GREEKSYMBOLS Ait
II
&
multicomponent Schmidt number, dimensionless = composition function in Equation 26. dimensionless = gravitational potential energy function, L2t-2 = 2m,’(m 1) = binary boundary-layer thickness in Equation 76, L = multicomponent boundary-layer thickness in Equation 81, L = position coordinates on S = eigenvalues of [ A ] , n - 1 in number? L-2t = distinct eigenvalues of [AI, J in number, L-2t = viscosity, h4L-lt-l = density, M L - 3 = local rate of entropy production, e.u. Lp3t-I = viscous stress tensor, = dimensionless interfacial velocity in Equation 64 = dimensionless interfacial velocity in Equation 65 = p.+,/pa:
+
p
6i ( >4
Ai.
A, P U
..
y.4B Q~
-
OVERLIYES
per mole
=
A
= per unit mass
-
=
-
=
transformed according to Equations 17a, b, c partial molar in Equation 88
SUBSCRIPTS A. . . B = species in binary system z, 1. k = species, transformed functions or transformed varia
ables in multicomponent system reference state = surface conditions =
0. 1
MATHEMATICAL NOTATIONS
[ ] 0m
= column vector or matrix, depending on context = negative side of zero = upstream conditions
literature Cited
(2) Cussler, E. L., Jr., Lightfoot, E. K., A.I.Ch.E. J . 9, 702, 783 11963). (3) Dunlop, P. J.: Gosting, L. J., J . Phys. Chem. 63, 86 (1359). (4) Fadeev, D. K.: Fadeeva, V. N., “Computational Methods of Linear Alcebra.” translated bv R. C. LVilliams. Freeman. San Franciscoo, 1963. (5) Frazer, R. A,: Duncan, LV. J., Collar. A. R.. “Elementary Matrices,” L-niversity Press, Cambridge, England, 1938. (6) Gibbs. J. LY.. ”Collected LVorks,” Vol. 1. Yale Universitv Press, k e w Haven. Conn., 1948. (7) Gilliland, E. R., Sherwood: T. K.; Ind. En?. Chem. 26, 516 (1934). (8) Halmos. P. R.: ”Finite Dimensional Vector Spaces:” p. 156, Princeton Cniversity Press, Princeton, N. J.. 1942. (9) Hildebrand. F. B.: “Methods of Applied ,Mathematics,“ Prentice-Hall, Englewood Cliffs, N. J., 1952. (10) Hirschfelder, J. 0.:Curtiss, C. F.: Bird, R. B., “Molecular Theory of Gases and Liquids,” Chap. 11, LViley: N e b 7 York, 1954. (11) Marcus, M.: ‘.Basic Theorems in Matrix Theory,” National Bureau of Standards Applied Mathematics Series, No. 57, 1960. (12) Mickley, H . S., Ross? R. C.: Squyers, A. L., Stewart, 11.. E., Natl. Advisory Comm. Aeronaut. Tech. Note 3208 (1954). (13) Onsager, L., Ann. >V.Y.Acad. Sci.46, 241 (1945). (14) Onsager. L., Phys. ReLm. 37, 405 (1931); 38, 2265 (1931). (15) Perlis, S.; “Theory of Matrices,” 3rd printing, Chap. 9, Addison-\Yesky, Reading Mass.. 1958. (16) Prober, R.: Ph.D. thesis, University of LVisconsin, 1961. (17) Prober, R.: Stewart, I V . E., Intern. J . Heat ‘Mass Transfer 6, 221 (1963); Corrigenda, Ibzd., 6, 872 (1963). (18) Stewart, LV. E.: A . I . C h . E . J . 9, 528 (1963). (19) Stewart, LV. E., Sc.D. thesis, Massachusetts Institute of Technology, 1951. (20) Stewart, LV. E.. Prober, R.. Intern. J . Heat ~ti‘ass Transfer 5 , 1149 (1962); Corrigenda, Ibid., 6, 872 (1963). (21) Tisza, L.. “General Theory of Phase Transitions,” Chap. 1: “Phase Transformations in Solids,” R. Smoluchowski, J. E. Mayer, and LV, A. LVeyl, eds., LViley, New York, 1951. (22) Toor, H. L., A.I.Ch.E. J. (to be published). (23) IVedderburn, J. H., “Lectures on Matrices,’’ Vol. 17, p. 28, American Mathematical Society Colloquium Publications, 1934. (24) W‘ilke, C. R., J . Chem. Phys. 18, 517 (1950). >
,
(1) Bird, R. B., Stewart, LV. E., Lightfoot, E. N., “Transport Phenomena,” 4th printing with corrections, LViley, New York, 1964.
