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THE LOCAL GROWTH RATE OF DESUBLIMATIONON THE SURFACE OF AN ANNULAR-TUBECOLD TRAP
Hiroaki UEDA* and Yoichi TAKASHIMA
Research Laboratory for Nuclear Reactors, TokyoInstituteof Technology, Tokyo, 152
Numerical analyses of the desublimation rate in an annular-tube cold trap were carried out.Dimensionless parameters necessary to obtain the thickness of the growing solid phase were
derived, and the effect of each parameter on the desired cooling length of cold trap was investigatedby using the reference distance.Experimental studies of the desublimation process also were made using water, benzene or
carbon tetrachloride vapor as a condensable gas. Various desublimated states on the cold surfacewere observed through a transparent outer tube, and the thickness of the growing solid phase wasmeasured along the axial direction. Somerepresentative results for the thickness were comparedwith those obtained by numerical analyses, and were in good agreement with them.Furthermore, somephenomenalistic studies were made with respect to the dependency on the
contents of inert gas to the apparent density of solid phase, the influence of free convection andthe occurrence of cracks.
Introduction
Cold traps in which condensable vapors desubli-mate onto the cold surface in separating from theflowing gas mixture have been used for various pur-poses such as the removal of organic impurities, thepurification of a volatile material or air conditioning.The desublimation rate, however, is not steady be-cause of the accumulation of the desublimated solid
on the cold surface. Therefore, its design has to bedistinguished from that of an ordinary condenser inwhich the vapor condenses as liquid.
Fundamental and phenomenalistic studies of de-sublimation have been made by many investiga-tors2'3'^, and are useful in obtaining some informationon crystal form, thermal conductivity and density.
However, there have been only a few studies6j8) aimedat the proper design of such equipment.Dunthorn5) calculated numerically the thickness ofthe growing solid phase for the removal of UF6.
Although the formulation of some basic equations forestimating the local growing rate was made and acalculation procedure was introduced, a systematicinfluence on the design could be hardly found from
this descriptive procedure.In the present paper, numerical analyses of the
desublimation process are dealt with for an annular-tube cold trap which may be one of the most simpleand basic types. The flow was assumed to be laminarin the cold trap and the desublimation to be controlledby diffusion, because these are general. Dimension-
less parameters which are useful for the design of thecold trap were introduced in the formulation of thebasic equations, and the necessary length of thecooling tube was evaluated by using the referencedistance defined here. On the other hand, the freez-ing state was directly through the transparent outertube of the cold trap, and the thickness of solid phasealong the axial direction was measured from time totime. The experimental results were compared withthe calculated results in some representative cases,and were confirmed to be in good agreement withthem.
1. Theoretical ConsiderationNowwe deal with an annular-tube cold trap inwhich the temperature of the inner cooling tube isuniform and the outer wall is thermally insulatedfrom the atmosphere.
The numerical analyses are performed in twosteps, that is, (I) evaluation of mass and heat transferat the initial condition (t=0)9 and (II) calculation ofthe thickness of the growing solid phase.
Received July 25, 1975.Presented at the 6th Autumn Meeting of The Soc. of Chem.Engrs., Japan, at Tokyo, November, 1972 and at the 8th
Autumn Meeting of The Soc. of Chem. Engrs., Japan, atTokyo, October, 1974.
VOL. 9 NO. 2 1976 121
1. 1 Basic equations in step (I)To obtain the equations for mass and heat transferrates, the following assumptions are used :1) The total pressure stays constant in the annular
tube.2) The desublimation rate is predominantly con-trolled by diffusion.
3) The fluid behaves as a continuum.4) Fog does not grow even if a region of super-
saturation exists.5) The flow pattern has already developed at the
entrance to the cooling tube.6) Diffusion and thermal conduction in the axialdirection are neglected.
