Ind. Eng. Chem. Fundam. 1986, 25, 450-452
450
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Effect of External Mass-Transfer Resistance on Facilitated Transport An analytical expression is derived for the facilitation factor in facilitated transport across a liquid film. This expression accounts for external mass-transfer resistances as well as diffusion and reaction within the liquid fllm. Evaluation of Sherwood numbers encountered in hollow-fiber membrane systems indicates the importance of
external mass-transfer resistance. A graphical method based on this equation is presented and compared to experimental results.
Introduction Facilitated transport is a process whereby a nonvolatile chemical carrier facilitates or augments the transport of a solute across a liquid film. (See Way et al. (1982) for a detailed discussion.) A measure of this facilitation effect is the facilitation factor which is defined as the ratio of the total solute flux with facilitation to the solute diffusion flux. There have been analytical solutions developed for the steady-state facilitation factor for the most common reaction mechanism A + B s AB. Here A is the solute, B is the carrier, and AB is the carrier-solute complex. Smith and Quinn (1979) obtained an analytical expression for the facilitation factor by assuming that the carrier concentration was constant across the liquid film. This assumption is equivalent to assuming a large excess of carrier compared to solute. The differential equations describing this process then become linear and are solvable. Their solution was originally introduced by Donaldson and Quinn (1975) for tracer diffusion through a reactive membrane solution, equilibrated with a constant partial pressure of untagged gas. The two solutions become identical in the limit of vanishingly' small tracer Concentration. Hoofd and Kreuzer (1979) obtained the same result using a different approach. They used a combined Damkohler number approach. A large Damkohler number solution is assumed to provide a first approximation to the solution for the solute concentration. This solution is assumed to be a function of the chemical reaction and, therefore, of carrier concentration. A correction is applied to obtain a small Damkohler number solution which is independent of carrier concentration. The two solutions are combined to yield a solution which is valid over the entire liquid film. It is interesting to note that this same solution has been obtained for other processes which have a similar mathematical expression. Danckwerts and Kennedy (1954) obtained the same result for gas absorption in a liquid film. They assumed a first-order reversible reaction with the absorbent and chemical equilibrium in the bulk of the solution. Aris (1983) studied simultaneous sorption and diffusion of a solute through a porous solid. His result showed that this phenomenon was similar to facilitated transport and not concentration-dependent diffusion, as had been proposed previously. The expression for a facilitation factor obtained by the above authors is based on concentration boundary conditions for the solute on each side of the liquid film. Any mass-transfer resistance due to boundary layers on the external surfaces of the liquid film is neglected. The
purpose of this study is to derive an expression for the facilitation factor which accounts for these additional mass-transfer resistance. A graphical method based on the resulting equation is also described.
Problem Description Referring to Figure 1, the differential equations which describe the steady-state solute transport are
The boundary conditions are
Equations 3 and 4 state that there is a constant source and sink for A, equality of flux of A on each side of interface, and that AB is nonvolatile. Smith and Quinn (1979) assume that CB is a constant based on reaction equilibrium. CT
CB =
f-7
1
+ Keq-m
(5)
A0
The above can be cast in dimensionless form (see Folkner and Noble (1982) and Kemena et al. (1983)) where t
= inverse Damkohler number = Dm/kJ2
(6)
= mobility ratio = (DmCTm)/(DACAO)
(7)
cy
K = dimensionless equilibrium constant = kfCAo/k,m (8)
C A * = dimensionless solute concentration = CAm/CAo (9)
CAB* = dimensionless complex concentration = CAB/CT(10) Sh = Sherwood number = & L / D A (11)
X = dimensionless length = x / L Equations 1-5 become
This article not subject to US. Copyright. Published 1986 by the American Chemical Society
(12)
Ind. Eng. C b m . Fundam., Vol. 26, No. 3, 1986 451
-L I-l
X=O
x = 1: where
I-x
-CB---
1 CT l + K The facilitation factor F is defined as total solute flux with facilitation F= solute diffusion flux
A+B=c\B Figure 1. Facilitated transport membrane.
(18)
Solving eq 13 and 14 with boundary conditions 15 and 16, one obtains
1
E-l
-
1+--l aK + K
X
+ (1 +
&)( & + &)
where A = - (1 1
2
+ (a+ l ) K
)
6
YAO
1/2
t(l+K)
It can be easily demonstrated that eq 20 reduces to the equation of Smith and Quinn (1979) and equivalent expressions when Sh = (no external mass-transfer resistance).
