Flow through charged membranes

Abstract

A theoretical analysis of transport through charged membranes is presented using a cell model. In an attempt to improve the capillary tube model, we have modeled the membrane by an array of charged spheres. Three sets of partial differential equations describe this system: the generalized Nernst–Planck flux equations, the Navier–Stokes equation, and the Poisson–Boltzmann equation. These equations are averaged over a cell volume to yield a set of ordinary differential equations on the gross scale, that is, a length scale of the order of the membrane thickness. It is shown for hyperfiltration that the cell model will reject salt more efficiently than the tube model, and in the limit of Stokes’ flow the present analysis reproduces the rejection coefficient previously obtained for the capillary tube model. Results for the electrodialysis mode of membrane operation are also presented. Comparison is made with the capillary tube model for the same total charge and equal volume to surface ratio. The effect of a spatially varying wall charge density on membrane performance is discussed.