RECEIVED for review March 9, 1964 ACCEPTED April 14, 1964
FRACTIONAL SOLIDIFICATION OF EUTECTIC-FORMING MIXTURES W
.
R
.
W I L C 0 X, Aerospace Corp., El Segundo, Calif.
The solute redistribution resulting from progressive freezing or zone melting of a simple eutectic-forming mixture has been solved analytically for pure diffusional mass transfer. Comparison with experimental results shows ,that constitutional subcooling usually occurs and produces considerable trapping of impurity.
zone melting work was done on high-temperature, For these materials the equilibrium ratio of solid to liquid solubility of impurity (the distribution coefficient) was often a constant. For this reason and for simplicity, nearly all theoretical treatments of solute redistribution in fractional solidification processes have assumed a constant distribution coefficient (3, 9 ) . Recently. however, a considerable amount of zone melting work has been done on organic systems (2, 9). Most organic systems d o not have a constant distribution coefficient but have phase diagrams of the eutectic type, as shown in Figure 1 ( 7 , 9). Concentrated mixtures of metals or inorganic compounds also usually have eutectic-type phase diagrams : either of the simple eutectic type or the limited solid-solubility type. I n this paper simple eutectic behavior (no solid solubility) is considered in detail. ARLY
E high-purity materials--principally semiconductors.
Assuming ideal conditions the concentration profile resulting from the zone melting of a mixture of composition ze, in Figure 1 can easily be predicted (9). Initially the solid freezing out would contain no impurity and so the solute \\.odd accumulate in the melt. M’hen the melt reaches the eutectic composition w e , the solid composition would j u m p in a step function to the original concentration, E,. Each additional zone pass of a semi-infinite solid would produce an added amount of pure material equal to that of the first pass. Experimentally such concentration profiles have not been observed (5. 6, 9). T h e composition deviates much earlier from pure material. the approach to E, is not a step function, but is gradual. and the zone composition is rarely E,. TIVO mechanisms could account for this. O n e is the occurrence of constitutional subcooling, which produces a rough freezing interface, which in turn tends to trap the melt (9). T h e exact effect of coiistitutional subVOL. 3
NO. 3
AUGUST
1964
235
I
0
I
WO
we
100
WEIGHT PERCENT SOLUTE
Figure 1 . Schematic solid-liquid phase diagram for binary systems forming simple eutectics wo. ws.
Original weight fraction Eutectic weight fraction
cooling on the concentration profile is not known a t present, even for mixtures with a constant distribution coefficient T h e second mechanism which may be proposed to explain the experimental concentration profiles is that of diffusion limited segregation. I t is this latter possibility that is considered in detail here. T h e limiting case of no liquid mixing (mass transfer by diffusion alone) is solved analytically for the concentration profile. Although the result does not compare favorably with the experimental concentration profiles. three other uses for the results are apparent: Prediction of the total separation and final zone concentration after passage do\vn a long solid. Previous comparable results for incomplete liquid mixing have proved useful in this respect (5, 7 7 ) . Prediction of the onset of constitutional subcooling--Le., the length of pure solid. Prediction of zone melting results for very dilute mixtures Ivith a constant distribution coefficient k < 0.1, with pure diffusional mass transfer. Previous results (70) are uncertain for k < 0.5.
