367
Ind. Eng. Chem. Fundam. 1986, 25, 367-372
I = current, mol/(s cm2) J = ion flux, mol/(s cm2) K, = dissociation constant for water, (mol/~m~)~ M = molar volume (crystal), cm3/mol PA = pore fraction, dimensionless (PA = 1for solution phases) CP = dimensionless potential, defined by eq 2 q = Bjerrum radius of bound ions = 3.56 X cm R = universal gas constant = 8.3143 V C/(K mol) r,,, = mean distance of approach of ion pairs, cm T = temperature, 298 K u = (8b)1/2 W = solution flux, cm3/(s cm2) XHoH = location of hydrogen-hydroxide shift, cm x = distance, cm x , = distance of co-ion from edge of repulsion zone, cm z = valence, dimensionless
Subscripts 1 = cation 2 = anion
3 = hydrogen ion 4 = hydroxide ion 5 = water A = fixed charge group in membrane L i t e r a t u r e Cited Breslau, B. R.; Miller, I. F. Ind. Eng. Chem. Fundam. 1071, 70, 554. Breslau, B. R.; Miller, I . F.; Gryte, C.; Gregor, H. P. Preprint Volume, MESD Biennial Conference; AIChE: New York, 1970; p 363. Dege, G. J.; Liu, K.J. U S . Patent 4024043, 1977. de Korosy, F.; Zelgerson, E. Isr. J. Chem. 1071, 9 , 483. Frllette, V. J. J. Phys. Chem. 1056, 60, 435. Keusch, P., RAI Research Corp., private communication, 1980. Nagasubramanlan, K.; Chlanda, F. P.; Llu, K . J . J. Membr. Sci. 1077, 2 , 109. Patel, R. D.; Lang, K. C.; Miller, I.F. I n d . Eng. Chem. Fundam. 1077, 76, 340. Vervey, E. J. W.; Overbeek, J. Th. G. Theory of the Stabilw of Lyophobic Colloids;Elsevler: New York, 1948; p 22.
Received for review August 1, 1984 Revised manuscript received July 29, 1985 Accepted October 24, 1985
A Study of the Separation Efficiency of the Batch-Type Thermal Diffusion Column with an Impermeable Barrier Inserted between the Plates Shau-We1 Tsal and Ho-Mlng Yeh’ Chemical Engineering Department, National Cheng Kung Universiv, Talnan, Taiwan, Republic of China
Instailation of an impermeable barrier between the plates substantially increases the separation efficiency by reducing the remixing effect. Theoretical considerations show that when the barrier is installed at the best position, maximum separation, maximum concentration of top product, or minimum concentration of bottom product may be obtained. Considerable improvement is obtained when the column is operated at the conditions leading to the best performance.
Introduction
The thermogravitational thermal diffusion column, introduced by Clusius and Dickel (1938), can be used to separate the mixtures which are difficult to separate by means of conventional methods such as adsorption, distillation, etc. The first complete presentation of the theory of the Clusius-Dickel column was that of Furry et al. (1939). A more detailed study of the mechanism of the separation shows that convective currents have two conflicting effects: a desirable “cascading” effect, owing to the countercurrent action of the thermally driven flow in the column, and an undesirable “remixing” effect, owing to diffusion along the column axis and across the column. At steady state, a dynamic equilibrium is established between these at which no further separation takes place. Thus, it is evident that any improvement in the equilibrium separation must be associated with either a suppression of the remixing effect and/or an enhancement of the cascading effect. Based on this concept, some improved columns have been developed in the literature, such as inclined columns (Powers and Wilke, 1957; Chueh and Yeh, 1967),wired columns (Washall and Molpolder, 1962; Yeh and Ward, 1971), inclined moving-wall columns (Ramser, 1957; Yeh and Tsai, 1972), rotary columns 0196-43131861 l025-Q367$01.5OlO
(Sullivan et al., 1957; Yeh and Cheng, 1973; Yeh and Tsai, 1981b, 1982), packed columns (Lorenz and Emery, 1959; Yeh and Chu, 1974),and rotated wired columns (Yeh and Ho, 1975; Yeh and Tsai, 1981a). Installation of an impermeable barrier between the plates may decrease the strength of natural convection, and therefore, both cascading and remixing effects are reduced. It is believed that properly reducing the convective flow may effectively suppress the remixing effect while still preserving the cascading effect and thereby lead to improved separation, such as in inclined, wired, and packed columns. The barrier should resist flow of liquid between the channels but must not permit molecular diffusion under the influence of the temperature difference between vertical walls. The purpose of this work is to develop the separation theory and investigate the separation efficiency for such an improved thermal diffusion column. Column Theory 1. T h e Open Column. Consider a flat-plate thermo-
gravitational thermal diffusion column filled with a binary mixture. The distance between the plates is WA + wB. At steady state, the horizontal mass flux of component 1, Jx, is related to the velocity, V,, by the differential mass balance equation 0 1986 American Chemical Society
388
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
The expression for Jx is in terms of two contributions, thermal and ordinary diffusions, as
2 h
and the velocity distribution of the naturally convective flow is
-$
Furry et al. (1939) derived the transport equation for steady-state batch operation from eq 1,2, and 3 by making use of the boundary conditions, J , = 0, at plate walls. The result is dC HC(l - C) = (Kc &)E (4)
CD
+
where Figure 1. Schematic diagram of a batch-type thermal diffusion column with an impermeable barrier inserted between the plates.
