Theory of concentration polarization in cross-flow ultrafiltration: gel-layer model and osmotic-pressure model

Abstract

A mathematically rigorous theory of concentration polarization in a steady-state cross-flow ultrafiltration is developed. The theory proceeds from the results of the analysis of laminar flow of a solution and convective diffusion of the low-mobile solute in porous channels with non-uniform wall suction. Two physical models of concentration polarization which are typically used in the literature are analyzed in detail, namely, the gel-layer model and the osmotic-pressure model. For either model, an equation describing the pressure/flux curve is derived. In the case of the gel-layer model, the theory leads to a simple analytical formula for the limiting flux. The flux turns out to be proportional to the cube root of the ratio of the gel concentration to the feed solution concentration, rather than to the logarithm of this ratio, as the simplified Michaels-Blatt theory predicts. On the other hand, in the case of the osmotic-pressure model, the rigorous theory allows the conclusion that at high applied transmembrane pressure, the permeate flux increases as a cube root of the pressure, so that the limiting flux is never reached. This suggests a simple criterion for the choice between the two models when interpreting experimental pressure/flux curves.