Assumptions T h e folloiving assumptions are made in order to solve the problem of solute redistribution caused by a single zone pass of a eutectic-forming mixture: Constant zone size, L Constant freezing rate, V Constant and equal cross-sectional area in solid and melt Constant. but not necessarily equal, density in solid and in liquid, p s and p l Lniform initial concentration, ze, Thermodynamic equilibrium according to simple-eutectic phase diagram (Figure 1) a t freezing interface Planar uniform freezing interface Negligible diffusion i n solid No convective mixing of melt Constant diffusion coefficient in melt The differential equation and boundary conditions have been derived and discussed (5. 70).
Solutions Only final results are given here, Details of the solutions are given elseivhere (7. 8). Initial Period. During the initial period. pure solid is obtained and the impurit)- content of the melt builds up. T h e concentration profile in the liquid is found by the method of Laplace transform to be: 236
l&EC
FUNDAMENTALS
For convenience 9 may be regarded as a dimensionless concentration in the liquid, '7 a dimensionless distance into the melt from the freezing interface. T a dimensionless distance down the charge (or dimensionless time). and p a dimensionless zone size. For p 2 5 the perturbation in concentration does not reach the melting interface, and the zone is, in effect, infinite in sizei.e., infinite progressive freezing. hlathematically a separate solution for p = m is also convenient. Thus the following equation may be found by Laplace transforms (7). or by letting k + 0 in the solution of Smith, Tiller, and Rutter ( 4 ) . For
P =
03,
I n order to predict the length of pure material obtained, it is necessary to find the time, T ~ at , which the concentration a t the freezing interface, q q= 0, equals the dimensionless eutectic composition, E . This is done by setting 1 = 0 in Equation 1 to obtain f o r p < 5
and in Equation 2 to obtain f o r p
> 5,
Equations 3 and 4 are plotted in Figure 2 for various values of zone size, p. The dashed lines shojv the approximate solution derived by a crude analysis of limited numerical calculations (5, 70)
I
’
2
l
l
l
l
’
4
6
I
“0
0
2
I
Figure 2. Relative solute concentration at freezing interface during initial period as function of relative time for various vallues of relative zone size Initial period ends a t rUwhen
--
- .. -
3
4
=t
c$,,=~
Figure 3. Relative solute concentration in solid during final period as function of relative time for various values of relative zone size
= c
From Equations 10, 12, and 15 for e
3 and 4 Approximate solution, Equation 5
24
E x a c i solution, Equations
was also derived. These approximate results turn out to be accurate for small values of T and t and will be shown to be invaluable for solutions to the terminal period. Useful approximations valid for q , = o 3 4 are found by letting i + m in Equation 1 to obtain f o r p < 5 , 0 =
Letting
T + m
Q
+ (1 +
T
- ’)e-?Q
- 2cP1R
in Equation 2 we likewise obtain for p Q
=
1 . + (1
+r
-
’)e-?
(7)
>5 (8)
Solutions for solid concentration ivith a constant distribution coefficient may be found by setting $s = k Q,,=o. T h e result will be valid for i < 0.5 f o r k = 0.1, for i < 10 for k = 0.01> and for T < 20 for k = 0.001 (7). Abovr these values of T the impurity content of the solid begins to be appreciable compared to re’,. Terminal Period. E:xact solutions to the terminal period could not be obtained. It was found necessary to use the approximate results, Equations 5 and 6>for e < 4 and Equations 7 and 8 for e 3 4 as initial conditions. By separation of variables \ye obtain for f 5, @ =
1
+
(c
- 1)c-V
in the liquid, and
+ in the solid.
in the liquid. and
Similarly for
e
< 4 there results
(10) in the solid.
