CONCENTRATION-DEPENDENT DIFFUSION COEFFICIENTS AND RATES OF MASS TRANSPORT SIR: In a recent paper (3), King has noted that mass transfer coefficients for the absorption of ammonia, acetone, methanol, etc., into water tend to be lower than those predicted by the equation
The same author has shown that for this equation to hold true, two restrictions are necessary. First, kl must be independent of k:! a t any point in the system, and second, the ratio ( k l / k l ) and m must both be constant a t all points of interface within the system. One reason why the second restriction is often broken is that the diffusion coefficient may be strongly concentration-dependent. For example, the liquid phase diffusion coefficient a t 25" C. of ethyl alcohol in water may be very well represented by the polynomial
D = 1.228 X lOV5(l - 2.741 C
+ 2.688 Cz)
(4)
(2)
I have shown (4) that the mass transfer coefficient, whether it is predicted by using the film (5), Higbie penetration (7), or any other type of mechanistic model (of which there are many) is strongly dependent on the concentrations a t the interface and in the solution bulk when the diffusion coefficient is concentration-dependent. I n the case of the transport of ethyl alcohol into a water phase, the film type of model predicts that when the difference between the interfacial and bulk concentration is 0.75 weight fraction, a maximum lowering of the mass transfer coefficient of 547, below that predicted for the diffusion coefficient a t infinite dilution is to be expected. The Higbie penetration model predicts similarly a reduction of 50.5% in the expected mass transfer coefficient when the difference between the interfacial and bulk concentrations is 0.80 weight fraction. The concentration differences between the interfaces and solution bulks are large in the above examples. Systems in which such differences occur are few. This, however, does not detract from the importance of the variable diffusion coeffi-
SIR: Any factor causing mass transfer coefficients to vary significantly over the range of compositions encountered in a contacting device should be considered in the analysis of mass transfer rates. As Rhodes points out, concentration-dependent diffusion coefficients produce concentration-dependent mass transfer coefficients. One result of this behavior can be a widening of the distribution of values of ( k l / k ~ )or (kl*/k**) from point to point of interface, with a consequent tendency toward greater deviations from the classical addition of resistances relationship ( 3 ) . Changes in concentration level can influence mass transfer coefficients through several factors. Other pertinent physical properties-such as viscosity, density, partial molal volumes, and surface tension-will vary in addition to diffusivity. I n the case of a small ACA driving force a t a relatively uniform solute concentration level one should clearly employ physical 146
I&EC FUNDAMENTALS
cient, as the "level" of concentration within the system will be very important. Thus the situation in which a very small driving force exists across the hypothetical surface film when the ethyl alcohol solution is very dilute will predict a mass transfer coefficient which is more than three times the coefficient predicted when the general level of concentration is 0.5 weight fraction. It is suggested that the tendency of measured mass transfer coefficients to be different from those predicted by theory may be due to the neglect of the concentration dependence of diffusion coefficients. The author hopes to show in a future publication that unless variable diffusion coefficients are allowed for in the theory, the use of increasingly sophisticated models in mass transport analysis will be pointless. Acknowledgment
Thanks are due to Department of Scientific and Industrial Research for financial support of the author, and to the Department of Chemical Engineering, University of Manchester, where this work was conducted. Nomenclature
C = concentration of ethyl alcohol (weight fraction) D = diffusion coefficient k* = average mass transfer coefficient for phase 1 (or 2) measured in absence or suppression of resistance in phase 2 K 1 = over-all mass transfer coefficient m = slope of equilibrium curve Literature Cited
(1) Higbie, R., Trans. A m . Inst. Chem. Eng. 31, 365 (1935). (2) King, C. J., A.1.Ch.E. J . 10, 671 (1964). (3) King, c. J., I N D . ENG.C H E M . F U N D h h r E N T a L s 4, 125 (1965). (4) Rhodes, E., Ph.D. thesis, University of Manchester, England, 1964. (5) Whitman, W. G., Chem. M e t . Eng. 29, 146 (1923).
E d w a r d Rhodes
Lrniuersity of I C'aterloo Tl'aterloo, Ontario, Canada
properties measured or predicted for the concentration level under consideration. In the event of a relatively large ACq driving force one must allow for continuous changes in all pertinent physical properties along the zone of primary resistance to mass transfer between surface and bulk. Another effect of concentration level comes about in the following way: I t is desirable for mass balance purposes and for considerations of interphase transfer to define mass transfer coefficients as the ratio of total interfacial flux of the particular solute (IVAo or nAo) to the appropriate driving force. Such a definition requires the inclusion of a concentration-dependent term of the form [l - zao(l S)]in the Sherwood group of a correlation for the mass transfer coefficient of component A. When Fick's law is considered in the form j , = - p D . h B C ~ . d and Awa is used as the driving force for k , z is kveight fraction and S is the ratio of mass fluxes across the interface ( n B O / n A O )
+