7) The gas properties are constant.Under those assumptions, the equations of conti-
nuity, mass balance and energy balance in the gasphase can be expressed by several dimensionless
terms as follows (see Nomenclature) :dV
dz+-R
(RU)=O
oZdRdW
dR~PeM\dR?n RdRJ1 /82W 1 dW\-' r>2H '
(1)(2)
d& do_ i fd'd idodZ +~UdR~PeT\dR* +R dRdwdo ...T^RjR (3)
where
Z=z/2(ra-rx), i?=r/2(r1-r1), V=v/vim, U=u/vim,W=(w-we)l(wt-wt), 9=(T- T.)l(Ti-T.),
PeM=2vim(r2-r1)/DAB, PeT =2pgvimCP(r2-r1)/kg,r=DAB(CpA - CpB)(wi -we)/2CI>vim(r2-ri)
The boundary conditions areV= G(R),
y=0,u=0,W
=l, 6=1, W=0, 0=0,
dW at Z=O
at R=R±
dRå =o,dR=0, atR=R2
where
G(R)=a1{l -(a,Rf+a3 \n (a2R)}a1=2/{l+«1-(l -K1)/ln(l/K)}, a»=2(l -/c)
a3=(l -/c2)/ln(l/«), K=ri/ri
The velocity distribution at the entrance, G(R), is
obtained by solving the Navier-Stokes equation underthe condition that the flow is fully-developed and thetemperature is uniform.The flow pattern has to be known to solve Eq. (1)to Eq. (4). Numerous calculations would be necessaryto solve simultaneously the Navier-Stokes equationand Eq. (2). Then the velocity in the radial directioncaused by the vapor condensation is assumed to be
u=- 1 1 dWPeM (W°-W) dR(5)
as used for mass transfer in a circular tube by Bassal^.By using Eqs. (1) and (5), Eqs. (2) and (3) becomerespectively
{w-i)Gdz -pgK^-wi ^+^m
dR2+
dw»)å ]R dRJ
(6)
KW l)lJdZ-PeT{>* *V\3R'+R dR
\PeM-r(W°- W)dW 38dRdR(7)
whereW°=(l -wt)/(wt-w.)
The radial mass and heat fluxes at the interfacebetween the cooling surface and the gas phase in the
interval from Z to Z+dZbecomer _ Pp (W°-\)\[R> WGR -V+-*
JAR-PeM RiW>{SZ) l)B^Wa-WfR\z (8)
_(W°-\)r[B* 6GR ,D1Z+á" /m
whereJAR=JArl{pgDAB/2(r2-r1)}
QB =qrl{kt{Ti - T.)IHr>-r1)}
WhenSherwood number and Nusselt number aredefined respectively as Sh=2hM(r2-r1)/pgDAB and Nu=2hT(r2-r1)/kg, the following relations are derived :
Sh= W°JAB/ WmNu= QB/9m(10)
(ll)
where
WVRdR \ VRdRR± I JR±
CR2 JCR2
=\ 6VRdR \ VRdR
1. 2 Basic equations in step (II)The growth of solid phase on the cooling surfacecauses the boundary conditions to vary due to thechange of surface geometry and temperature. How-ever, the mass transfer rate could be considered to bea quasi-steady process, because the mass transfer rateis not very fast.
To obtain the time variation of the thickness, thefollowing assumptions were taken :1) The Sherwood number and Nusselt number do
not vary with time.2) The thermal conductivity of solid phase doesnot vary.
3) Thermal conduction through the solid phase istaken account of only in the radial direction.
Thus, the dimensionless radial mass flux, JAR, andheat flux, QR, at the interface can be expressed by
JAB=Sh(Wm- W,)/(W°- Wf)(12)
122 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
QB=Nu(Om-ef)(13)
whereW/=(H'/-M'.)/(w4-w.), e/=(r/- 7;)/(r4- r.)