( P D B ( ~+ AW
B ) ~(6)
All the physical properties defined in the Nomenclature section are evaluated at the reference temperature ii = T, (7) Morgado et al. (1981) solved eq 4 with the boundary condition = 0, c = Cb,o (8)
exchange, except for the shared temperature gradient. At the beginning of operation, both sides of the barrier are filled separately with a binary liquid mixture of uniform composition. At steady state the compositions in each will therefore depend on the composition of its initial charge. Following the same derivation as that performed for open columns, we obtain the separation equations for steady state as follows. 2.1. Section A on the Cold Side. = ct,A - Cb,A exp(-XACf) - 1 Ct,A = exp(-XA) - 1 AA
and a mass balance for the whole column
(14)
(15)
J L C dZ = C,L The resultant equations obtained at steady state were = ct,o - Cb,o (10) exp(- XCf) - 1 (11) ct,, = exp(-A) - 1 Cb,o
where T
XA
=
I
LA'
+ Kc,A/Kd,A
exp(XCf) - 1 = exp(X) - 1
where Ct,oand Cb,o are the weight fraction of component 1 at the top and the bottom of the column, respectively, and Kd,A = fPDBuAhA 2. A n Improved Column. Consider a batch-type flat-plate thermal diffusion column. An impermeable barrier, the thickness of which is comparable to the distance between the plates, W A + 9,is inserted between the hot and cold plates, as shown in Figure 1. The use of a truly impermeable barrier, such as a metal barrier, has the effect of producing two open columns side by side, the two having equal heat fluxes and a common temperature at their junction. Since the two sides, sections A and B, are separated everywhere by a barrier permitting heat flow but no diffusion, each will operate independently with no mass
(21)
In the above equations, all the physical properties are evaluated at the reference temperature of section A FA= T m , ~ (22) The temperature difference and the reference temperature for section A are easily calculated by assuming the thermal resistances between the plates to be in series. Hence, we obtain
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 368
FA = Ti + (AT)A/2
(24)
2.2. Section B on the Warm Side. All the equations in section B are the same as those in section A, except that the subscript A is replaced by B and
1.0
0.9 0.8
0.7
TB = Tz - (AT),/2
(26) 0.6
Since the quantity of the product in each section is proportional to the thickness of the section, we may define the average products as
20.5
+ WB)
(27)
0.4
Cb,av = (uACb,A + WBCb,B)/(aA + WB)
(28)
0.3
Ct,av
Aav =
%.
=
(wACt,A
+ wBCt,B)/(wA
(WAAA+ WBAB)/(WA+ WB) = Ct,av - Cb,av
(29)
It is convenient to define dimensionless groups as a1
= 6/(WA
a3
+ WB)
(30)
= k/kl
(31)
= WB/WA
(32)
If the temperature difference between the plates, i.e., AT, is not large or the products of physical properties in the transport constants are not sensitive to the reference temperature, eq 27,28, and 29 can be rearranged to eq 33, 34, and 35 by making use of eq 5-7, 13-26, and 30-32 Ct,av
=
(ct,A
+ a3Ct,B)/(1
+ a3)
~
Cb,av = (Cb,A + a3Cb,B)/(1 + a 3 1 Aav
= (AA +
+
*3A~)/(l
= Ct,av - cb,av
- L'=0.5
o,2L--20
0.110
30
60
50
40
70
1
Kc / K d Figure 2. Optimal position of the barrier for obtaining maximum concentration of top product. To obtain the position corresponding to minimum concentration of bottom product, Cf is replaced by (1 - Cf).
