dissipation effects
are negligible.
b) The heat source is of infinite leneh and zero
diameter (line source ). This assumption will "ca
removed as more refined solutions to equation (8) are
derived.
0) The fluid medium is externally unbounded, and the
limiting value of the temperature, sufficiently far
away from the heat source, is zero.
d) Physical properties of the test fluid aro temperature
independent.
e) Power dissipation, ql, per unit length of heat
source is constant. The last two assumptions will also
be subsequently removed. Lastly
f) Radiative heat transfer is negligible.
11
(8)
The solution of eqn. (8) under these conditions is
well known (Ref. 5a).
ql T(r,t) = El 44-0t
-u where -tai (-x) =J 2 du x u
1 (9)
and K = . Clnarly, T(r,t) is referred to the initial
temperature. For large t. eqn. (9) reduces to
qa T(r,t) — Loa L 21 ar2
where C= exp ('y) and y is Euler's constant.
If the heat source is assumed to be a cylinder of
infinite length, uniform diameter and infinite thermal
conductivity the solution becomes (Ref. 5b)
va,t) [ 1:_r 2
( ge ) 1r; in 11,:r . .1 (10 kra C T
where a = 2(plepl/p Cp) and T = Kt r2
In obtaining equation (11) surface resistance to heat
conduction is assumed to be negligible.
(10)
Temperature Dependent Plagioal Properties
We now remove assmption (11) as wall as (b), and assume
linear temperature dependences for thermal conductivity,
density and specific heat.
X = Xi(1 +X2 T) ; x:.1 a (12a) 2 X1 dT
p = p1(1 + pi, T) ; q = 1 dp
2,3)
Pi dT (1
* C
P = C
Pi (1 + Cp2 p T) ; C*2 - = 1Cp1 (12o)
dT
whore the subscript / denotes the properties at the initial
temperature. The problem can now be stated follows:
p Cp at = Q ' (1. V T) (13)
with initial conditions
T(r, 0) = 0 2 (14a) r a
dr (r , 0) =n v (114b)
and boundary conditions (Ref. 6) aT(a,t) a2(a,t)
' (15) 2naX + = 7; a2 131 Ct ar at
where p' and Cr: are the density and specific heat of the
wire respectively, and
T(art) = f(t), (16a)
lam r9 t) = 0 for finite t. (16b)
13
14
f(t) is obtained as data during the experiment. Given equations
(12), the problem stated above is nonlinear.
We will now use a set of linearization transformations
(Refo 7) to obtain the linear analog of the problem, the
solution to which, of course, is similar to eqn. (11),
and then invert the solution.
Consider the specific enthalpy, as referred to the
initial temperature
dh = C dT
and define through h, the quantity such that
= - Jo X dh.
tkCp
Using the last two equations
dO dh =XdT
frau which it follows that
= XdT and v 2 15 = • (x T) dr dr
Furthermore, we define the quantity g , such that
= J. L-- dt ; ag = X dt . (19) PCP pCp
Through equations (17) - (19), equation (13) can be transformed to
= 20 (20) a g
(17)
(18)
15
The initial and boundary conditions are also transformed:
0 :r, 0) m 0 ; (r,0) = 0
(21)
4 ce 0 (r g ) = 0
(22)
and
+ -2
al r'" dr Il or al r = a (23)
Here a is assumed to be constant with temperature. This is
further discussed in Appendix 2.
So far we have used a set of transformations to linearize
the nonlinear system of equations (12) - (16). The
linearized problem has been solved by the Laplace Transform
technique (Ref. 5b):
0(r, §) = [in C +12T y + (54--g ) 2T In + (24) 4n
where T=t/r2. With the exception of T and absence of %,
this is identical to equation (11). We must now perform
the back-transformation.
Combining equations (12a) and (18) we have
Elk _ dT dT - X — + X T , dr i dr 2 dr
Where
(25)
X2 =Xi )12 = dX dT
16
Integrating both sides from j to co
0(00, g)_gr, [12(c°!t)—T2(rtt)) (26)
Using equations (150 and (21)
0(r, g ) = xj(r,t) +2i2 T2 (r,t)
(27)
where gr, g ) is given by equation (23). In particular
0(a, g ) = X1T(a,t) + 23. T2(a,t).
(28)
Throughout the treatment g is given by
g = 0 dt. Pp
Using equations (12a) -(12c) and letting r = a
I! X2 T (a,t)
[1+ p2T(a•t))[1+ Cpl T(a,t)]
The second terms on the right hand side of equations (28)
and (29) constitute the total correction for the case of
variable physical properties, under the given set of
assumptions. Neglect of these terms would reduce the set
of equations-(24), (28). and (29) to equation (11).
An upper limit to the value of the integral in the
rhs of equation (29) can be calculated by making use of the
physical properties of toluene and the magnitude of the
temperature rise in the cell. This calculation is presented
xi = + P1CP1
dt (29)
in Appendlx 10B, and shots Thai: neglect of the integral
introduces an error of the older of .01% into the thermal
conductivity measurement, For this application then
g = (X1/P1Ci)1) t = Kt
(30)
where z: is the thermal diffusivity. Combining
equations (24), (28) and (30) we get
ql I In 2T+ (5 ) 01 ln(46T )+ ....] C
= X1T(a,t)+ 22 T2 (a, t),
where
T = Kt a2
5. Thiagagpstr a Time Dependent Heat Source
In the previous section the heat conduction equation
has been solved with the assumption that heat input into
the system is constant throughout the experiment. As
the current romains substantially the same, and the wire
resistance changes by about .01 to .02 ohms during a
run, we know that qi (=12 R/1 ) is a weak function of time.
We will now assume this dependence to be of quadratic form,
(11 = ql 4. (12 t c13 t2
(31)
and Polve the heat transfer problem of equations (12 )- (16)
1?
with the altered boundary condition:
where
/77
qi -
Q pe 2 ) = 27r do 2
+ -3 dg r = a (32a)
(32b)
It must be noted that in equations (32a) and
(32b) the approximation of equation (30) has
been used.
The derivation of the temperature profiles
under these conditions is rather lengthy and will
be presented here only in outline form; the full
treatment may be found in appendix 2.A.
After linearizing the problem as before, the
Laplace-transform of the heat conduction equation
and the boundary conditions are taken:
a2 0(r ye) 4. 1 Oi(ro) (33)
61-2 r or
27Ca aj6 (a, 0) + + Q2 Q2 + 2Q3,
ar s2 d3 a S 0 (a, s) (34)
lim 0(r,$) = 0. (35) r-9 00
19
Here s 5' the complex variable of the transforar.t101i,
0(r,$) ;he Laplace traAsn..,..m of 0 (r,t), mite, Ls before,
i' asz'amed to be lonstant. 13017ing equations (33) -
(35) for -95 vo obtain
ig(r, 6) = (41 Q.2 / + 2cs /8 a ) i(0(r 5 ) 27cas tgl(0a)+ as Ko(fla)]
By inverting equation (47) back into the real plane,
using equations (28) and (30), dropping small terms in the
second and third brackets, and rearranging, we have
X T(a,t) + T2(a,t)
qi 4Nt a2 rqs 0 a2 44
411. Ca2 a t-n
(37)
0 0
.• q2 r t tln 4a -1 +1 • • • • 1
IC 737
▪ q3[ t2 E In 114 3/2 -j +• 7t
.
Ca.
Again by setting q2, q3 and N2 equal to zero, equation (11)
can be obtained.
In deriving equation (37), some of the simplifying
assumptions of section 3. have been dropped. We must now go
on to examine the remaining ones.
( 36 )
ItE g a2)2tTio(xxl)li(xn)—YcAcn)100
n ( xnxna)).:2
b T(a,t) = 1C 1 + 41c1
20
6. Effect of the External Bounder
hci,lon (37) has been derived with the assumption
that an infinite medium surrounds the central cylinder. The
duration of the experiment, then is limited to times beyond which
heat loss to the outer walls significantly distorts the
temperature profile in the cell. The equation of heat
conduction has been solved (Ref. 8) for a bounded medium,
under the following conditions:
a) A non..convecting medium is bounded internally and
externally by two infinitely long concentric cylinders, with
radii a and b respectively.
b) Power is dissipated from the central cylinder,
at the constant rate of qipita for t 0. •
c) The external cylinder is held at the initial temperature.
d) Physical properties are assumed to be temperature
independent over the temperature rise in question.
The solution of the heat conduction equation is then
given by
b — >1'
where a = a n is the running index n = 1, 2, . . . •
and the xn satisfy the equation.
Joi:(7 )S1)11(xn) - Yo (a xn) J1 (xn) = 0
where Jk and Yk are Hassel functions of the first and second
kinds, of order k respectively. For large values of t.
equation (38) reduces to
2 exp-aa)t T(a,t)-z in I) .. 2 ;
irxr15- 'r
2 Ta zwl ..
t a xn [( Jo 3-774
When h>> a, using (Ref. 8 and Ref. 9).
Z4 0 lira RIk(z)-4(1z)ki r (mil)
1 Ykcol. -(;) r (k)(1z)
-k lim
z+ 0
where for k an integer
(k) = (k-1)! ,
along with equation (39) leads to
Jo(axn) = . n Yo( a xn)
z xn
and
J (x ) Y (x ) = - 1 — 1 n 1 n n
Combining equations (39) ... (42) leads to
21
(39)
1 .
•
(40)
OW
(42)
T (a, t)=2 ql 7r7%.
, b °° exp - (Kxd a)
2 t
E n — a - 2 n 21 F Li ffo(m`n) f - xn
22
(43)
where in this approximation xyla satisfy the transcendental
equation
J(xncr ) = 0.
(Ref.8).
Equation (43) will be referred to, when cell design
requirements are considered.
7. End. Effects.
The assumption of infinite wire length must now be
examined. Heat generated in the small diameter, high
resistance section of the wire is conducted away by the
current and potential leads. Heat loss through the former may
be rendered negligible by leaving a suitable length of heating
wire between the ends of the section where voltage changes are
monitored, and the thick leads leading out of the cell.
The following method has been used to estimate this length.
(Ref. 10a and. Ref. 5c).
The two ends of the wire, of length 21,, are assumed to
be kept at zero temperature, and also surrounded by an
23
enclosure at zero temperature. The temperature distribution,
in the steady state, along the length, Ot5x ts2L, for
constant power, qi, supplied to the wire is then given by
Tf(z)=.21_ 1-
2na H cosh M(L
z)
cos ML (44)
where H is the heat transfer coefficient given by (Ref. 4c)
H = 2X /a in.1.4"-i—r T r= Kt/ a2 (45)
and
(2110w a )2
(46)
where X is the thermal conductivity of the wire. In w
equation (45) the steady state temperature at the wall has
been approximated by the "line source" solution of
equation (10). The value of 6 for which
Tf CL) T (214 -8)
is much less than .01% of Tf(L) has been calculated (Ref.10a)
to be about 1 am; the error rises rapidly to .6% for 5= .2 am.
(Ref. 4d).
Heat conduction away from the potential leads had initially
been treated (Ref. 12) as cooling fins and the error neglected.
Pittman estimated the error in the thermal conductivity
measurement due to these losses, on a scaled up model (Ref. 4e)
24
for an essentially stationary fluid. The resulting
calculation shows that for toluene measurements, errors would
be about .3 . .4% over the pressure range. This seems to be an underestimation, for reasons to be discussed in the
next chapter. Still, however it is possible to
minimize this error by extrapolating 'ors. t data to zero
time, as the error due to end effects grows with time.
8. Convection.
While in the reduction of the equation of energy
transfer, convective heat transfer and viscous dissipation
were neglected, it is to be expected that the presence of a
radial temperature gradient along only the middle part of the
fluid, will give rise to free convection.
By using steady state methods it was previously found
(Ref. 10b and Ref. 13) that for the Rayleigh number,
R < 1000, convective effects could be assumed negligible.
R = g poiATd3/10: < 1000.
Here g is the gravitational constant
p,a , 114 K are the physical properties of the medium and
d = b-a is the charachteristic dimension of the systom
av t
(47)
25
The velocity profiles of the convectinu fluid have
subsequently been investigated, Pittman (Ref. 4a) numerically
solved the equation for the velocity distribution (Ref. 3)
within the cell, with the following assumptions:
1) A cold front of fluid begins to rise from the lower
edge of the heating section, as soon as heat is supplied to the
central cylinder.
2) The radial temperature distribution is assumed to be that
of the stagnant fluid, and to give rise to density gradients which
determine the velocity field.
3) The velocity along the r.-axis is zero.
14-) Heat conduction from the warm region into the cold front
neglected.
The equation describing this system has been solved
numerically by Pittman (Ref. Lid)
a2 v 1 av+,a a T 1
+ — — 1- - ar2 r a r --"Ir = r
with the initial condition
v(r,0)=0
and boundary conditions
v(a,t) = 0
r4 (r ,t) = 0 co
26
The calculated velocities are 10 to 1 lower than those
observed (Ref. 14) by interferometric techniques (Ref. 4f).
As mentioned in section 2., the viscous dissipation term, is
calculated by using velocites obtained by this method.
9. Heat Transfer by Radiation.
So far, radiative heat transfer has been neglected. For
systems where radiation effects are important equation (8)
becomes
p + en = 7t, -
8T at
where, e n is called the net emission,
e n bb (r. t)e
and
K is the absorptivity coefficient
e bb is the black body emission function.
s the absorption function. (Ref. 15)
The difficulty in the analysis of radiative heat transfer
is twofold: firstly the absorption function is a very complicated
function of the geometry of the system, and has been constructed
here only through use of simplifying asemptions; secondly even
the simplified form of en involves integrations over the
(48)
(L9)
27
temperature and equation (148) then becomes an intractable
integro..differential equation, as will be presently shown.
Approximations at various levels have been made to
overcome these difficulties, a good survey of which will be
found in Ref. 4g. In relation to the hot wire experiment
it had been suggested (Ref. 10c) that the anission from a
central cylinder into a black enclosure would yield a good
estimate of the radiative heat losses from the central wire;
the error calculated in this way had been found to be of negligible
magnitude. Other estimates of these losses for measurements
on toluene were as high as 2%. (Ref. 11, 16.48).
It is assumed here, that since the central cylinder
is of infinite length (see section 3) radiation
emission from it and from concentric shells of surrounding
fluid may legitimately be taken as having no component along
the axis of the central cylinder. The equation for
monochromatic radiation intensity, I. in cylindrical
coordinates, is then given by (Ref. 15)
dr
+Ky.]; t) (50)
where
Ey = EY(T) = niCy Ibb,v (T) (51)
Kv= Kv(T)
n = the index of refraction
2 c2h
I = bb,v vlexP ;KT -J
and
= speed of light.
h=Plancles constant.
v= wavelength of radiation.
k = BOltzmannos coast., and finally
T = the absolute temperature.
Equation (50) can be solved by the integrating factor
method:
(r,t) = ulwro Ev (T) u(r)dr (53)
where the integrating factor u(r) is given by
u(r) = exp (Iv+ Kv ) dr° (54)
ro Here ro is the radius of the central cylinder and r° is the
variable of integration. All terms in equation (53) are
functions of position, wavelength and temperature; the
latter in turn is a function of position and time. An
obvious simplification of equation (53) is to adopt a suitable
average value of Ks., over the infra red region which is
relevant for radiative heat transfer. We may then write
I(r,t) = f E(T) u(r) dr (55) u(r)
2.6
(52)
29
Since radiative transfer has been assured to occur only
radially, e= I(r,t).
Also using the Stefan•aotzmann law
(S. • ebb(r't) = 2 7n2a (A T)4; AT = (r,t) (56)
dr Hence
= K(T) E "aria T - u(r)
fro E(T) u(r9) di-a] (57)
where
u(r) = exp[ fr ro
(70+ K(T)) dr° J. (58)..
Due to the form of equations (57) and (58), equation (48)
is mathematically intractable. The difficulty could be
removed by assuming K, the absorptivity coefficient to be
temperature independent. This however would impose tine
independence on the radiative transfer problem, which
would in turn lead to lower levels of outward emission .:oin
concentric fluid shells and hence to a smaller correction
to the thermal conductivity. Pittman (Ref. 4g) solved the heat
conduction equation with a heat sink term representing absorption
of radiation by the liquid, with a constant absorptivity
coefficient and the boundary condition,
artaX air= ql oq(t) ar
where
6q(t) = 87ra a e TA T(a,t)
a = Stefan-Boltzmann const.
e = emissivity coefficient
TA = Absolute temperature
T(a,t) = Temp rise in the wire, from the start
of the experiment ,
with the conclusion that for toluene, the error involved is
of the order of 1 to 2%. This would seem rather high. The
boundary condition assumes a temperature gradient equal to
T(a,t), which would lead to overestimating the error;
neglecting absorption of radiation by concentric fluid
shells would have the same effect. Though due to its
approximative nature, this treatment, like previous ones
cannot be used for actual corrections to thermal conductivity
data, a practical aspect does emerge. The error decreases as
time goes to zero; thus azero time extrapolation of data
for each run would lead to a value less affected by
radiation losses. (Fief. 4g)
CHAPTER.
Apparatus, Procedure and Data Handling
1. Introduction
The theory of the hot wire cell presented in Chapter
2 was applied in designing a thermal conductivity cell
adapted to high pressure measurements. Here a description
of the cell, the pressure system and temperature control
system will be given, along with the working equations
used in analysing the data. The experimental procedure
will also be described, and finally the method of
handling and processing the data will be given.
it
2. The Thermal Conductivity Cell
The coil (Fig. 1) was constructed of EMS stainless steel
with dimensions of 1 cm I.D., .98 inch O.D. and 15.7 cm
internal length. Four 26 SWG thermopuro platinum leads
were connected from the sealed electrodes on the high
pressure plug to the cell; the two current leads were
spotwelded to needles at the top and bottom of the cell,
which were insulated from the main body by baked pirophylite
beads. The two potential leads which ran down the side of
the cell were insulated from it by pyrex capillaries drawn to
3/1660 OD. In the cell a spring (Pt + 10% Iridium) was
placed between the top current lead and the heating wire
in order to prevent the wire from sagging as the experimental
temperature was raised. The spring was shunted on both sides
with .001° 1 thick, /16° wide platinum foil (aef la), as the
spring material is of rather high resistanco. The heating
wire, .00160 diameter, die drawn high alpha grade platinum
wire supplied by Johnson Mathoy timitedowas then connected top
and bottom to the spring and lower current lead respectively.
The potential leads were then spot welded, about 1 cm away
from the spring at the top, and 3 cm away from the current lead
at the bottom. After completion, the welds and the heating
32
c A C 1.-98"---1
F /G. / THERMAL CONDUCTIVITY CELL
28 T PI TAP 14 DIA. N -..1 1-i-lie:'
Ti
i .5 0"
F'- 31-...1 16
A 1/8 DIA. HOLE FOR INSULATED POTENTIAL LEADS.
V DIA. HOLE FOR INSULATED POTENTIAL A'i LEADS.
SCALE 1" .7. 2"
kA
r- t B
V B
SECTION BB (ENLARGED)
SECTION CC (ENLARGED)
.24-1
"
.75" ID ON UPPER RING
720"
8 BA CLEAR COUNTER SUNK
60°
c r-V C
AA
300 30
EQUIDISTANT TAPPED 8 BA 98" DIA.
6 EQUIDISTANT HOLES-CENTRE 1 282" FROM A CENTRE OF DISC.
30°
30°
30°
30°
34
wire were inspected by and found free of kinks or
distortions. The potential lead weld regions were found to
be partially flattened, up to a distance of 3 or 4 wire
diameters.
The design specifications arise from considerations
fully discussed in references (1), (2) and (3). These
arguments will now be briefly summarised.
i) Effect of the External Boundary.
It has been found (Ref. lb) that, comparing, in the
first approximation, equations (9) and (43) of the previous
chapter, agreement is within .01% for Kt/62 .4: .12. For
toluene at 90°,K, the thermal diffusivity, is about .7 x 153.
This implies that over 30 seconds are necessary for wall effects
to become significant in a 1 an I.D. cell.
ii) Distance Separating the Potential Taps from Ends of the Heating Wire.
The distance between the top potential tap and the spring
is determined by considering conduction away from the heating
section by the current load (and spring). Here leaving
approximately 1 am between the bottom of the spring and the
potential tap was found sufficient (see section 2.7).
Other considerations enter in determining the length
of wire to be allowed between the voltage tap and the bottom
35
cuireent lead. Free convection starts as soon as the central wire
begins to heat up. Shells of liquid, with the same
temperature distribution as for the case of no convection00
rise followed by the cold front described in section 2.8 (Ref 4).
The experiment can be continued as long as the cold front
remains .5 cm away from the bottom end of the heating wire.
As mentioned in section 2.2, the duration of the experiment is
limited by the length of wire allowed below the bottom potential
lead. Clearly the velocity of the cold front also depends on
the heating rate: low heat rates would allow longer experiments
but lead to low voltage changes and hence loss accurate results,
where as high heat input rates would necessitate short
experiments due to convective and wall effects. It was found
(ref 3a) that for a distance of 3-4 an below the bottom
potential lead, heating rates corresponding to 17-25 milliamps
cell current would allow the experiment to last up to about
30 seconds. In fact all measurements were completed within the
first 20 seconds.
iii) Diameter of the wire - Analysis of initial specific heat
effects, end effects and radiation losses, indicate the necessity
of using heating wires with the smallest diameter with which
it would be possible to build a cell. .001 inch diameter was
found to be adequate for this purpose.
-46
iv) Wire length. Two cathetometers, set at right angles
relative to each other were used to measure the length of the
heating section of the wire, while the cell was clamped dawn
vertically. The wire length was found to be 8.100 + .002 cm.