The dimensionless thickness, A, is expressed bydt+ ~Jab
(14)
wheret + = t/{4ps(r2-rif/PgDAB}
The variations of the dimensionless average massfraction, Wm,and the temperature, 0m, with axialdistance become
dW.=-K-
sh (w°-wmy(wm-wf)
(15)
dZ "PeK (W°-l) (W°-Wf)
d6m_ (W°-Wm){Nu Sh (Wm-Wf)\dZ (W°-l) \PeT PeM (W°-Wf)\
X (©.-©,) (16)
whereK=2{R1+ d)/{R\ - (R1 + J)'}
K is a geometric factor of the effective cooling area.The energy balance at the interface is
E1JAE+QB=Qt=Ei3R
R=Rl+d(17)
whereEi=ptDABL/kg(Tt - T.), Et=k.lkt
In the left-hand side of Eq. (17), the first term repre-sents the energy transfer by latent heat and the secondterm that by sensible heat. The right-hand side rep-resents the heat conduction through the solid phase.The temperature distribution in the solid phase is
9.= Qt(A +Ri) ln (RIRi)IEi(18)
where=&f atR=R1+A
Wf is a function of 0f, which is derived from theequilibrium relation between the vapor and solidphase and from the operating conditions, as
W/=F(©/)= - W+ l/Jff1 (19)
^1= (Wl-w.)[l +{MBjMA){- 1 +(P/ai)exp(H2)}]H,= -a,l{Tc- Tt){6f+6°)
whereW'=wtKwt-wt), S'=Tc/(Tt-Tt)
and the constants ax and a2 are determined by theequilibrium relation, Pf=a1 exp (-a2/Tf).If Wf can be expressed approximately as Wf-c6fiwhere c is a constant, the dimensionless interface
temperature becomes from Eq. (17)f=(Y2-V Yl-2Y1Y,)/Y1
(20)
where
Y1=2c(Nu+ Y4), Y2=c(E1Sh+Nu0n)+ W°(Nu+ 74)Y3=E1Sh Wm+Nu6m W°
(start)
Pen. Per,W°etc.
Calculationotw.e
StepI 1Sh'N"1
StepI1 lB.C.aJT+=0|
jB.C.at Z=01
CalculationWf=c9fl °f/ lwt=F(e0
I EvaluationDetermination 1 of ®f I
I of ©f I ~JF
*-=I /^sCalculationI ofW-m.©«, |
å ^^^^C^l(STOPy 1WRITE1
Fig. 1 Block diagram of numerical calculation
F4 =M(^1 + ^)ln{(i?l + ^)/JR1}]
2. Calculated ResultsA block diagram of the numerical calculation is
shown in Fig. 1. The parameters required to calcu-late the dimensionless thickness of solid phase arePeM, PeT, W°, r> Ex, E2, F(6f), Rx and R2.
To give a design standard for the effective lengthof such a cold trap, we shall define a reference dis-tance necessary for desublimation, zd9 within whichthe volume of the solid phase corresponds to 90%ofthe total desublimated volume. Assuming that thepartial pressure of the vapor in bulk becomes the
saturation pressure corresponding to the temperatureon the cooling surface before the flow is out of thecold trap, the dimensionless reference distance, Zd(=zd/2(r2-r1)), should be determined to satisfy
SZd 09
{(R1+ Jf-Rl}dZ=-^å PeM-t+-(Rl-Rl) (21)
The variation of the reference distance with time isshown in Fig. 2. The figure shows that the increaseof the reference distance becomes larger as E1 in-
creases and E2 decreases. As seen in Eq. (17), thesame value of E2/E1 would give about the same ref-erence distance if J5i>l. Accordingly, in Fig. 2,
the curve for Ex=60 and is2=30 overlaps with thatfor E1=20 and E2=l0.The influence of W° on the reference distance isshown in Fig. 3. As it approaches 1, the referencedistance at t+=0 becomes shorter due to the higherdesublimation rate near the entrance. However, as
VOL. 9 NO. 2 1976 123
0.51 1 1 1 1 ^^U-
Wf =3©f r=o ^^^\
Ei=20 R«=a156 \Ei=30
ai " Ei=60 Rz=0.656
0.0 1 1 1 1 1 10.00 0.01 0.02 0.03
t+ (-)
Fig. 2 Variation of reference distance with time inW, =39f
0.51 1 1 1 , 1
Wf=30f
7"0.4 - Ei=60 ^^^00^^
40.3- ^^^^^^^ W°=2 I ~
y\/\ \ w°=2 W°=1.25
W°=1.25 Ra=0.656
0.0 1 1 1 I I I0.00 0.01 0.02 0.0 3
t+ (")
Fig. 3 Effect of W° on reference distance
a5 rn-i-~p-'-~7~[-
^ [/// ^^^ I I F |ETg[0L2»// /^ J_ ~T0~ -
\\/ JL 3®f 20|f/ / _3 3 0_0.1\f -A- 10 -
U _5_ 4©f 20
0 0.01 0.02
ZIPeu (-)
Fig. 4 Effect of E1 and E2 on maximumthickness
I I I I IQ - -4° O exp.
