Extended ion partitioning in ion-exchange membranes: Approximate models with implications in electrodialysis and reverse electrodialysis

A thorough understanding of electro-membrane processes, including electrodialysis (ED) and reverse electrodialysis (RED), requires detailed knowledge of ionic concentrations at the inner boundaries of ion-exchange membranes (IEMs). To this end, the Donnan equilibrium equation at the IEM-solution interface, which is based on the equality of the electrochemical potential in the two phases, must be analytically or numerically solved. When the number of involved parameters increases, as occurs in multivalent systems described within the extended framework, the use of approximate expressions with simple functional dependences enables a more efficient interpretation of the ion transport process. In this work, a set of approximate analytical expressions for the ionic concentrations at the inner boundaries of a permeable IEM, including activity coefficients and steric parameters, are derived for multivalent electrolytes. We use a general procedure to obtain approximate expressions for the ionic concentrations from small perturbations with respect to total or partial ideally selective behaviors. The results are applied, for the first time, to analyse the variation of concentrations inside the membrane in an ED stack with the 1:1 electrolyte, the extended ion partitioning for mono and divalent cations in multi-ionic electrolytes, and the uphill transport for the divalent cation in a RED stack. The simplified analytical expressions provide a practical tool for the design, optimization, and performance assessment of electro-membrane processes operating with multivalent electrolytes. The highlights for the manuscript “Extended ion partitioning in ion-exchange membranes: Approximate models with implications in electrodialysis and reverse electrodialysis” by PRIVATE A.A. Moya are: • Extended ion partitioning equation at the IEM-solution interface is studied. • Approximate concentrations are derived for multi-valent binary electrolytes. • Approximate concentration variations inside the IEM in ED stacks are obtained. • Approximate concentrations are derived for multi-ionic electrolytes with co-ion exclusion. • Approximate analysis is used to describe uphill transport for divalent cations in RED.