(33)
(34) (35)
where XA and AB in the concentrations are rewritten as XA
=
L'(1 1 + (Kd/Kc)(1
+ a3)4 + *J8(1+ .1..2)'
(36) b f . O . 1
0 10'
If we let
"
'
'
'
I . ' " ' ' '
IO2
'
l.,"'
IO'
'
'
I , , , . ' ' '
10'
'
lo5
Kc/Kd
Figure 3. Comparison of the concentration of a top product obtained at optimal conditions in the improved column with that obtained in the open column.
a3*= l/r3
then it is easy to show that Ct&3--w3*
= Ct,B;
Cb,Alr3-r3* Cb,B; Ct,avlrpr3*= Ct,av; AavIr3-r3*
Ct,Blr3-r3*
=
ct,A
(39)
Cb,Blwpr3*
=
cb,A
(40)
Cb,avlr3-r3*
=
Cb,av
(41)
=
Aav
(42)
Consequently, we conclude that eq 33-35 are symmetric around 7r3 = 1.0, and only the values for a3I 1.0 need to be considered.
Optimal Position of the Barrier Equations 33-35 can be used to obtain the optimal position of the barrier for a maximum degree of separation, maximum concentration of top product, or minimum concentration of bottom product. Differentiating b V ,C, and CbCvwith respect to the channel width ratio a3in eq 33-35 and setting the derivatives to zero give the necessary conditions for the optimal position of the barrier. The resultant equations are shown as follows. For simplicity,
I t is easy to show that this equation is also symmetric around a3 = 1.0.
370
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
In general, the thermal conductivity of the barrier is much larger than that of the mixture. Accordingly, we may let (1 + 1. Some graphical solutions of eq 43 have been calculated and are presented in Figure 2 with initial concentration Cf and dimensionless column length L ' as parameters. It is shown that as the value of Kc/Kd increases, the value of r3,0p will approach unity, i.e., uA= wB for all the cases. The top-product concentrations for both the open column and the improved column, operated at optimal conditions, have also been evaluated and are presented in Figure 3. 2. Minimum Concentration of the Bottom Product.
100.9-
080 7-
0.4 03
011
10
20
30
40
50
60
70
3
Kc / K d
Figure 4. Optimal position of the barrier for obtaining the maximum degree of separation.
It is also easy to show that this equation is symmetric around 7r3 = 1.0. Moreover, it is found that when Cf in eq 44 is replaced by (1- Cf),the resultant equation has the same form as eq 43. Accordingly, we may find the optimal position of the barrier in this case from the solution of eq 43 or from Figure 2. 3. Maximum Degree of Separation. 4L'(1 + r3)4 X
t
(1- Cf) exp[-X.A(l
[
L'=O5 L
- - L' = 1.0
I
10'
+ (1 - Cf) exp[X.A(l + Cd1 + Cf exP(XACf)+ e x p h ) [exp(X,) - 112
I-
+ Tl*2)2(1 + *&' x ( [1 + (Kd/Kc)(1 + *1*2)2(1 + T3)']' {[(l - Cf) exp[-XB(l + Cdl + Cf exp(-XBCf) exP(-XB)l/[[exP(-XB) - 1121+ (1- Cf) exp[XB(1 + Cf) + Cf exP(XBCf) + exp(Xd 1 - (Kd/Kc)(l
t
L'=I O
for improved c o l m n
for open column
+ Cf)] + Cf exp(-XACf) - exp(-X,) [exp(-X,) - 11'
[exp(X,) - 112
+ *lT2)'(1 + 73)'I + (Kd/Kc)(l + r1*2)'(1 + *3)'12
*34[r38- (Kd/Kc)(l [*3*
0.6-
11
1
Ix
-
exp(-XACf) - 1 - exp(X,C,) - 1 - exp(-X&,) - 1 + exp(-XA) - 1 exp(XA)- 1 exp(-XB) - 1 exP(ABC,) - 1 (45) exP(XB) - 1 We find that this equation is also symmetric around x3 = 1.0. Moreover, if Cf in eq 45 is replaced by (1 - Cf), the resultant equation is not changed. Hence, we need to consider only the cases with Cf 5 0.5. Some graphical
IO2
10' Kc / K d
I0 '
IO'
Figure 5. Comparison of the degree of separation obtained a t optimal conditions in the improved column with that obtained in the open column.