1^Z a-r^L-<b) _2
I 1 PeM 1 PeT 1 W° 1 y I E, I E2 I
Q 21.0 29.8 2.01 -0.0015 17.4 25.1"b| 6.8 9.1 4.52| 0.0195167.1 |54.5]
Q| 1 1 1 1 1 lo0 0.01 0.02 0.03
t+ (-)
Fig. 5 Comparison of reference distance between calcu-
lated and experimental results
time proceeds and the thermal resistance of the solidphase increases, the effect of W°becomes reverse onthe reference distance, because a higher total desub-limation rate would be required for a smaller valueofW°.
WhenWfreaches 1, no more vapor condenses andthe thickness of the solid phase never grows. Such
a state (Wf= l) extends further from the entrance withtime. As 0m and Nu vary with Z, the maximumthickness, ^max, vary with the distance, Z. By solvingEqs. (16) and (19) under the condition that Wf=l,@mand 0f can be obtained, and then 4max can bedetermined from Eq. (17). For the determination ofJmaxj it would be appropriate to use the value ofNu at W°=oo. Typical results of Jmax are shownin Fig. 4.
Some results by these calculations were compared
with the results obtained by the experiment describedlater. An example is shown in Fig. 5, where twocases are compared: (a) benzene-nitrogen and (b)water-nitrogen. Each Wf in (a) and (b) becomesrespectively
Wf= -4.0i+ l/H1 for (a)#i=0.1053+exp(-23.27+ 109.9/(©/+5.030))
Wf= -0.693+ 1/#! for (b)^1= -0.1067+exp(-22.08+ 1 17.1/(6>/+5.1 19))
The physical properties at a cooling surface tempera-ture were used for the above cases. Theresults inFig. 5 show good agreement with each other.
3. Experimental ProcedureThe flow diagram is shown in Fig. 6 and the detailsof the cold trap are shown in Fig. 7. The cold trapis a type of horizontal annular tube in which the innercooling tube is made of copper and plated with nickel.A cork rod 10cm long is attached to the head of theinner tube to insure the development of flow pattern.The outer tube consists of a glass annular tube, andtherefore the condensing state can be directly ob-served. The annular section of the outer tube iskept at a pressure of 10~5 mmHgby an oil diffusionpump for the purpose of thermal insulation. Thebrine, a propylglycol solution cooled by a refrigerator,was used for cooling the inner tube under the ex-perimental condition of surface temperature no lessthan -23°C. It was comfirmed that the temperaturerise between the inlet and the outlet of the inner tubewas below 0.5°C by detecting with two sheathedresistance thermometers. In the case of surfacetemperature below -55°C, direct cooling by freon-22wasadopted. The temperature of the cold surface,Tc, at z=2, 30 and 60cm and the inlet of the coldtrap, Tu were measured by thermocouples (CC)0.2mmin diameter. The total pressure was meas-
124 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
ured by D.O.P. manometers at z=0 and 45 cm.Flow rates of both the vapor and inert gas were
controlled through each critical nozzle by adjustingthe pressure at a vapor generator or the pressure ofan inert gas in front of the nozzle. The range of gasflow rate was 0.5 to 3 mol/hr, and the mole fractionof the vapor 0.2 to 1.0. After the gases were mixed,they were introduced into the cold trap, and theinert gas was exhausted by a rotary vacuumpump.The vapor that could not condense in the cold trapwas captured by a liquid nitrogen trap.The thickness of solid phase was measured with anaccuracy of 0.05 mmby a travelling microscope on aslider 1 m long. The positions measured were 7 to 12points in the axial direction and the time intervalswere 10 to 15 minutes. If the desublimation rate inthe upper surface of the cooling tube differed fromthat in the lower surface, both thicknesses weremeasured.
4. Operating Conditions and Collection EfficiencyAs the Reynolds number (Re=2pgvim(r2-r1)/iLi)
evaluated by the flow rate at the entrance was below100, laminar flow was maintained in the cold trap.The operating pressure was 0.55 to 12.5mmHg.
