Ind. Eng. Chem. Fundam. 1983, 22, 139-144
primary decomposition products from decomposition of saturated species. The few elementary parameters have been estimated beforehand with general thermochemical kinetic methods. Generally speaking, the agreement on experimental data has already proved to be either fair or good along with these preliminary estimations. It was hard to find experimental information so accurate as to allow significant changes of these hypotheses. A practical application of these results such as to produce detailed kinetic schemes should require, at least, the following features: (1)kinetic description of “secondary” reactions; (2) further reduction of complexity of the resulting kinetic model (in terms of both species and reactions); and (3) numerical methods to solve the whole set of balance equations. As far as the first item is concerned, it seems important to point out that also radical addition reactions on alkenes (for instance, hydrogen, methyl, and ethyl on alkenes) give rise to saturated heavier radicals whose decomposition and isomerization reactions may be described again by the same elementary kinetic parameters. Suggestions to overcome the remaining points (that constitute the nucleus of the mathematical modeling of pyrolysis reactions) have been presented in a recent paper (Dente and Ranzi, 1982).
Appendix Decomposition and H-Abstraction Reaction of Cz-C4Radicals. A simplified kinetic scheme describing the final products of ethyl, 1- and 2-propyl, 1- and 2-nbutyl, and primary and tertiary butyl radicals is reported in Table XII. The following reference values for the removal of a single primary H-atom are assumed: primary radicals (ethyl, 1-propyl, 1-butyl, and p-butyl),K = 108.3exp(-14000/RT); secondary radicals (2-propyl ancf2-butyl), K, = exp(-16000/RT); tertiary radicals (t-butyl), Kt = exp(-17 000/RT). Dependence of the H-abstraction rates is not only con-
139
fined to hydrocarbon partial pressure but also to the nature of the attacked hydrocarbon (number and type of hydrogen atoms). Obviously, due to bimolecular H-abstraction reactions, an increase of hydrocarbon partial pressure will provide a higher amount of saturated species (ethane, propane, and butanes). The amount of lower saturated species (H, and CHI) will increase with temperature owing to lower activation energies of H-abstraction in comparison with decomposition reactions.
Literature Cited Aiiara, D. L.; Edeison, D. Int. J . Chem. Kinet. 1075, 7 , 479. Appleby, W. 0.; Avery, W. H.; Meerbott, W. K. J . Am. Chem. SOC. 1047, 69, 2279. Benson, S. W.; Kistiakowsky, G. B. J . Am. Chem. SOC. 1042, 64, 80. Benson, S.W. “The Foundations of Chemical Kinetics”, McGraw-Hili: New York, 1960; Chapter 13, p 344. Benson, S. W. A&. Chem. Ser. 1070, No. 97, 1. Benson, S. W.; O’Neal, H. E. “Kinetic Data on Gas Phase Unimolecuiar Reactions”, NSRDSNBS-21, U.S. Government Printing Office: Washington, DC, 1970. Benson, S. W. “Thermochemical Kinetics”; Wiiey: New York, 1976. Chrysochoos, J.; Bryce, W. A. Can. J . Chem. 1065, 43, 2092. Davis, H. G.; Williamson, K. D. A&. Chem. Ser. 1070, No. 183, 41. Dente, M.; Ranzi, E. M.; Barendregt, S.; Goossens, A. G. 72nd Annual Meeting of American Institute of Chemical Engineers: Sen Francisco, CA, Nov 1979; Paper 38b. Dente, M.; Ranzi, E. M. “Pyrolysis: Theory B Industrial Practice”; Academic Press: San Diego, 1982; Chapter 7‘. Doue, F.; Guiochon, 0. J . Chem. Phys. 1068, 36, 395. h u e , F.; Guiochon, G. J . Phys. Chem. 1060, 73, 2804. Frey, F. E., Hepp, H. J. Ind. Eng. Chem. 1033, 25, 441. Kosslakoff, A.; Rice, F. 0. J . Am. Chem. Soc. 1043, 65 590. Ilks, V.; Pieszkats, I.; Szepesy, L. Acta Chim. (Budapest) 1073, 79, 259. Ilies, V.; Weither, K.; Szepesy, L. Acto Chlm. (Budapest) 1073. 8 0 , 1. McNesby, J. R.; Drew, C. M.; Gordon, A. S. J . Chem. Phys. 1056, 24, 1260. Murata, M.; Saito, S.;Amano. A.; Maeda, S. J . Chem. Eng. Jpn. 1073, 6, 252. Murata, M.; Saito, S.; J . Chem. Eng. Jpn. 1074, 7 , 389. Rice, F. 0.; Herzfeld, K. F. J . Am. Chem. SOC.1034. 56, 284. Tanaka, S.; Arai, Y.; Salto, S. J . Chem. Eng. Jpn. 1075, 8 , 305. Tanaka, S.;Arai, Y.; Saito, S . J . Chem. Eng. Jpn. 1076, 9 , 161.
