A unified model for quantification of concentration polarization (CP) of particles during cross-flow membrane filtration

Brownian diffusion, inter-particle interactions, shear-induced diffusion and hydrodynamic lift forces all contribute to restricting the formation of concentration polarization (CP) during cross-flow membrane filtration of particles. In this study, a unified CP model was developed for the quantification of the CP formation of colloidal particles. The basic methods adopted in previous models such as mass balance equation accounting for particle diffusion and convection, disintegration of the solid pressure and osmotic pressure and derivation of particle drift velocity are kept. In the new model, the hydrodynamic lift force is taken into account in the force balance equations and shear-induced diffusion is included in the particle transport process. The simulation results show that the inter-particle interactions and the hydrodynamic lift force are predominant for relatively small and large particles, respectively. More specifically, while the inter-particle interactions are important for relatively small (e.g. 2a = 10 nm) particles, the hydrodynamic lift force dominates for relatively large particles (e.g. 2a = 1 μm). Neither inter-particle interactions nor hydrodynamic lift forces play a significant role in CP formation of sub-micro particles (e.g. 2a = 100 nm), for which the solid fraction buildup within the CP formation is most severe. For filtration of relatively large particles under a fixed hydrodynamic condition, it appears that there is a critical flux, over which the particle deposition rate will increase dramatically. Similarly, at a fixed filtration flux, there exists a critical shear stress induced by the cross-flow, over which the membrane fouling rate will decrease sharply. Use of a higher shear intensity can effectively alleviate membrane fouling caused by particles of all size ranges.