Thus, the flow was in a viscous region as the Knudsennumber (Kn=A/2(r2-r1)) was below 0.01. The pres-
sure drop between the two measuring taps was negligi-bly small, and so the average axial velocity should
decrease in the flowing direction.All the experiments satisfied such a condition thatthe partial pressure of the vapor in bulk becamealmost equal to the saturation vapor pressure cor-responding to the temperature of the cold surface
before reaching the end of the effective cooling section.In this case, the collection efficiency, rj, that is defined
as r}=(GAi-GAn)/GAi can be given by
7=1-
GB(PelP)GAi l-(PJP)(22)
where GAi [mol/hr] is the flow rate of the vapor at theentrance, GAn[mol/hr] is the flow rate of the vaporuncaptured in the cold trap and GB[mol/hr] is theflow rate of the inert gas. In Fig. 8, rjcai calculatedfrom Eq. (22) was compared with 7]exp that was ob-tained by measuring the mass of the vapor caught inthe liquid nitrogen trap. The two values agree well,and the loss of vapor was always negligibly smallcompared to the total mass even though fog forma-tion might occur in some cases, as far as these ex-periments wereconcerned.
5. Macroscopic Aspects of the Growing Solid StateTwotypical photographs of the growing ice on thecold surface at Tc=-55°C are shown in Fig. 9.These examples are quite contrasting concerning the
I pi© 1®^^© [frQn1.cold trap 2.inert gas cylinder 3.vapor generatorA.critical nozzle 5.liq. nitrogen trap 6.rotary vacuumpump 7.diffusion pump 8.refrigerator 9.brine tank
10.gear pump ll.mercury manometer 12. D.O.P.manometer 13. McLeod gauge 14travelling microscope15.slider 16. soap flowmeter
Fig. 6 Flow diagram of experimental apparatus
. 100 X 850 -
pressure tap
T-r=*=^±=l=M',1 -rL^*1}^-J c -- ---- -J ---.^---- -----.mA
I I V*>^*-«sv^i djisvLlated rod /coolingtube | ^
mixed gas inlet chain ' vacuum ,insulation exhaUst
Fig. 7 Detail of cold trap
l.Or-| | [ 1 1 Q
Key System J&?Q H2O y~*
I 0ICCg4 I /I0.8 I I /-
» \-^\0.5 (B'8
'Y\ 1 1 1 1^05 0^6 0?7 078 0?9 LO
Fig. 8 Comparison between rjcai and iqex
macroscopic aspect. In (a), the desublimated phaseis very porous and consists of rough and needle crys-tals, whereas in (b) the phase is very compact and hasa smooth surface. Various aspects can be realizeddepending upon the contents of the inert gas beforeentering the cold trap. The apparent density ofsolid phase, pa, varies with the mole fraction of theinert gas at the entrance of the cooling section, xBuasshown in Fig. 10. The greatest density obtainedunder the condition of relatively low mole fraction ofthe inert gas was about the same as that of a com-pletely crystal ice at 0°C, 0.917 g/cm3. Biguria2) andBrian3) showed experimental results on the density ofsolid phase in the case of much higher fraction of theinert gas than that in this case, and obtained densitiesdown to 0.1 g/cm3.
VOL. 9 NO. 2 1976 125
rc=-55°C(a) H2O-N2
GAi=0.841 mol/hrGB=1.31 mol/hrP-2.15 mmHg
(b) H2O-HeGAi= 1.68 mol/hrGB=0A65 mol/hrP=0.831 mmHg
Fig. 9 Photographs
^0.8
u
CT0.6
Qs0.4-\ or J»v
^81Tc=-55°C
H2O-Het =30min
0.0 0.2 0.4 0.6 0.8 1.0Xbi(-)
Fig. 10 Effect of xBi on apparent density for H2O
9^S (mol/hr) (mol/hr) (mmHq)
O H2O-N2 0.759 0.265 0.971~®IH2O-N20.8411,312.15
0.6 ir-
I o.Aen
^0.2 -
["Tc = -55°c l0.01 ! 1- ' ' -'
0 20 40t (min)
Fig. ll Time dependence of apparent density
The apparent density will increase slightly withgrowing time if it is porous, because the vapor willdiffuse into the colder position through the porouspart. Examples of time variation on apparent densityare shown in Fig. ll.Photographs for desublimation of water vapor atTc=-23°C are shown in Fig. 12. In this case, thevapor desublimated in a granular form during theinitial period and soon the surface was covered bythese grains. Once a smooth surface madeup, thesmoothness was maintained to the last. The influenceon the macroscopic aspects of solid phase by eitherchanging the kind of inert gas or changing its molefraction was not appreciable. The density of solidphase obtained was about 0.91 g/cm3 in all cases at
TC=-23°C.