The measurement was repeated after the thermal conductivity
runs and no significant change found.
3. The Cell Casinq, and Bellows
The case which envelopes the cell is made of three parts
(Fig 2). The main body of the case screws into the pressure plug.
The middle part, a short cylinder, is argon are welded to a
bellows; this assembly screws into the main body of the case.
The middle part has been designed as a separate piece from the
main body for easier handling and replacomont) The lower end
of the bellows was welded to a plug, with a tapped hole for
filling. All parts were made of stainless steel and screwed
joints were sealed with teflon flat rings.
The bellows, made by Teddington Aircraft Controls Ltd.,
had + 1 inch axial movement. This corresponds to about 32%
compression of the fluid confined in the cell agembi7 which is
sufficient to raise most liquids to 7000 atmospheres, over the
temperature range of 30-95°C.
The assembly (Fig. 3) was filled with the test fluid,
under its own vapor pressure, following a bulb to bulb
Pressure plug
20 TPI UN Form FIG. 2
BELLOWS AND CASING
Mat. EMS Stainless unless specified
BELLOWS DATA
ID 1.00 i OD 1.37" THICKNESS .005/.0055 MOVEMENT AXIAL ± 1" FREE LENGTH 3.47'' 30 CONVOLUTIONS
PTFE Square ring Iii; xlhes -1" ID
Pressure vessel
Case V' ID, 1.37 OD
PTFE Flat ring iiix 1/16'; 1.2 ID.
Argon arc weld over backing ring
/ ,
1 1
1
PTFE Flat ring
.3/e. Whit; stainless
CD
1;Iii#1)11) *pl c-
39
vacuum distillation, where liquid nitrogen was used as coolant.
A glass system, Fig. 4, was designed for the purpose, and
.Anohed to n standard vacuum system, including a bathing pump
and an air cooled oil diffusion pump.
4 The Electronic
A system previously developed (Ref.3b) for
atmospheric pressure determinations of thermal conductivity
was used for the high pressure thermal conductivity measurements.
(Fig. 5).
Basic requirements in the high pressure experiments were
simi)ar to those at atomospheric pressure:
i) 20 mA current stable to 1-2 ppm,
ii) Eight readings a second with 1 microvolt resolution in
about 20 millivolts,
iii) Stable backoff facility,
iv) Automatic data logging.
The major parts of the curcuit will now be briefly described.
1) Power supply. D.C. voltage standard made by Cohn:
Electronics, Kintel Division, U.S.A., with voltage range 0-1000'.%
' provides currents up to 50 mA. This instrument was placed in
series with a 10,000 ohm resistor, in a thermostatted oil
bath, to provide the cell current. The latter was measured by
monitoring the voltage drop across an NPL calibrated 10 ohm
40 FIG. a VACUUM DISTILLATION AND FILLING APPARATUS
TO VACUUM --13."'
1 5.5"
9.
5
RUBBER 0-RING SEAL HELD BY BRASS FLAT RING
SCALE 1 cm. =1" MATERIAL = GLASS
D.C. SINVARD _J r
FIG. 5 CURRENT SUPPLY AND VOLTAGE MONITORING SYSTEMS
"\nse• 10 KQ 10
STANDARD -POWER
SUPPLY ---0-i000 V
10V
BALLAST
RELAY 4,_. _ - TIMING
, — CONTROL
8 ris t 100 KC V
_ _ _ 1 _AMP I IDVM I I -._=_z_- 7.—. 1.__=_____I--: -. I I r-- 1 1
CELL 1 1 i 1 r-- 1 ' 1 1
I 1 1 ' 1 ' I 1 1
I 1 1 1 1 I i -j r I 12502 j 1 1 I I I I I i 1 I 1 1 1 I _
1 I LL 1
25x 50Q
„TELETYPE PUNCH
I I
__J
Tinsley standard resistor.
2) Backoff Circuit. A Model 735A D.C. Transfer standard
made by Hewlitt-Paclard Co., U.S.A. provided the bachoff,
through a voltage divider of 2500 ohm total resistance in
steps of 50 ohms.
3) Amplifier. Hewlitt-PacItard Model DY 2441A
amplifier, with gain selection of 1,10, and by-pass, was
used in series with the voltmeter. The gain is specified
accurate to ± .007% with temperature coefficient of less than 10
.5 microvolts /oC. Input impedance was 10 ohms.
4) Voltmeter. A Hewlitt-Packard Model DY 2401 C Integrating Digital. Voltmeter was used in series with the amplifier.
Maximum resolution of the instrument was 1 part in 300 000.
When used with a set sampling period of 1 second,resolution was
.1 microvolt, and for a sampling period of .1 second, 1 microvolt.
Accuracy is better than + .025% of the reading. Both the
amplifier and voltmeter are guarded and have high common modo
re jection.
5) Data-Logging Equipment. Monitored voltage changes across
the cell were logged by a Hewlitt-Packard DY 2545 high speed
tape punch set with a ERPE II tape punch made by Teletype Corp.
,I3
Before a run, the system was allowed to stabilize by
passing current through the ballast. This is a 100 ohm,
continuously variable resistance introduced in order to avoid
transients in the power suppl-s- due to changes in the load,
when a run is triggered. This was done by a push button which
activated a mercury wetted relsy. A Zenor diode was provided
in parallel with the 10K ohm resistor (See Fig. 5), so as
to avoid large voltages appearing at relay contacts. The
falling edge of the pulse generated across the relay triggers
the opening of the gate to a counter circuit available in the
voltmeter with a 100 kilocycle signal. This signal provides
the timing control for reset pulses externally generated and
fed back into the voltmeter. The first integrating period can
be delayed by up to 1/8 sec. after the opening of the gate; the
the delaying period is followed by a reset period of 9.7 milliseconds
which is the gap between the reset pulse and the beginning of an
integrating period. The integrating periods are 100 milliseconds
each separated by 25 milliseconds from each other, the start of each
being triggered by an external reset pulse.
The sequence of operations and voltage measurements loading
up to a thermal conductivity run, will be described in a
late• section of this chapter.
5. The Voltmeter Ii.tegration Period
Each voltmeter reading is an average value cvor a
100 millisecond integration period. Hence
t T = f 2 TR (t) dt t2-ti
where TM tho measurml temperature change is tho average value
over At (= t2-t.;) of TR 0 the real tomporaturo. Equation (37)
of tho previous chaptor must now bo modifiod to take this
integration into account. Dofining from oqn. (37)
4Kt 2 2 2 , g Kt L.A.n o -a72 + 2-: + -cy wt• -a02 +....
.44
( 1 )
c02 4n ln Cat - 1] t +
cp3=4 [ Inca - t2+...
wo may writo
X1 TR + A2 TR2i2 c°1 + cP2 cP3 •
Integrating
and t2
A tit j
equation (5)
and dividing by
rt2 X2
with rospoct to
at, wo got
ft2 , L. t TR ' dt At =
1ti
-t + t, TR d 2At
and hence
t botwoon t1
( colicP24(P3) dt
t2 TM = X1At
[ J ti (pi+ cp2 cp3 _ .12.?TR dt
As tho variable conductivity correction is mpoctod to bo
loss than .5% it is pormissiblo to calculate tho average
value of T2 from the lino sourco mothod. Thus by oqn. (10)
of tho provious chapter
T2 r ql4Kt 12 4rry n Cat •
Integrating equations (2)-(4) and (7) with respect to t
we got
TM = 1 [ c-61 c-132 c.153 -?-x whore aftor dropping terms containig (At)2 (At)3 otc.
4K(tilist)
= 4n17 ZKAt [ti +
In [i+ tLt
1 ] + ln Ca —
45
1
(6)
(7 )
(8)
4. a[e=2„. 4K] At L
r [in plyCa +At ]2 rin 1-1-K4 -4 12 t C a
- _ 1 c12 r 2 92 - 877; Lti ln[141] + 2t At In 4K t• Ca 3tiAt+..
I
5, [1 . ti At N 4K(WAt) 11,2 1 (11) EL ,-. At n + 3ti At In Ca .0.1 At +..1 12
116
and
T2. fiL-1 2 ,t 4.At [in iti±g1]2 - I-4%X J 1
At 1 ' Ag Ca.2 iL t rl n tatil2
1 -2t1 in(1t) - 2At In 14g!rl'At) + 2At
(is)
Hero At = .1 second.
6. Tho Working Equation For fixed a, A2 and a equation (8) givos the measured
temperature changes as a function of timo, at tho wire-fluid
boundary
T1(t) - _ l F(t) .
For any interval, At. thon
Al = AF(t) / ATM(t).
For small
ATM temperature
AR whoro
changes (Rof. 1) AV V = AI . I 12 dR
Thoroforo
and
AT, -;ri2 Al ]
dR Al - dTm
r1 AV V Al 1-1 L IAF -12 AF J (13)
From equation (8) it can be soon that F(t) is a weak function
of Al . Hence an itorativo calculation is called for. Tho
computor program written to porform thoso calculations will bo
doscribod in a lator suction.
70 21222121212E22YAS
The pressure system wss designed to reach pressures up
to 7,000 atmospheres. The accompanying flow sheet, Fig. 6,
illustrates the layout.
1) The pressure vessel was manufactured by Pressure Products
Inc. (UK) Ltd.,; design details may be found in Fig. 7. The
vessel was made of EN 25 stainless steel, with le bore and
12° working length, fram the tip of the electrode head to the
bottom of the vessel. The electrode head was also made of
EN 25 steel with initially a beryllium copper-teflon half
Bridgman main seal and four electrode seals. The latter consisted
of Hilumina insulators (Smith Industries, Ceramics Division) and
brass canes, successively lapped in.
It was found during the experiments, that beryllium-copper
work-hardened sufficiently to scratch the vessel bore when the
plug was being extracted from the vessel. Copper was tried and
found to flow too easily and fill the gap between itself and the
plug (see Fig. 7). Phosphor bronze was then tried with a larger
angle, 6 degrees, between the ring and the plug and found satisfactory,
provided a groove was turned off on the outside, as shown in
Fig. 7, This groove prevents the 0.D. of the ring from flouring
flush to the vessel, bore which would have increased surface
I0
35Z
g 1
h.p. line I. p. 1 in e
fig 6 .
flowsheet of the pressure system
F/ G. 7
PRESSURE VESSEL AND PLUG
6 „
3 va
HILUMINA PACK I NG CONE BRASS ELECTRODE
A
EN 25
EN 25
EN 25 PT FE PHOSPHER BRONZE
11
22 'k
PHOSPHOR BRONZE MAIN SEAL
51Ct'',
ID 1190 OD 1.565
lie WEEP HOLE
T :1
yv 12"
D iA
SCALE 1"=
50
contact and causA. leaking. This design was found sufficiently
durable over the pressure al.d. tempeeature cycling that took
place during the experiments.
2) 10,000 ataosphere gauge, made by Budenberg Gauge Company,
rated to 1f accuracy of full scale deflection.
3) Letdown Valve, with nonrotating spindle, rated to 100,000 psi,
was made by Pressure Products Inc.
4) Valve, with same specifications as (3).
5) Gauge, with same specification as (2).
6) Intensifier, rated to 200,000 psi, with intensification
factor of 15; model 112.5) made by Harwood. Engineering Co., J.S.A.
7) Non-rotating spindle valve, rated at 30,000 psi, made by P.U.I.
8) Pump, stainless steel body, rated at 60,000 psi, made by
MoCarnay Manufacturing Co., J.S.A.
9) Valve, with some specifications as (3).
JO) 40,000 psi gauge, made by Budenberg Gauge Company.
11) Non-rotating spindle valve, rated at 60,000 psi, made by
12) Hand pump; same as (8).
13) Valve; same as (11).
The whole pressure system was enclosed in a steal frame and
shielded with -t° thick mild steel plate (Fig. 8).
The system is betiaily pumped up by (12) to about 2,G00
atmospheres. Valve (9) is closed off to isolate the low pressure
side, and the pressure fUrther raised by pumping (8) on the low
5/
FIG. 8
52
side of the intensifier (6) until the piston reaches the end of
the stroke. Valve (4) is then closed off, valve (7) opened and
the pressure between (4) and (9) dropped to about 2,000 atmospheres.
Valve (9) is then opened and the intensifier piston pumped down by
(12); the pressure is raised again to 2,000 atmospheres, and
values (7) and (9) turned off. The pressure between (4) and
(9) can then be raised by (8) to the vessel pressure, valve (4)
opened and pumping continued. Vessel pressure can normally be
raised to the designed maximum during the second stroke of the
intensifier. During experiments, (4) is shut to isolate the vessel
from the rest of the system.
The pressure calibrations were carried out against an N.P.L.
calibrated 'dead weight?, standard pressure gauge over the range
0-5000 atmospheres. The zero-error of the two 10,000 atmosphere
gauges remained, as pressure was raised, and no error exceeding
the quoted accuracy (1% of fUll scale reading) of any of the three
gauges was found. The calibration was repeated by reducing the
pressure from 3,000 atmospheres, with the same result (Fig. 9).
The results were assumed to hold for pressures above 3,000
atmospheres.
8. The Temperature Control S stem
As the temperature rise of the central wire during the
ciAuci aI c10000 ATS)
A GAUGE 2. (10000 ATG)
CfAUGZ 3 (2.5o0 AT5)
•••••••lar,
1••••••11.
.11•••••••,
53
UI
0 5
0 7
Q 3
el cr/ 2
at a.
48, 0
/
o /0 I I I I I I
0- i 2 3 4 5 G 7 PRESSURE 10 ATMOSPHERES
Plc.,. 9 PR essuree --,Auci cAt-teRATiow cutzvE
5E4'
experiment is .3 C, maximum allowable temperature drifts in the
cell are on the order of .,05°C/hour. In order to achieve this
temperature stability the pressure vessel was placed in a
temperature controlled, stirred oil bath. (Fig. 10).
1) Tubalox immersion heater, rated 230/50 volts, 3 kw, with no
heat dissipation above the surface of the oil. Power to this heater
was supplied through a 15 amp variac.
2) 1" thick blockboard case, housing vermicullite insulation.
3) 3° thick layer of vermicullita insulation around and beneath
the galvanized iron tank.
4.) 1/30 HP induction motors made by Klaxon Ltd; 1425 rpm
slowed down by gearboxes and with shafts mounted with 1209 long
vertical fins as well as 4P diameter brass propellors at the
tip.
5) i" thick 2.11-11d si:eel plate, supporting pressure vessel. 1
6) /8" thick duraluminium sheet oylindev!, provided. lagging by
trapping 1" of oil between itself and the pressure vessel..
7) Pressure vessel.
8) 10 gauge galvanized iron tank, 20" long, VP wide, 35 9 deep
and with 4P flanged top.
9) Tubalox immersion heater, rated 230/250 volts, 500 watts,
supplied through the temperature controller.
55
28
front
Oft
1
.fig 10
Thermostat Bath plan view
1 cm 2"
56
The whole system was provided with a lid made of -14' thick
syndanyo plate and 3n layer of vermicullitz, enclosed in cardboard,.
The heat transfer avid used was Shell Voluta Oil 45, a
mineral oil based fluid, which can be used at temperatures up
to 300°C. The oil was pumped in and out of the thermostat bath,
using the oil handling system shown in Fig. 11, which was also
used for cooling the system.
A temperature controller was designed and constructed in the
departmental electronic workshop for the purpose of providing the
required stability, and will be briefly diseussed here.
The sensing element, Degussa, 100 ohm (nominal) resistance
themaneter, was used as one arm of the bridge with a PYE,
variable resistance box, av the pre-set arm. The bridge was
driven by an 18 -felt stabilised pcuer. The out-e-ba5.anc3
voltage from the bridge is fed into a pre-amplifier, and then into
a 3-term unit gain amplifier, the output of which is used .to determine
what proport&oi of a second the heater will be turned on.
This signal drives a gate which allows the output of a
mains driven zero voltage pulse generator to reach the triau
controlling the heater. The latter shuts power off when voltage
across it passes through zero, thus breaking circuit at each
1 Reservoir 2 Heat Exchanger 3 By Pass 4 Pump
5 Temperature Gauge 6 Draining Line 7 Filling Line 8 Thermostat Bath
fig 11 Flowsheet of Oil Handling System
58
mains half cycle, and being reactivated by the zero voltage
reset pulse, as long as the gate remains open.
Temperature stabtlity in the cell was followed over time
by measuring the cell resistance, and the system described above
was found to drift at rates of less than .05°C/hour after a
settling down period of 12 to 18 hours. Furthermore, the vertical
temperature distribution in the vessel bore was investigated by
vertically moving a resistance thermometer. In the absence of the
bath lid and the pressure plug, it was found that a vertical
gradient of about .05°C existed. It was assumed that the gradient
would be reduced to negligible proportions when the vessel
closure and bath lid were replaced
Finally in older to measure the axperimenUl temporaturo, an
N.P.L. calibrated 25 olza (nmainal) resistance thermometer, made
by H.Tinsley and Co. Ltd., was immersed in the bath between the
pressure vessel and the cylindrical shell surrounding it.
(see Fig. 10). Thermometer resistance was measured to .0001 ohm.
The constants of the calibration polynomial were used by a
computer program , in order to evaluate the temperature.
(see section 10). Ekperimental temperatures will be quoted here
to the nearest tenth of a degree centigrade.
9. Procedure The thermal conductivity of toluene was measured, at
each temperature, first at atmospheric pressure and then at
various elevated pressures.
As the pressure is raised by pumping, work is dano on
the compression oil (DDT 585, Shell) and the test fluid;
consequently the temperature in the vessel and the coal rises
above the temperature at which the system is being controlled.
More than three hours were allowed for this heat to be dissipated.
Experiments were carried cut only after temperature drifts due
to cooling :sere observed to be indIstiriguishable from controller
drifts for about half an hour,
The fl.AtevA:-r; -,v),7-17,111t7- for erz,ors due to tee7e7re
drifts in Nto cD31 is thr:-.E. drifts shou lass thsn .1% -;_ho
temperature rise over the duration of the experiment. As
experiments lasted approximately 20 seconds, and temperature
rises wore never larger than .3°C, this condition corresponds to drifts of about .07°C/hour. In fact no experiments were carried
out with observed drifts above .05°C/hour. These drifts were
followed by measuring the cell resistance at short intervals. If
before a run cell resistance changes larger than those allowed for
59
60
controller drifts were found, this was attributed to cooling due
to pressure leaks and the set of runs cancelled.
In theory, two factors must be taken into account in
identifying a pressure leak through cell resistance measurements.
The first, as indicated above, is the changes of temperature
undergone by the system, as the pressure is raised or lowered.
This must be distinguished from resistance changes that Pt wire 0
will undergo due to changes of pressure. At 50 C, the pressure-
resistance relationship for platinum may be represented by the
empirical equation (Ref. 5)
R = R (1+a P + b P2)
p
0
whore -12
a = .1.949 x 10-6, b = 7.86 x 10 and R0 is the resistance
at atmospheric pressures.
While, clearly, the two effects change the wire resistance in
opposite directions, this in practice does not pose a serious
problem as the magnitude of the resistance change duo directly to
pressure changes, is much smaller than the temperature change due
to work done on the liquid.
At each setting of the temperature and pressure, the pressure
was raised to a slightly higher value than the desired one, in
(14)
61
order to compensate for drops duo to heat dissipation. At each
pressure and temperature, three thermal conductivity measurements
wore made. In addition a run was simulated where the voltage drop
across the 10 ohm standard resistor was measured in order to
calculate tho current change.
An experiment is set up as soon as temperature and pressure
stability criteria are satisfied. The power supply is allowed to
stabilize at 10 volts output, where the current drawn is about
/ milliamp. The following operations are then carried out.
1) Total cell resistance is measured and the ballast resistance
set to the same value.
2) Current is switched into the cell and the voltage drop across
the two potential leads measured. Heating due to the 1 milliamp
current is negligible.
3) With the current going through the ballast the potential
drip across the standard resistor is measured. This allows the
calculation of the current flowing in the call in (2) and conse-
quently the calculation of the cell resistance.
Output from the power supply is then raised to 200 volts,
drawing approximately 20 milliamps; the system is allowed to
stabilize for five minutes.
62
4) Necessary backoff voltage is calculated on the basis of
allowing about 20 millivolts across the voltmeter; this is set on
the voltage divider.
5) With the current flowing through the ballast the potential
drop across the standard resistor is measured in order to
calcUlato the initial current. Tho voltmeter is then set to the
.1 second sampling period, the tape punch activated and the
current switched into the cell. 50 minutes was allowed between
runs.
63
10. Processing of Data
In order to calculate the thermal conductivity from
V v.s. t data, it is necessary to know the temperature
coefficient of the cell wire resistance. To this end the cell
resistance was measured as a function of temperature at each
experimental pressure, over the range 30..95°C.
A computer program was written to execute the following
operations:
1) Thermometer resistance data was converted to temperature
readings.
2) For each pressure the temperature v.s. cell resistance data
was fitted to a straight line.
3) Using the T v.s. Re fits, pressure v.s. cell resistance data
was cross-plotted and fitted to a quadratic.
4) These curves, in turn, are used for fitting straight lines to
T v.s. R , for each experimental pressure. dR/dT values
relevant to the experimental states wore then computed.
The source program listing may be found in appendix 3B.
The least-squares polynomial fitting subroutine used, was a Program
Library deck written by C. Ho, Computatitn Laboratory G.P.D.,
Rochester, hinnosota, The same subroutine was
used in the main data analysis program , which will noNrbe
described.
G4
Data in the form of paper tape produced by the Teletype
punch in Atlas autocode was first loaded, without code translation
on to magnetic tape by the IBM 1401. The main programme written
in Fortran IV was then loaded on the IBLI 7094 to perform
the following operations.