Received for review May 27, 1981 Revised manuscript received August 4 , 1982 Accepted September 23, 1982
Shape Factors in Facilitated Transport through Membranes Rlchard D. Noble Center for Chemical Engineering, 773.1, National Bureau of Standards, Boulder, Colorado 80303
The steady-state flux of permeate is calculated for spherical and cyclindrical membranes by use of a nonvolatile carrier to facilitate transport under two limiting conditions, reaction equilibrium and reaction-limited conditions. This result is used in conjunction with similar results for flat-plate membranes to obtain a shape factor for each geometry. The shape factor demonstrates the limits of transport in spherical and cylindrical membranes compared to flat plate membranes of equivalent thickness under identical conditions. For reaction equilibrium, the shape factor is found to depend only on geometry. For the reaction-limited or “frozen” condition, the shape factor is a function of transport and kinetic properties as well as geometry. The results can be used to predict the change in facilitated flux of the volatile species with a change in geometry. Since experimental flux measurements are often performed in flat membranes, the results of this work can be combined with experimental results to predict the total flux obtained in a tubular or a Spherical configuration.
Introduction Facilitated or coupled transport in liquid membranes is a promising separation process. This process combines the selectivity of conventional membrane processes with a high flux of permeate due to the facilitation effect. This is accomplished by nonvolatile chemical carriers within the
membrane which transport the permeate. Modeling of this process has been confined in most cases to steady-state transport in a flat-plate or slab geometry. Very little consideration has been given to the effect of geometry on system performance. Some of the most promising applications of liquid membranes involve spherical and cylindrical geometries. Unsupported liquid
This article not subject t o US. Copyright. Published 1983 by the American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
membranes tend to exist as spherical emulsions, and porous cylindrical hollow fibers offer high surface area opportunities for supported liquid membrane applications. Olander (1960) presented an analysis of one-dimensional mass transfer with reaction equilibrium conditions existing for flat-plate geometry. He considered four different reversible reaction schemes in his analysis. Friedlander and Keller (1965) also studied mass transfer in reacting systems near equilibrium. Goddard et al. (1970) discuss diffusion with near-equilibrium reaction to finite membrane systems. Matched asymptotic expansions are used to demonstrate the existence of an equilibrium core with boundary layers existing at each end of the membrane. Smith et al. (1973) also demonstrate this result. Goddard (1977) provides an extensive review of facilitated transport theory with applications to near-equilibrium regime. He also discusses corrections to account for departures from equilibrium. Recently, Goddard (1981) provided a model for facilitated transport in two-phase dispersions using the limit of reaction equilibrium. Schultz et al. (1974a,b) provide an extensive review of the entire field of facilitated transport. Mechanistic aspects, experimental systems, characteristic regimes, and mathematical analyses are all described. One area which was indicated as deserving attention was analytical results for cylindrical and spherical geometrics. Recently, Halwachs and Schugerl(l980) wrote a review which is general in nature and provides a brief mathematical analysis. Ward (1970) studied the steady-state transport of nitric oxide through liquid membranes containing a ferrous chloride solution. He developed analytical solutions for the steady-state nitric oxide flux under two limiting conditions, either reaction-equilibrium, diffusion-limited or reaction-limited transport. The nitric oxide flux was perpendicular to a flat membrane so the membrane was characterized as a one-dimensional flat plate. Cussler (1971) describes a general mathematical model for two solutes diffusing through a flat liquid membrane. For the case of the carrier reacting with only one of the solutes, the result is analogous to Ward’s (1970) for the reaction equilibrium case. Smith et al. (1973) analyzed steady-state facilitated transport by a perturbation technique using matched asymptotic analysis. They use the concept of film thickness to describe the limiting transport mechanisms. For thin films, the transport is reaction limited. For thick films, the transport becomes diffusion limited. For thin films, they found that their result for the flux of permeate was the same as Ward’s (1970) to first order. For thick films, their