H2O-He
TC=-61°C
G^=1.87 mol/hrG*=0.913 mol/hrP=2.85 mmHgt=3 min
(b) /=20min
Fig. 12 Photographs
(a) C6H6-Herc=-61°C
GAi =0.751 mol/hrGs=1.14 mol/hrP=3.51 mmHg
(b) C6H6-N2
TC= -2VC
GAi= 1.29 mol/hrGB =0.560 mol/hr
P=9.71 mmHg
Fig. 13 Photographs
1.21 1 -i 1 1 1
£ ,0 -^8^^^-^-
<£0-8"IKey|System] | |O C«Ht-N* Tr=-61°f
II3lak-Hel | ,L0T I I I I
0 0.2 0.4 0.6 0.8 1.0XBi (-)
Fig. 14 Effect of xm on apparent density for C6H6
Typical photographs of desublimated benzene areshown in Fig. 13, where (a) was operated at Tc=-61°C and (b) at Te=-23°C. The crystal was
opaque and white at Tc= -61°C, and transparent andscaled at TC=-23°C. The surface became a little
more rough in the later part due to the higher molefraction of inert gas. However, the apparent densitywas not significantly dependent on the mole fractionat the entrance if it is less than 0.8, as shown in Fig.14 for rc=-61°C. For all the data at rc=-23°C,the density obtained was about 1.02 g/cm3, which isthe density of the complete crystal4}.
The desublimated carbon tetrachloride was trans-parent and had a smooth surface at Tc=-70°C.The apparent density obtained was 1.8 to 1.9 g/cm3,which maybe the same as that of the complete crystal.The solid phase often had cracks during the operatingperiod, as shown in Fig. 15. After the operation
126 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
ended, muchfiner cracks began to appear every-where. Because the supply of latent heat for de-sublimation ceased, the solid phase would have be-come colder very quickly, losing the sensible heat,and would have been constricted significantly. Sucha phenomenonwould makeit possible to remove thesolid phase easily by a simple mechanical scraper.6. Reference distance for desublimation in the early
stepAt 7=0, the mass fraction wd at zd is given by
wd=0.1(GAi-GAn)+GA
OA(GAi-GAn)+GAn+(MB/MA)GB(23)
Equation (23) is rewritten, in normalized form, to :Wd=O. l W°/(W°-0.9) (24)
By solving Eq. (15) under Wf=0 and K=K0 (=2R1/(Rl-Rl))9 the dimensionless reference distance, Zd9 isderived by
1 l V{(^°-l)ln(10)+0.9} (25)PeM ShavK0 W
where
Shav = (YdSh dZy2
The variation on Zd/PeM with W° is shown in Fig. 16,where experimental results were plotted for severalcombinations of condensable gas with inert gas andthe numerical analysis is expressed by a solid line.Both experimental and theoretical results showedfairly good agreement, though some exception due tothe influence of free convection was observed as willbe explained later.
7. Time Variation on the Desublimated RateAs the solid phase is growing on the cold surface,the local desublimation rate varies with the time.An example of growing rate of thickness is shown inFig. 17.
The desublimation rates in the upper and lowerparts of the cooling tube were observed to be differentin such cases as C6H6-He at Tc=-61°C and CC14-He at rc=-70°C in Fig. 18. The large difference ofmolecular weight between the vapor and inert gasproduced free convection around the cooling tube,where a remarkable concentration gradient will takeplace and the free convection flow will rise around thecooling tube and fall along the outer wall. The freeconvection due to the temperature gradient causes areverse flow in such a case. However, the tempera-ture effect is relatively smaller than that of the con-
centration gradient, that is, the Grashof number formass transfer, GrM9is larger than the Grashof numberfor temperature, GrT.