1) Eaeh data batch consisting of several thermal conductivity
runs and a current run, were translated into BCD code*, assigned a
decimal point in units of volts and read into the neiory.
2) The following information was fed in from data cards, for
each run:
- the temperature coefficient of thermal condu-ttivity at the
initial temperature
- time delay before the start of the first integration period
- mire and fluid densities and specific heats
• ex,Jerimental temperature and pressure
- backoff voltage value
- temperature coefficient of wire resistance at the initial
-.11 -ra...••••32.111c..1:••••41,-
* I am indebted to Mr. Richard Beckwith of C.C.A., Imperial. College of Science and Technology for the translation routine.
am.inrAci
65
temperature
- voltage drop across the 10 ohm standard resistor with current
flowing through ballast and power supply set at 10 V; this value is
denoted by SV1, and is used for calculating the current at the
10 volts setting.
- voltage drop across the heating section of the wire,
10 volts across the circuit, this is denoted by RIVT, and is
used for calculating the cell resistance.
- voltage drop across the 10 ohm standard resistor, with power
supply set at 200 volts, SV2, used for calculating the initial
current in the coll.
3) Calculation of 01, qz
and q3. The heat dissipation per
unit length in the active part of the cell is given by
q(t) = V(t) I(t)
where V(t) is the sum of VB the backoff voltage (measured
accurate to 10 microvolts) and the voltmeter reading Vv(t)
(accurate to 1 microvolt).
V(t) = V (t) + V . v B
The current changes are measured by monitoring the voltage changes
across the standard resistor (the resistance of which is known)
during a simulated run. This data is fitted to a quadratic,
I(t) = io + t-l- I 2t2
(15)
66
where io is the value relative to the backed off voltage. The
initial current is then calculated from SV2 and the resistance
of the standard resistor. Then
I(t) = I0 + t + 12 t 2
The power dissipation can can now be written as
Iq(t) = I(t) V (t) + I (t) v (16)
The upper limit of the total current change during the
experiment is 4, ppm. Taking VB 300 millivolts, A V;
= 500
microvolts, we see that the change in .he second term on the rhs
of the last equation is much smaller: •
) V
A q
here If = I0 + A I. Hence measuring VB to 10 microvolts produces
negligible error, in the calculation of q2 and q3.
Combining equations (15) and (16), with voltage change data
as a function of time is sufficient to fit q(t) where the
first term Ali is calculated from the measured initial current
and cell resistance. Hence
q1 (t) = q(t)/L = qi + q2t4-q3j
q. + q,
2
CI. =- LI
67
The time averaged power dissipation is given by 2
raix.rz jr 1̀3 t max 3
') Calculation of X I. The string of voltage changes measured
as a function of time are smoothed Ly fitting to a quadratic in
in t averaged over the integration period.
V(t):A + Bg (t1) + Cg(ti) *
where
g(t i ) = t1 ' [1+ nt h + (t1 + 6 ti
t
Data taken during the first half second where specific heat effects
are important is ignored. The first conductivity value is, found
by processing readings 5-15 and a conductivity is calculated through
an iterative procedure by using 6 more readings each time.
An iteration is initiated by calculating the first
approximation to the thermal conductivity by the lino source
method. This value is used for calculating, through equation (13)
a new value, which is fed back into equation (13). The process is
(=tinned until the change is successive iterations is less than
.01% of the value of X . 1
A series of apparent conductivities as a function of
experimental time are thus calculated, along with corresponding
68
standard deviations, and differonces from the line source method
arising from finite mire diameter, variable power and temperature
dependent thermal conductivity.
A listing of the source program may be found in appendix 3A.
69
CHAPTER 4
DATA AND DISCUSSION
Intrn:4,1ction
The thermal conductivity of toluene has been measured
at three temperatures between 30 and 90 degrees centigrade
over a pressure range of up to 6250 atmospheres. In dIl,a
series of approximately 50 measurements has been made. No
data could be found for comparison with the high pressure
thermal conductivity measurements of this work. Atmospheric
pressure measurments were within about 0.5% of previous work
by Pittman (Ref.5).
In this chapter, the data will be presented along with
results of hot wire dR/dT calculations. A sample calculation
for the conductivity will also be given.
2. Toluene Properties
Analar grade toluene (Hopkin and Williams Ltd.) was re-
fluxed over sodium wire for about six hours and distilled.
A middle cut was separated and it°s refractive index
measured. This was found to be 1.4942 + .0002 at 25 ± .1thC.
70
No high pressure eLata on the heat capacity and density
ofttausine are at present available. The heat capacity
values at high pressure were taken to be those at the
corresponding temperature at i atmosphere (Rea). Sinco,
in the calculation of the conductivity the heat capacity
appears in terms which are rather amail, the error arising
from this approximation is negligible (see Chapter 3).
Density data for toluene had to be estimated in order
to relate the pressure to the compression of sample volume.
Results of this calculation were used to compute the thermal
diffusivity at high pressure, The method for estimating
high pressure densities (Ref.2) from atmospheric data on
P v.s. T , is based on the assumptions that isochors
(constant volume lines) are straight, i.e. that (dP/dT)v
is constant and that the (dP/dT)v v.s. P relationship is
linear.
r P2 P1 1 [LP] = Ap + B dT V L T2 - Ti Jv
By using the few available data points (Ref.3)
(1)
71
PRES8.(ATK) Vrel.
1810 .885
2930 .853
4400 .824
where Vrei is the ratio of/ and the density
temperature relationship from the International Critical
Tables ,
p(T) = .88448 - .9159 x 10-3 T + .368 x 10-6 T2 (2) (0< To C‹ 110)
the density can be estimated over the relevant temperature
and and pressure range (Fig. 1) •
The method was checked by calculating n-hexane densities
and comparing with previously measured values (Ref.4); it was
found to be within 1% up to 2000 atmospheres, deteriorating
to about 5% around 6500 atmospheres, over the 0-1000C
temperature range.
1 1
5 G a 3 4- PrZESSLIRM 7 P 1000 ATS
1
.0 .....01.....
• 00
1
za_
Fir . t. Tot.. ue-Nt E. i,eziSiTI ES "T p-iico-I P P. E.SS UM E. (E. ST %MA:M)
The lake
The length of the wire between the two potential
taps was measured before the toluene runs. The catheto.
meter measurements, which were described in the previous
chapter yielded
8.103
8.100
8.099 Haan : 8.100 am.
The measurement was repeated after the conductivity
determinations.
8.098
8.105
8.102 Lean :8 .102 cm.
This change of less than .02% is .within the accuracy
of the measuring instruments, and in any case negligible.
The calculation of the temperature coefficient of wire
resistance has been described in the last chapter. At each
pressure, dR/dT was taken to be constant over the experi-
mental temperature range, as the latter was rather small.
73
PRESS. WK) dR/dT(ohm/°C)
1 .059875
1:500 .059734
2250 059663
3250 ,059565
4850 .059407
6250 .059264
4. Col ene Thermal Conductivities
Results obtained in the experiments are given
below:
74
TABLE I
75
PRESSURE (A911)
MP, (°C)
mw /. X(moan)
mw/cm.0C Av. % Deviation from the mean
28.2 1.297
1.300 1.298 0.1
1.297
30.8 1.282
1.266 1.285 0.2
1.286
1500 30.8 1.633
1.631
1.630 1.633 0.12
1.636
61.4, 1.589
1.598 1.595 0.25
1.598
91.5 1.537
1.543 1.538 0.5
1.544
TABLE I Cont..
PRESStRE (ATM)
TEMP. (0C)
X X (mean) mw / cra_oc
mw/em-°C Av. % Deviation from the mean
2250 30.8 1.824
1.814 1.815 .34 1.808
61.4 1.822
1.776 1.793 1.1
1.780
91.5 1.770
1.800 1.791 0.8
1.803
3250 30.8 1.917
1.911 1.914 0.1
61.3 1.937
1.943 1.938 0.1
1.935
91.6 1.940
1.935 1.938 0.1
10939
76
TAME I Cont..
X X(moan) FRESSURE . • A-re % Deviation (ATH) (oc) mw/cm.0C from the mean
77
.1A50 q 2408
2.125 20110 0.35
2,123
61.3 2.170 2.183 2.173 0.3
2.170
91.6 2.183
2.187 2.186 .1
2.189
pler,1•1•Me,A.M.
30.8 2276
2.295 2,202. .4
2.276
91.6 2:,369
2.404 2.383 .6
2.375
6250
These results are plotted in Fig. 2.
0 I 2 3 4-
78
Pfac-SsuRa 1000 ATS
Pick. 2. THEM MA 1_, CON DUC T IVITY OF TO LueN E
5. Samele Calculation.
Run 23. Date: 1/8/196
Temperature = 30.8°C; Pressure = 4850 Ats.
Temperature Drift = .01 °C/hr
Measurements Before Rim.
DC supply : 10V
Voltage Drop Across Cell: 27.5970 mV
ca ca co Ballast: 27.5971 ml!
S2 St C?
Standard Resistor: 9.8302 mV
CI SI
Heating Section of Wire: 16.9312 mV
Back off Setting: 314.04. mV
DC Supply: 200 V.
Voltage Drop Across Standard. Resistor: 196.494 mV
79
Summary from Computer Output.
Operating Current: 19.6478 milli Amp
Initial Resistance: 17.2250 ohms.
dR/dT (approx. value used): .058954 ohm/°C.
No. Pts Included -.
Standard Deviation
15 •2 - 23-x 10.2
.64358 x 10-6
33 .21145 x 102
.60107 x 10-6
51 .21126 x 10-2
.75573
x 10-6
69 .21123 x 10-2
.76917 x i0-6
87 .21165 x 102
.74863 x 10-6
105 .21154 x 10.2
.81744 x 10-6
Linear fit extrapolated to zero time gives
X= 2.110 mw/cm.°C
Smoothed dR/dT = .59407 ohmeC.
Corrected% = 2.125 mw/cm. C.
80
6. Sources of Error
During the experiments. temperature drifts in the
cell were foilgxal by ru-.3slit,ing the cell resistance at short
intervals. As each set of determinations was completed
within the span of about two hours and the accepted drift
rate was 0.03 C/hr thti measurements can be averaged with
a maximum error of .1%, over the pressure range. On the
other hand, as mentiond in the pl-evious chapter, the
experimental temperatoro is neasursd in the oil bath
between a hollow cylinder surrounding the vessel, and the
pressure vessel itself. While this arrangement somewhat
shields the thermometer from temperature fluctuations in
the oil bath (4- .1°C), the temperature smem by the
thermometer is expected to be loss stable than that in
the cell, which is surrounded by the thermal mass of the
pressure vessel. As the time lag between the exterior of
the vessel and the cell is about 3 to 4 hours. the
temperature differenon (assuming continuous drift in one
direction) could be as much as 0.3 to 0.4 degrees centigrade.
This gives rise to two types of error.
1) The measured thermal conductivity, depending on the
magnitude of dx /dT at the given pressure, would be in
error, by 0.2 to 0.5%
81
82
2) In the evaluation of dR/dT, the cell resistance would be
matched to a temperature different than that in the cell.
The resistance value would be in error by about 0.2%. The
error in dR/dT with an upper limit of about 0.5% due to this,
might be expected to decrease through the double smoothing
procedure of R v.s. T data outlined in the previous chapter.
Furthermore, the shortness of the temperature range would be
expected to affect the accuracy of the temperature
coefficient. This error could be estimated to be in the
range 1.0 to 3.0%. and closer to the larger value at the
ends of the temperature range.
Errors in the thermal conductivity measurement due to
convective effects are due to two types of mechanisms. The
first is due to heating of the central part of the wire with
the result that at the extremities vertical temperature
gradients aro set up. It has been assumed that the
radial temperature field remains unchanged until the `cold
front approaches the bottom potential tap. To this must be
added heat losses from the potential lead into the convecting
fluid; this clearly ties in with and aggravates the problem
of heat loss from the central wire by conduction into the
potential loads, with the result that the radial temperature
field is distorted. These errors, however, may be
83
minimised by extrapolating the data to zero time, provided
that the fluid is assumed to be initially stagnant. This
brings us to the second type: free convection at the start
of the experiment due to vertical temperature gradients.
The observed fall in apparent conductivity with time
observed in many runs is consistent with hot fluid rising
from below the heating wire in the cell. It is difficult
to estimate the error arising from these initial vertical
temperature gradients in the cal, since these would
depend on the magnitude of the gradient. This effect is
somewhat smaller at higher pressures, as would be
expected, and the error arising from it would be estimated
in the range 0.2 to 0.5% after extrapolation to zero time.
This initial free convection would also imply that
potential tap losses are not completely eliminated by
extrapolation to zero time. The resulting error could be as
high as 0.5%. Errors due to radiation losses may be ignored
with little error when the data is extrapolated to zero
time. Finally, as can be seen from percent deviations from
the moan an error of 0.1 to 1.0% must be accepted from
random factors, such as electrical noise, and scatter in
the extraplated lines.
Summarizing, the error duo to averaging the results of
each run should give rise to 0.0, uncertainty in the
temperature measurement to 0.4 to 1%, the initial convection
effect to 0.2 to 0.5%, uncertainty in dR/dT to 1 to 3%,
end effects to 0.1 to 0.5%, and random errors to 0.1 to 1%.
84.
' CHAPTER 5
THE BEAT TRANSYPIt EQUATION WITH VARIABLE
PHYSICAL PROPERTIES
1. Introduction
In Chapter 2 a set of linearization transformations,
proposed by Kudryashev and Zhemkov, were applied to a problem
where the inversion procedure was rather straightforward due
to g reducing to Kt without significant error. Here the
evaluation of those transformations as a tool for solving
certain types of partial differential equations will be
attempted. For a chosen simple geometry, the solution obtained
by this method will be compared with two other ways of solving
the problem. It will be seen that while all three methods
agree for small nonlinearities, the differences for large
values of the temperature coefficient of the thermal conduc-
tivity are significant.
2. The Infinite Slab
For purposes of comparison, a problem solved with a
variational technique (Ref.l) has been chosen.
85
86
An infinite slab of thickness 2L is held at uniform
temperature Ts , and the boundaries -L and I subjected
to T(-L,t) = T(L,t) = 0 , for t > 0 . The temperature
profiles are symmetric with .respect to the plane x = 0 .
The density and the heat capacity arq assumed constant and the
thermal conductivity is given by
X(T) = Al ( 1 + X2T ) where A2 X =1i dT •c-11 (1)
The problem can now be stated as
T(x,0) =Ts ; -L x I
T(-L,t) = T(L,t) = 0 ; t > 0
with the partial differential equation
PC aT _ a [ x(T) aT ] . (2)
1- at ax at In order to simplify handling, the problem is put in dimensionless
form. The new variables are defined as :
X2 t _ T ; =p7-1,2 ; e - y
s • 6 = Ts X2 (3)
and the problem can bo re-stated in dimensinless form as follows:
6(p,0) = p < 1 (4)
e( ) = e( 1.er ) = 0 ; T > 0 (5)
ae — a [ ( +me ) ae] (6) aT a$ a$
87
For this particular problem. the Kudryashev and Zhemkov
transformations (ref. 2) can be re-cast as follows.
The enthalpy/mass, referred to the initial temperature
is defined as
dh = dO and the quantity 0 as
0 =111 X d2
Differentiating (8) and using (7)
d0 = X 0
which immediately leads to
=a de d1 dP and , V20 = v (X VG)
Furthermore, we define the quantity
= • dg = x dT 'o
Using equations (9)-(11) we get
V20 = ag
and by (7) and (8)
0(1,0) = 0 .
88
In order to obtain the boundary conditions,
0 h = rdh, de'- e
At 0 = 0 h = -1. Therefore -1
0(-1,,§) = 0(1„g) = f [1+ a( 11' 41 ) dh'-= -( 1+ ;3 ) . (14)
The solution to the problem stated in equations(12)-(14) is
well known (Ref. 3) •
‘ 22 x(x,g) = 11-2.0. ,(-011 exP[-(2n+1) ] cos (2n+1, 1n3 (15)
nomk2n+1) 4 2
`where x=0 +(1 +i),and x0 =(1
The transformation must now be inverted in order to got
e(13,T) • If equation (10) is integrated with respect to 13
R.: di, °dre x g2200 • B
By using (5) and(14)
0(01g) + (11) = 0(5,T) +i 02(e•T) •
from which it immediately follows that
X(O•g) = e(o,T) e2(a,1) .
This equation, coupled with eqn. (11)
g ( 1+ ae ) ci,r 9 0
will be used to obtain 0(0,T) .
(16)
(17)
(18)
89
Solution of Equations (1,(17) and (18).
In order to solve the system of equations (15),(17) and
(18), it is necessary to resort to numerical procedures. g
is taken to be the polynomial of the form
g = T f Ai eXP(-Yi T) (19)
where each pair of constants Ai and yi are computed from
successive approximations. For any given pair of p and T
we first integrate under the curve e(p,T) by setting g=r
Then
g—r+alecPre =
+ Ai AXIA.-Y1 T) • (20)
Clearly, in the first approximation
Al exp(-yi T) = aledT°
(21)
Taking the natural logarithm of both sides and differentiating
with respect to T we get
9l (T) I = T Yi (22)
0 dT°
Substituting this result in (21) and solving for Al
Al = a [exp(Nti T)] J e .
0 (23)
The second pair is obtained similarly;g now is given by
g = T + exp(-YiT) .
A new value for the integral 0 dT'
Ai exp(-yiT) +12 exp(-Y2T)= o j 0 aTo 0
where Ai and yi are known. Proceeding as before r
Y2 =-4Yi - 0(T) / jor e dT° and
Az = exP('111T) + 1: dro exp(v2 T)
It can easily be shown that the n-th pair of coefficients
is given by -1 r n E 0(Y) 1
Tn L k=1 Tk 0 die J and
(24)
ng[a j 0 de Ak exp(-yk T)]exp(yni). 0
The iteration was stopped when the last two calculated
values of 0(T) were less than 10-5 apart. The calculation
was repeated for each given pair of 0 and T A 16 point
Gauss-Legendre quadrature was used for performing the inte-
grations.
4. The Finite Difference Approximation
For the purpose of comparison, the problem of equations
(4) - (6) was solved using the finite difference technique
(Ref.4); as only a general outline of the way to deal with
the problem was found in the literature, the essentials of
the solution will be given here.
90
can now be calculated
An (25)
If we divide the one-dimensional space-time plane into
a grid whore the space variable incrment is h, and time in-
crement k, the first derivative of u [i(xlt) is
apprcozimated by
6u & [ u(x4h/2) u(x.h/2)
Then
o(x 6u)& t [A (x+h/2)u• (xih/2)-k(x.h/ 2)u• (x-h/2)] (2?)
where utia given by (26). Denoting the central grid element
by ujo rather than u(x,t) we have
v.(xvu)g-- {1/44(ui4i-ui) - X (u - u )] -2 1 14 i-1 h (28)
and the time derivative of u
au L ( u _ u4 ) 8t • k i.j+1 -0
By defining r= h2 , an= rAn and X= 1+ au,
and using the Crank-Nicholson method (Ref.5),
u. a = -(2r+2) + 2ra
91
• (26)
(29)
(30)
where rar 2 ar 2 AL-2(2r+2)2 44rar7 uifi 4:rui41.4E u3,-1 +rui_1+2bi]. (31)
Here all les are understood to denote the grid point j +1.
In equation (31)
be ui (ui41 -ui) - (urui_i)
(32)
where Ws are understood to denote the grid point j.
Using Gauss-Seidel iterations (Ref.5), the n+ 1°th approx-
imation forixi,141 is given by
n+1_ -(2r+2) +MIT (33) ui 27r
where
.n+1 . ra 2 A = k2r+2)44ra[ -T-= u -Fru u2 2 i+1,n i+1,n+ 2 i-101+1
+ rui_i 0141+2bi]
(34) where bi is given by (32) with the uos denoting the n + loth
approximation. For faster convergence the iterations were
carried out with successive over relaxation (SOR). -14 n n Defining A= ui u. , in SOR
n+i _ n wb A
where the SOR coefficientwb is defined by
wb = 2/(14' 4t)
andµ= l;cosh. (Ref.5).
92
5. Results and Conclusions
Temperature profiles were calculated by the two methods.
Figure 1 shows that fora = .1 agreement is quite good. On
Figure 2 results of the solution of the problem by the varia.
tional technique (Ref.l) are plotted as well as the two
previous methods for or = 1. Assuming that the finite
difference solution is the °° correct one, it will be seen
that for a = 1
a) the variational technique is inaccurate for small times,
and gets progressively better for longer tomes, and that
b) the method resulting from the Kudryashev-Zhemkav trans-
formations shows that better agreement for short times but
.'rapidly deteriorates as,' gets large.
That the error should grow with time is probably in.
dicative of instahlitt7 in the integration procedure, as the
set of equations (15), (17) and (18) is exact.
The results for a = .1 can be taken as justification
for the use of .this method in the analysis of the experiment
presented in earlier chapters,since the corresponding a is
less than .01. Thus in the present form this method is
93
94
15
PICO. COMPAgisom OF SoLuTtoisis FOR cy.::: •1
.4
:2
0 '4 a 1-0 .2 .6
•2 •4 •6 •S
•8
•4
•2
0
95
FICI 2 COMPAMISON1 OF THE THREE METHODS .
applicable to a large number of problems. It also lends
itself to a straightforward evaluation of the errors due
to temperature dependence of the physical properties, by
estimating tho'uppor.Umit of the'intogral
A , =•1t p t cp a
as illustrated in appendix 1.
Computer programs for the above calculations will be
found in appendices 3C and 3D.
