Generalized Nernst–Planck and Stefan–Maxwell equations for membrane transport

Abstract

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.