The thermal conductivity of solid phase, ks, was
CCl4-He
7c=_70°C
GAi =0.756 mol/hr
Gs=1.51 mol/hrP=4.86 mmHg
Fig. 15 Photograph
Q5i 1 1
I0.2 ^4^°
^ ^^> systemMsil3 o H2O-N2
0>1"SL * H2°~He ~W& 3 C6H6-N2
/à" ° C6H6-He
I e CC14-N20.05à" 1 aI CCU-He I-
1 2 5 10W°(-)
Fig. 16 Relation between Zd/PeM and W°
I I I I I0.8- H2O-N2 Gai=0.841(mol/hr) Gb=1.31(mol/hr)
P=2.15(mmHg) Ti=23(°C) Tc=-55(°C)
^"=^\ (^^^^ ^^^' z=5
°0 20 40 60
t (min)
Fig. 17 Time variation of thickness of solid phase
system Gai Gb P TcO CeHe-He 0.549 1.1A 3.46 -61B CCU-He 0.857 ~0.569| 2.85| -70|
_°-8" lower ^"^ ~'P upper J^'^^
l°-6-2=2,cm,̂̂ ^ "
ol^e^ I 1 1 10 20 A0 60 80
t (min)
Fig. 18 Effect of free convection
chosen so that the profile obtained experimentallymay be consistent with that obtained from the nu-merical analyses. For example, the value of thermalconductivity of desublimated benzene was estimatedas 0.00060 cal/cm-sec °K at TC=-2TC for benzene-nitrogen gas mixture, and the comparison of solidphase profile obtained from experimental results withthat from the numerical analyses is shown in Fig. 19.
VOL. 9 NO. 2 1976 127
6I 1 1 1 1 1 1 1 1 15 L (a) GAj=0.932(mol/hr) t(min)"_
Att?^ GB=0.519(mol/hr) |-|f-_3P\x %l P=9.30(mmHg) |^| gQ |_
^5O ^^^^-yn (b) GAj=1-31 (mo(/hr)
^A Q^^O^oJ^1^) GB = 0.269 (mol/hr)
cm 3(^ ^O^. "^^(D P=8-97(mmHg)
5 _a, (c) GAi=1.31 (mol/hr) _
^J ^l^g^ Gb=0.5A5 (mol/hr)
^P "vQ-Q^o ^"CD< P=9.71 (mmHg)
. 0 5 10
z (-)
Fig. 19 Comparison of experimental results withthose obtained from numerical analyses for C6H6-N2
6I 1 1 1 1 1 1 1 1 1S0r (a) (*j=0.741(mol/hr) t(min) _
4]\ %-0.997(mol/hr, |-f--
3I \V^<D-^ P=2.96CmmHg) |q| 60 |_
o -i-i-i-i-T P^T rH^^^5 - ^b) GAi=0.731(mol/hr) -
^-^(f\^ GB=0.257(mol/hr) -
^ 3ySs^O-Z5S^>3r-^_ P = 1.03 ( mmHg )
6iV^NN>N\ ^C) GAj=0.933(mol/hr) ~
I O\^\ GB=0.260(mol/hr)
\. O N. q\ P=1.58(mmHg)
0 1 2Z (-)
Fig. 20 Comparison of experimental results withthose obtained from numerical analyses for H2O-N2
These profiles are consistent with each other, and thevalue of thermal conductivity is a little lower than thevalue, 0.00067 cal/cm-sec-°K at T=0°C or 0.00075cal/cm-sec-°K at r=-78.5°C7). About the same
results were obtained in the case of water-nitrogen gasmixture at Tc=-23°C, as shown in Fig. 20. The ther-mal conductivity evaluated was 0.0025 cal/cm à"sec à"°K,which is muchsmaller than that of the transparentice, 0.0058 cal/cmà"sec à" °K.
Conclusion
1) Numerical analyses of the desublimation ratein an annular-tube cold trap were carried out. Fromthe basic formulation, the dimensionless parametersnecessary for the calculation were derived.2) The variation of effective length of the coolingtube by these parameters can be evaluated by usingthe reference distance defined here.
3) The analytical results were in good agreementwith the experimental results.