96
CHAPTER 6 97
The Harmo.ac Oscillator Model
for Liquids of Spherical and
Chain Molecules
La short account of the ocill model for simple 1LcralcIs,
and its extension to the harmonic oscillator approximation
will be given, followed by the application of these models
to chain molecules. The assumptions involved in each case
will be briefly discussed.
1. The Basic Equations
The Helmholtz free energy, A, of a system is defined
by the equation
A = - kT In
where k is the Boltznsnn constant, T is the absolute
temperature and Z is the partition function of the system,
defined as the sum over all states of the Boltzmann factor.
exp ( Ei/RT), Ei denoting the energy of state i.
In the classical approximation, the translational
partition function can be expressed as an integral
= NION exP[-H42,1,...1244, 1 r - , ...,s0]dp/..4541..dsg
where H, the Hamiltonian is given by (2) N 2mi
H = + 0 ( r ILINT)* .(3)
Pi = m dr vector of molecule i, which has the mass m.
Here and Si is the position
On the assumption that the Pi and ri are independent of
each other, the two sets of integrations, over the.?.j and r-
oan be performed separately and the partition function wi'llten
as a product of the two. The set of integrations over the
momenta yields: 3N/2 [32_111ell
h2
The integral over the positions of the molecules is called
the classical configurational integral defined as, 1 .
ci(N.Tonr.T exp(— uNT) 4E1 .... 41:6 (4)
where U is the total configurational energy, (potential
energy) of the system. If one assumes this energy to be
pairwise additive, then
U = u(Rii), (5) i,.j
where Rij, denotes the intermolecular separation, and u(Eij )
the intermolecular potential. The translational partition
function for thQ system is then wiitten as,
z =L h2 1 Q(N.T.V) .
F27,,,ilert-o/4 (6)
2. The Equations of the Cell Lattice Model
For dense systems, composed of molecules with attractive
forces, the potential energy is approximated by N
U = Ua + L [0(ri) e(0)i (7) i=1
93
99
under the following assumptions;
a) That the volume of the system be subdivided into cells
of equi-.1 volume,
b) that each of these cells contain one molecule, and
o) that each molecule move in its cell independently
of molecules in neighbouring cells. Ref (la).
In equation (7), Bois the potential energy of the
system when ail molecules are positioned in the centers
of their respective cells, ri in the displacement of
molecule i from the center of its cell and x(ri) x(0)
is the potential energy change involved in this displacement.
Equation (4) then takes the form 1
Q(NiT•10 = NI exp OAT) X (8) •
where
7
exp [- t- (r) - '(0)3 /kT) dv (9)
the integration being performed over the volume of each cell.
3, The Smearing Approximation
In order to calculate the mean displacement energy,
given by equation (9), (Ref (2) and (3))we take a molecule.
A4fixed while the second, B. is allowed to move about a
sphere of radius r. The center of the sphere is a distanc% 4(ip
from molecule A. The distance between molecule A and molecule
molecule B is given by
100
r 2 d LeFr 2 -2 ar cos OP
where 9 i:, angl p20. 3r:f.', The avorago mutual
potential over the surf ace. of -one sphere can be written as 2n r
e.Q.,51mea...ay oo
f2n Lo sin 0 dO dO
If e(r) is taken as the Lennard-Jones 12-6 potential
e (r) =1(-17-2) —2(-°) 12 6
where e is the potential minimum, and 1.0 the intermolecular
separation corresponding to that minimum, and if we donote the
nearest number of neighbours by z, the average potential
per cell is thon given by:
4 rv°12 ;(r) = 4ze 1.1i-3] rl(y)+1] 1; J [m(Y)+1]) (12)
where
vc) r
= 0 and a 12V .Y - r2 •a
1(y) = (1412Y425.2 Y2+ 12 Y3+.Y1-4 ) (1-y) 1° 1
(13)
m(y) = (14y)(1-y)-4 - 1 . (14)
Using equation (12) to derive 'e.(0), we get
r.„ e(r) — e (0) -
0 z 1(Y) — [irrf 2 m(y)i ' (15)
4. 1s....._11T:ElamlatallEalllaiall:421:0)dinatio_______
By expanding equation (15) as a Taylor series about
the center of the cell, in terms of y, we obtain an
(r) (10)
101
expression for small oscillations of the molecule
F3r)-7(0)=7"eq0)--3 4"0(0)+ . • • • by trunc.---,,;_,11,;.; relf; ; S113 i afte-t tr.31.7), the
harmonic approximation to tae cell potential acting on a
molecule which performs small oscillations about the center of
its lattice cell. Ref. (4a).
(r)-; (0)= ze [ 22n4 - 10 12 (16)
Thus, the configurational integral can be expressed in the
harmonic oscillator approximation as in equation (8), with
X r , [22 101-4j2 [V dv • (17) J v
By making use of the expression for the restoring force constant for small vibrations, it=-4n2v 2n, the neon frequency of
oscillation can be given by
1 2z e v° 4 v° 2 v = , r 22 (-3r ) -10 (7; ) (18)
2n f; a' L
5. Discussion of the L-J-D Theory:
A. Kirkwood's Treatment . (Ref.5) In an attempt to establish a firm theoretical basis for cell
lattice theories, Kirkwood shows how the assumptions mentioned in paragraph two of this chapter, arise in the mathematical
treatment. The classical configurational integral is written as
Q=S. • • . at,, (-L/kT) dv1 . . .dv (19)
102
where v is the total volume. By assuming N lattices each of
volume A, and expressing tho integrals over V of each molecule
as a sun: .pf 073" the toalvidual cells, (assumption (a)
in paragraph 2.1 C4 ca.:1 ilewl-A.1
Id N 611 (4= E . . . .E . exp(-UAT)dvi . . . dvg. (20) lid 1N=1
The NN integrals of equation (20) can also be written in terms of
integrals 21/(41 mN) where the m are the number of
molecules occupying each cell i:
Q = E Ng
ZN
1. . . . mN) (21) N mi.
a941 147:° (Ins!)
. . m
s=1
Here there is one integral ZN corresponding to the case of single
occupancy of each cell, Z(1'1' 1). we can now define
the parameter a by the relation
aN
= z(mi • • • mN)
mi. . li(ms!) z (1 . . . . 1)
and .,N
Z(1'I
1) N: (23)
At high densities where the single occupancy assumption is
reasonable, a approaches unity, whore as, as the density tends
to zero a will tend to the value e. The entropy calculated
on t'tie single occupa::.-.7 assumption is low as a is assumed to be
equal to unity. Tho fact that more of the
(22)
rat N
103
total volume than javet one cell is available to each molecule,
and hence that a lies between unity Inde, gives rise to the
concept of communal entopy.
Kirkwood continues his treatment by considering only the
single occupancy integral. In order to write the free energy .
esplicitlk the thitd major assumption is introduced: the
relative probability density in configuration space is written
in the form of the product of probability densities of each...
cell, assumed,to be independent of 444i-other.
PN m(rs) (24) s=1
where. rs , the displacement of molecule s from the origin in
its cell .
The subsequent minimisation of the free energy by Kirkwood,
seems to yield lower energies than those of the L-J-D theory.
However there is no"theoretical justification for expecting the
lowest calculated free energy to be closest to the real value.
"It is much nearer to the truth to regard the variational theory
as justified insofar as it approximates the L-J-D theory"
(Ref. ib)
B. Barker's Critique
i) The smearing approximation: For rigid spheres, free
volumes calculated by using the smearing approximation are about
thirty per cent lower than those calculated by detailed analysis
k
of the volume distribution. The pressures calculated by this
approximation are not very different from those calculated from
the "correct' free volumes, but the entropy is considerably lower.
Similar behaviour is observed for potentials with attractive
forces, the error passing through a maximum in the vicinity of
the critical density. As the density approaches the triple point,
and further increases to that of the solid, the error tends to
zero. (Ref 10).
ii) Correlation effects.
Calculations assuming that only first neighbour motion is
significant has yielded good results about the critical density,
and this result should be valid at higher densities as
Most of the error due to assumption (c) of paragraph (2) can be
accounted for by taking into account binary and ternary correlations.
(Ref 1d).
iii) Multiple occupancy of cells: This remains as an
essentially unsolved problem: Attempts have been made to modify
the simple occupancy configurational integral, in order to take
multiple occupancy in to account, such as:
Q (tti1 (uin )1(i . . . . i)
(25)
where multiplied by a factor <Di given by eqn.(25)
for each cell that is occupied by i molecules; various ways to
calculate the W1 3 have also been put forward, Refs. (6),(7),(8).
These models however have so far failed to take into account
105
the altered correlations that arise from having multiple
occupancy. (Ref 10)
iv) The Harmonic Approximation. The approach is similar
to the. Einstein model of the solid and is a justifiable
approximation only at high densities. The error arising from
the assumption that the-molecule vibrates in a cell where,, all
neighbouring molecules are fixed* can be dealt with by taking into
account short range correlations, as in paragraph (ii).
comprehensive treatment of these correlations will be found in
Ref (1), Chapter 6. The second major departure of this model from reality is
the assumed constancy of the vibrational frequencies throughout
the system. Clearly one expects to:observe - a whole spectrum of
frequencies; the Debye model is relevant for the analogous problem
in the solid. •
6. The Cell Lattice Model for Pure Polymer Solutions. (Ref 4b)
This treatment consists of a cell model approach applied
to long chain molecules, for the calculation of the configurational
partition function in a manner that is essentially independent of
chain length. The liquid is charachterised by three parameters,e
the attractive energy minimum, r° the intersegat separation
corresponding to the energy minimum and the 30, external degrees
of freedom. The latter are independent of valency forces:
intrtmolecUlar frequencies are at least one order of magnitude
larger than the external frequencies, and the influence of external
106 '
factors on these frequencies (and there-by the internal degrees
of freedom) need not be considered in the first approximation..
The 3c external degrees of freedom are determined altplrally
through a corresponding state treatment of a homologous series,
in this case normal alkanes (Ref.4c). Here only odd numbered
chains are considered since from x-ray data, the volume of a
CH2 -.CH2 segment is known to have about the same volume as the
monomer of the series, CH4, 'Thus the segment number R is defined by
R = 2(n+1)
where n is the number of carbon atoms in the chain.
The major assumptions involved in the model are the following:
The chain molecule is treated as a set of point centers, each of
which moves in a sphericallymmabtrie force field. The potential
energy between two point centers of different r-mers (chains) is
taken as a two parameter law
e(r) = e m(r.)
where r is the point center separation, and p is most
commonly taken as the Lennard-Jones 12-6 potential.
The criterion for the existence of lattice is that the mean
distance between point centers be equal, whether the point centers
belong to the same r-mar or not; i.e. that
a= d 6 r e
where °aP is the mean distance between two neighbouring chain
segments, belonging to two different chains, and d is the distance
between two successive elements of the same chain. At absolute
zero, a = r and the r--marsaro perfectly ordered on a regular
107
lattice. As the temperature is allowed to increase, the lattice
will be distorted by the expansion of the liquid, and the model
will progressively cease to represent the liquid structure.. The
treatment assumes that the distortions can be ignored if the
volume expansion is less than a few per cent.., Then the volume per
segment is
v = L; 3 y a3 (26a)
whore for an f.c.c. lattice Y=tri , and the reduced volume per
molecule is given by
v = 14 [A )3 33 y 1.0 ( 26b )
It then follows that, assuming the Lennard-Jones 12-6 potential
between two point centers of different chains, in the smearing
approximation yields an expression analogous to equation (15):
i(r) - Z(0) = r 1(y) (f)2 02(y)
whore only the external number of contacts St is different R'
from the analogous expression for the monomer. Hero z is the
coordination number of tho f.c.c. lattice as before and (7q /R),
defined as:
R = 7,2 +(2/11). (28)
is seen to be a weak function of chain length (Rei 4d).
In the harmonic approximation, the mean frequency of oscillation
can be derived andlezously:
v = 217c_ n [ 2( ) [ 22(r)4 10 6r)2 ] ,*-9)
(27)
108
7. The Heat Capacity
The partition function for this model can now be given
by
Z = exp (oallo /kT) xNc/R (30)
For spherical, harmonic oscillators, there exists three
translational ( kinetic energy ) and three configurational
(potential energy ) degrees of freedom. The translational
degrees of freedom remain unchanged for R-mer segments;
however,the configurational degrees of freedom have to be
modified to take into account the additional limitations
imposed upon the segment by the intramolecular contacts.
The surface around a segment is only partly free for
intermoledular interactions, the remaining part tieing
blocked by the adjacent segments in the same molecule
(Ref. 9). Hence the number of configurational degrees of
freedom per segment is 3c/R ; the coefficient 3 is absorbed
into x since the latter is a volume integration
hence the exponent Nc/R in eqn. (29).
The derivation of the heat capacity at constant volume
from eqn. (29) is straightforward and yields
Cv = 1 k(1+ c/R ) • 2 (31)
109
CHAPTER 7
The Harmonic Oscillator Model
of Thermal Conductivity
In this chapter the theory of Horrocks and McLaughlin
will be briefly discussed and then extended to pure liquids composed
of R-mer molecules. Results of calculations will be presented and
compared with experimental data for normal allanes.
1. The Harmonic Oscillator Model of Thermal. Conductivities of Simple Liquids (Ref la)
Heat transfer down a temperature gradient occurs by two
molecular mechanisms: a) vibrational, b) convective.
a) Vibrational mechanism: The rate of heat flow can be written as
-2n. . vi ALI (1) dt. dx
where n = number of molecules per unit area of the liquid
quasi lattice.
v = mean vibrational frequency of the molecule given by
equation (18) of the previous chapter.
P = the probability that heat transfer occurs when two
vibrating molecules collide.
1 du = the energy difference between successive layers dx of the liquid quasi lattice.
The expression is multiplied by 2 since the molecule crosses
a plane perpendicular to the temperature gradient twice for each
complete vibration. Using
dU = dU dT c d_r six dTdx vdx
and the one dimensional Eourrier equation, we get
110
Avib = 2nPvl Cv (2)
where Cv is taken as 3k. Assuming the virtual absence
of holes n 1/a2 ; also 1 = /ra/2 and P= 1 . Hence
Avib = a `'.17 v • (3)
b) Convective Contribution.- In the absence of a temperature
gradient, the frequency of movement J of molecules from
one adjaeent layer in the liquid. to the next is given by
nh 1 J= exp(-e0/ kT) 1. vf
where of is the free volume of the liquid, eo the energy
barrier to be overcome for molecular convection, and nh
is tho ratio of the number of holes to the total number of
molecules. Then
Xconv. = 21.01 frii •
Without going into further detail it can be said that
X Xvib Xcony '
and that for simple liquids, up to their boiling points. Xvib
is by far the dominant term in equation (5), and that the .
convective term can be dropped as a good approximation (Ref. lb).
Thus
x = 1-11 cv v (6) a
2. Application to Chain Molecules
As intercellular convection for individual chain segments
is less likely than for spherical molecules, the assumption that
° X Avlb
is retained. Thus as before
A = Cy v a (7)
where a is the length of a side of the cal, confining a chain
segment, Cv is the vibrational specific heat of the same, and
v the mean vibrational frequency as defined by eqn. (29)
of the previous chapter. Using equation (28), (29) and (31),
the thermal conductivity of liquids composed of chain molecules
can be 'written as
X 2 1'2 (1 +i) [(2,--2+ 12) e[22 (1174)4 -10 (v)2
3. Comparison with Experiment
The above extension of the theory of Horrocks and
McLaughlin to pure R-mors was compared with experimental data on
normal alkanes. The values of ro and € used throughout the
homologous sories are those of tho monomer, mothane, (Ref. 2)
as was indicated in paragraph 6 of the previous chapter. Density
versus temperature data were obtained from Ref. 3, and the
thermal conductivity data
112
for the same temperature intervals from Ref. 4 in the form
of correlations of existing data. Calculations were executed
on a computer; the relevant program for these calculations
will be found in appendix 3.
A summary of the calculated results is given in
Table 1 , along with the experimental data. It win be
seen that the calculated and experimental slopes of the
/. vs. T curves are in good agreement, certainly within
experimental error, but that the absolute values differ
by an amount which does not seem to change significantly
over the homologous series.
TCarbonTTemR Rangel Number 'C Imw/cml
5 -80t0+40 .649
7 -120to+20 .633
9 -40to+120 .591
11 -30to+130 .633
13 +40to+200 .697
15 60to220 .734
17 60to220 r .758
19 60to220 1 .771
!
SLOPE(DATA31 cm/ eita 2
SLOPE(DATA) mw/ cmcli2 /'DIFF ' A
.0033 .0037 9.5 -
.0039 .0041 3.2 -.016
.0023 .0028 16.5 -,042
.0022 .0026 13 ! +.042
.0023 .0025 6.5 +.064
.0022 .0023 3.9 +.037
.0020 .0023 9.7 +.024
.0020 .0022 8.2 +.013
TABLE I. A is the difference between X and X averaged exp calc•
over the temperature interval.
113
It has been shown by Harrocks and McLaughlin (Ref 5) that
the principal factor controlling the temperature dependence of
thermal conductivity is the coefficient of thermal expansion.
This remains unchanged for chain molecules, where the equation
X = a • Cv v
again leads to
(9)
the expressions for v and v being modified as in section 6
of the previous chapter. That this predicitin is a good one is
reflected in table 1. The results indicate however that a second
term, which would be additive and of the magnitude of about
0.65 x 10'3 mw/cmoK is missing. In the absence of further evidenoe
two reasons may be put forward as contributing to thiS second
term.
a) Heat transfer down the chain: Relative independence of
chain length could be expected as the average length of chain
parallel to the path of heat flow need not increase with the number
of carbons in the chain.
b) Degrees of freedom associated with the hydrogen atoms
attached to the carbons: As the average number of hydrogen atoms
per segrent can be taken as constant, this contribution would be
independent of chain length. Further as the heat capacity arising
from this vibrational contribution is expected to be insensitive to
v X LdTJp L 3 d ln v
the tomperatwo changas, this term would be independent of
temperature as well.
114
115
CHAPTER 8
Trans•ort Coefficients of Pure Hard Shore Fluids al•••••••••••.....
1. Introduction
In this chapter the dorivations leading up to the thermal
conductivity and viscosity of pure hard sphere fluids for low pressures
and dense gases will be summarised. These formulae will be used in
the following chapter in the analysis of dense mixed fluid transport
coefficients. As the latter have been derived only for the case
of hard elastic spherical molecules, no other intermolecular
interaction potential will be considered.
While the thermal conductivity coefficient is of primary concern
here, the viscosity coefficient has also been considered, as the
two derivations are very similar. Also, because thermal conductivity
data for binary liquid mixtures of spherical molecules is lacking,
corresponding viscosity data has been considered for comparing the
model with experiment.
A word on notation should be added. Vector quantities in this
and the next chapter will be written with a bar under the letter,
and tensor quantitites with two bars. For the stress tensor, the
Chapman and Cowling notation has been retained:
A ZO .
116
2. The Heat Flux Vector and the Pressure Tensor
It can be shown (Ref la) that if X is any molecular property
which is a function of molecular velocity, the value of X averaged
over all the molecules within a small volume element dr, during
a short time interval dt, is 1
=n1Xf(c, r, t) dc (1)
where n is the number density in the defined region, c is the
molecular velocity vector and f (c, r, t) is the velocity distribution
function; here f(c, r, t) do dr defines the probable number of
molecules with velocity in the interval c to c + de in a region of
space bounded by the volume element dr. In equation (1), the
integration is carried out over the whole of velocity space.
If, in particular, the relevant molecular property is heat, the
heat flux vector can be written as (Ref lb).
q = m$ C2 C f dc
here m = mass of the molecule
C = - 20 denoting the mean mass velocity of
the gai which can be obtained from equation (1), and C = the
magnitude of C.
Clearly, if f is known, q can be derived explicitly . and then
combined with Fourrier's Law of heat conduction.
q = - X dT dr
to yield the thermal conductivity.
For the case where X denotes molecular momentum mc, the pressure
It/
tensor can be written as
E =p J CC f dc (3)
where p = n The pressure tensor is the sum of the
hydrostatic pressure and a second term composed of the stress
tensor multiplied by the coefficient of viscosity, vs.
Thermal Conductivity and Viscosity at Low Pressures
Boltzmann's Equation for a non-uniform gas is
+ c • + F at — Or — 8c LatJcoll
Where in addition to previously defined quantities, mF is
the external force acting on the particle, and (Wat)coll
is the time rate of change of f due to collisions. Eqn. (4)
can be written as
(
= 0 5)
where 0 operates on f . It is assumed that
a) f= + f2 +.... where f° turns out to be the
MaxwbIlian velocity distribution function ( i.e. that for a
uniform gas) and fi (i>0) are successive correction terms,
and that
b) the operator 9 can be broken down such that
0(f) = o°(e) +01(1.1)+ 02(f2) + (6)
Where the Di(fi) satisfy the separate equations
Do(fo) = 0 ; Etl(fo,f1) = 0 ; p2(fo,fi,-2% r ) = 0 ,etc. The quantities Or can now be defined (REf. ic) as • fr = fo Ar
(4)
(7)
such that the rth correction term to the average value of
property X now becomes
1 r r TEr =1 j Xfr do = n j Xf° Or do,
Where the Maxwillian velocity distribution function f° is
eXplieitly given by:
fo = ni m oxp( metakT)'.
In this and the next chapter, we will work with only the first
and second terms of equation (6) as, duo to the increasing
complexity of the successive approximations,, the formula: 'or
mixtures, with which we are ultimatalY concerned, have not been
developed beyond 01. Thus, we will take
1 = 10 + X1 = xedc + nxf°01dc (10)
Since the first integral makes no contribution to the heat flux
vector. .