solutions of eq 45 have calculated and are presented in Figure 4 with L' and Cf as parameters. In general, as Kc/Kd increases, the optimal position of the barrier will approach r3,0p = 1.0, i.e., the center of the column. The degrees of separation for both the open column and the improved column obtained at optimal conditions have also been calculated and are presented in Figure 5 .
Improvement of Separation The improvement in separation by placing the barrier at the optimal position is best illustrated by calculating the percentage increase in the separation relative to an open column without a barrier. Let
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 371
Table I. Comparison of the Degree of Separation and Concentrations of Top and Bottom Products in the Improved Column Operated at Optimal Conditione for Each Case with Those Obtained in the Open Column (for L' = 1.0) improved column for max improved column for min improved column for max open column C,, Cbav 4" C,o, Cb.o, 4, (Ct,ev)mru, 11, (Cbav)min, 12, (Aav)max, 139 KJKd % % % *3,0p % % *3,0p % % *3,0p % % Ce = 0.1 14.6 1 x 10' 6.4 8.2 0.26 16.5 13.0 0.23 5.6 12.5 0.25 10.9 32.9 14.9 24.9 5 x 10' 6.2 8.7 0.73 67.1 1.00 2.4 61.3 1.00 22.4 157.5 15.0 1 x 102 6.2 36.5 8.8 1.00 143.3 1.00 0.6 90.3 1.00 308.0 35.9 15.0 6.1 8.9 1.00 1 x 103 72.0 380.0 1.00 0.0 100.0 1.00 72.0 709.0 Ce
10' 10' 102 103
40.0 40.8 40.9 41.0
21.2 20.5 20.4 20.4
18.8 20.3 20.5 20.6
0.26 1.00 1.00 1.00
43.6 58.6 74.8 97.8
0.3 9.0 0.24 43.6 1.00 82.9 1.00 138.5 1.00
18.9 9.4 3.2 0.1
10.8 54.2 84.3 99.5
0.25 1.00 1.00 1.00
24.7 49.2 71.6 97.7
31.4 142.4 249.3 374.3
1 x 10' 5 x 10' 1 x 102 1 x 103
61.2 62.0 62.2 62.2
38.8 38.0 37.9 37.8
22.4 24.0 24.2 24.4
0.25 1.00 1.00 1.00
64.6 78.6 90.4 99.8
Ce 0.5 5.6 26.8 45.6 60.5
0.25 1.00 1.00 1.00
35.4 21.4 9.6 0.0
8.8 43.7 74.7 100.0
0.25 1.00 1.00 1.00
29.2 57.2 80.8 99.8
30.4 138.3 233.9 309.0
1x 5x 1x 1x
10' 10' 102 103
78.8 79.5 79.6 79.6
60.0 59.2 59.1 59.0
18.8 20.3 20.5 20.6
0.24 1.00 1.00 1.00
81.1 90.6 96.8 100.0
Ce 0.7 2.9 14.0 21.6 25.6
0.26 1.00 1.00 1.00
56.4 41.4 25.2 2.3
6.0 30.1 57.4 96.1
0.25 1.00 1.00 1.00
24.7 49.2 71.6 97.7
31.4 142.4 249.3 374.3
1 x 10' 5 x 10' 1 x 102 1 x 103
93.6 93.8 93.8 93.9
85.4 85.1 85.0 85.0
8.2 8.7 8.8 8.9
0.23 1.00 1.00 1.00
94.4 97.6 99.4 100.0
0.26 0.73 1.00 1.00
83.5 75.1 63.5 28.0
2.2 11.8 25.3 67.1
0.25 1.00 1.00 1.00
10.9 22.4 35.9 72.0
32.9 157.5 308.0 709.0
1x 5x 1x 1x
c, = 0.9 0.9 4.1 6.0 6.5
Table 11. Comparison of Separation for the Benzene-n -Heptane Mixture in the Open Column and Improved Column open column improved column Ce
Ct,o,
Cb,o+
A09
%
%
%
(C*,,V),,
*%OP
%
11,
(Cb,ev)min,
129
(Aav)max,
131
%
%
%
%
%
0 0 0 0.8 20.3
100.0 100.0 100.0 98.6 76.1
79.7 99.2 100.0 99.2 79.7
687.8 381.6 309.8 381.6 687.8
0 0.3 1.8 9.1 45.1
100.0 98.8 95.9 85.9 48.5
54.9 90.6 96.4 90.6 54.9
1120.0 762.9 677.4 762.9 1120.0
L'= 1 (L= 118 cm) 0.1 0.3 0.5 0.7 0.9
15.1 41.0 62.2 79.6 93.9
6.1 20.4 37.8 59.0 84.9
9.0 20.6 24.4 20.6 9.0
1.0 1.0 1.0 1.0 1.0
0.1 0.3 0.5 0.7 0.9
12.4 35.4 56.2 75.1 92.1
7.9 24.9 43.8 64.6 87.6
4.5 10.5 12.4 10.5 4.5
1.0 1.0 1.0 1.0 1.0
79.7 99.2 100.0 100.0 100.0
423.8 142.0 60.8 25.6 6.5
L'= 0.5(L= 59 cm) 54.9 90.9 98.2 99.7 100.0
