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 DAB = 1.60
mol/cm3)
mol/cm3) 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
L i t e r a t u r e Cited Aerstin, F.: Street, G. Applied Chemical Process Des&; Plenum: New Ywk, 1978. Aris, R. Ind. Eng. Chem. Fundam. 1083, 2 2 , 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 approximationprocedure 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
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
I I
I
The requirement for continuity of mass flux of both species at interface yields
I I I
PAakAa(cA2 I
I I
I
I
1
453
- CAai) = PAbkAbCAbi
(8)
= PBakBaCBai
(9)
P B b k B b ( c B b o - CBbi)
where the reaction factors PAb and P B b in phase b are defined as PAb
=-*( x) dCAb
CAbi
PBb
= CBbo
- CBb
x=o
(11) x=o
The expressions defining reaction factors PAaand PBain phase a may be obtained from eq 11 and 10, respectively, by the replacement of subscripts B and A by A and B and of b by a. Equations 1-4 are nonlinear and cannot easily be solved analytically. However, note that this problem gives the same set of differential equations and relevant boundary conditions as the problem of simultaneous absorption of two gases which react between themselves in a liquid (Hikita et al., 1977), when one pays attention to the diffusion of both species in the individual phase. Thus, the approximate analytical solutions for reaction factors @Ab and P B b in phase b are given by
where
Coefficients l b ' 0' = 0, 1, 2, ...) are those for the terms in the following knominal series In phase b: (3)
with
(4) The boundary conditions for eq 1-4 can be written as
The corresponding expressions describing mass transfer
454
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 ' ' " - _ _ _ _ Exact sol., Eq.04) '
20
'
' I
vb
-Resent appl: sd,Eq.(lZ)
10 -
Sadaet al.,appr. soI.fl977) Rod,appr. so1.(1974)
----Numerical sol.(Sada et aL(1977)) -Present appr. sol.,Eq.(lP) ----Sadaet al.,appr.sol.(l977)
8 -- Ya = 0 6 - HA@AbkAb/@AakPa> 1, HAPAbkAb/@Aak&