solution for the flux is analogous to Wards in the core of the membrane. They demonstrate the existence of boundary layers at each end of the membrane. Yung and Probstein (1973) used a similarity transformation method to develop a single mathematical equation for steady-state transport in flat membranes which could be numerically integrated over the entire range of transport limitations. Their results also agree with Ward’s in the limiting cases. Smith and Quinn (1979) describe a method for predicting steady-state facilitation factors for flat membranes. Their results for the flux of A reduce to the values obtained by Smith et al. (1973). Stroeve and Eagle (1979) developed equations for the facilitation factor in reactive slabs, cylinders, and spheres. They studied dispersed phase shapes in a continuous phase and provide a graph which shows the effect of geometry when subjected to a unidirectional flux of the diffusing species. One system which has been studied is oxygen transport
in hemoglobin solutions. Kutchai et al. (1970) modeled steady-state transport in a flat-plate membrane. Spaan (1973) models oxygen transport under two conditions, transient transport in a stationary film and steady-state transport in a laminar flow. Stroeve et al. (1976) developed a model for dispersed spheres in a continuous phase using the method of Friedlander and Keller (1965). It was applied to oxygen transport in hemoglobin. Ulanowicz and Frazier (1970) discuss transport of both oxygen and carbon dioxide through hemoglobin solutions. Otto and Quinn (1971) describe the steady-state transport of carbon dioxide through bicarbonate solutions. Suchdeo and Schultz (1974) investigated this same system for the purpose of determining gas permeabilities in reacting solutions. They determined that very accurate data were required for separate estimates of diffusion and solubility. Donaldson and Quinn (1975) discuss enzyme carriers for carbon dioxide transport. Smith and Quinn (1980) described carbon monoxide facilitation using steady-state conditions and flat-plate geometry. Ion transport in more general terms has been discussed by Caraciolo et al. (1975) and Le Blanc et al. (1980). Goddard (1980) developed a theoretical model to describe electric field effects in facilitated ion transport. One objective of this paper is develop an analytical solution for the steady-state flux of the volatile component (A) for both reaction-limited and diffusion-limited transport in spherical and cyclindrical membranes. The spherical membranes are analogous to emulsion liquid membranes where the membrane can be envisioned as a shell separating the continuous phase and a separate phase within the sphere (Li, 1971). This is not analogous to the homogeneous spheres used to model red blood cells for oxygen transport (Stroeve et al., 1976). The cylindrical membranes are equivalent to those reported by Hughes et al. (1981). An additional objective of this work is to demonstrate the limits of transport in spherical and cyclindrical membranes compared to flat membrane performance. These limits are described by a ratio of the flux for each geometry under identical operating conditions. Also, the limit of operating parameters is demonstrated. A third objective is the use of these results to predict operating performance based on experimental data. Often, experiments are performed in planar geometry (for example, see Ward (1970) and Donaldson and Quinn (1975)). Therefore, permeate flux information is based on this configuration. Cylindrical and spherical geometries offer higher surface area to volume ratio and are often more mechanically stable than a planar membrane. The results of this study can be used to predict the change in facilitated flux with a change in membrane geometry.
Mathematical Analysis The most common reaction mechanism is k,
A+BF!AB k2
where A is the permeate, B is the carrier, and AB is the carrier-permeate complex. This mechanism will be used in the analysis below. The steady-state condition can be approximated under two conditions. The pseudo-steady-state assumption can be used if the concentration on each side on the membrane is varying slowly with time. Secondly, if a rapid chemical reaction converts the volatile component (A) to a nonvolatile product at one boundary, then the concentration of A a t that point is zero. This, combined with a constant
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 141 CT
= CB
+ CAB = constant = amount of B initially placed in membrane (10)
The equilibrium constant for eq 1 is given as (assuming an ideal solution)
Combining eq 10 and 11 CAB=
KegCTCA
+ KeqcA
Equations 8 and 9 can be integrated directly using boundary conditions (5) and (6). Equation 12 is then used to remove CAB from the solution. The solution for eq 7 then becomes
Figure 1. Membrane cross section.
