Critical Flux in Particle Ultrafiltration of Variable Sizes: Using Interacting Brownian Dispersions

Cross-flow ultrafiltration (UF) is a pressure-driven separation process used, e.g., for water purification, blood treatment by (artificial) kidneys, and protein enrichment. In this process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes solvent to flow out of channel so that a particle-enriched diffuse layer is formed near the membrane walls. This so- called concentration-polarization (CP) reduces the permeate flux and hence the filtration efficiency, due to osmotic pressure built-up counter-acting the TMP. The particle concentration inside the CP layer increases with the applied TMP. When it reaches a freezing concentration where the particles are immobilized, an unwarranted cake layer is formed at the membrane walls. The permeate flux value signaling the onset of cake formation is called the critical flux. Predictions of the critical flux are commonly made using standard film theory which, however, applies only to colloidal particles and proteins of a diameter smaller than 10 nm. When the diameter roughly exceeds 100 nm, the predicted critical flux is an order-of- magnitude smaller than the experimental one. This is referred to as the critical flux paradox.We present theoretical results for the UF concentration and flow profiles, and the critical flux for dispersions of various size of particles. The results are obtained using a recently published modified boundary layer approximation (mBLA) method of cross-flow UF. The semi-analytic mBLA method provides an accurate description of UF concentration and flow profiles, on accounting for the concentration dependence of dispersion transport properties and osmotic pressure. The considered model dispersions encompass impermeable and permeable hard spheres and charge-stabilized particles. For hard-sphere dispersions, the mBLA method is extended to the microfiltration regime where shear-induced migration becomes important.