Modeling Ultrafiltration of Interacting Brownian Particles

Membrane crossflow ultrafiltration (UF) is widely used for the enrichment and purification of colloids and proteins. In this continuous process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes the solvent to flow out of the membrane channel. The TMP-induced permeate flux of a colloidal dispersion is commonly smaller than that of pure solvent. This unwarranted permeate flux reduction is due to the following effects. First, the permeate flux transports colloidal particles towards the inner membrane surface where they form a particle-enriched diffuse layer. This so-called concentration polarization (CP) layer increases the osmotic particle pressure which counteracts the applied TMP. Second, membrane-fouling mechanisms are operative, including the formation of adsorbed particle layers partially blocking the membrane pores, and the formation of a cake multi-layer of immobilized particles. Fouling mechanisms decrease the hydraulic permeability of the membrane and possibly cause irreversible damage to the membrane. We have developed a semi-analytic modified boundary layer approximation (mBLA) method to calculate concentration and flow profiles in the UF of colloidal dispersions from knowledge of the concentration-dependent bulk suspension properties [1, 2]. The mBLA is an accurate and fast method, producing results in full agreement with numerically expensive finite-element calculations [1]. Moreover, the semi-analytic mBLA expressions provide the important insight into the influence of dispersion properties such as the collective diffusion coefficient and shear viscosity, which helps to improve the UF efficiency [1, 2]. We present mBLA results for the UF of dispersions of solvent-permeable rigid, and of charge-stabilized colloidal particles in a broad range of particle sizes. A simple cake layer model is used to unravel the relations between critical and limiting permeate fluxes and applied TMP [3]. References[1] G. W. Park and G. Nägele, Journal of Chemical Physics, 2020, 153, 204110.[2] G. W. Park and G. Nägele, Membranes, 2021, 11, 960.[3] G. W. Park, M. Brito, and G. Nägele, manuscript in preparation.