This work presents an analytical solution to the differential equations that govern the electrically coupled transport of three different ions through membrane barrier layers in which the Solution-Diffusion–Electro-Migration model applies. This model disregards convective coupling between the trans-membrane flows of ions and solvent and uses composition-independent single-ion permeances to quantify ion transfer through the membrane. The limited number of adjustable parameters should make their unambiguous determination feasible from a limited set of experimental data. The solution presented here enables rapid calculations that explore the effects of spontaneously arising electric fields on rejections in nanofiltration (NF), volume flows in pressure-retarded osmosis (PRO) through NF membranes, and NaCl rejections in forward osmosis (FO). For NF, ion flux simulations confirm that large differences in the permeances of mono and divalent ions, e.g. Mg2+ and Cl− or Na+ and View the MathML source, can lead to negative rejections or enhanced rejections of trace monovalent ions, depending on the ion charge. Calculations also show that rejection of divalent ions by NF membranes can lead to significant osmotic flow, even in the presence of some excess of NaCl. Osmotic flow first decreases and then increases as the NaCl concentration increases in a draw solution containing MgCl2 or Na2SO4. In FO using MgCl2 as a draw solution and NF membranes, modeling reveals 3- to 4-fold enhancements in NaCl rejection relative to a pressure-driven process under similar conditions. These results demonstrate the importance of modeling coupled ion transport when designing membrane-based ion separations.

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