source at the other boundary, would yield a steady-state flux condition. A typical membrane is shown in Figure 1. It is assumed that a constant uniform concentration of A (CAJis present at the outer surface (r,). Likewise, the concentration of A at the inner membrane surface ( r = r,) is uniform and equal to CA. This would be the situation under the steady-state condition stated above. The active carrier B and the carrier complex AB are both nonvolatile within the membrane. Steady-state differential mass balances on each component are
DA DAB KeqCT(CA1 - cA2) N& = L ( C A 1 - CA,) + L (1 + KEQcAl)(l+ KeqC.4,) (13)
”
( v 2 )
KeqCT(CA1
r2(rz- rl)DAB(l+ KeqcAl)(l
-
cA2) KeqCA,)
(15)
Equation 13 was presented by Ward (1970). To determine the effect of geometry on system performance, the membrane thickness is kept the same for both geometries L = r2 - rl (16) where j = 0, 1, 2 for planar, cylindrical, or spherical geometries, respectively. These are subject to the following boundary conditions ~ C B CAB r = r,; CA = CAI; - = -dr dr - 0
It is now possible to look at comparative performance where the only alteration is the membrane geometry. Evaluating eq 14 and 15 at the outer membrane surface
The total flux of A is given by (7)
Reaction Equilibrium The first limiting case to be described is the reactionequilibrium or diffusion-limited case. For this case, eq 2 and 4 become
It is commonly assumed that DB = Dm It can then be shown that
L/r2 In [1/(1 -L/r2)l
(17)
NA~/N& = 1 - (L/r2)
(18)
N A-~ -
(5)
Nb
-
This defines the shape factor based on the outside surface area. One could also define a shape factor based on the inside surface area. Equations 17 and 18 provide one bound on the comparative performance of a cylindrical or spherical membrane to a planar membrane of the same thickness under identical operating conditions. The bound is the case of reaction equilibrium within the membrane. This shape factor is limited to the steady-state condition previously stated. Equations 17 and 18 can be tested for limiting cases of r2. As L l r , 0, the flux ratio becomes 1 as one would expect since this infers that the cylinder sphere has become a flat plate. For L/r2 1, the flux ratio becomes zero. At this point, the entire cylinder sphere becomes the membrane and there is no outlet for A.
-
-
Frozen Condition The second limiting case is for slow reaction. For this case CBand CABare considered to be “frozen”,that is their profiles are flat and C B and CAB are constants CB and CAB
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Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
throughout the membrane. Equations 2 and 4 are solved using boundary conditions 5 and 6. This allows evaluation of eq 7 for this case. NAo= [K,’(cos~XL - 1) + Kl’CA, Kl’CA, cosh X L ] / X sinh XL (19)
XL - 11 - K cosh XL NA2 - = 1-
($( [
K
z
(
9
)
11
(25)
+ 1-
N A O NA1
=
(9)]/[ + Kz(coshXL)
1 - cosh
XL
It
(26)
To evaluate eq 25 or 26, it is necessary to know CAB/CB. Solving eq 2 and 3 using eq 5 and 6 under the conditions of this case ( C B and C A B are constants, C A I = 0)
Kl’CAl Xr, cosh X(r, - r ) + rX2 sinh XL KI‘CA, r2 r1 - sinh X(r2- r ) + - sinh X(r - r l ) r rX2 sinh X L r
N A z=
1
[
Xr, cosh h(r - r l )
1
I
CAB), = K( CB
XL
- cosh
Xrl cosh X(r2- r) - Xr, cosh X(r - r l ) + -sinh X(r, - r ) + r
r2 - sinh X(r - rl) r
DA
U]/{
= k1CA2/k2 (22)
= K1’ -
X
+
2 ( l - ~ ) - [ l + ( l - ~ ) ] c o s h ~ ~ )(28)
’ I
rl
(1 -
(y) (E)’ (9)
-
J
rX2 sinh
[ t) + (E)
(24)
Here Io, 11,KO,and K 1 are modified Bessel functions. To evaluate the shape factor for the “frozen” condition, there are two cases to consider. In the first case, the flux of A enters the membrane at r = r2 and moves inward (negative r direction). For this case, CAI = 0. In the second case, the flux of A enters the membrane at r = r1 and moves outward (positive r direction). For this case CA, = 0. Setting the concentration of A equal to zero at the membrane exit is typically done in practice to allow the reverse reaction in eq 1to proceed at the maximum rate at this point. This can be accomplished by a high sweep rate across the membrane exit or a rapid irreversible reaction to consume A at the membrane exit. Case I. Flux of A enters membrane at r = r2. CAl- 0. For this case, the shape factors evaluated at r = r2 become
(29)
Yung and Probstein (1973) showed that for the “frozen” condition in flat plate geometry, eq 25 should reduce to K / 2 . Substituting L l r , = 0 into eq 27 and 28, one can verify this result. For the limiting case L / r 2 = 1,eq 27 and 28 become equal to K . Therefore the ratio C A B / C B will be between K / 2 and K for this case. By -substituting the appropriate limiting values of Cm/CB into eq 25 or 26 with the appropriate values of L / r 2 , one can verify that the flux ratio reduces to the reaction equilibrium case in each limit. For intermediate values of L / r 2 ,eq 25 and 26 will not generally reduce to the reaction equilibrium limit. A trial-and-error method is required to obtain the values of XL and CAB/CB which are to be used. XL can be written as
where
5= [ sinh M(K[IO(Xrl)K1(Xr2)+ Il(Xr2)Ko(Xr,)]+
NAa
Yung and Probstein (1973) demonstrated that the “frozen’! condition exists when c 2 10. The ratio of diffusion coefficients (DABID*)will normally be between 0.1 and 1.0. Case 11. Flux of A enters membrane at r = rl. CA?=
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
143
1.o
0. For this case, the ratio of eq 21 to eq 23 evaluated r = r2 becomes
0.8
0.6
0.4
XL - 11 + K
11
0.2
(33) 0 0
0.2
0.4
0.6
0.8
1.0
L/r2
Figure 2. Shape factor plot for a spherical membrane.