= al = ZmSC2 cfo 01 dc (11)
The solution of the Boltzmann equation, leads to an expression
for Al, which can then be evaluated for hard spheres. The
derived dilute gas thermal conductivity coefficient for a fluid
of hard elastic spheres, X0 , can then be written (Ref 1d) as:
25 1 ( k5T) 2 o = atr:2 (12)
113
(8)
(9)
119
where .k is the Boltzmann constant and the molecular diameter.
The viscosity is obtained in a similar manner from the difference
between the pressure tgnsor and the hydrostatic pressure: S.L. = 1 0:- 16 (12)
4. Thermal Conductivity and Viscosity for Dense Gases. In the above discussion, the transport coefficients have
been derived by assuming that both momentum and energy transfer
take place by the motion of molecules, between collisions, through
the available volume. As the density is increased and the mean
fru° path becomes comparable in magnitude to the molecular
dilny-tere collisional transfer takes on increasing importance.
The collision frequency is increased by a factor g(cO, the contact
radial distribution function. The details of g(a) are best
considered outside the mainstream of the discussion of the
transport coefficients; it will be used implicitly and defined
in a later section.
It has been shown (Ref le) that the velocity distribution function
for a non-uniform dense gas can be solved for, in a manner analoa.ous
to the dilute gas; f° again turns out to be the Maxwellian
distribution function, and makes no contribution to the heat flux
vector.'. Three contributions to the latter arise (Ref 2), (Ref lf):
1) from heat tt ansfor by molecular motion between
collisions, i.e. the kinetic contribution
1 P C-17 = 1 (1 + 12b* g) g 5
X k =
x° (1 + 12 b* g)
.5 (14)
120
where b* = (c/6) na3.
2) from that part of collisional transfor which can be looked
at as taking place with a locally Maxwilliam velocity distribution:
-Cy (7) a T = b* b* g 2,511
kmalor =5.1.1 b*2 g X 0 257a
(15)
where Cy = (3k/2m) and.; is tho bulk viscosity given by
; = (4/9) g n2 (74 (nmkT)i;
3) and finally from the distortion of the locailyMaxwellian
velocity distribution function.
6 b* gP C2 C - 12 b* X 0 (1 + 12 b* g) dT -- 3 5 5 dr
Dist. = 12 b* X 0 (1+12 g). 5 5
In the first approximation then, the thermal conductivity of
a dense fluid of hard spheres is given by
X=4X0 b* 1E 3
+ 6 + + 22_ ) b* g -4D7 25 25n
(16)
Cloarly,A 0 is the thermal conductivity of tho dilute gas at the
same temperature.
The viscosity of tho dense gas is mado up of contributions of
the same origin (Ref 2):
1) Tho stress tensor arising from molecular motion botweon
collisions is 0 MMOI.WWW1041W VIII1101.100.010
121
p C C $.4o 8 * a ( g) 2— co 5 ar
which gives rise to
=( 1 + b# g) . 5
2) The stress tensor arising from the locally Maxwellian
velocity distribution function is 0
•••••••Masalam.
2 ..5 g n2 cr4 (itmler)i ar co
which leads to
.21_6 *2 . li'Maxw 25% g Ib
3) Finally the stress tensor arising from the locally
Maxwell/an velocity distribution function
3 5 b p 20CC -- gives rise to
ftist. = 5 b* g wk •
The first approximation to the viscosity the is:
0 = 4 06 b* 1+37 152 +4(1k + 483n ) b*g ]
(Ref. ig)
(17)
(18)
(19)
(20)
122
5 The uations of Lonyuet-Higgins and Pea's) 7Refr3
It has been shown that the purely collisional contribution to
the transport coefficients can be derived via the assumptions:
a) that the spatial pair distribution function depends only on the
temporlture Arnsitz* and not on the temperature gradient or rate of strain,
and b) that the velocity distribution function is Maxwellian with
a mean equal to the local hydrodynamic velocity, and a spread
determined by the local temperature. The resulting equations, in
our notation can be written as
A = 5.2 0 b*2 25 n
and
µ =z6§µ o b*2 g• 25n
These equations are identical with the locally Maxwellian contributions
to the expressions derived through the Chapman-Enskog theory,
equations (15) and (18) respectively. (Ref 2).
As indicated by Dahler, the corrections to the collisional
terms arising from the distortion of the velocity distribution
function are not negligible; the contribution of the distortion term
to the thermal conductivity is quoted at over 50% of the collisional
term, and the corresponding correction to the viscosity is
reported to be above 20% and increasing with density. (Ref 2).
6. The Contact Radial Distribution Function
The radial distribution function g(r) is defined as
g (21)
g(r) n(2) (rt. z2) / n2 (23)
where n is the number density and o(2) z:) dri dry is the
probability that molecule 1 is in volume element dri about the
point ri and that molecule 2 is in tho volume element dr2 at r2
imultaneously. Are now define the correlation function
h(r) g (r) - 1 . Tho limiting value of g(r-) is expl-u(r)/kT1 where u(r) is the .
4intermolectlarpotential. Hence es g(i) tends t :.7.11A) viluy
-for low•denaitios, h(r) tends to zero. The latter is °a measure of the total influence of molecule 1 on
another, molecule 2, at a-distance rd. (Ref 4).
h(t) can be split into two terms (Ref S):
h(ri.2) =4,C(r12) + n C(r13) h(r23) 41/3
where the first term on the tight hand side is a direct correlation
function representing short range interactions, and the second is
the long range interactions propagated from molecule 1 to molecule
3, which in turn exerts its total influence on 2. Defining two
further functions
F(r) =[exp ( -U(r)/kT)) -1 (25)
and
y(r) = g(r) exp [U(r)/kT] (26)
and making the Percus-Yovick approximation (Ref 6) that
C(r) = F(r) y(r), (27)
we can derive the equation ,
123
(21 )
124
Y12 = 1 nIF ` 13 Y13 h23 3 (28)
from equations (24) - (27).
This equation has been solved, (Ref 7), for the hard sphere
potential, to obtain y12 at contact, i.e. when the intermolecular
senaration isa ,(the molecular diameter) where g(a) = y(cr). The
result is:
g(a) = 71:1X)2 (29)
As before b* = na3. This then is the expression for the
contact radial distribution function for hard spheres, in the Perms-
Yactek approximation, which can be used in the evaluation of the
transport coefficients.
The equation of state of a denso fluid can be derived in two
ways, leading to the2 pressure equation, through the virial theorem,
r 1,1,
P= nkT - 6 Jr du r) g(r) dr (30) dr
and the compressibility equation
kTon = 1 + n.1 [ dr) - 1 ] dr (31) ap
derived from fluctuation theory. In the Percus.devick approximation,
these two equations( can be reduced, by using equation (29), to
P = nkT E (1 + 2b* + 3b*2) / (1-b*)2
and
P = nkT [ (1 + b* + b*2) / (1-b*)3
respectively. (Ref 7).
(32)
(33)
125
CHAPTER 9
Tranwr Coefficients for Dense Binarj
1111 1P1191.72-----121"1-.11
1. Introduction
In this chapter, the equations for the transport coefficients of
dense hard sphere mixtures will be given, and combined with the
corresponding radial distribution functions, (Ref 10 derived
through the Percus-Yevick approximation. In order to compare
with experiment these equations will be reduced to ratios: the
mixture transport coefficient divided by that for pure species 1.
The equations will also be factorized into the purely kinetic, dis-
tortional, and locally Maxwellian collisional terms, and the
Enskog minimum will be shown to exist. Comparison will be made
with experimental data.
2. Transport Coefficients for a Binary Dilute Gas Mixture TRef la)
The method is analogous to that for pure systems. The
Boltzmann's equation for the first gas is
aft 01 a f + F1 afl r eefi at — a r- L at Jcoll
and a similar equation can be written for the second gas, by changing
the subscripts to 2. For the non-equilibrium case, the equations for
f1
and f2 are solved, as before, by a method of successive
approximations. Again the first approximation is a Maxwellian
function, and again the f's can be written in the second approximation
=
f = f0 2 2
(2)
Here, however, the Drs contain cross terms of the properties and
velocities of the taro species,
The heat flux vector is now written as,
= i mif 1 + i m 2J f 2C 22C2 d.22 (3)
where ail quantities have been defined in the previous chapter.
The Xaxwellian part of the distribution function makes no contribution
to the heat flux vector. Of tho terms arising from the integration
over the )7 01 , the ono representing the heat flow duo to the i i
126
(1)
127
temperature gradient is simply
q = dT mixo 510
where X ,o is the binary gas mix'eure thermal conductivity for low
pressures. The expression is cumbersome and as it will not bo made
direct use of, will not be reproduced here (Ref ib).
The coefficient of viscosity is derived along similar lines.
(Ref lc). We need only deal with the first correction term, as the
baxwellian part gives rise to a contribution to the pressure tensor
that reduces to the hydrostatic pressure.
Than,
041 (i)
- _ mlj fiP121qicisi 171 .. 1°242222 c1.9.2 • (5)
which is equivalent to
(1) = 111111 9121 ne2224
Solving the Boltzmann equations to obtain the rhs of equation (5)
leads to the viscosity coefficient.
These methods for the derivation of the transport coefficients of
mixed fluids have been extended by H. H. Thorne to the case of denro
binary mixtures, (Ref 1d). In the first approximation, the thermal
conductivity is found to bo 2
mix 8 12 (AXXX 2+ lixxxYx cxYx Dx (6) where
X = 1 + 2 7m1613g1/5 81iM1l12Y12g12(.5 (7)
(8) Yx. = 1 + 2nn2c1;3g2/5 + BuMp2n10123gi2(5
141 =LIA/('141312) ; M2 =D1 2/(T7'14112) mo = m1442 a-1-1 )1.1 (a11 a_l_i -a21_1)-1 mi x2
(n1m2)I
Ax =
Bx (a11 a-1-1 -a21-1)-1
cx = 2. (all a_i_i -ai_i) -1 7i
(12)
= (2/3) n2 (nOT)1 tx1 gi + 2(8K1vivi xi x2 g1 2 924. 4. (13)
2 m4 m2 g2 "2
where xi and x2 are mole fractions of species 1 and 2 respectively,
ali = a;i + 3
2a £1 ail x2 g12
(iA)
aii = 5k1![ *(64+912) *24.ii1912] /M1E (15)
E (ItTA 4 6142 ; '712 = (c7141/2) (16)
aii m 5kT/2'10 (17)
where 010 is the dilute gas viscosity coefficient for species 1.
a_i_i = a°1_ + 2C2 Z2 alt 1 x1 gi2 -1-1
(18)
where 04_1 corresponds to equation (1s) with species numbers
129
interchanged and 01_1 to equation (17) with pio replaced by 11.20
Finally
ai_i = -27 kT (y12)2/4E0 (19)
and the radial distribution functions g12' gi and g2 will be
defined in the next section.
The first Approximation to the viscosity of a mixed dense
hard sphere fluid is given by (Ref id)
=-51I y, 2+ +CY2 +Di
(20) Unix 2g12 V; µ 11 0 0 -I
where
= 1 4% n1ai3g1/15 + 8n M2n2cr123g12/15
Yp = 1 + 4% n2a23g2(15 + 8% Minia123g12/15
x 2 .-1 = ;O. , bii - bi_i ,v2
B 11= -2 b1-1 (b_1_1 )-1
b b 2 ri C = b11 -1-1 11 1-1
i 2 4 = (4/15) n2 (7;k1)2 4 +2(2MoillyidiXiX2g12Cfli
2 4 ]+m2 x2 g 2a2 (26)
and
(21)
(22)
(23)
(24)
(25)
1 131.1 = by + b„ i 11 2 E12
with bY11 = 51GTrq + 5-20
130
(27)
and b" 11 i - a . Further,
= b:1-1 +2C2x1 £4.2
where
by = 5kT +fil)/E --1 2
and b" = a_0 i_1.
b1-1 = -4%T/3E.
The radial distribution functions givgl and g2 will now be defined.
3 Equilibrium Properties of Dense Hard Sphere Fluids42212)
The methods for the treatment of pure and mixed dense hard
sphere fluids are analogous. The compressibility equation assumes
the form ap
1— i niPii(r) kT ari (31)
and the direct correlation functions C.. for an m component ij
liquid can be written as
[gij(E)-1] = Cii(E) iE nbi[gij(r-y)-1]Cij(y)dy (32)
where gij is the radial distribution function. In order to obtain
the gij's we need another equation relating the two sets of
functions:
(28)
(29)
(30)
131
gij() eXpr-u. (00] -1 = exp[-uij(t)/kT]Cij(0 (33) ij - This is the Percus-Yevick approximation for mixed dense fluids. The
essential implication, as before, is that the direct correlation
function is of the sauG range as the intermolecular interaction
potential.
potential,
0 = Q12
where
= gi L
and
g2
Here,
=
vt =
Thus, assuming a binary mixture and the hard sphere
it can be shown that
:(a2g4ag2)/2 12
2
+ .1 n nag (o1-a2)J (1-g)-2 2 2 .g
na12(a2-a ) (1_0 2
(niai3+ n2c723) = -
17 (x1 + x2 r3 )
ne13/6 ; r = a2/a1 • v= 1/n
(34)
(35)
(36)
(37)
whore n is the number density. Using the expression for the
radial distribution function, the compressibility equation for a
binary mixture can be written as (Ref 5)
c * p v 1 = g 14.g+e) - ....224x2(1-r)2e (38) T (1 -g)3(xl+x2r3) (xitx2r3)2
+x r2 (14-0-Frg (111g75 )
(1-03
•
The pressure equation, the pure analogue of which was given in
132
equation (30) of the previous chapter, can be treated in the same
manner and should yield identical results with equation (38) if
the distribution function were exact. Though this clearly is not
so, results from both expressions are sufficiently close (Ref 5)
for us to work with only one of these expressions. The information
obtained through the compressibility equation will not be substantially
different from that of the pressure equation.
4 Reduction of Equations (6) and (20)
Equations (6) and (20) have been derived for mixtures of hard
sphere fluids. One expects the error due to this simplification to be
reduced if the ratios Xi/Xi and vindx/il , are considered rather than
the absolute values.
The denominators X, and 119 denote the transport coefficients
of pure dense species 1. Thus by (6) and equation (16) of the pr-yious
chapter.
• - Arnix = fX [ fc + B2G` X -
'‘.A
+ Dx 1 (39) Xi
where
2r1 Xx = 1 + 12g1710/5 +6111M2g12 2(14r)39/5 ; co = v P (40)
Yx = 1 + 12 g2=c2r2y/5 4.61y129.2X1(1+r)3T/5, (41)
ffX = 1411p [4g± + 5 + 4(i + ild g 9P ] , (42)
and equation
— = 1 mix U41.
f
(20)of the previous chapter.
F A 7 X 2 + Bo X •X 4Coy 2 Do 1 (49)
133
and
X
B° X
C;
k .11- • • 1 • r i /kg - icii 2 ••
• "12 2
1 142) cif-1/(9 q1-1 )
(i" q -q ) r41 M2 ' -1 11 1-1
[ N.all T2 • xi
2 gl (8-2)2 xix2612(14r)4
A -4-- 2 h (1K)2 rLfa;, 1 2
n
and
= ., )2r4.1-
1-1 4 x2 g12 '14T1 '1%2/
where
.0 = tli),1121t (614 2+ 5% 2) - M 2+ 14
' 4 2 - 5 i 5 i 2 I. 2
can be obtained by interchanging subscripts 1 and 2 in
equations (47) and (48). Firm -1-1Y (3 1_1= -(54/ 1112 Likewise. the =nix/t4 can be obtained by equation (20)
g12 --i.r1;1)2
L k ( g1)4 g12 "2
2
512 25,t
(43)
(44)
(45)
(46 )
(47)
(48)
where
P /(P P -P 2 ') -1-1 -1-/ 11 1-1
P /(P P 2 1-1 -1-1 11
-P1-1
= 8Txigil5 2P142612x2(147)315
t y = 1 + 2 r3g2 + Pcp11 .1.
g,4 1.
kl4r1, 3/e
3Z1 (_+i'2 (1. )1- x2
= -16(11)1 ( 72+37)2
e,2 = 82s2 P /(P P P 2. ) r+1 11 -1-/ 11 1-1
134
(50
(51)
(52)
) (53)
(54)
Do = 2§§ [ x1.2gi+ (12 figi2x1x2(14r)4+ Vgex22] 25n
and
P = p + 8x1 g2 M2 (-L2)2 (6) -1-1 gi2 l+r
where
fou = 40(3. 5
(N,112 A 3 M' 2 (57)
P11 can be obtained by interchanging subscripts 1 and 2 in
equations (56) and (57). Finally 1-1 = (32/3) ( .111":2 At and
, (55)
f = 4, [--i- + 11 + 4(1-1.25 25 + 1-18- ) to g ] • "(58)
V, P 4get)P 5 7: P ,
In both equations (42) and (58) cep is the value of vtiv
for pure species 1. This value has been computed from equation (38)
by setting xj. = 1.
135
In order to evaluate the ratios of equations (39) and (48),
the ces corresponding to each pressure, composition and diameter
ratio must be known, This was done by solving equation (38), by
fixing (P wi/kr), r, and x. The solution of (38) for cp at x = 1
was used to evaluate the contact radial distribution function and
the transport coefficients of pure species 1, in the compressed
state. Calculations ofvt/v and the Xna.21/il and OndA were
made for the fallowing sets of values:
(Pi T) = 1, 2, 4, 8, 20, 301 r 1, and
R (= NA) =4, 1,2, over the Pall composition range. Three
Fortran IV- programmes were written for the execution of these
computations, the texts of which will be found in the appendixes
3F , 3G , and 3H for (71*/v ), XnixAki. and 0 /01
respectively. A sample set of curves, from those computed have
been presented in Fig. 1. for the case R (Pv 1*/KT) = 4 over
the composition range, for four values of a-. Results for both
Xudixi (greater than 1) and 0 4'1 (less than 1) reproduce
the quadratic type dependence on composition that is observed
in simple liquid mixtures.
At this stage it mould be desirable to split equations (39)
and (48) into their respective kinetic and collisional terms. In
connection with this, it is relevant, first, to look at a purely
collisional model of the thermal conductivity of mixtures.
/36
• 5 10 • G 0
X I Fig. i. ComPo%rrioN PGPRAID Mt,JCZ OF A/ Ni AI. 9/ rii
FOR A DUNS E. HARD SPHERE. FL-latC, PAPKWIZE. R zi pi. /KT 2:4
137
5 The C0111.11214LATT0ximation to Thermal
gStrfibigtiXi1X..d-421120=31=(Rer By assuming that the two particle density function can be
taken as that of the fluid in equilibrium and that the velocity
distribution function for each species can be taken as a
Maxwellian function, the heat flow vector due to the temperature
gradient can be written for the case of equal diameters. for the
two species as (Ref 2b)
= - riTE E n AnB , 23,ABIer A nom. A B memB % •
and hence the thermal conductivity of the binary mixture as
_.25/cv E nA nB (2,3,ABILI\ a Alslx k (55)
. • . ig 1 . •
*here 742 P I and SAB mimBNA+ ger :
X in this approximation is purely collisions." since the kinetic mix
term vanishes for a Mixmellian velocity distribution. and clearly
there is no distortion term involved. Equation (57) can be written
as
X 2 mlxi ( 8R A 2 .1 10
= nx2 (iiR)3 + x2 R a (60)
where k10 is the thermal conductivity of pure component 1, for
low pressures.
138
6. The Collisions' and Kinetic Contributions in Equations (39) and9E2).
It can easily be shown that if 107t 1 given in equation (13),
is divided tyr 1/2 • given in equation (15) of the previous
chapter, and al = a2, one obtains equation (60). The origin
of the term suggests that equation (13) arises from . eollisional
heat transfer duo to the locOly Maxwellian velocity distribution.
The obvious step then is to write the locally MSImellian
collisional term for alt
'mix- )2 l. [xi[g1+
[ 8 ( 1%)3 ] ixix2g12(i+r)14+R.442r4 (61)
where vio is the molar volume of pure species 1, v is that of the
mixture, and g is given by oqn. (29) of the previous chapter.
Thou 6-11 Languet-Higgins, Ppple and VS1leau did not extend their
treatment to the viscosity coefficient, analogous expressions can
be written for the viscosity, using (26) of this chapter and eqn. (18) of
the previous chapter. o n
41;11.4 ) j ( X1281+ 16. X1X2g 1 2( l+r)14: (-1.2.4 )14R+414g 2 ) (62) g
and for a1 = a2
mix/
1 = x12 + (NI 2 X1x2 x221/-11- • (63)
139
This analysis can be extended to the ti-,-ee initial terms
of equation (6). The kinetic contributionto the heat flux vector for dense fluid mixtures is defined similarly to that
of mixtures at low pressures, i.e. eqn. (3) which immediately
leads, by definition, to
22 2 - pi CI + x P2 C2 92
The total flow of heat (Ref le) is given by
q IP + (7x - 1) i P1 C127;
+ P 2 CC-2 (TX 1) * 2 C:C2 "X aT
where IX, Yx. and D are given by equations (7), (8) and (13)
respectively. While a rigorous factorization to separate the
kinetic contribution from the distortional one (i,e, the term
arising from the distortion of the locally Maxwollian velocity
distribution) is called for, it would be extremely laborious.
