Ind. Eng. Chem. Fundam. 1982, 21, 189
189
CORRESPONDENCE Comments on “Ultraflltratlon in an Unstlrred Batch Cell” Sir: Trettin and Doshi, in their recent publication (1980), investigated the ultrafiltration of macromolecular solution in an unstirred batch cell. However, their conclusion was based on an erroneous solution. They adopted a similarity transformation to solve the solute mass balance equation, eq 1,with the boundary and initial conditions, eq 2,3, and 4. One of the original independent variables, t , appears in the time-dependent parameter, V, (defined in eq 7) in the transformed eq 6. Therefore, the similarity transformation cannot be applied to eq 1without the assumption that V , is constant, which is not justified. The closed integral, eq 12, which is presented as the solution for the dimensionless permeate concentration, 0, is incorrect. At best it is only an approximation. The solution can be obtained readily without the assumption by applying Laplace transform
where all the notations are those of the authors. The above solution should be substituted for eq 12 and compared with the data for the physical validation of the formulations, eq 1, 2, 3, and 4.
Literature Cited Trettin, D. R.; Doshi, M. R. I d . Eng. Chem. Fundem. 1980, 79, 189.
Laboratory of Renewable Resources Engineering Purdue University West Lafayette, Indiana 47907
Juan Hong
Response to Comments on “UltrafiltrationIn an Unstlrred Batch Cell”
Sir: It appears that the correspondent has not understood the problem. The main objective of solving eq 1to 4 in our paper is to determine the variation of permeate velocity, u,, with respect to time. Thus, u or u, is a dependent variable which should not be assumed to be constant as the correspondent has done. It follows from the similarity transformation, eq 5, that u, should be proportional to t-O.S. This is also found from an integral method solution, eq 18, and the film theory solution, eq 27. Therefore, the final solution, eq 12 of our paper, is correct.
0196-4313/82/1021-0189$01.25/0
We feel that the correspondent has solved an unrealistic problem by assuming D, u,, C,, C,, and C, to be independent of time. Consequently, the reported solution is inconsistent as these time-independent parameters become a function of time. [Note that dimensionless V, is prosince u, and D are assumed constant by portional to the correspondent.] The Institute of Paper Chemistry Appleton, Wisconsin 54912
0 1982 American Chemical Society
Daniel R. Trettin Mahendra R. Doshi*