-
Results Kemena et al. (1983) demonstrated that the optimal values of K for maximizing F are in the region 1 I K I 10 for many orders of magnitude in CY and t. Table I shows the resulb of calculating F,via eq 20, for a few values of CY,K, and t within the above region. The results were also compared to numerical results and the differences were insignificant. Table I shows that facilitation is decreased as the Sherwood number is decreased. This result is expected as increased external mass-transfer resistance will reduce the solute concentration at the (X = 0)interface and will decrease the forward reaction rate. At the (X = 1) interface, the solute concentration is increased, causing the reverse reaction to decrease. Some values of Sh for hollow-fiber membrane systems can be estimated. Hughes et al. (1981) used a reverseosmosis unit impregnated with silver nitrate solution to study selective olefin removal. Estimates of the tube-side values of Re and Sc are Re = 3.01 X loa and Sc = 0.533. Aerstin and Street (1978) provide j factor grapha for shell and tube heat exchangers. Assuming that the j factors for heat and maas transfer are equal, the resulting Sh = 8.11. Cianetti and Danesi (1983) used an apparatus discussed by Danesi et al. (1983) to study nitric acid removal from aqueous streams. The membrane contained a tertiary amine in an organic solvent. Re = 208 and Sc = 385 for
Figure 2. Plot of inverse enhancement factor vs. gas phase for C02-EDA system. Table I. Some Rsrultr for Facilitation Factor (F),Showing Effect of Shrrwood Number Sh F Sh F K = 3.0,u = 10.0, K = 5.0,a = 15.0, f = 0.001 f 5 0.1 2.0 1.76 2.0 1.62 5.0 2.61 5.0 2.21 10.0 3.56 10.0 3.77 20.0 4.65 20.0 3.30 40.0 5.62 40.0 3.70 100.0 6.51 100.0 4.02 200.0 6.88 200.0 4.15 500.0 7.13 500.0 4.23 W 7.31 4.28 (D
the tube side of their system. Again, if one uses j factor graphs in Aerstin and Street (1978), the resulting Sh = 24.7. Referring to Table I, one can see the effect that Sh valuea of this magnitude can have on the facilitation factor. A rearrangement of eq 20 can be useful for analyzing experimental results. For reaction equilibrium, diffusion-limited transport, A-' tanh X 0. Equation 20 can be rewritten as 6'= (F-1)-1 I
[ (&)( + &) + -4+ 1
(1 +
(22)
Sh has been assumed to be equal on each side of the membrane. The term in brackets is a constant, and CY-' is proportional to the feed solute concentration. Therefore, a plot of 6'VS. CAo (or YAo) should be a straight line if the system is diffusion-limited. Also, the slope and in-
Ind. Eng. Chem. Fundam. 1986, 25, 452-455
452
tercept can be used to determine values of Sh and DAB. If aK >> 1, the intercept is a direct measure of Sh. The slope can then be used to determine DAB. An example of this graphical analysis is shown in Figure 2. The system is COP transport across an aqueous film containing ethylenediamine as a carrier. The liquid is immobilized in a microporous polypropylene film. The details of the experimental procedure are reported elsewhere (Bateman et al., 1984). A least-squares fit of the points in Figure 2 yields a slope of 13.04 and an intercept of 0.06 (r2= 0.98). Assuming aK is large, the small value of the intercept translates to a large Sh and negligible external mass-transfer resistance. An effective diffusion coefficient for COz through the membrane equaled 7.92 X lo+ cm2/s and was obtained from diffusive experiments in which no carrier was present. The slope can then be used to obtain an effective diffusion coefficient for the carrier-solute complex (DAB) DAS DABCT (7.92 X lo4 cm2/s)(2.64 X
slope =
___
DAB(1.0 X
mol/cm3)
mol/cm3)
DAB = 1.60 X
= 13.04
cm2/s
where CAO/m= SUA, and aK = 202 for this system. The major points of this graphical method are that the influence of Sh can be determined from the intercept and DAB can be estimated from the slope. DA can also be determined from diffusive experiments. If E-' vs. CAoor an analogous measure of solute feed concentration is a straight line, the system is diffusion-limited. In conclusion, eq 20 can be a useful tool for determining the effect of external mass-transfer resistance on experimental measurements. Also, design of process equipment could be improved by using eq 20 to predict actual performance. Hollow-fiber (cylindrical) membrane systems could also be analyzed by using eq 20 in the model of Noble (1984) for these systems. Since numerical simulation gave virtually identical results with the analytical solutions, the analytical expression can provide a quick and accurate method for prediction of facilitation factors. Acknowledgment
This work was supported by the US. Department of Energy Morgantown Energy Technology Center under DOE Contract No. DE-AC21-84MC21271.
