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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.

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.

Membrane crossflow filtration is widely used for the enrichment and purification of colloidal and protein dispersions. In this pressure-driven process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes advection of the dispersion toward the membrane, and formation of a particle-enriched diffuse layer near its surface. This so-called concentration polarization (CP) layer increases the osmotic particle pressure counteracting the TMP. When the particle concentration reaches a solidification limit, an immobilized particulate layer, termed cake layer, is formed next to the membrane surface. The cake layer adds to the hydraulic membrane resistance and lowers thus the filtration efficiency. A key problem is to understand quantitatively how cake formation is related to filtration operating conditions, and to dispersion and membrane properties. In this study, we theoretically analyze how permeate flux and cake layer formation are influenced by the size, charge and feed concentration of dispersed particles [1]. We consider dispersions of neutral and charge-stabilized colloidal particles. Under conditions where shear-induced migration matters, empirical expressions for shear-rate dependent transport properties are used. Our results for concentration and flow profiles under filtration are obtained using a recently developed modified boundary layer approximation (mBLA) method [2, 3]. The mBLA is a numerically efficient and accurate method for predicting filtration properties. A thorough dicussion of the so-called critical permeate flux related to the onset of cake layer formation is presented. Moreover, an analytic expression for the critical flux is derived and compared with standard predictions by film theory and mass transfer coefficient calculations. References[1] G. W. Park, J. K. G. Dhont, and G. Nägele, manuscript in preparation.[2] G. W. Park and G. Nägele, Journal of Chemical Physics, 2020, 153, 204110[3] G. W. Park and G. Nägele, Membranes, 2021, 11, 960