The constants are given by mj cot mj
+ P2
- =
0
(11)
VOL. 3
NO. 3
AUGUST
1964
237
0.4
0.2
1
0.q-
O . 1 0.1
I
I
0
0
I
Figure 4. Plot of values of p
3
2 +B
I
1
I
I
4
rt as a function of r f for various
From Equations 10, 1 1 , 13, and
17
with e = 3
in the liquid, and 'The concentration profile for the final period is found in Figure 3 as the curve forp = a , Soting that the concentration in the solid us = C # J ~ ~ and ( . ' ~that , distance z = z, ( T & / V ) pi/^,)^, the final concentration profile is obtained and shown in Figure 6. For comparison the corresponding curve (a step function) for complete liquid mixing is also shown in Figure 6. This does not compare favorably with the few available experimental results (5, 9). Hence it is concluded that constitutional subcooling must significantly affect the concentration profile resulting from the fractional solidification of a simple eutectic-forming mixture. Apparently theoretical and experimental studies are now needed to enable prediction of the quantitative effect of constitutional subcooling on mass transfer in zone melting and related processes.
+
in the solid. T h e foregoing results are plotted in Figures 3, 4, and 5. Comparison with Experiment
To compare these results with experiment, a solid-concentration profile is calculated. T h i s also illustrates one possible use of the results. I t is supposed that a single zone pass is made of a eutecticforming organic mixture under the following typical conditions: Initial concentration, zc,, 0.1 \reight fraction Eutectic composition, w ; , 0.4 weight fraction Zone length, L: 1 cm. Diffusion coefficient, D,0.1 sq. cm. per hr. Zone travel rate, V , 1 cm. per hr. Ps = PZ
The resultant concentration profile in the zone refined solid is desired. First p is calculated:
For values o f p over 5 it is known that the result is the same as for a p of infinity. Next E is calculated: E, 0.4 e = - = - = 4 Wo 0.1
T h e initial period, during izhich pure material is obtained, ends when q 1 = 0 = e. From Figure 2, usingp = a and e = 4. this occurs a t T~ = T = 2.2. Using the definition of T the length of pure material is 238
I&EC
FUNDAMENTALS
I
O b '
1
3'5
1
IO
I5
1
(
1
I
20 25 C STANCE, z(cm1
3'0
35
4'0
Figure 6. Comparison of calculated concentration profiles for complete mixing and for pure diffusional mass transfer e = 4 p = 10
Nomenclature
p
eigenvalue in Equation 9, given by Equations 12 and 13 = molecular diffusivity of solute in melt, sq. cm./sec. = index. 0 i 1. + 2, etc. = index. 1 . 2. etc. = equilibrium dktribution coefficient, ws/wr - 0 = length of zone. cm. = eigenvalue given by Equation 11 = dimensionless zone size, p = ( L V / D ) ( p , / p J
Q
=
r
= e/(€
R
=
C,?
D
i 1
k
L mj
7,
= duration of initial period during which solid contains
r1
9
= dimensionless time from beginning of terminal period, 71 = ( T - 7,) = dimensionless composition of melt a t 17 and T, q =
&-o
= dimensionless composition of melt a t freezing inter-
no solute-i.e.,
=
Tat which pq=ofirst equals
w/w,
face a t
T,
=
wx=o/wo
= dimensionless composition of solid a t
T ? p,,
= m,/w,
m
Literature Cited
F-~P
i=o
C
- I)
m
ipe-iP
*=o
t x
I’ u’
we u, zc,? w,,n z
= = = = =
= = = =
time from initi.ation of freezing: sec. distance from freezing interface into melt, cm. rate of solidification, cm.:sec. weight fraction of solute in melt a t x and t weight fraction of solute in eutectic Lveight fraction of solute originally in melt (uniform) weight fraction of solute in solid a t z weight fraction of solute a t freezing interface distance down solid from first solid frozen out, z = tV: cm.
