VELOCITY PROFllLE IN THE STEFAN DIFFUSION TUBE SIR: For diffusion in a relatively large tube-e.g., I-inch diameter (2)-at atmospheric pressure, it is reasonable to assume that the mass average velocity, u, is zero a t the wall. Equation 2 of the original communication (5) is correct. This assertion is supported by experimental measurements of the pressure drop resulting from flow induced by diffusion in capillaries (3, 4, 7). The latter two investigations have shown that the measured pressure drop and bulk flow in two-bulb cells agree with Poiseuille’s equation. Slip occurs a t lower pressures and for smaller tubes than those considered in this discussion (7, 6, 7). The contradiction pointed out by Whitaker (8) remains, however. I t is explained by the tacit assumption (5) that the molecular weights of the two diffusing components are equal. The previous analysis (2) was also based on this assumption. Unfortunately, the reader is given the impression by Equation 3 of (5) that the molecular weights of A and B differ. This equation should be merely 5 = u, and then Equations 4 et seq. lead correctly to results given in the tables (5). Although uB is zero when averaged over the tube radius. it is of course not zero at the wall or a t the center of the tube. T h e mass average velocity has no radial component, but uA and uB certainly do. Component B diffuses downward at the wall counter to component A , so that the mas3 average velocity here is zero. If the molecular weights of the two components differ, 5 is thus not zero a t the wall, and Equation 3 of the original paper (5) is then only a n approximation. T o satisfy the equation of continuity, there is in the reported solution a flow of B u p a t the center of the tube, down at the walls, radially inward i n the lower part of the tube, and radially outward i n the upper part of the tube. The boundary conditions require that u B Z be zero for all r a t z = 0 a.nd z = L. This requires that u A = u a t the ends of the tube, and thi? velocity distribution is then what creates radial concentration gradients; A diffuses outward radially in the lower part of the tube and inward radially in the upper part of the tube.
SIR: \$’hitaker makes two points in his letter: first, that the no-slip condition at the rube wall is not valid in the Stefan diffusion tube; and, second, that the velocity profile is uniform, provided entrance and end effects are negligible. These points have received attention in the literature of the past (7-5). I n 1962 Heinzelmann et al. (7, 3 ) examined the assumption of no slip at the tube wall. PL perturbation analysis was attempted to obtain the rigorous solution of the momentum and species diffusion equations. This, analysis showed that the assumption of no slip violated the species continuity equation at the wall. Thus Whitaker’s result agrees with Heinzelmann’s. Heinzelmann et al. also considered the question of entrance and end effects. Their analysis showed that a typical sized diffusion tube was all entrance and end effects in that the velocity profile is continually developing from a flat distribution near the liquid surface to a parabolic distribution. Experimentally, Heinzelmann (7) also found that end effects were appreciable. I am aware of these refinements to the plug flow assumption which account for the progressive change in velocity distribution. When we wrote OUI- article (2) we dealt with the maximum departure from this simple model-i.e., fully developed
If the molecular weights of A and B are significantly different, the problem takes on an appalling complexity. T h e molal average velocity becomes
and in this equation the vectors are not colinear. where u is zero,
At the wall,
Elsewhere in the tube, ti has both radial and axial components, so that a term to account for radial flow must be added to the left-hand side of Equation I in (5). Equation 8 of ( 5 ) really holds only a t the ends of the tube and not a t every point in the tube. This equation was actually applied only a t the tube extremities, as was discussed in relation to Equation 13 in the original work. I n summary, the analysis in (5) appIies strictly only to fluids of constant density. literature Cited (1) Evans, R. B., Watson, G M., Mason, E. A., J . Chem. Phys.
35, 2076 (1961). (2) Heinzelmann, F. J., Wasan, D. T., Wilke C. R., IND.END. CHEM. FUNDAMENTALS 4, 55 (1965). (3) Kramers, H. A., Kistemaker, J., Physica 10, 699 (1943). ( 4 ) McKartv. K. P.. Mason, E. A.. Phvs. Fluids 3. 958 (1960). (5) Rao, S.’S., Bennett, C.’O., IND. ENG.CHEM.FUNDAMENTALS 5 , 573 (1966). (6) Scott, D. S., Dullien, F A. L., A.Z.Ch.E. J. 8,113 (1962). (7) Volobuev, P. V., Suetin, P. E., Souiet Phys.-Tech. Phys. 10, 269 (1965). ( 8 ) Whitaker S.,IND.ENG.CHEM.FUNDAMENTALS 6, 480 (1967).
S. S. Rao University of Connecticut Storrs, Conn.
C. 0. Bennett
parabolic velocity profile-since it was found to yield negligible error (1.4~o)it was felt that there was no need to go into any of the complications of the intermediate cases. Thus, although the velocity distribution does not i n fact follow plug flow, it happens that the error in assuming it is negligible. Thus whatever the shape of the velocity profile, it has been conclusively shown by Heinzelmann et al. and, both by theory and experiment, that diffusion data calculated from Stefan diffusion tube experiments with the plug flow approximation are substantially correct. Literature Cited
(1) Heinzelmann, F. J., M.S. thesis, University of California, August 1962. ( 2 ) Heinzelmann, F. J., Wasan, D. T., Wilke, C. R., IND.ENG. CHEM. FUNDAMENTALS 4, 55 (1965). (3). Heinzelmann, F. J., Wasan, D. T., Wilke, C. R., “Concentration and Velocity Profiles in a Stefan Diffusion Tube,” University of California Radiation Laboratory, Rept. UCRL-10421 (1962). ( 4 ) Lee, C. Y . ,Wilke, C. R., Znd. Eng. Chem. 46,2381 (1954) ( 5 ) Rao, S. S., Bennett, C. O., IND.END.CHEM.FUNDAMENTALS 5 . 573 (1966). D. T.Wasan Illinois Institute of Technology Chicago, Ill. VOL. 6
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