Abstract
Cross-flow ultrafiltration is a pressure-driven separation and enrichment process of small colloidal particles where a colloidal feed dispersion is continuously pumped through a membrane pipe permeable to the solvent only. We present a semi-analytic modified boundary layer approximation (mBLA) method for calculating the inhomogeneous concentration-polarization (CP) layer of particles near the membrane and the dispersion flow in a cross-flow filtration setup with a hollow fiber membrane. Conditions are established for which unwarranted axial flow and permeate flow reversal are excluded, and non-monotonic CP profiles are observed. The permeate flux is linked to the particle concentration on the membrane wall using the Darcy-Starling expression invoking axially varying osmotic and trans-membrane pressures. Results are discussed for dispersions of hard spheres serving as a reference system and for solvent-permeable particles mimicking non-ionic microgels. Accurate analytic expressions are employed for the concentration and solvent permeability dependent dispersion viscosity and gradient diffusion coefficient entering into the effective Stokes flow and advection-diffusion equations. We show that the mBLA concentration and flow profiles are in quantitative agreement with results by a finite element method. The mBLA results are compared with predictions by an earlier CP layer similarity solution, showing the higher precision of the former method.
Highlights
Membrane ultrafiltration (UF) is a pressure-driven process for the concentration and purification of dispersions of particles undergoing strong Brownian motion, having the major advantage of low energy consumption
We further show that the modified boundary layer approximation (mBLA) results are in distinctly better agreement with the finite element method (FEM) data than a similarity solution prediction for the CP layer employed in an earlier work
We have presented a generic model describing CP layer effects in the cross-flow ultrafiltration (UF) of dispersions of solventpermeable, hard spherical particles inside a hollow fiber membrane
Summary
Membrane ultrafiltration (UF) is a pressure-driven process for the concentration and purification of dispersions of (colloidal) particles undergoing strong Brownian motion, having the major advantage of low energy consumption. On the basis of this similarity solution, the CP layer and permeate flux profiles in cross-flow UF were calculated in two studies by Roa et al. for dispersions of charge-stabilized particles and for a model of solvent-permeable particles mimicking non-ionic microgels, respectively In these studies, accurate analytic expressions were used for the concentrationdependent gradient diffusion coefficient, dispersion viscosity, and osmotic pressure. We present a modified boundary layer analysis method, referred to as the mBLA method, and an associated finite element method (FEM) of calculating the CP layer and flow profiles for cross-flow UF of dispersions of non-permeable and solvent permeable particles inside a hollow fiber membrane. Salient details of the mBLA method are summarized in Appendixes C and D
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