GREEKLETTERS = dimensionless eutectic composition, e = ZL~,/LL’, = dimensionless ‘distance from freezing interface into melt, 17 = x V p , / D p l p l , p s = density of liquid and solid, respectively, g./cc. 7 = dimensionless time. T = ( t V z / D ) ( p s / p l ) = ( z V / D ) X
e 7
(PSlPJ2
(1) Friedenberg, R. M., “Ultrapurity and Ultrapurification of Pharmaceuticals by Zone Melting,” Ph.D. thesis, University of Connecticut. Storrs, Conn., 1963. (2) Herington, E. F. G.: “Zone Melting of Organic Compounds,” Wiley. New York, 1963. (3) Pfann, W.G., “Zone Melting,” Wiley. New York, 1958. (4) Smith. V. G., Tiller, W.A., Rutter, J. W., Can. J . Phys. 33, 723 (1955). (5) Wilcox. 1.V. R., “Fractional Crystallization from Melts,” Ph.D. thesis, University of California, Berkeley, 1960. (6) 1t:jlcox. \V. R., J . A f f l . Phys. 35, 636 (1964). (7) Wilcox. M’. R., “Solute Redistribution during Solidification of Eutectic-Forming Mixtures,” Aerospace Corp., El Segundo, Calif.. Rept. TDR-169 (3240-10) TN-2 (1963). (8) IVilcox. W.R.? “Zone Melting of Eutectic-Formining Mixtures.” Aerospace Corp., El Segundo, Calif., Rept. ATN64(9236)-Z(1963). (9) \Vilcox, I$-. K., Friendenberg, R . M., Back, N., Chem. Reus. 64, 187 (1964). (10) Lt’ilcox, I V . R., It’ilke, C. R., A.I.Ch.E. J . 10, 160 (1964). (11) Nilcox: h’. R., U’ilke: C. R.: “Ultrapurification of Semiconductor Materials,” p. 481, Macmillan, New York, 1961, RECEIVED for review December 20, 1963 ACCEPTED March 30. 1964
KINETICS OF ABSORPTION OF CARBON DIOXIDE IN MONOETHANOLAMINE SOLUTIONS A T SHORT CONTACT TIMES J .
K. A . C L A R K E
Warren Sprzng Laboratory, Defartment of ScientiJic and Industrial Research, Stevenage, Hertfordshzre, England Work was carried out to measure the absorption rate of a gas, a t short contact times, with a fast chemical reaction taking place simultaneously, and to test the relation between the absorption rates observed and the diffusion coef Ficient, the physical solubility of the gas, and the kinetic rate constant for the homogeneous liquid phase reaction. Measurements are reported of rates of absorDtion of carbon dioxide in laminar jets of aqueous monoethanolamine solutions a t contact times of 3 to 2 0 msec. and gas pressures of 1 and 0.1 atm. Rates c f gas take-up observed a t the lower pressure are in agreement with the “penetration” theory for pseudo-first-order reactions developed by Danckwerts. Absorption a t atmospheric pressure corresponds to a less ameriable kinetic condition, since the concentration of unreacted monoethanolamine at the interface becomes seriously depleted during even the shortest attainable contact time of gas and liquid, and heat of reaction appears to influence the observed rates of absorption.
of considerable interest to relate rates of gas absorption with simultaneous fast chemical reaction in liquids to the physicochemical parameters of the system concerned. For this purpose the laminar liquid jet technique (6, 24,25) is particularl>- valuable. Contact times of gas and liquid of 3 to 30 msec. can be achieved: and simple hydrodynamic conditions in the jet permit the c’ontact rime to be directly related to the flow rate of liquid. T h e technique was applied by Nijsing. Hendriksz, and Kramers (25) to a study of the absorption of COz in laminar streams of aqueous alkali-metal hydroxide solutions
I
T IS
Present address, Department of Chemistry, University College, Dublin.
and agreement with the theory of diffusion Lvith chemical reaction was found. Empirical methods ”err used to evaluate the diffusion coefficient and the physical solubility of C 0 2 in the liquids. A number of studies have been made by different techniques, of “transient” absorption rates of CO? in monoethanolamine and its aqueous solutions which are important in industrial practice as carbon dioxide absorbents (2, 73: 75. 76). The present work was designed to permit measurement of rates of COS absorption in aqueous monoethanolamine under conditions such that the “penetration” theory of diffusion accompanied by pseudo-first-order chemical reaction (8)could be applied with VOL. 3
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239
