A numerical model of steady-state permeate flux during cross-flow ultrafiltration

The mass transport mechanisms during cross-flow ultrafiltration (UF) are mathematically expressed using the two-dimensional convective diffusion equation where the axial diffusion term is neglected for an axial Peclet number much greater than the transverse Peclet number. A numerical scheme is presented to solve the steady-state two-dimensional convective diffusion equation for the case of known uniform permeate flux. However, in the actual cross-flow UF process, the permeate flux along the axial direction is unknown and usually decreases with axial distance. Therefore, an iterative algorithm is developed to predict the steady-state permeate flux based on the assumption that the concentration at membrane surface cannot exceed a certain limiting value. Using the numerical model with an effective diffusion coefficient, which is considered to be the sum of molecular diffusion and shear-induced hydrodynamic diffusion coefficients, the effects of particle size, feed concentration, and axial velocity on the steady-state permeate flux were investigated.