Modeling crossflow filtration: Effect of shear on particle-enriched polarization and cake layers

Crossflow filtration is a pressure-driven separation and enrichment process of colloidal particles, where a feed dispersion is continuously pumped through a hollow cylindrical membrane channel permeable to the solvent only. During the filtration process, particles are advected to and accumulate at the membrane, forming a fluid-like concentration polarization (CP) layer and a solid-like cake layer. Based on an accurate semi-analytic method developed in this paper, we determine spatially resolved flow and particle concentration profiles in the transition regime between ultrafiltration and microfiltration, where the effects of shear flow compete with Brownian particle motion. The results are presented for model dispersions of colloidal hard spheres, wherein we account for the shear-rate and concentration dependence of collective diffusion and viscosity, and for a thin, permeable filter cake. In particular, a non-monotonic shear-rate dependence of the axial flow across the CP layer is found. We show that the shear-rate dependence of particle transport properties gives insight into the origin of a long-standing apparent paradox related to the critical permeate flux characterizing the onset of cake formation.