Generalized integral and similarity solutions of the concentration profiles for osmotic pressure controlled ultrafiltration

A mathematical model describing the concentration polarization phenomenon during osmotic pressure controlled ultrafiltration is presented. Generalized integral and similarity solutions of the concentration profile in the mass transfer boundary layer are obtained. The parameters governing the shape of the concentration profile vary with time in case of a batch cell and axial distance in a cross flow cell. The model is used to predict the permeate flux and the solute rejection simultaneously during unstirred batch cell and cross flow UF. The results obtained by integral and similarity solutions are compared with the results of detailed numerical solution of the governing equations for both the systems. The predictions of permeate flux from the generalized integral method are also compared with some approximate solutions in order to assess the limitations of the various approximations. UF experiments were performed with Dextran (T-20) in cross flow system and with PEG-6000 and Dextran (T-40 and T-20) in unstirred batch cell. Predictions of the model are in remarkably good agreement with detailed simulation as well as experimental results. Moreover, the integral solution can also account for the variation of diffusivity with solute concentration. Comparisons show that (a) while the generalized integral method is much simpler than the detailed numerical solutions, it is much more general and accurate than other analytical and semi-analytical solutions, and, (b) the proposed solution predicts the osmotic pressure controlled flux decline accurately over a wide range of operating conditions. The expression for gel layer governed UF (constant membrane surface concentration) is found to be an asymptotic case of the present solution.