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.

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A generalized treatment of fluid transport in a porous membrane is developed on the basis of the ’’dusty-gas’’ model, which is extended to include electrical forces due to charges on the solutes and the membrane. The Stefan–Maxwell diffusion equations are augmented to include viscous flow, and then phenomenologically generalized for the case of any fluid mixture. The resulting generalized transport equations are cast back into the original Stefan–Maxwell form. From these a set of generalized Nernst–Planck equations is obtained that includes convective flow and solute–solute interactions. It is demonstrated that viscous flow effects can be properly incorporated into the Stefan–Maxwell equations by augmenting the diffusion coefficients with viscosity terms. It is also shown that the main source of error in the usual simplified form of the Nernst–Planck equations lies in the treatment of convective flow terms. The generalized Nernst–Planck equations approach the proper limit for gases in both the Knudsen and continuum regimes, and can be easily reduced for liquid solutions to the simple form proposed by Schlögl and others.