where (CQV)-, (Cb,av)min, and (AaJmaare the maximum concentration of a top product, the minimum concentration of a bottom product, and the maximum degree of separation obtained at the optimal conditions, respectively. Some results are presented in Table I for L' = 1.0. For the purpose of illustration, let us assign numerical values for the separation of a benzene-n-heptane liquid mixture as follows: CY = 1.2,g = 980 cm/s, j3 = 1.263X g/(cm3 K),p = 0.7602 g/_cm3,p = 0.4576 cP, D = 2.736 X cm2/s, AT = 42 K,T = 301 K,B = 10 cm, and W A + uB = 7 X cm. Substituting these values into eq 5 and 6,we obtain H = 6.890 X g/s, K , = 8.138 X lo-' g cm/s, and Kd = 1.456 X g cm/s. With these values, results were calculated from the corresponding equations and are shown in Table 11.
Conclusions (1)The equation of separation has been derived, applicable to the whole concentration range in a batch-type thermal diffusion column with an impermeable barrier inserted between the hot and cold plates. Since intro-
342.7 156.8 74.7 32.8 8.6
duction of a barrier can substantially reduce the remixing effect, it may effectively increase the degree of separation and concentration of a top product and decrease the concentration of a bottom product. (2)Installation of an impermeable barrier in thermal diffusion columns might decrease the strength of natural convection. Therefore, properly adjusting the location of an impermeable barrier will effectively reduce the remixing effect while still preserving the cascading effect and thereby lead to improved separation. There exist optimal positions of the barrier for obtaining the maximum degree of separation, the maximum concentration of a top product, or the minimum concentration of a bottom product. The equation for evaluating the optimal position of the barrier for each case has been derived and represented in Figures 2 and 4 by taking Cf and L' as parameters. Of course, the choice of optimal conditions depends on which case we choose. (3)Further investigation of Figures 2 and 4 shows that qOp will approach unity when the value of KJK, increases. The behavior is more evident as L' increases. Conse-
372
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
quently, for the practical cases with K,/& 2 80, the installation of an impermeable barrier at the center between the plates, Le., qOp = 1.0, results in obtaining (Aav)", (Ct,av)max, and (Cb,Jmin simultaneously. (4) We have compared the degree of separation and the concentration of a top product obtained at the optimal condition for each case with those obtained from the open column. Some examples are represented in Figures 3 and 5. From the figures, the benefit of the improved column is obvious. ( 5 ) The improvement in a separation based on the open column has been evaluated and presented in Table I for L' = 1.0 and with K , / K d as the parameter. A numerical example for the separation of benzene-n-heptane mixture was also illustrated, and the results are shown in Table 11. It is found that a considerable improvement is obtained as K,/Kd increases. Therefore, it is recommended to use the improved column developed in this work because there would be little, if any, increase in the fixed and operating costs. (6) During the derivation of the equation, the most important assumption is that AT is not large or that the products of the physical properties in the transport constants are not sensitive to the reference temperatures. Modifications may be required when the above assumptions