Hence the following method has been used:
Consider the first three terms of equation (6), and assume
the existence of two unknown kinetic contribution terms, U, and
U2, such that, by dropping the subscripts,
AX2 + BXY + ce = xu1 +YU2 (65)
where U1 + U2 =7,
k
and. Xk is defined as tho purely kinetic contribution to the
thermal conductivity of the mixed dense fluid. The form of equation
•
(64)
Imo
(65) necessitates
U = AX
U2 = CY + FR (66)
where A and C are those of equation (65) and G and F
are as yet unknown: Hence
U1X + U2Y = AX
2 +( G + F) XY +CY2 (67)
Clearly B =G+F. The symmetry of equation (64) suggests
that it would be reasonable to assumo G = F = B/2. This
assuTption has been checked by calculating the kinetic;
distortional and locally Maxwellian collisional contributiond
separately and comparing the sum against the unfactored
equation, over the full range given in paragraph 4.
The identical argument applies to the viscosity and
for both transport coefficients the kinetic part, in the
brackets of either ono of equations (6) and (20), has the
form
AX + (B/2) (X + + CY
(68)
and the distortion term
(Z-1) [AX+(B/2)Y] + (Y-1) [CY + (B/2) X]. (69)
Computations, of the various contributions, have been
carried out and the sums v /X 10 v 14') and ( vAr /ploy 1*)
plotted against the dimentionless pressure y = (P/nkT) .1.
Figs (2) and (3) show a set of representative results for
the thermal conductivity and viscosity respectively. These
have been calculated for constant composition Xi = .5, and
r = 1.5, R = 2. On both of these graphs curve 1 gives the
kinetic contribution, curve 2 the sum of the distortional
and locally ha. Tian collisional. and curve 3 gives the
sum of the two curves. These results are similar to those
obtained for a pure substance (Ref 6). As the density is
increased, at the same temperature and constant composition,
the influence of heat transfer through molecular flux
decreases (as molecular convection decrease, for increasing
density) and heat transfer through collisions increases.
Also both graphs show that the transport coefficients of
mixed fluids also go through the Enskog minimum as the
pressure is increased and the collisional contribution takes
over linearly.
In these calculations, (7°7 A10 vi*) and (Pv 10v1*)
have been obtained from programs identical to those of
paragraph 4. of this chapter simply by letting f = viqv
in equations (42) and (58). The splitting of the collisional
and kinetic contributions introduces minor differences, and
hence these programs will not be reproduced here.
141
20
1- KINETIC 2- COLL( 510mAL 5 - ToTAL
15 -1_
/42
10
5
0
I 3 _ Fig. a . THERMAL. CONDUCTIVITY RATIO V.S. Li; RELATIVE
CONTRIBUTIONS OF THE 14.10.1E.T= AWE, COLLISIONAL MECHANISMS
/43'
Ftq viscosrrY RATIO V.5 JRELATIVE CONTRlt3uTioNS OF
(GIPS UT' I C. Akio COL.LISIONAL. INC.-.CHAN rvi
7 Comparison witheriment
Equations (6) and (20) have been derived for hard
sphere fluids, and while one could obtain values of the
transport coefficients, directly, by combining these
equations with (34) and (35), one would not expect
good agreement with experimental data. The effect of
intermolecular forces should be reduced however if ratios of
the transport coefficients wore taken as in equations (39)
and (49). Clearly the most suitable systems for comparison
are mixtures of simple liquified gases. No data on the
thermal conductivities of the latter were found and hence
data on the system carbon tetrachloride + benzene (Ref 7)
was used for comparison with theoretical calculations of
both ratios, while data on the viscosity of the system
argon + methane (Ref 8) was used for comparison with theory.
i) The system carbon tetrachloride + benzene. Density
data on the system (Ref 9) at 30°C and the molar volume of
carbon tetrachloride at absolute zero (Ref 10) were used for
the calculation of ( vi*/v ) over the composition range.
NA v* = 55.24 cc/mole
NAv = :-2-0.2)finax
144
where 7 denotes the mole fraction, gnix density of the
mixture and NA Avogadro7s constant. r was taken as
r = vo 2
3"
v° 1
where ve denotes the molar volume of pure species i
at the given temperature, and R = .5077 from the molecular
weights. The values of the ratios of eqn.s (39) and (49)
were computed using this information. Also, similar
calculations were executed on equations (60) through (63)
and the results plotted in fig. 4, along with experimental
data.
For the thermal conductivity, all three theoretical
curves conform to the general behaviour of the experimental
data. Agreement between calculated and experimental values
is within 10%, and gets even bettor for equation (39). A
similar situation is observed for the viscosity ratios.
Equations (62) and (49) give practically identical results.
ii) The system argon + methane.
Calculated results were compared with data (Ref 8) taken
at 90.91°K. Density data for the puie components (Ref 11)
and excess volume data (Ref 12) were used to compute the
molar volumes over the composition range, r was taken as
145
-".
."5 r----------------------r
,- Z
,. I
'-0
·s
, .. DATA (R.EtF 7) Z· E9 GO
3 .. &:.9 61
4-" Eo, "9
\- DATA (Re.f" 7)
z.. E9 G; '3 - eCJ 62-
4 -E'1 4 ')
-6 1~.O-------·8~·--------GL--------~4-------·~Z-------10 Xl
/46
Ii FIG- 4. CAl..CUL-ATED AtoJD Ei<PEJGUVlE{rAI- \ll'S C.OSITHi.'S> ANI> THER.MAL
CON pUC."vtTIES FO~ THE. S'IS'E.M cAP .. BoON T£:.T[aAC.HL-OJ;!tOe.(') + R,!"':~~I""r;;,r'>fr:, ,.,'\ .f'>"l. i7,nc~_O!"""'.~~ _____________ ....J
1 • 0
•
I— DATA (REF 8) 2- Eci, G•5 3- Eci, 62
Eq, 49 3,4
I I I • • 4
S. CALCULATED AND EXPEIZimENTAL VISCOSITIES P012 THE. SYSTEM AegoN(s) t METHANE (Z)
AT 90•9I°K
/ 47
• a
=1.067
1)4c
and v*1 from the no? ^Y vol mq cf solid argon (Ref 13) at
0°K. Fig. (5) shows that agreement of experiment with
any of the three theoretical curves is not as good as that
of the previous system.
149
Thermal Diffusion in Dense Hard Sphere Fluids
1. Introduction
•••• •
A treatment that of the previous trap chapters,
has been applied to thermal diffusion. Using the Lebowitz
radial distribution functions, theoretical calculations
have been executed with both the Chapman-Enakog and the
Lonsuet.Higgins, Pople and Valleau theories. Results
from these two theories were compared with experimental
thermal diffusion ratios as a function of composition and
pressure.
2. ailMo rdtglss
Diffusion of one component relative to the other
takes place if the mean velooities of the two sets of
molecules in a binary mixture are not the same. Then
in a small volume about r, between time t and t +at.
- it2 = .71i , 421 . TT:a f29.2 d 22 # 0
(i)
where the second approximation to f1and f2 are taken as
fi = fl° [1 ;61(1)] fg = f2° [ 1(O].
(2)
and the quantities 0 have been defined previously (Ref.la).
Substituting for 0 (1) in eqn. (1) explicitly leads to
an expression which indicates that diffusion takes place
a) in ,ho direction tending to reduce inhomogeneity
in the mixture,
b) when accelerative effects of forces acting on
molecules of the two gases are non-uniforms
c) when pressure is non-uniform;
d) the velocity of diffusion posesses a component
in the direction of the temperature gradient. This thermal
diffusion produces a non-uniform steady state in a gas parts
of which are maintained at different steady temperatures.
(Ref. la).
If the abseroa of external forces and pressure gradients
is assumed - -
+ DT aln (3) C C - 2 -1 -2 n nD12
12 n ar az
where n is the number density of the mixture
ni =
D12 = mutual diffusion coeffloiimt.
DT = thermal diffusion coefficient.
Equation (3) can be rewritten as 2 n
-1 -2 nin2 D -- c i an1 12 1n ar a In T (4)
ar
where kT ( = VD112) is called the thermal diffusion ratio, or using Ili = x.n
150
a 1xi T ar
151
where a ( = kT/x x2) is called the thermal diffusion factor.
(Ref. 1b)
The Chapman-Enskog derivation of 1c1 for dilute gas
mixtures has been extended by Thorne to donse fluid mixtures,
In the first approximation, the thermal diffusion ratio of
dense hard sphere fluid mixtures is given by (Ref.1c)
= (Ax + Br)/c
(5)
where i
A = -10 x1 [ ( 3
) F+ 3:11 -, (6)
m4 B =10 2 [ (
3 711 )a G +y
2 (7) • 2 /14121
C = g12 [G F- y (8)
G = e + 8 X., 12 (itr-) 2 ( ) (9) g 1 12 2
2r 2 A.. F = a + 8x1 ) (it.; )2 (10)
2 g
X = 1 +le gi xi w5 2 g12 x2 (1403 N . (1i)
Y can be obtained by interchanging subscripts in equation
(n.). Finally a , and y have been given in the previous
chapter, and tho quantities gi, g2 and g12, as before are
the radto7_ distribution functions arising in the Perms-
Yeviek approximation (Ref. 2 and 3).
With ;the assumption that the velocity distributicvn
function is locally MaxweIlian. and that the pair
distribution function is that at local equilibriums,
Longuet-Higgins, Pople And Vaileau (Ref. 4.) have derived
an expression forkT for an isotopic binary mixture. In
the absence of pressure gradients and external forces yr i*
kT = x1 '2 (a2 — H1) g, (12)
where all quantities have been defined in the two previous
chapters.
Fig, 1 compares values calculated from the
Longuot-Higgins, Pople and Valleau theory (HPV) and from
Thorne's extension of the Chapman-Euskog theory (CE). It
can be soon that for both low and high values of the reduced
pressure p* (= prrista), results of HPV rise more sharply
than those of CE, towards the middle of the composition range.
It is also seen, though not very clearly on the graph. that
While HPV predicts symmetric behaviour of IcT about x1=.5,
this is not the case for CE. For m2/m1 = 0.5, 0"2/'44; = 1,
and p* = 1
152
.....m.•
/ / \
I. 5 — / \ __
/ \
/ \ I
\ /
0 P / \
/ \ IL IL i • 0 — d. I a
/ \
1 I
\ Xul i CHAPMAN - EN sg.oq
1- I
1 THEORY
— — L- HICictINS) POPLE \ _
I AND VALL EAU
I
I
I P*--: 20 _ P-1
\ , ,
0 • 5
1.0 X I
MOLE FRACTION
Fici . 1 CompARtsom OP kr FROM THE. CHAPMAN -C.NIsKOci THEORY WITH THAT OF L- f-licic.o.,1S OT. AL..
k (CE) k, (HPV) 0.1 0.030917 0,072832
0,2 0.052982 0.12948
0.3 0.067168 0.16994
0.4 0.074298 0.19422
0.5 0.075064 0.20231
oc6 0.070040 0.19422
0,7 0.059696 0.16994
0.8 0.074/411.03 0.12948
0.9 0.02W:2 0.072832
Fig. :;:! shows :density dependence on these two
theories, It is seen that for r = .5 a falls monotonically,
that for r = 1, -c1 goes through a shallow minimum
ce(cE)
.04 .27454
.07 .27433
.1 .27455
.2 .27676
and that for r = 2, a rises slowly. In contrast HPV (r = 1)
predicts the rather sharp rise of a with density.
153
/ 7
rz"-5
y 7
7
y y
V z z
z
• 1 /
1
/
CHAPMAN Er451.0C THE.0.17-"(
- - - L.. - HICICt i,..15 PopLe A1.40 VALLEAU
'2
/ I I I 0
•••7 il! 0 J-•
LL
1
• G
0 (7) 3 .5 u LL_ J
1 • 4
g 1 1--• V • 3
•2 .4. .6 P°
S t•O
Flq. a. 0(.. v.s. PReSsuRE )11z/rat = - 5 ; K = - 5
3. Comegrison vitt} F. ment
Results obtained from the CE and HPV theories were
compared with experimental data. The composition
dependences of the calculated a os were compared with data for
the system CC41,.0yelohexane (Rof.6) and the pressure dependence
with the system X0 - CH4 (Ref. 5). Calculations were similar
to those of the previous chapter. The Fortram IV program
written for this purpose will not be presented in the
appendix as it is very similar to the thermal conductivity
program.
a) The system CC14(1) + Cyelohexane (2). As can be seen
from fig. 3, there are considerable differences regarding
values of ce, between measurements of different experimenters.
While results of Horne and Beaman (Ref. 6) go through a
minimum around xi= .5 those of Thomaos (Ref. 7) exhibit
quasi linear behaviour over the composition range.
Furthermore agreement of both theories with experiment is
poor. The* calculated from HPV rises with the mole fraction
of CC14; all values are about 100% higher than the
measured ones. a calculated from CE, on the other hand seems
relatively insensitive to composition changes.
154
0 -2. -4- -G •8 I•O Xi
MOLE FRACTION OF CCI4
Fig 5. COM POSITION DEPENDENCE OF cc POR. MIXTURES OF CCI.4 (I) + CYCL.01-1EXANIE (4 AT 25°C-
155
Density data used in these calculations wore taken from
Wood and Gray (Ref. 8) and the N vi* from Blitz and Sapper
(Ref. 9)
b) The system Xe (1) +CH4 (2)..0 has been measured for
xi = .0015, at 25°C, as a function of pressuro, the latter
going up to about 100 atmospheres. Two different ways (Ref. 10
and 11) of analyzing the same sot of data give widely differing
results. Those resulting from the method of Drickamer, Tung and
Mellow seem internally more consistent as seen in fig. 4.
There, it can also be seen that HPV theory is very sensitive
to pressure changes and rises rapidly to values much higher
than the ones likely to be the correct values. This behaviour
could be expected from results plotted on fig. 2.
Although not apparent on Fig 4, the CE results go
through a shallow minim= around 10=
p
.01
a
.002 .74471
.006 .74455
.008 •74454
.01 .74458
.015 .74488
A A
A ANALYSIS OF ONSAC.IER ET.AL .
0 ANALYSIS OF DIZICKAWIER el* AL.
CHAP MAt...I-E NISK.00i THEORY
--- L.- I-4 t ;COWS POPLE. AMC> VA 1-.1- EAU THEORY
LA A
g
2 O
tL O
lil I -
0.00 0.01 0.02 o•05 0.04 0.05 0•0G. 0.07 0.08
DENSITY m( cc
Fici • 4- PRES.SuRE DEPENDEN.ICE OF ot. FOR THE SYSTEM Xe (1) CH4_(2) AT 25 °C. 14 1 = • 0015
156
If the Drickamer et. al. treatment of the data is accepted,
CE does qualitatively predict the behaviour of a' as a function of
pressure.
Here density data for the caleatations was obtained
from Ref. 5, and N7 1* from Ref. 1.2.
APPENDIX I
E3rauattoliSt— —Tg_TCP,a)t,
For the case of axial motion only, DP/Dt reduces to
DP = op aP nt at vz az
Ihe pressure at the bottom of the cal is given by
P = PE + p gh
-*ere PE
g
h
p
is the equilibrium vapor pressure of the
liquid at the given temperature
is the density
is the acceleration of gravity, and
is tho height of the cal.
(A2) then becomes
43 1- + vz pg • Here has been assumed constant. Also
p_prz 2zE aT at dT at
hence
Bt aT at 'z pg DP = 2EE aT
(A5)
157
(A3)
For toluene at 90° C
-4P2, 1 1.3 x 104 dT gm / cm-sect - °C ,
fl t: 1 le °C / sec at t=10 sec
assuming the uniform distribution of temperature; also
vz pg = 98 gm/cm-sec%
r,7')9efore
]P t 110 gm/cm-sec3
and DP •
T art — 40 orgs/cm3 - sec
which is much smaller than
P CP at - aT 10 ergs/cm3- see •
158
159
B. Comparison of vs. Kt
We assume at 90 °C, a temperature rise of .5 °C in 15 seconds.
Choosing the maximum C as K(90.5 00t, we insure that the
calculation overestimates the error.
K(90 °C) = .76498 x 10-3 K(90.5 °C) = .76376 x 10-.3
Using
X/ T(a,t) ii2r2(a,t) = .T.• r "Th L g2 a2 Cam
and comparing values of Al obtained from the calculations
with K(90 °C) and K(90.5 °C) , the error is .015% . For
t=1 sec. the error is less than .01% ,
160
APPENDIX 2
Derivation of Equation 37 of Chapter 2e
A. Solution of Fourriees Equation in Cylindrical Coordinates
w!_th Time Dependent Heat Input and Temperature Dependent
Physical Properties.
Fourrier's equation can be written as
pcy, aT [ et aT (i) .r at - r ar ar
tti=. ire
Al + A2 T P =P 1 2 T
CP = Cpi + Cp2 T
are the thermal conductivity, density and heat capacity respec-
tively, of the medium surrounding the central cylinder. The
initial and boundary conditions are:
T(r,0) = 0 (2)
dT(r,0) = 0 (3) dr
T(a,t)= f(t) (4)
where f(t) is obtained in digital form as the data. Further,
2xaX aT(act.) (q1+ q2 t + q3 t2)=Ica2PC°P aT(a,t) ar at
lim T(r,t) = 0 (6) r49- co
(5)
161
Equations (1)-(6) are linearized by making use of the
transformations listed in chapter 2.
72 0. it (7)
gr,0)= 0 ; r>a (8)
Bin gr,g) = (9) r- 0 2ma + c1/2 4Q2g4Q3t2 = 2 ; r=s (10)
dr a dg
vLare Qi = qi/Ki , and K is the thermal diffusivity. Transformation
of q c114. q2t 4' q3 t2 to q Qii. (/2 4' °I3 g2 is straightforward because g#Kt as in chapter 2.
Using the initial condition, the Laplace transform of the
boundary value problem is taken:
a2 i(r.$) ar2
i sailli2-la a kr99) r r (u)
ata a 4. cii+ Q2+ = 2sa a ar 8 7 a
lim "kris) = 0 r+ co
where 0 is the Laplace transform of
variable of the transformation.
r =a (12)
0 and s is the complex
162
Here .2ed APC P P where p and 9p are the density
mad specific heat of the fluid medium respectively and cY
and C' are the corresponding proper+ies of the central 12$
cylinder. As mentioned in Chapter 2, a is assumed cmetknt;
justifiaatton of this assumption is given below.
The solution of the heat transfer equation where in
addition to the assumptions listed in Chapter 2, the physical
properties are accepted to be temperature independent, and
::he power input constant is
T(r,t) = Qi Fin + + In + (13) L C 2T Ci 2T C • • •
Kt whore T=;:f , C = exp(y) andy is Eulees constant. For
times longer than .5 sec in the thermal conductivity experi-went, both the second and the third terms become negligible
i.e. (<.1%) in comparison to the first. Hence the dependence
T(a,t) on a is confined to the early part of the curve, which
in the experiment is discarded. While it does not seem possible
at this stage to show rigorously that the temperature dependence
of does not sensibly (i.e.> .1%) alter the solution of
equations (7) to (10) of this section it is reasonable -co
assume a= is constant as the term containing a, itself is
rather small.
163
Equation (11) is now an ordinary differential equation,
the general solution of which is well known:
kr,S) = C1/0(00 + yo(Or) 14)
Iihen and C2 are arbitrary constants, 13= /t and Io and
E0 are modified. Bessel functions, of zero order, of the
first and second kinds respectively.
As Io increases with r, using equation (13) C1 = O. Then
(r,$) = C2 Ki(130 dtz.$)= -C2f3K1(13r); (15)
clearly. d6(a,$)= -C20E1(0a) dr (16)
and wiry equations (12) and (i6) at(a,$) + 21,C21C0(ea) = -C20K1(0a) . ar 2nas a (17)
Solving. (17) for Cp and substituting in (15) 0(r.8) Sir + .92 .1:223. p(er)
2zas s 82 s) [OKI (0a ktgo Oa)] ( is) He will now use two identitieq which mil be :rived in the
last paragraph of this Appendix.
Ko(er) = ..[1n(03r) 4'04[1:10CW-1] ...] (19) and
BKi (Oa) =1 [1+ -102a2 [121((C0a).4] +... a
and by long division + ii(res)= L2 c k.2 kicar)+182r2in(lcor) (21)
442a2[1n(ICOr)]2 -12a21n(1041r)tln# 2
+a% [ln(. Cor)]2+ 9111n(l‘qq.r)•ln!- 4132r2 a a
41-14.1n(i0Or)]2+1,Ls21n(03r)Elnii -1]+... 4a
(20)
164
All that remains to be done now is to invert equation (21)
from the complex to the real plane. In doing so, we till make
WO of the theorem that,* in the Class of problem under con-
sideration P011.
(0= et 0(S)(15
1 r, 0(s) ds 121-21 (22) 2ni v60 to j.
when y is real, and i = -1 Hence we can make use of **
L'1 [lii(ks)] = (23)
and ri con in(ks)] = (.1)n+1 cri p / (tn+1) (24)
where k is any constant,
and ri is denotes the inverse Laplace transform operator,
riq 16(8)] ail fest - i(es) ds = 0(t) (25) We also need the following inverse transformations***:
* Caralaw, H.S. and Jager, J.C. "Conduction of Heat in Solids°, p 370 Oxford University Press. 1959.
