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
e E = 0 3687 MCunr 8 Wtlholm (19491 E = 0 3547 MCuw 8 Wilholm 119491 A
E = O 436 Wlll$arnson
01 01
(19631
Theoreltcal resulis EI-Ka#ssy 8 Homsyll9731 E i 0 2 6 Karobelos 01 a1ll9711 E = 0 41 Jolls 8 Honrally(1969) a E = 0 26 ThoeMS 8 Kram~rsl1958lPack~ng lb 0
These data along with those of Thoenes and Kramers (1958) for packing l b at high Schmidt number and intermediate Reynolds numbers are plotted in Figure 1 as the j factor us. (Re)mod where
and y = (1 - €)1’3
Figure 1. Plots o f j factor us. (&)mod at different z .
Sh = 1.26
[?
agrees very well with the experimental measurements by Karabelas, et al. (1971) of the overall mass transfer rate to an active sphere in a close packed cubic array of inert spheres ( 6 = 0.26). Expression 2 was also found to be quite good for high values of e. Inspection of Figures 2 and 3 in Pfeffer’s (1964) article as well as Figure 4 in El-Kaissy and Homsy’s (1973) article show that the latter’s analytical expressions (which reduce to that of Pfeffer for creeping flow regime) are significantly above the data in the creeping flow range. This is because the data used by these authors should not be used for testing the theory for reasons already mentioned. We will now show that, in the intermediate range of Reynolds numbers, where Red is greater than 10 but less than Re, which defines the transition from the steady to the unsteady flow, conclusions drawn by El-Kaissy and Homsy (1973) about the range of usefulness of their analytical expressions for transport in packed beds ( e . g . , on page 88 of their paper “The theoretical curve is seen to overpredict the experimental data for all Red less than about 60 above which agreement is excellent,” or “The agreement between theory and experiments is good in general and becomes better as R e d is increased”) are somewhat confusing. It is known that in the intermediate range of Reynolds numbers, measurements of overall mass transfer coefficient to a single active sphere in a packed bed of inert spheres indicate that the Sherwood number no longer varies as (Pe)ll3. Rather Sh varies as Sc1l3 Redn where the exponent n varies between $ and 2/3. For example, consider the following correlations of overall mass transfer data on single active sphere in a packed bed of inert spheres. (i) Karabelas, et al. (1971) Sh = 2.39(Red)0*56(SC)‘/3 (3) The conditions of measurement were Pe ------t m; Red Re,; = 0.26, dense cubic
-
It fits the data over the Red range of 30-1000. (ii) Jolls and Hanratty (1969) Sh = 1.44(Red)0*58(SC)‘/3 The conditions of measurement were Pe “0; 140 > Red > 35; E =
-
(4 )
0.41, random packing 74
Ind. Eng. Chem., Fundam., Vol.
14, No. 1, 1975
In the same Figure 1 are plotted the analytical results of El-Kaissy and Homsy (1973) and the experimental data that these authors have used for comparison with their theoretical solution. It is obvious that their theoretical expression for mass transport to a sphere in a packed bed at high Schmidt numbers underestimates the available mass transfer data on single active spheres in the intermediate Reynolds number range mentioned in their paper. Further, their analytical solutions indicate a value of the exponent n of Reynolds number quite different from what the proper experimental data indicates. In fact, the correct Reynolds number dependence of the Sherwood number is displayed quite interestingly by the following interpolation formula of Karabelas, et al. (1971), for pure forced convection Sh = [(4.58Re‘/3S~‘/3)6+ (2.39Re0.56S~”3)6]1/6 (6)
which describes their single active sphere data for e = 0.26 (dense cubic) from the creeping flow regime to Rec. Finally, it should be pointed out that the extent to which their analytical solution for a spherical cell model underpredicts the experimental mass transfer data seems to be comparable to that in the case of drag prediction by the same authors in packed beds a t intermediate Reynolds numbers. Acknowledgment The author wishes to thank Professor George M . Homsy for providing him with a large size copy of Figure 5 of their paper. Nomenclature d = diameter of the active sphere Di = diffusivity of the ith species K = mass transfer coefficient Pe = Peclet number (Usd/Di) Rec = Reynolds number (U,d/u) at transition to turbulence Red = Reynolds number ( U,d/v) (&)mod = Reynolds number def.ined by eq 5 Sc = Schmidt number ( u / D , ) Sh = Sherwood number ( K d / D i ) U, = superficial velocity in the pa‘cked bed
Greek Letters y = (1 - e p ’ 3 t = void volume fraction u = kinematic viscosity
Literature Cited El-Kaissy, M. M., Hornsy. G. M., Ind. Eng. Chem., Fundam., 12, 82 (1973). Happel, J.,A.I.Ch.E. J., 4, 197 (1958). Jolls, K. R., Hanratty. T. J., A.I.Ch.E. J., 15, 199 (1969). Karabelas, A. J., Wegner, T. H., Hanratty, T. J.. Chem. Eng. Sci., 26, 1581 (1971). Pfeffer, R . , Ind. Eng. Chem., Fundam.. 3, 380 (1964). Sirkar. K. K., Chem. Eng. Sci., 29, 863 (1974). Thoenes, D.,Kramers. H., Chem. Eng. Sci., 8, 271 (1958)
Department of Chemical Engineering Indian Institute of Technology Kanpur-208016, U.P.,India
Kamalesh K. Sirkar
