Friend, J. P., Leifer, R., Trichon, M.. J. Atrnos. Sci., 30, 465 (1973). McLaren, E., Yencha, A. J., Kushnir, J. M., Mohnen, V. A., Tellus, 26, 291 (1974). Mascorello, J., Auclair, J.. Hamelin. R.. Pglecier, C.. Proc. Arner. Power Conf., 31, 439 (1969). Matteson, M. J., Fox, J. J., Preining, O., Nature Phys. Sci., 238, 61
Schumann, H., 2.Anorg. Chem., 23,43 (1900). Vian, O., Iriarte, F., Melcheso, S.,U.S. Patent2,912,304 (1959). Yencha, A. J., Kushnir. J. M.. McLaren, E., Mohnen, V. A,, Chessin, H., Tellus, in press.
110791 \‘“.L,.
Received for review May 20,1974 Accepted November 4,1974
Preining, O.,Boscoe. G. F., Matteson, M. J., J. Aerosol Sci., 5, 71 (1974).
CORRESPONDENCE
Use of Infinite-Dilution Activity Coefficients for Predicting Azeotrope Formation at Constant Temperature and Partial Miscibility in Binary Liquid Mixtures
Sir: Brandani (1974) has proposed using infinite-dilution activity coefficients to predict partial miscibility in binary liquid mixtures. For a binary mixture which is described by the van Laar equation, Brandani states that a miscibility gap will exist if
where A = In T i m , B = In Y 2 m and Q = A/B. The symbol [Q] represents “the integer part of Q” and A and B are defined so that Q 2 1. Inequality (1) was apparently derived from thermodynamic stability theory. However, the result is unnecessarily slack and a precisely rigorous form is available. Rigorously, for A and B such that
A
>
1
-[2(1 2Q
- Q
+
Q2)3/2
-
(2
- 3 8 - 38’ +
2@)] (2)
the van Laar equation predicts a miscibility gap. If the inequality in (2) is replaced with the equality sign, the value of A calculated corresponds to critical solubility conditions. The consolute composition may be cal-
culated from
Equations 2 and 3 follow from critical solubility conditions appled to the van Laar equation as presented by Treybal (1963). They have an additional virtue in that they apply for any positive Q (not only for Q > 1). If A is less than the limiting value implied by eq 2 the van Laar equation will predict no miscibility gap. Equation 1cannot be used with this assurance. Limiting values of A calculated from the inequality of Brandani (1974) and from eq 2 are presented in Table I for comparison. Brandani calculates that for Q = 5.581 (phenol, water), there will be a miscibility gap for A > 3.538. In fact, the van Laar equation predicts a miscibility gap forA > 3.080. I should like to point out also the way in which Brandani has dealt with the data of Brian (1965). The five systems involved are indicated by an asterisk in Table I. Brian in the first place calculated A and B for these five systems beginning with mutual solubility data. Nothing a t all is proved by comparing the resulting values with ei-
Table I. Prediction of Partial Miscibility
Acetonitrile Ethanol Aniline 2-BUtanol 1- Butanol
Phenol Propylenoxide Ethanol Methyl ethyl ketone 1-Propanol Nitromethane Ethanol 72
Benzene Benzene Water* Water* Water* Water* Water* n- Hexane u- Hexane Water Carbon tetrachloride Isooctane
Ind. Eng. Chern., Fundarn., Vol. 14, No. 1, 1975
1.163 2.361 4.223 3.807 3.795 3.691 2.557 3.045 1.301 2.642
1,059 1.581 3.022 4.112 4.487 5.581 1.430 1.305 1.114 2.152
2.361 3.314
1.138 1.393
2.060 2.719 3.245 3.406 3.538 2.509 2.346 2.120 2.772
2.057 2.420 2. a44 2.979 3.011 3.080 2.336 2.255 2.106 2.648
2.146 2.459
2.127 2.313
3.011
ther inequality (1) or (2). The rigorous result must apply to any A and B values calculated in this way.
Treybal, R. E.. “Liquid Extraction,” 2nd ed, p 82. McGraw-Hili, New York. N. Y.. 1963.
Literature Cited
Department of Chemical Engineering University of Calgary Calgary, Alberta, Canada T2N IN4
Brandani, V . , Ind. Eng. Chem., Fundam., 13, 154 (1974) Brian. P. L. T.. Ind. Eng. Chem., Fundam.. 4, 100 (1965).