** ibid, p.341
*** Erdelyi, A., Editor, Tables of Integral Transforms°, MCGrawmHill Book Company, New York, 1954, p.251
L 1̀ [s"1 ks)2 ] = 1.n Ct ]2 -n2
[ 5-2 (ln ks)2 ] = t[ (1.'.n it)2 + i - ,2] (27)
and
/7,[s-n" inks] = 11120 (28) k
The inversion of equation (21) follows directly from
(22) - (28). By dropping terms containing 1/t2 or higher
powers of (1/t) we obtain 2
3(alg)= [ Zg 2 + in
4n a n a n CC
[-g+in a5 (§1112) +(sin A.5 )2 a ..2 itsn Ca2 2 2 Ca2 a
f,2 2 2 lag.. a 2 +. • •
4 24 a
(43 2 2 I. Nag
2 + -2 (l-3_ )+ a2) 4 2 a 12 Ca 2 a
g[ln ]2 -2 +....1 Ca2 -4 a
1.65
(26)
( 29 )
166
where g = Kt. The time dependence of power is weak and the
sum of the second and third brackets of equation (29) is less
than 1% of the first. Thus, without introducing any sensible
error into the analysis of the experimental data, terms that
contribute less than 1% of their respective brackets in the
second and third brackets can be ignored. For the second bracket taking toluene at 900C,
K = 7.6 x 10'4 orn/sec, a = 1.27 x 1or3cm ,ah 3.67 at 1 sec. 2„ Lit = 5.3 x 0-3 an Ca Kt = 7.6 x 10-4
Kt In(Z) = 5.3 x lfr3 Ca"
a..2 a [111 Mt? & 10-5 a "4 ae a2_ 8 x 10-7 _ 2 2 2 az-2 = 5 x 10-7
a
.21-12E22
Et= 7.6 x 10-3
in 4Kt - -572, - .07
a2 lnict -
) 7x 10-6 -2. Ca
[ In Kt] C
2 = 3 x 10'5
167
So the major terms in the second bracket of equation (29) aro
-Kt + Kt ln(-41Ct ) . Cat
(30)
The same comparison of the relative magnitudes of terms
can be carried out for the third bracket of equation (29).
After 1 sec.
(Kt)2 = 4.4 x 1077 4
Kt a2a,-2 [1- 12 3 = 2 a 12
a 2 t) 1n 4Kt = 2.04 x 10-6 2 UP
Kt 1.2 In Ea. = 8 x ir9 ce a.4
Kt [lnIASI 32 cx-2 a2 = 2.5 x 10-8 Cit. a
1.3 x 10710
After 10 sec.
(Kt)2 = 4.3 x 10 -4 4 Kt a20L2 (1-n2) = 1.33 x 10-9
2 a 12
t 2 (Kt) in LACI = 2.7 x 10-4 2 Ca'
Kt a2 In 1 t = 1.2 x 10-7 a Ca
Kt inLai a-2 a2 = 2.2 x 10-7 Ca" a
The terms that are large in the third bracket then are
hft . ..2 (K02 4. et2 in 4 2 -67.2
Combining equations (29),(30) and (31), with
we get
4
0(ait) ., :11 + ..a +c a2 In +.... =
L Ca' 2Kt 1c72 2Kt Ca'
4g Ft ( in '
1) + 4% L Ca q3 p(14Kt 3)4. .... ] 4n "EP 2
where0 is now related to T by equation (39) of Chaptor 2.
168
(31)
= Ql/Ki,
(32)
gait) = X1 T(a,t) + X2 T(a,t)
K(Pr) =2E I—‘)(Pr) - Iv(Pr) v 2 sin vit
(35)
B. Derivation of Equations (V) and (201*
The modified Bessel equation
+ cly i + v2 . 0 dzz z dz
where z is the independent variable, y = y(z)
specified constant, is satisfied by (z)v+2r
( ) 46 471(vir+1) Iv z = r
where for v an integer r(n) = (n-1) I
Clearly for v = Q, and z= Pr Io(Or) = 1+ (3Dr)2 +
(24)2 +
•
Kv(r3r) is now defined as
169
and v any
(34)
then
Ko(r3r) = -fln + Y Io(Sr) + ) 041, 111 and
Ko(r_kr) = -c ln((ar) + S2r2(3.n tar -1) +....] (36)
Also by making use of the expression
24(z) -.%ffv(z) = -6C\141(z) for v = 0, we have
400= - K100 (37)
* Carslaw, H.S. andJitegor J.C., "Conduction of heat in Solids." p.448 Oxford University Press, London, 1959. (2nd Ed.)
Hence
OKI(Pa) = - d K (ra.) UM:
leads to
170
01C1 (Pa) 1 1+ C2a2 icna a
)
171 APPENDIX II IA
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_ CHANGE st (ABS CON,04kONIM-14),W_CON_04)___
Kze II )_+1 THCONCONIfill CONSEAKI WANOV( Jj)
• - 11 I 014=7.• •-• SUB L 1_
zgAsSiV142-_-V_I 11))
=- - - St15224148a01221a SVE332ABS (1/3421-V3 ( 1 ) )
175, CRR34_K)=SU83/SUBO
_-,-_241, 1•EL-OPETAtt ,IFLOAS_LIL-)
535
------RR4=44i=rarSUB4oeStee NTR-C-ICLAIM
C _ THCOND,EIL_BEGINS
LMAX-u-3 5L9 L*L+1__
NOOT=.1
cALL=P_OIXF_TIZ_t__THC LIC-ORACALCatAANOUT• STOVALSMO_TH_4 FITRES )
TCONO1LI=CO
C
20_0EORMAT(1H1 • 35X407HRUN NO * /4 )
_ 201_ E ORMAT_Ce/6X_t_17_14EXPE R I MENT_TEMP= El 36_6 6H DEG C_,_20)11 SHE XPER IMENT
WR I TE ( 6 I 202 ) SETCUP•CELLR • BOVOLT I ) -11
1OHM___i_e***_3ACKOFFs • E13_•6_,ASH____VOLIS
-2=3.-1=03Toopya2pSolV 20_3_ E ORMAT( 6X 9_12HPOWER_C dICM /SEC Vs E 13 • 6 11H+ T SEC) * • E13 z6 LI9H +
WRITE16,204)URoT —
WRITEU•25(1) 72 i I ) DLALIDA RCLAr=rEkaiveciE12FL17__ U***--=-CP=E=1,9tt===
_l___J/G/D**** 01-AMDAw •E139_6.17H J/cM/SEC/DEG**2) VOtrWri_i2 2Q6 FORMATLEKLEMICIREROPTE139_6•12H **** CpPT= .2130019H J/G
UNI11,11! 1. s. .s-WM"~~r73 WRITE16_, 2 0_71_44J •JJF I N
_ !RITE(6c208) -e fa:FORttkr iNOttriAtTW
WRITEA6t2q9) — • , - _ 211 F1V siSE.T.4
101_00 601__L=1.LmAX
— WRVrEt )
60-cots1riNtle WRLTE ( 69_2_11) LMAX
210 F ORMAT 18X • 12 5X to El 3 • 6_,_5X
; 176
WRI=TEI=Effar1C0 2 V_2=FORI4AZt6Xa6)4LAt4Chs—eE-12-.6 )
ETEt642.4allt4CAK) =
60_2_,ONT_INUE
t6X_.64Tt-ACW:141Xa6HSTANDILLI2X_s_314RC.8 •
LASTs_t_LitjFjjj_j_z_
CRR31 _ 1)
215 F OpiA AT 1 "IXAE6.1_•2Xia,3312X tE1.3•_6_12X tE13 • 6, 2X * El3 • 6_921(1,E13_•_6_t2X•E-13_, 6 )
-----60ctitattilm 451 CONTINUE
IF NOBAT*NRATCH1QQ9 1999t_1_000
END
SUBRcitaiNE_BRA IA SECA DIEEtAOKALRAT • FZERO9RAT_WONEALW OAFTI-IR ) ---C wrta = =leas-7g " 441 g = az
C LOG (T_LAOK) CALLED RLT1M
-c - 177
C BETA IS LINE SOURCE PART OF FZERO C.- _ AMtatAiS=5M
PERnia T---- EXILEREXmcalorntUURY494_ RLTOTuALOGIC4.*(SECAPER11/(AOK*EVLER))
RI1,DALOGkI.+iPEP/SgC) t _
--_13LMW-CSEO-eFIERTItoR ERT) GA MM Ask( A OKL ( 2 wiliPER-)4*R1.D1+ALRAT-111Agici34.4teRV_ittiRLTDriume)._.-t( RLT_IM *4244 RATaGAMMAZBETA
F ONE: (ISEC**2 ) .11A_RLDZIW_Ri_14. ( 2 .*SECARLIDT_) nt3_831SECI_ • f...."77.=.A.„ (_ EC1
FTHR I t t SEC+PF_RnEtRLTD_T**2 ) 1--LSECIIIRLTIPI*4121)_.A2.41SECIIRLD4-AZ. *PER
RE-TURN ND
$ IBFTC DECK3 - -
-ISUBRO KEIT-X-*COPMEC4-CC)WW4CoV -T-4E-STD_V*Silf00=TBA-RE
LHrN51ON FO ARG ciyENsioN x(250)•y(250).A(400).c_i2p 5o)•RE_s( 25o)
C C_ MEN_S_LON FOR SELE.T_GENERATED___AV LUES
.COLZIS 41r.
PTS N
C C ustiTAALLZAIILoN
SmyX 1 1 gs 0.0
DO 2 I 1 *KTOR
SUMYs_ 0 0
C NoRMAL SPECT TO XMAX
DO I 001#2_,N__
7 11- t.Tagf4A AIEAl 101 XMAXx• X
Do _t_o2 N
C
C FORMULATION OF NORMAL EQUATIONS
DO 3 1 =1 eN
_ 178 ' ML-AlitisaljPIWPTS
KOIR --AmEAPtsurot us
DO-8-14s I sle.012
DO 8 J=IWOR
IJ st(J....1)*KOR +I
b4ti• 11.1116-71M•t-1,1-0,1111 '1711111: DO II 1=2WOR
11____A(I11) a A(11I)/A(I)
Kt4x___J.1
Apim goo
------00145-47LA-Ort4X114- _AK -,_a_tte 0R+_A
114 API si AP1 + A( /K) *A IKJI -- -
_14 ACIJI AALJI___AP1
I EAJP-•-K011_444_2444 • 445
APi 0,0 UEM1=1-____101
JK__21( *_ KOR__+_J --
116 APIAAPI+A_WKI*AAKL)
_ -2--- -= 445 DYMMYR_O. 0
CAI/ IBCC')/A(I)
AF_q_s_fik14 0
_ DO 118_(111'M
118 APIs:API +A(IK) *C(K), IEUEFE la_c_l ik( )
IF (KORM) 122_. 12_34122 PI. APIs 0.0
DO 1.21—KoMPAK0f4 179 mic_licn44*KOK7-4.m
121_APItARAWOCIBLIalg1 P_
12.3_APIA! 0.0 PI PI
24 API— inAP1—+AMEANW (j 1 *C(1) PI _XCU4EIANIaiwAPI Pi
SPIE.S1•_0_•_0
SMOOTHIJI=C0
27 SMOOTH(I) "'SMOOTH( I )4c._( J )*X ( I )**J
SRESIISRESARESI_WHE2
C
C DENORMALIZAT1ON_WITH_RESPECM_TO_XMAX_____
- 77 CONTINUE _ =-L--STCAritS421
Do Pa tit/stomp
RFTURN
PROGRAM_WRITTEN_FOR__SMOOTHING__DR/DT DATA
• 119 WAS USEDINLTHELPROGRAM_FORLPROCESSING_VOLTAGE_MS T1ME_DATA
DINENS I ON_ X(50) • Y 9A4_40_0_1_t_CISQ LtPRESS(-10 )_ s_ NC ( 201_•_THERMRU_0120)
n INENSION_CC120 I • T T(20 ) IF:30(1_0 ) • PP ( 10.5) •R(IQ) • RO ( MARISA 0 ) 2, • ..,-.:.'CZMEI.1661.1-27 14"---71i=
=NORMAXA3 R0241Z95-
ALFAit0s00392669
__ DELTA "0_4_4936
555 READ 501_0__q1NP
1300 DO 3°0 L!-IsNP
101 FORMAT(F9s2) ---IEEriCoLWA402444"1"----
180 _
SVOL-r-CL•=a)
- • - - = ^
I
T=t KrwC114ERM R 400--K,S, R0=LEAARMELTAIV4k-UK/,4LO0s ta1'104/
STErigAs
140 TO-400
CEL4RtLtJkCEL4-1J*ANRIL$V0tJW J)
300 WNT-INUE
1-1MAXat7
- 1304 _ 0 30_44*1 1 • NP NNC*44C-C14
Laos_nn 305 Jig I • NNC
- TTA J /*TEMP ( *..11 - - -
NOUTig.1 =7,15 -215.Se =1-7.77--2751-.1=2,221.227+2. 2-7. 3 2
P0( I )*CO =1=306= PPA I_,K /t_g_CIK)
- -
30 CONT INUE - - .32C-11-4E-Vul-T-MA ' 132 NP
1_307 1)0-307 Kztl-aLL ZROSSielliCY)
307_ CONTINUE _
NP
_NC/UT:2..01
ROA II /_11C0 111_
R2 C 1UC_I (23 TC 32Q CONTINUE /NUE
TCRCRSC 1_1A30 •
RRII 1 ) R0( 1 I )+141 ( I I ) *PRESS ( I ) +R2(I1) * ( PRESS ( I )**2 )
zr, —1VP1:51rie
CA = — a a: . . - • al4 =
1 poLy.Norc_oRDER_tax LI 2 ) - =__WRITETOWE2701W---
- =1 V= E "tf ' E - al +771.1rb Ea,- 1 4_
- 309==CON N= LI MAX 181
201__FORMATt//12,X4_4HTEMPA_11X.._4HURIDI.UKcatCELL R)
NCC=NNC+1 KIAICCR
P4U.AiNC 13 Lo_p_o_a
- ----"TEMIVENA42=311. TEMP-41-.14111=60_0
QRO_T:Ct-1=) - I 111111111%111FasMST ai CWT.
- ---1312 DO- 312, Kiaa•LL_____
6'9 ' • 1 j itilICK_)
404__WALTE16120213EMPLL.ILIDRIIIARF SF I T
_ 31_0___CONT NVE -J- 301 -17.,NTANUE
302_CONTINVE -=-G-O-T0=b5b
__END
APPENDIX It I C -t& ,F,Ta •:4 - EmpERA_TuR L-ES--
I NF_INI_TE_SLAB •
art-- --jarr:re,*•=:7,1,:=11 aizma Loa •Bara =4:1 _ au OVER-RELAXATION.
raMENSIAZ - __DIMENSION_TIMEASI
cikAtiHELtifimalAnmfAqN5---- smol
B5=1.1
JJ =0
PI-Email-4-159265358
182
4 4.-
NHic4,0 NW14314H4-1
_1414224NW*41-2--- FLICUECNNIWMW1W410
R5=_R/P r
FR* P/(241*RP1) :
RMUse1R/RP1 ) *( COS tpt.E.L1 FLOATINHLYW
----u4-1=st.tra.0-443 -300—001*T-INUE_ •
_130 1Z0G 303*1-aNt41 - --3_000NTINUE
NT_TriNIT+1
1302 DO=302-_ _I *20011_ _ _ _ _ r - _ _ _
302 coNtiNuF 424FOOPI 1303100 301 ImPINH
--L-14MA2— R*S5TWlt-Vi4COUMMWILF+7L04 _BRA3nAR*S5•1_41_111_11 l_e_N+.1_)_**2•1-+R*UAN+1_)_+2 • *BJ_LI _
)---- -- ---DEL-TAwCIR00-
DOigW*DELTA
QABSiQOLM!1N+1)
NL-013Pk_t2 _ I___=14/41
BRA214RIP$B4*(U(I+1•N)**21+R*U(I+19N ) - ef-= - - ,s.m 9—
BRA23_•_*S*Rit_CBRA2+_BRA3 ) :Lata*s" aZTAI:almos x
DELTA14_t( ROOT•,..B ) / (9 *S*R)).-.VA111111
U(1.414+1 )=U( I'41+0110
NL0opm 2 183
1304
GO_To___420_
TA.44itU_C_I_*N+ }
30-4=C:Dtt_T4NUE 2
ttP2=4Q3==1 Q519&i 98
wR r_TE16_•2004T4mE (4.1) Ala z
WRITE 1 6 20 1 )
*r.-= :Flaffs
• .-t-.--- -.
_306 CONTINUE 08=414
, - - - .11.,-T • 3!_400-T_O_ 555
END
APPENDIX 1110 .-.0:-T.'1=7•1' • — s'" - .'"= 2.12-- • -
IN AN INFINITE SLAB, US_ING_THEA<VDRYASHEY-ZHEMK0V_TRANSFoRmATLOt
tz•ZMY Jr=reE.We..
c OmmoNIFVBB • SIGMA
SIGMAg*1 )4" --L44) *
TAvolig•3_ _
Bl_LIfiOALO
—101!4W6___ _ e_rastol
B 91i1 • 9 - 1V1114W95-=-- 1 30 o_pp 3 0 CIA
130_1_00-101____JA214
v4-1421-TENP4-T4 184 -
400— A-LL-8105UM-42-•.2•1sS)
I V-00-
e —' • . ar: CZ .2a • . ••=.• • .• -V= ;
•••••••••••. •
—WASS
I Fl
„___21302=0_0_302 L P eLL
DUMAKCLUNWAA(L )*( EXP C—C; ( L )11T_Alkt../) ) ) ) ---301414--NUE
_ _ _399_ G4K) riDUMG
GO TO 400
XS
_ _ _201=EARMATILIXt51
= _ __202 F_ORMATA 2X,1_414TA_Uske F_9_•_5 )
Wasit WR TE,169203 )1( f_V K T IN
203-#0144-,WP GO__T0_4_03
204__EORMAT(7XiI2•111TH_AERNIk_21E15.8_0X)13X13N***)
3cliTINuE STOP
SUBROUTINE BIGSUm(T•Tm•S) TaA'7 .-ar.
COMM_ON__LE2/K • G 50 ) t A ( 50 ) T.-
U ( 2 ) sr-104457502
V14475540441
U B )_gs.s. • 4580_16TR
U CB) __O 9501251
U(ll ) w.0 (6 ) u_t=LakEtza 185 (_13)=-4( 41 A.,1 1011410111- U(15):—U( 2)
uvr6ttVF) RkLL * • 0271524_6_
RAZ ) = 09515851
12.1_51_=_•_14959599
R f 71_=_•18260341._
WO) as R(8)
R R-U
DUA4mY.0 .0 DL
1300 DO 300 L*1•16
XS IsACM 1* C EXP --G t )410 I+XS I
_T_T_tL Lat_TEt4P(XS I)
COMMON_LF1,188_, S IGMA FUE-- 17iTt54 55a58
AO x2167_4013_0
t (14* 22. 20660990 A t6802 1 A ( 3)=120.90265391
A( 5 ) =298.55553313
A ( 7 ) =555. 16524755 Aicti a9
_k(.9 Y=890 •_73179719
4 17. T1.4 ; ;
B-Cg Vw4V2V5 - 186 13A-51, 8759595
t 74;2Z.56LF14442
EitcW-*--9.11454-3,0249
C4-24-o-i•-•-P00
C-t6-_+-e0-Z-69?-3077
C 8) at+. 058a.e1524________ ---C • E05263I59 SUmiti_EXp_(.AAO*T_)_)__)A(COS1804oBB) )
T7
SUN=SUM+AC1K))*IEXPk-(A_tK)_ii_T4.WacOSit3tK)4_H3a)_)
Sat 4.4_17124E)ASUMILLI_. + SIGMA/2,0 ) 1-FfS1-4:0
TEMPS
600 TEMPI' (-1.+S_ORTAA • + (2,WIS ) )/S IGMA Ekr -
END
APPENDIX I I IE
OF_CHAIN_MGLECULES FROM_TI3E_CELL_MODEL_EOR_PURE_POLYMERS___
IN TESIER—PRESS
a - • . - al7= • 17=1' al"12:a -4-7- f..;:2. 1/ sV],_ DI MEN.5 LON TgMPI tO fr45
vtl:11-4-5pRESEELL—is)
; TA ett 1”2 g==t-v. =1.3.8
N425
DEP-ENDS-014—PRESSUPF
• 1=--..--ra.-.7.4::!*3 1- 'g 2: & - - cTNCHN
_ C-V
C- trisiRm_m90_, tit pur s 200 t__Dos_300, IFS 600_e__ARI_T_7 GO = = =
or READ(5, 102 'FPS( I)
600 READ15t_1041N( I) a r.4
NJ=MJi I 1
_READ(5,110) NK( Lt.)) *PRESSW)) m 111,410 9
1_03_F_ORMAT__(F1P.5•Ei2.q) 187
00_304_1_41.N I EN EL:0- WQLjfl
_41 DEsPia *_ENEPEPS__11 WM061.1/IL_NOSEG*N__*VIX.UMIBMSTAR liQA•WX1=4
120-302JAL1•14.)