fail. (7) The results obtained in this work can be extended to the concentric-tubethermal diffusion column (Jones and Furry, 1946; Yeh, 1976) or other improved columns. Acknowledgment We express our thanks to the Chinese National Science Council for the financial aid. Nomenclature B = column width, cm C = weight fraction of component 1 in binary mixtures Cb,Cf,C, = C of bottom product, initial charge, and top product, respectively D = ordinary diffusion coefficient, cm2/s g = gravitational acceleration, cm/s2 H = transport constant defined by eq 5, g/s I,,12,I3 = improvements defined by eq 46,47, and 48 Jx = mass flux of component 1 in the horizontal direction, g/(s Cm2) J X . o D , J X . T D = mass flux of component 1 in the horizontal direction due to ordinary and thermal diffusions, g/(s cm) K,, Kd = transport constants defined by eq 6, g cm/s k , kl = thermal conductivitiesof the mixture and the barrier, respectively, cal/(cm s K) L = total column length, cm
= reference temperature, K
T,,T2 = temperatures of the cold and hot plates, K
V , = velocity distribution in the vertical direction, cm/s
X,2 = Cartesian coordinates, cm Greek Letters a=
thermal diffusion coefficient defined in eq 2
P = thermal expansion coefficient defined as -(t3p/t3T), g/(cm3 K)
A = degree of separation defined by the difference of con-
centration between top and bottom of the column AT = difference in temperature of the hot and the cold plates, K 6 = thickness of the barrier, cm p = viscosity of the mixtures, cP h = dimensionless group defined by eq 13 7rl, 7r2, 7r3 = dimensionless groups defined by eq 30, 31, and p w
32 = density of the mixture, g/cm3 = thickness of the section, cm
Superscripts and Subscripts * = defined as the inverse of the original quantity, i.e., 7r3* = l/r3 ' = dimensionless group defined in eq 13 and 18 A = for section A B = for section B
m = at the arithmetic mean o = for the open column av = average quantities defined in eq 27, 28, and 29 op = at the optimal condition max = maximum value obtained at the optimal condition min = minimum value obtained at the optimal condition Literature Cited Chueh, P. L.; Yeh, H. M. AIChEJ. 1967, 73, 37. Clusius, K.; Dickel, G. Nahnwlssenscheften1998,26,546L. Furry, W. H.; Jones, R. C.; Onsager, L. m y s . Rev. 1999,55, 1083. Jones, R. C.; Furry, W. H. Rev. Mod. phvs. 1948, 78, 151. Lorenz. M.; Emery, A. H., Jr. Chem. Eng. Sci. 1959, 77, 16. Morgado, H. F. L. S.;Pinheiro, J. de D. R. S.; Romero, J. J . E.; Bolt, T. R. S e p . Sci. Techno/. 1981, 76,897. Powers, J. E.; Wiike. C. R. AIChEJ. 1957,3 , 213. Ramser, J. H. Ind. Eng. Chem. 1957,49, 155. Sullivan, L. J.: Ruppel, T. C.; Willingham, C. E. Ind. Eng. Chem. 1957,49, 110. Washall, T. A.; Molpolder, F. W. I d . Eng. Chem. Process Des. Dev. 1962. I , 26. Yeh, H. M. Sep. Sci. 1976, 7 7 . 455. Yeh, H. M.; Cheng, S. M. Chem. Eng. Sci. 1973,28. 1803. Yeh, H. M.; Chu, T. Y. Chem. Eng. Sci. 1974,29, 1421. Yeh, H. M.; Ho, F. K. Chem. Eng. Sci. 1975. 30, 1381. Yeh. H. M.; Tsai, C. S.Chem. Eng. Sci. 1972,27,2065. Yeh, H. M.; Tsai, S. W. J . CY". Eng. Jpn. 1981a. 74, 90. Yeh, H. M.; Tsai, S. W. S e p . Scl. Techno/. 1981b, 76, 83. Yeh, H. M.; Tsal, S. W. Sep. Sci. Techno/. 1982, 77, 1075. Yeh. H. M.; Ward, H. C. Chem. Eng. Sci. 1971,26,937.
Received for review August 13, 1984 Revised manuscript received May 17, 1985 Accepted July 22, 1985