Robert A. Heidemann
Use of Infinite-Dilution Activity Coefficients for Predicting Azeotrope Formation at Constant Temperature and Partial Miscibility in Binary Liquid Mixtures
Sir: I agree with Heidemann that the inequality (2), reported in his letter, is the rigorous form deriving from the van Laar equation. However, bearing in mind that, as was pointed out for symmetric systems by Renon and Prausnitz (1968), “the minimum activity coefficient a t infinite dilution required for phase instability is lowest in the van Laar equation” (In y m = 2), while it is variable in the case of the NRTL equation increasing from In y m = 2 to In y = 2.94, I think that my simple empirical equation is of use for a preliminary estimation of the occurrence of partial miscibility. In fact the A m i n values calculated by my equation present an average deviation of about 6% compared with those obtained from the rigorous equation and presented by Heidemann in Table I (the largest deviation of about 30% one could have for Q = 1.999). Moreover, if the data (A = 2.533, Q = 1.686), reported by Smith and Robinson (1970) for the ethanol-benzene system at 25”C, are considered, while Heidemann ( A m i n = 2.471) calculates that will occur phase splitting, my equation predicts phase stability in agreement with the available experimental evidence. Another question raised by Heidemann in his letter is
that his equation must be applied to any A and B values calculated beginning with mutual solubility data. Leaving out of account the uselessness of the immiscibility gap prediction from mutual solubilities, for the furfural-nheptane system the value of Q , calculated from the mutual solubility data reported by Pennington and Marwil (1953), is 1.005 and both equations give the same value 2.005 for A m i n . Finally, the correct expression for calculating the critical consolute composition is XI = [Q - (1 - Q + Q z ) l l z ] / ( Q - 1) whereas with eq 3 it is possible to evaluate the mole fraction of the other component.
Literature Cited Pennington, E. N., Marwil, S.J.. Ind. Eng. Chem., 45, 1371 (1953). Renon, H., Prausnitz, J. M., AIChEJ. 14, 135 (1968). Smith, V . C.,Robinson, R. L., J. Chem. Eng. Data, 15, 391 (1970).
Istituto di Chimica Applicate e Industriale Uniuersita Degli Studi Dell’Aquila L’Aquila, Italy
Vincenzo Brandani
Transport in Packed Beds at Intermediate Reynolds Numbers
Sir: In a recent paper, El-Kaissy and Homsy (1973) had developed analytical expressions for mass transport to a sphere in a packed bed of spheres for high Peclet numbers and for the intermediate range of Reynolds numbers beyond the creeping flow regime. The axisymmetric velocity profile around a sphere in the packed bed was obtained from the full Navier-Stokes equation by using the free surface cell model of Happel (1958) and the technique of regular perturbation around the creeping flow solution. This velocity profile was substituted in the convective diffusion equation and expressions for Sherwood number were derived for high Peclet numbers. Excellent agreement was claimed between the theoretical results and the available experimental data on mass transfer in packed bed of spheres. However, the data used for comparison correspond to the overall mass transfer coefficient to a sphere in packed beds where all the spherical particles are taking part in the mass transfer process. The data that should have been used for comparison is the overall mass transfer coefficient to a sphere which is exchanging mass with the fluid in a packed bed in which all other spheres are inactive or inert. Jolls and Hanratty (1969) have pointed out that the data on overall mass transfer coeffi-
cient for a sphere in a packed bed where all the spheres are active are significantly lower than those for a single active sphere in a packed bed of inert spheres. Further, in Figure 4 of El-Kaissy and Homsy (1973), the theoretical results are consistently and significantly above the quoted experimental data up to a Red of about 50, beyond which there is a reasonable correlation with the data plotted. This behavior may be explained in the following manner. The present author (Sirkar, 1974) has recently derived the following analytical expression for creeping flow mass transfer to a single active sphere surrounded by a random cloud of inactive spheres at high Peclet numbers
{8(1 Sh = 0.992
-
E) -
3(1
- E)2}i’2
~ ( 2- 3(1 - E))
(1)
This expression is useful for packed beds only at high values of t and it agrees very well with the mass transfer data on single active spheres for t = 0.476, Red < 10, large Sc (Thoenes and Kraemer, 1958, packing 4b). It was further shown by the present author that Pfeffer’s (1964) analytical expression for the same problem Ind. Eng. Chern., Fundarn., Vol. 14, No. 1, 1975
73