_ - 302 Kis 1 *MK DE
VOLUMaVSTARI_I )
2050 FORMATAIK0,2X12Ht1e • 13 • 5)(.2H.P• • 13 • 5X s_amokiALL31______„
TCOND__(__L•_J_•K)__ticoNsT
304CONT1NUE
_ 650- WRL-T_E_Cd t-202,-LL•KI ( Ii • ( I_Leep.$14,4 _ - -• = P iFFIf--3>IFOSFINOSEG*ITI2X-WHEFES-IL:014
&
DO 305 JILLINJ
2O4 _FORMATJAHOA2X102KIHt_ie_lepRFSSURF = • T •1)(.2X_e_.3HBARl_OX_•_L2_, IXt5HTEMPS )____ _
205 FORMAT( 1H0 • 4X • 4HTFmP s 7X • 7HTHILCOND . 9X • 7HDENSITY 13X • IHI • )5_(_1_11-1.j.5)(•_1 ANK1-
VI R LIE} )
_ MIONK ( I • J) 007-17-305_11K-7- WRITE(6•207)TEMPC I •J•K)ITCOND( I •JeK)9R0( I •J•K) I •J•K
=207'-'EO 0)W712_,-4 _1a4
305 CONTINUE fiff lEt-kit)
217 FORMAT (I HO • 2X • 15HTHATS ALL FOLKS)
END
EGWMC "-I. "t122.25,111 m.:-.95.17rai=1 2.4 1' B = Ls 38_00-23 F3*-s5
SRE-SDIFM;214)- - CY*3
A_V 0=6 s_02252E+23
WSZG* j__AY_OlIFLOAT ( NOSEG) I
ACU_Bo_VAVIIST12_ -===C77_7•7—ALF-
ALE1__* CALOG ( ACUB ) ) /3. 4.74 i
XWZnICI
Y-10.56 188
V M_VOLUMZ_VAV EIRA-tar—X*- 44+ 2-)
E-RACSIOIF-LNE ZELDATA,NOSECL),144k--(101-4VJ -2- --15141krynotriV2—
011501*MAVSEGF)---- BRA12A - . •
BRIO* (So2*cV/A)/C2**PIE*WS01 TfiCC*BRATLFOURI5 WRITE (61207 ) NOSEGIMSEG1VAV-1SR2-tACUalALEAA•VcWAOL -20-FFORP4* rz - rbg, er: -
-I 1E12ALS1 3X 1E12 fa5
"=" 11.--7. 1 Pir— .r.%
LS.1 1•726-=11111' -'7#-73;.-21: =-- -z1".7.7- 9-21-1-"Tr
208- FORMAT_C3X/iX e 3X • F1 2•5t 3X t E12.5* 3141E1 P•,_ e_fiXiE12_e_ait_3X • E1.2•_5_,3X tE124, 5__
_RETURN_
APPENDT-1-14, PROGRAKWRIXTEN—FOR=CALW_LATING XS1 ANDVWV
THESE_RESULTS_WERE_USED 1 N-_CALCYLAT_I NG_VLSCOSI_TIES AND _ THEEIMASONDUCIMMIAE- --FCARE -
DIMENS 1 ON P(20) .14.12 0_1_, X21 2_0_1
eia)=.• =
P CaLe: • 3
P ( 7-) 1I•5
ptc4)•.7
P CL11_1: 9
_
_PC4Y12L1.5 X2 XI ( 1 )*0.0
TTi t300DO3 X1 ( I ) AIX 1
300 CONTINUE
C- F=LOOP=VKT C J LOOP R
...Nt=U N41:4_ NK 1211
r30=L—DA2
200 FORMATUKI-A5X_2_3HP**iF6•24 189 =
13Q2D0 302 Js 1 t NJ
2AL_FORMTAAii3X111).i5'._.9)31!SI .11X *311Pkil •12X • 4}1Ao13&, INB_LI3X11171C413X11±0 k*BX*3KNT
WRITEA6+2024RAJ4 - 2aimapoxts
r p+
_ 223: ( 1 • +R (MA - ---- A ---- aUllillrAV4rIll r'21.1? r r -wm--5,-.1-erery-
VI. !re-! .1 =sue.
rf. • 3---FS 7,-. T.
0941141( G**2 ) 1331.244-•__4_42-4-*A*C-)
1F(DISC .LT. 010AM TO 499
DD*--D
F2= • *B--SORT ( ( 4 .11 ( B**2 ))-( 12. *A*C_Ill/A6 • *A )
FRE*N24K, __RAT=
IF tEl___ALGT AMA, Fl • LT a__ La )G0 TO 40_0
N2GOit 2 4010--/FIF2s-v- ;-----s., _L._ _
NGO=NG0+1
14280a4 40 .1_1EING0-1 ) 404940_3_9_402
402 NRit 1
C22-* 3'
CALL TRIALS(019_02-911A9XNAPHI reNTR )
C
-44.4951W Q1-.3
I:8 GO TO 999
It -•"7" : ULM " -4:iif'i
C 4O-PR*2
3.90
. • — t
1 -7_
_ - 40 5 Ng fr2
- •• • - • 2„.. '44. • -
-
406 CONUE • •
1N-as2
_ 4Q7u iNset_
— 4OCONiUE F a:ism& I NELTAI4 el • - . - - -• ^ • -2-2
—409- FI:faPTWCOM41~1
CA1.1,--_TRIALS- XX4 j_. 4a9(10 E42999
C-IIHHHE-RotTrE__
Fiflet Alt_tE rl**23+C*FT+D
g•T. 1=- - •E•
__FEE FF2*FFI - -
- -art ti WWI. ef 1UPW1.3.. C
= C
= GO To 999
C**** ROUTE P
- - - —.429—CONT4N_VE - - - - tfatir2 1-F-LIFF O- 0-421
B22821 -=1N112—
_ GO__TO_422____
_
___LE___(02FDT2__• LT •__Q_•_)_GO TO 423
424 F3RF-1+ CV* (DELT A/4* ) ) - 191
F4 =fr 2
CA1 L TR 1ALS(F4 aF34 N•XX•PHI *NIP)
Its Lc— — .C").7.T7E'r.E1 1"Mr. 71-7
_ _ _ 203 FORMA:MI.4 X_6 • 3•_612X • F1_0_5_)_•_3( 2X•_13
30P CONTINUE
STOP
---=SAMEIROUTLNF.,_-_TR1Al tsti-XX•PH1 NTR )
40 .• "... .3.: 7 = .1•.="3 —.7 71" .-
IF IN *LT • 2 )G0 _ TO—_450--- A2 girlif-
TETI 450 _A2311__A
--
czar =Dale
451 CONT_1NUE
_NTRet 1
Y1 ) +132*_(P1_*_112._)_+C2_41431+D2
_XXottP1•.-(Y1/S)
IPA DEL_ • LT•____•_0000 LIGO____LO 452 141:14NTR41_____
Pp P_27sXX
r __s P2**3)_+132_*_( Paii-11_21 +C2t1122+02______
XX_ • G_T_a_ 0 a___eAND • XX • LT • 1 • ) GOT_O__453
PH 1_32_1_0 RERN
TRE_ _UR.) N
ApPENaiX-144G
192
C "" • • 71::= %," = ;
PIE2a3.141592654 Airr.rma="u•
2.
12(2)st 4a
P_(_5)Ar20 -Ptewiet.
RM )la .5 =
Rt413) 42, R-41-17•—.
X2-t-11-a 0
_300__C,CitsiT1INNE
C**** _LOOP___PM-44A_SS—RATTO
C.****-1-1-00 01MRACLLON___
NL1 I • - f ------D00-11=-11 PPnPtJI
READ15,100)PHI (K. U 'PHI (1021 'PHI (K*3) •PHIAK14) epHIAKAL5)
_ 1004_FORMNYM11_9C, READ ( 50 1011PHIJK16 ) 11:31-1140_7) •PHISK•81_1__P__H_LiK191A2HI (K, 10)
IT 1O*AT45(2X ) REAoi19_21EHISKILI )
_7_i_10FVRt4
304_ CONTINUE
RRWIRMIJ) . — —
BM2uRRM/(1.+PRM) .1 = zt..st, 2;E law'
.4•21V1.... . .
At MOMS ••••
zm15lit_4GRT(t./Bmi) 193
__,ZMU_31g... ( 3, e/3 • )* ( SORT I BM 1 *BM? I /O. L)
WRi _ ( - UD - (
1303 DO 303 K-1.NK
alai-%=:7==,7"---.3,"-.2." - : -_—__- 1.1-..—.Th* 5'17.= re.."1 .7.!=
Gm(24b+PHIP)/(24*([14..PHIP)**2))
P Rtmi_Cas*RRW-146_,WR41-**2 Rau 1-24 -VV-ipnR*V-4141t2-- NNL Ni 1
RXmX2ILI/XIALJ_
_ OMR*1s..RR
ZXRs X1(L)+ZR3
14:1111-uK-g-L---WfxR FaiT
7fr_WEIGOMPEWW7e- Gis(l_efEt,5*T*ZXR44.41.5*ZR3*T*(CAR/RR)))/ZD
G212 1 • 4( • SIFT*ZXR ) 1 1115*M ( ) 41T-*() Z0 C_TERSCEE-4EtURREVLiGZE_____
GGII*G1/412
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cs(8.*RX*ZMI5*R2*CB)/DENOM .44 -'7•1"
FIRST*G1*(X1(L)**2) _=7,5ECONDOSC)12jAz kEtW_ RFtrat _
THIRD= (SoRT(Bm2/em1))*G2*(RR**4)*(x2fL)**2)
COLL*D/F •••.- :SA
ringiPP/T)-1.
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DI-STist 014.024 fig- DCS)Lmat-S=T+COLL 194
Cr _ _ _W R ITTECeec202-) )(1-AW,.11RATIETAITAL—E-CiN,e0C01.-4,Wr - 20 .."--v.tc. —
==305CONTINUE
302 CONTINUE
STEP ENO
PROGRAM__WRILTEN___FOR_CALCUMRMAL_CONDUCItLVIrLO_E_DENSI- mARt.__Sv
szTr.1—. •=• - IENSIVN___ TO_TNE_CNAPMAW-ENSKOG THEORY*
c**** TNERMAL_CONOUCT TICT
tUblEbts- CLV3M k2 o *PHI ( P0,20 ) P IERJ.t 41 "7,92 P(1)22 16
p(3)21 4 3-
P( 5) 20 ..y.
R M ( 1) 5 ". I 1 O.= -1:7_ -
BM (311_12. .666667
* I • 5
X2(1)460 - - 1-300 00
XI t I 1'0(1 t
300 CONTINUE 1C***OOFiP PVT =-- C**** J LOOP PM MASS RATIO
ffIC****LXit= _ C**** L LOOP XI MOLE FRACTION
NJ =3
=14
PP Op(I )
R laf=1:41,-;
/ 195
100 FORMAT_(5(2X•E12.5) ) -y - -5 B 171 .1.1" V = =7- -27" . 1"a" 4:17.Z.71=1:=="3283
101 FRORMAT ( 542X • El2 .5 )1
,21.731iMV2 _ - - -7-.==g1741r111"
-7-Vera .7-.:74 77-.1.-;r1Y5-% =.5 - BRAX=lo+12•4*G1*X1(1-)*T1+(14,2*BM12*G12*X2(L)*T*4 (14b+RR)**3))
RRI3Ht2flfr
196 zmt2511—zmP5,7m1 's
ELat___1-__6_**14211ZM2_5_11WN c"6i42 4125211c8il:Rx3Dtriclom COEFF-atT*42 125.•_*P4E4_4
_Tfif = .7.:/"P.y..11:1 sEcoNoloc_i_tt_likx2=1.1—)_*_Gi21_81- mt*csaRTA_Evitazi_a_444-*_t_t_i_a_+RaleiHk4)
--0E-kmaer CRATs )4EIRAStaRt RAYA_KIZ) +D I /F
CINallA*BRO(W(BI2-4.--)A4BRAX+BRAY))+IC*BRAY4)/F XDTSTSBRAA.
YDISTAB
0220(DIST*((C*BRAY)+((B/2.)*BRAX)1
C,COORDIST+COLL
wRITE(612021*) (LI.cRAT+ToTAL.c1N+DcoLtYY ,
305_CONT/NUE
302 CONTINUE
STOP
•
•
CHAPT4 1
1) Ziebland,H., "Thermal Conductivity," Edited by R.P.Tye,
Academic Press, London, 1969. p.65.
2) Pittman,J.F.T., Ph.D. Thesis, University of London, 1968.
3) Horrocks,J.K. and McLaughlin,E. Trans.Far.Soc.,
p.206, 1960. 4) Prigo&i.ne,I.,. The Molecular- Theory of Solutions," N..Ho}and•
Publishing Cog, Amsterdam, 1957, Chapter 16. 5) Horrocks,J.K. and McLaughlin,E., Proc.Roy.Soo., MA,
p.259,1963. 6) Chapman, S. and Cowling,T.G., "The Mathematical Theory
of Non.Uniform Gases," Cambridge University Press,
1953.a ) p.292. 7) Lebowitz, J.L. Phys.Rev., 1.12, pA895, 1964.
8) Percus,J.K. and Yevick,G.J., Phys.Rev., 100, pl, 1958.
9) McLaughlin,E., Jour.Chem.Phys., p1254, 1969.
10) Longuet.Higgins,H.C., Pople,J.A. and Valleau,J.P.,
"Transport Processes in Statistical Mechanics," Interscience
Publishers Inc., London,1958, p.73..
(5)
CHAPTER Z
Bird,R.B.,Stewart„WZ. and Lightfoot,W.L.,
"Transport Phenomena," J.Wiley Inc., (1960)
MeLaughlin„E., Chen.Rev., 64, 389, 1964.
Goldstein,P.J. and Briggs,D.G., Trans.ASME (H3at
Transfer), 86, p 490, 1964.
Pittman,J.F.T., Ph.D. Thesis, University of
London, 1968.
a) 232 d) 135
b) 80 e) 290
c) 99 g) 111
Carslaw,H.S. and Jaeger,J.C., "Conduction of Heat
in Solids", 2nd Edition, Oxford University Press.
a) 261
b) 343
c) 152
(6) Jaeger,J.C., Jour.and Proc. of ioy.Soc.
21E, 1940, p 3420
(7) Kudryashev,L.I. and Zhemkov,L.N., Izv.Vyssh.
Uchhebn.Zay. Priborostr, 6, p 100, 1958.
(8) Fischer,J., Ann.Phys., Lpz., p 669, 1939.
CHAPTER 2 (continued)
(9) AbrmnovitiL,MI. and Stegunr i., Editors, "Handbook of
Mathematical Functions,' N.B.S., Washington,D.C.
p360.
(10) Horrocks, J.K. and McLaughlin,E., Proc.Roy.Soc.,
A 273, 1963,
a) 264, c) 269
b) 259,
(11) Leidenfrost,W-G, Paper presented to the Thermal
Conductivity Conference, Gatlinburg, Tennessee, 1963.
(12) Briggs,D.G., Ph.D. Thesis, University of Minnesota,
Univ. Microfilms Inc., Ann Arbor, Michigan.
(13) Van der Held,E.F.M. and Van Drunen,F.G., Physica,
865. 1949.
(14) Gebhart and Dring, Trans.ASME, (c), p 274, 1967.
(15) Viskanta,R., Ph.D. Thesis, University of Illinois, 1959.
(16) Poltz,H., Int.J.Heat and. Mass Transf., 8, p 609, 1965.
(17) Ibid, 2, p 315, 1960
(18; Poltz,H. and Jugei,R., Int.J.Ht.M.Transf., V10,
p 1075, 1967.
(1)
(2)
(3)
CHAPTER 3
Horrocks,JrK. and McLaughlin,E., Proc.Roy.Soc.,1963,
Ag7.22a)y 259r
b) p 261
Fischor,J., Ann.Phys., Lpz., 34, p 669, 1939.
Pittman,J.F.T., Ph.D. Thesis, University of London,
1968.
a) Chapter 3
'o) p 140
Gold stein ,R 0 and Briggs , D.G. Trans &ASS
(Heat Transfer), 86, 490, 1964.
International Critical Tables, V6, p 137.
CHAPTER 4.
1) International r4.:Uicali_ Tables, V , p115.
2) Francis,A.W., Chim,Eng.Sci., 10, p37, 19590
3) Toohey,A.C., Ph.D. Thesis, Imperial College of
Scienco and Technology, 1961, p208.
4) Bridgman, P.W., °The Physics of High Pressure`', p129,
G.Bell and Sons, London,1949.
5) Pittman,J.F.T., Ph.D. Thesis, University of London,
1968, p185.
CHAPTER 5
(1) Hays, D.F °Wonelqvilibrium Thermodynamics
Variational Techniques and Stability°,
p.17, University of Chichago Press, 1966.
(2) Kudryashev, L.T, and Zhemkov, L.N., Izvayssh.
Uchhebn. Zay. Prihorostr, 6, p.100,1958-
(3) Caralaw, H.S. and Jaeger, J.C., °Conduction of Heat
in Solids",
p.97, Oxford University Press, 2nd Edition,
(4) Richtiyer, R.D., "Difference Methods for Initial
Value Problems°,
Interscience Puo, Co., N.Y., 1957; p.101
(5) Smith, G.D., °1717.merica1 Solution of Partial Differential
Bquatione,
Oxford University Press, London, 1965,
CHAPTER 6.
(1) Barkor,J.A., Static() Thoorios of tho Liquid Stato,"
Porgamon Pross, 1963.
a) p 48 d) p 80
p 66 0) p 84
0) p 78
(2) Lonnard.Jonos,JZ. and Dovonshiro,A.F., Pft d AoyeSoc.
OLc p 53, 1937.
(3) Ibid,,A 165, p 1, 1938.
Cu.) Prigogino,I., "Tho BblocUlar Thoory of Solutions,"
N. Holland Pub.Co., Amstordam, 1957.
a) p 130 c) p 337
b) Ch. 16 d) P 331
(5) Kiikwood,J.G., J.Chom.Phys., 18, p 380, 1950.
(6) Barkor,J.A. Pocaloy.Soc., ,A 237, p 63, 1956.
(7) Poplo, J.A., Phil.Mag., p 459, 1951.
(8) Janssons,P. and FAgogino,I., Physica, 16, p 895, 1950.
(9) Hijmans,J., Physica, p 433, 1961.
Chapter 7
1 Horrocks, TS:. and McLaughlin, E., Trans, Farad., Soc.,
1960.
a) 206
b) 209
2 Prigogine, I., The Molecular Theory of Solutions",
N. Holland Pub. Co., Amsterdam, 1957, p 339.
3 International Critical Tables. V3 , pp 29-33.
4 I. Chem. E., Physical Data and Reaction Kinetics
Committee, 1967.
5 Horrocks, J.K. and McLaughlin, E, Trans. Farad. Soc.,
52,p 1709, 1963.
CHAPTER 8
1) Chapman, S. and Coding, T.G., "The Mathematical Theory of
Non-Uniform G•ases, teralstlyi elge University Press, Second
Edition, 1952, London.
a) P. 28
b) P. 112
c) P. 111
d) P. 169
e) Ch. 16
f) P. 287
g) P. 286
2) Dallier, J.S., Journ. Chem. Phys, , az, P. 1428, 1957.
3) Louguet-Higgins, H.C., and Pople, J.A., Jour.Chom.Phys,
al, P.884, 1956.
4) Rawlinson, J.S., Rep.Prog0Phys., 28, P.174, 1965.
5) Ornstein, L.S. and Zernike, F., Proc.Acad.Sci. Amsterdam,
12, Po 793, 191)-,-,
6) Porous, J. , and ovick G. J.. Ph,yscRevo , 110, P. 1, 1958.
7) Thiele, E., 9 39P P.474, 1963.
CHAPTER 9,
( 1 ) Chapman,S. and Cowling,T.G, The Mathematical Theory ,
of Non.Uniform. Gases,• Cambridge University Press,
Second Edition, 1952, London.
a) Chapters 8 and 9.
b) p 166
c) p 167
d) pp 292..294
e) p293
(2) Longuet-Higgins,H.C., Pople,J.A. and Valleau,J.P.,
1958, "Transport Processes in' Statistical Mochanics."
(London, Interscienoe Publishers)
a) p80
b) P 83
(3) Lebowitz,J.L., Phys.Rev., , p A895, 1964.
(4) Lebowitz, J .L. and Rowlinson, J.S. , Jour.Chem.Phys.,
41, p 133, 1964.
(5) McConalogue,D.J. and MoLaughlin,E., Mol.Phys., 1969
to be published.
chapter 9Icontinued)
(6) McLaughlin,E., Chem.Rev., 64, p 403, 1964.
Bearman,R.J. and Vaidhyanathan,V.S., Jour.Chem.Phys.,
p 264, 1961.
Boon,J.P. and Thomaes,G., Physica, 32, p 123, 1963.
vlood,S.E. and Brusie,J.P., Journ.Amer.Chem.Soc.,
p 1891, 1943.
Elitz,W. and Sapper, Ae Z • Anorg.Allgera.Chem.,
p 277, 1932.
Rowlinson,J.S., "Liquids and Liquid Mixtures& 4/ p 50
and 56, Butterworths, London, 1959.
Lambert,M. and Simon,M., Physica, 28, p 1191, 1962.
Dobbs,E.R., and Jones,G.O., Rep.Prog.Phys., 2O p 560,
1957.
HcLaughlin,E., Jour.Chem.Phys., iq, p 1254, 1969.
CHAPTER 10
1 Chapman, So 9.n: OclainF T,G, °The Mathematical
Theory of Non-Uniform Gases," Cambridge University
Press, Second Edition, 1964.
a) p. 142
b) 13. 144
0 P. 293
2 Lebowitz,J.L., Phys.Rev., 1.21, .18951 1564.
3 maaughun,E., J.Chem.Phys., 59.1 12549 19690
4 Longust-Higgins,H.C., Pople,J.A. and Valleau,J.P.,
"Transport PI-messes in Statistical Mechmriost° p.80.
Interscience Pubatd., London, 1958.
5 Tung,L.H. /And Drickamer,H.G., J.Chem.Phys.,
p 1031, 1950.
6 Horne F. and Beaman J.Chem.Phys., p 2842,
1962,
7 Thomaes,G., PITsica, 17, 885, 1951.
8 WoodIS.E., IImy,J.A., III, J.Am.Chem.Soc.,
2a, 3729$ 1952.
Blitz, w. and Sapper,A.,Z.Anorg.Allgera.Chem., 2(2) ,
p.277, 1932.
10 FurryX.H., Jones,R.C. and Onsager, Phys./Roy..11.
p 1083, 1939.
11 Drickamero H.G., Frellow,E.W. and Tung,L.H., J.Chem.Phys.!
P 945, 1950.
12 Dobbs,E.R., and Jones,G.D., Rep.Prog.Phys., 20,
P 516, 1957.