(E As in the prior case, it is necessary to know CAB/CB to solve eq 33 or 34. Proceeding as in the prior case with the appropriate boundary conditions on CA for this case.
\
9
L/r2
Figure 3. Shape factor plot for a cylindrical membrane.
5)2 CB = K( [ (1
-
E)
- (1
-
E)
&(
r2 1 - 9 ( y ) ] / { 2 ( r2 )
+
]
1 cosh
XL +
cosh XL -
1-
(E )(
E) [ -
(1 -
)I)
(36)
Equations 35 and 36 have a value of K/2, as expected for L/r2 = 0, this corresponds to the planar case. For L / r 2 = 1, eq 35 and 36 equal zero. Therefore, for this case
CAB K
05-5CB
Substituting these limiting values of Cm/CB into eq 33 and 34, with the appropriate values of L / r 2 ,one obtains the reaction equilibrium solution. For intermediate values of_L/rz,a trial-and-error solution will again be required for and X L to solve eq 35 or 36. Equation 30 is still valid for XL with y and K defined as = CT/CA~ (37) =
k1CA1/k2
(38)
Discussion Figure 2 demonstrates the bounded region for the spherical shape factor. Although the equations for both case I and Il are different, the shape factor for this example was the same in both cases. In general, this may not be the case. The shaded area in Figure 2 is the allowable region for the shape factor. Figure 3 demonstrates the bounded region for the cyl-
indrical shape factor. The line corresponding to the “frozen” condition limit was calculated for case 11. The shaped area in Figure 3 is the allowable region for the shape factor. As with the spherical membranes, it would be possible to predict performance limits with a change in geometry by drawing a diagram such as Figure 3. Figures 2 and 3 illustrate the fact that the highest facilitated flux under identical operating conditions will be obtained with a planar geometry. Since this is the geometry often used in experimental studies, flux data obtained in this manner will correspond to a maximum with respect to geometry. At this point, an example can aid in illustrating the use of these results. Ward (1970) presents experimental data for NO transport through a ferrous chloride solution. His NO flux results are for planar geometry. Using his data, K = 2.08, y = 19.1, and DAB/DA= 0.133. To compare his results with a hollow fiber (cyclindrical) geometry, it would be necessary to use eq 17 and 33. The gas mixture containing NO would pass through the interior of the hollow fiber and the NO gas would be transported radially outward. For Ward’s data, eq 17 and 33 collapse to the same line. Since eq 17 is only a function of geometry, the reaction equilibrium line in Figure 3 can be used. For L / r 2 = 0.3, the facilitated flux in the cylindrical geometry would be 0.84 of the value measured in his experimenta using the same physical properties, concentrations, and membrane thickness. Kutchai et al. (1970) published data for oxygen transport using hemoglobin. Their results were for planar geometry. A specific application could be the encapsulation of oxygen with a spherical membrane containing hemoglobin for controlled oxygen release into a aerobic biological system such as a wastewater system. Using their data, DAB/DA = 8.33 X K = 4.55, and y = 1.18 X lo2. Analysis of the effect of spherical geometry on the facilitated flux requires solving eq 18 and 34. The allowable region in
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Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Figure 2 for this system collapses to the reaction equilibrium line under these conditions. This line is only dependent on geometry so it can be used here. For L / r z = 0.4, the facilitated flux will be 0.6 times the value shown for their results. For this example, the interior will shrink with time and this calculation would be valid for a portion of the total operation. An alternate description of the facilitated flux is the facilitation factor. It is defined as the facilitated flux divided by the diffusional flux in the absence of any carrier. For reaction equilibrium, the facilitation factor does not change with a change in geometry. For the “frozen” condition, the facilitation factor is close to 1 and any change in geometry will not affect the value much. In general, there is some change in facilitation factor with geometry (see, for example, Folkner and Noble, 1983). It is important to note that the facilitation factor and the shape factor presented here are different. The shape factor is a measure of the change in absolute flux through the membrane with a change in geometry. The facilitation factor is a ratio of facilitated flux to simple diffusional flux for a given geometry.
