Membrane Sieving of a Continuous Polydisperse Mixture through Distributed Pores

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Abstract
A mathematical theory is presented to describe membrane sieving based on the relative sizes of pores and solutes. A polydisperse liquid mixture with a continuous distribution of solute radii undergoes hindered transport through distributed membrane pores. The hydrodynamic theory for rigid spheres in cylindrical pores provides expressions for the hindered diffusive and convective fluxes. Continuous and discrete (including fractal) distributions of pore sizes are considered. For a steady-state ultrafiltration process with a well-mixed upstream concentration distribution, the theory predicts how much the downstream concentration distribution shifts toward smaller solutes as larger solutes are hindered or rejected by the membrane. Sieving coefficients, which are related to permeation rates that depend on relative pore and solute sizes, are presented for gamma distributions of solute radii and for continuous and discrete pore-radii distributions.