F = facilitation factor (see eq 18) K = dimensionless equilibrium constant = ( k f / k , ) ( C A o / m ) Keq= equilibrium constant, m3/mol k_ = reaction rate constant k = mass-transfer coefficient, m/s L = liquid film thickness, m m = solute partition coefficient = (external-phase concentration) / (membrane-phase concentration) S = solute concentration in membrane phase for Y = 1, mol/m3 Sh = Sherwood number = k e / D A x = distance from one edge of liquid film, m X = dimensionless distance = x / L Y = solute mole fraction in external phase Subscripts
A = solute A0 = solute (external phase) AB = carrier-solute complex B = carrier f = forward r = reverse T = total carrier (carrier plus carrier-solute) 0 = position where X = 0 1 = position where X = 1 Superscript * = dimensionless concentration Greek Letters = mobility ratio (see eq 7) 6 = inverse Damkohler number (see eq 6)
CY
Literature Cited Aerstin, F.: Street, G. Applied Chemical Process Des&; Plenum: New Ywk, 1978. Aris, R. Ind. Eng. Chem. Fundam. 1083, 22, 150-151. Bateman, 8. R.; Way, J. D.: Larson, K. M. Sep. Sci. Techno/. 1084, 79, 21-32. Cianetti, C.; Danesi, P. R. Solvent Extr. Ion Exch. 1083. 7 , 565-583. Danckwerts, P. V.; Kennedy, A. M. Trans. Inst. Chem. Eng. 1054, 32, s49-s59. Danesi, P. R.; Chlarlzia, R.; Castagnola. A. J. Membr. Scl. 1883, 74, 161. Donaldson, T. L.; Quinn, J. A. Chem. Eng. Sci. 1075, 30. 103-115. Folkner, C. A,; Noble, R. D. J. Membr. Sci. 1082, 12, 289-301. Hoofd, L.; Kreuzer, F. J. hfath. Blol. 1070, 8. 1-13. Hughes, R. D.; Mahoney, J. A.; Stelgelmann, E. F. Presented at the American Institute of Chemical Engineers Meeting, Houston, TX, Aprll 1981. Kemena, L. L.: Noble, R. D.;Kemp, N. J. J. Membr. Sci. 1083, 15, 259-274. Noble, R. D. Sep. Sci. Techno/. 1084, 79, 469-478. Smith, D. R.; Quinn, J. A. AIChE J. 1070, 25. 197-200. Way, J. D.; Noble, R. D.; Flynn, T. A,; Sloan, E. D. J. Membr. Sci. 1082, 72, 239-259.
National Bureau of Standards Center for Chemical Engineering 773.1 Boulder, Colorado 80303
Nomenclature
Richard D. Noble* J. Douglas Way Laurel A. Powers
Received for review December 3, 1984 Revised manuscript received November 24, 1985 Accepted February 18, 1986
C = concentration, mol/m3 D = diffusion coefficient, m2/s
Mass Transfer with Chemical Reaction In Two Phases The rate of mass transfer accompanied by chemical reactions of general order proceeding in two contiguous phases has been analyzed on the basis of the two-film theory. A previously proposed linear approximation procedure was shown to be useful in the present analysis, yielding satisfactory accuracy. Introduction
Numerous solutions are available for the problems of simultaneous diffusion and chemical reaction occurring in one phase. However, a number of industrially important organic reactions may take place in two liquid phases, in 0196-4313/86/1025-0452$01.50/0
principle, since many reactants have some solubility in both phases. Some examples of such reaction systems have been summarized by Doraiswamy and Sharma (1984). A few investigators (Rod, 1974; Mhaskar and Sharma, 1975; Merchuk and Farina, 1976; Sada et al., 1977) worked with 0 1986 American Chemical Society