Conclusions The steady-state flux of the volatile component in facilitated transport has been derived for membranes with spherical and cylindrical configurations. The calculations were done for the two limiting reaction situations of reaction equilibrium and a ”frozen” condition. By obtaining the ratio of these results with expressions for the flux under similar conditions in flat-plate geometry, one can obtain a shape factor to relate the performance of membranes of equivalent thickness under identical operating conditions. This can allow one to perform experiments in one configuration and estimate performance when geometry is changed. The results for the shape factors give upper and lower bounds on the flux ratio for the volatile component. For reaction equilibrium, the shape factor is a function of
transport and kinetic properties as well as geometry. In addition, equations were derived for the ratio of the nonvolatile components under “frozen” conditions. Limiting values for this ratio were obtained. The limit as the flat plate geometry is approached is consistent with previously published results. It was shown that the planar geometry will yield the largest facilitated flux and a change from this geometry will correspond to a reduction in facilitated flux. Examples were presented to demonstrate the use of the results.
Literature Cited Caraccioio, F.;Cussler, E. L.;Evans, D. F. AIChE J . 1075, 27(1), 160. Cussler, E. L. AIChE J . 1071, 17(6), 1300. Donaldson. T. L.; Quinn, J. A. Chem. Eng. Sci. 1075, 30, 103. Folkner, C. A.; Noble, R. D. J . Mem. Sci. 1083. in press. Friedlander, S. K.; Keller, K. H. Chem. Eng. Sci. 1985, 20, 121. Goddard, J. D.; Schuttz, J. S.; Bassett, R. J. Chem. Eng. Sci. 1070, 25, 665. W a r d , J. D.; Schuttz, J. S.; Suchdeo, S. R . AZChE J . 1974, 20(4), 625. W a r d , J. D. Chem. Eng. Sci. 1077, 32,795. Goddard. J. D. AIChE Symp. Ser. 1080, 77(2), 114. Goddard. J. D. Chem. Eng. Commun. 1081, 9 , 345. Halwachs, W.; Schugerl, K. Int. Chem. Eng. 1080, 20(4), 519. Hughes, R. D.; Mahoney, J. A.; Stelgeimann, E. F. Presented at AIChE Meeting, Houston, TX, April 1961. Kutchai, H.; Jacquez, J. A.; Mather, F. J. Biophys. J . 1070, IO, 38. Le Blanc. 0. H.. Jr.; Ward, W. J.; Matson. S. L.;Klmura, S. G. J . Memb. Sci. 1080, 6, 339. Li, N. Ind. Eng. Chem. Process Des. Dev. 1071, IO, 215. Olander, D. R. AZChE J . 1060, 6(2), 233. Otto,N. C.; Quinn, J. A. Chem. Eng. Sci. 1971, 26, 949. Schuttz, J. S.; Goddard, J. D.; Suchdeo, S. R. AZChE J . 1074, 20(3). 417. Smith, D. R.;Quinn. J. A. AIChE J . 1070, 25(1), 197. Smith, D. R.;Quinn, J. A. AIChEJ. 1980, 26(1), 112. Smith, K. A.; Meidon, J. H.; Cotton, C. K. AIChE J . 1073, 79(1), 102. Spaan, J. A. E. Pfiugers Arch. 1973, 342, 289. Stroeve, P.; Smith, K. A.; Coiton. C. K. AIChE J . 1978, 22(6), 1125 Stroeve, P.; Eagle, K. Chem. Eng. Commun. 1070, 3, 189. Suchdeo, S. R.; Schultz, J. S. Chem. Eng. Sci. 1074, 29, 13. Ulanowicz, R. E.; Frazier, G. C., Jr. Math. Biosd. 1070, 7, 111. Ward, W. J., I11 AIChE J . 1070, 76(3), 405. Yung, D.: Probstein, R. L. J . Phys. Chem. 1073, 77, 2201.
Received for review January 26, 1982 Revised manuscript received September 7 , 1982 Accepted October 4, 1982