4) The apparent density varies with the fraction ofinert gas at the entrance. The reduction of density ismore remarkable for lower cooling temperature.
5) The occurrence of fine cracks was often ob-
served for the desublimation of carbon tetrachloride,especially just after the feed of gas mixture ended.
6) Asymmetric desublimation was sometimes ob-served due to the free convection around the coolingtube, especially when the density of condensable gaswas significantly different from that of inert gas, forexample, the gas mixture of C6H6-He or CCl4-He.
Nomencl ature
Cp = specific heat of gas [cal/g- °K]Cp = average specific heat of gas mixture [cal/g-°K]
Dab = diffusivity [cm2/sec]E, = PgDABLlk^Tt - Te) [-]E2 = kjkg [-]
G = gas flow rate [mol/hr]hM = mass transfer coefficient [g/cm2 - sec]hT = heat transfer coefficient [cal/cm2 « sec - °K]Jar = dimensionless flux of condensable gas
from gas phase to interface [-]jAr = flux of condensable gas from gas phase
to interface [g/cm2 à"sec]K = 2(i?1+J)/{i?! -0R1+J)2} [-]
k = thermal conductivity [cal/cm* sec - °K]L = latent heat [cal/g]M = molecular weight [g/mol]Nu = Nusselt number, 2hT(r2-r1)lkg [-]P = total pressure [mmHg]PeM = Peclet number for mass transfer,
2vim(r2 - r1)IDAB [-]
PeT =å Peclet number for heat transfer,2pgvimCp(r2 - r1)lkg [-]
p = partial pressure of condensable gas [mmHg]QR = dimensionless sensible heat flux from
gas phase to interface [-]qr = sensible heat flux from gas phase to
interface [cal/cm2 à"sec]=dimensionless radial distance [-]=dimensionless radius of inner tube [-]
= dimensionless radius of outer tube [-]=radial distance [cm]
= radius of inner tube [cm]= radius of outer tube [cm]
= Sherwood number, 2hM(r2-r1)lpgDAB [-]= average Sherwoodnumber,
(£dsh dz) izd [-]=temperature [°K]=time [sec]=dimensionless time [-]
= dimensionless velocity in radial direction [-]= velocity in radial direction [cm/sec]= dimensionless volocity in axial direction [-]= velocity in axial direction [cm/sec]= average axial velocity at entrance [cm/sec]= normalized mass fraction of condensable
gas [-]
128 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
W° = (l -Wc)l(Wi-Wc) [_]w = mass fraction of condensable gas [-]x = mole fraction of condensable gas [-]Z =dimensionless axial distance [-]Zd = dimensionless reference distance [-]z =axial distance [cm]
zd = reference distance [cm]
T = DAB(CpA- CpB)(wi -wc)l2Cpvim(r2-r1) [-]A = dimensionless thickness of solid phase [-]d = thickness of solid phase [cm]0 = dimensionless temperature [-]k = rjr2 [-]
X = mean free path [cm]in = viscosity [g/cm- sec]
P = density [g/cm3]
<Subscrip ts>A = condensable gasB = inert gasc = cold surface
/ = interface between gas phase and desubli-
mated solid phaseg = gas phasei = entrance to cooling sectionm = average in cross section of annuluss = solid phase
Literature Cited1) Baasel, W. D. and J. C. Smith: AIChEJ., 9, 826 (1963).
2) Biguria, G. and L. A. Wenzel: Ind. Eng. Chem. Funda-
mentals, 9, 129 (1970).3) Brian, P. L. T., R. C. Reid andT. Shah: ibid., 9, 375 (1970).4) Cox, E. G.: Rev. Mod. Phys., 30, 159 (1958).5) Dunthorn, D. I. P.: Oak Ridge Report K-L-6220 (1968).6) Dyer, D. F., R. W. Mayne and J. E. Sunderland: Brit. J.
Appl. Phys., 18, 297 (1967).
7) Eucken, A. and H. Englert: Z. fur die gesamte Kdlte-Ind.,6, 108 (1938).
8) Kawamura, Y.: Dr. Eng. Thesis, Tokyo Institute ofTechnology (1961).
9) Yamakawa, N. and S. Ohtani: Kagaku Kogaku, 36, 197(1972).
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