Simple Analytical Approximations for Donnan Ion Partitioning in Permeable Ion-Exchange Membranes Under Reverse Electrodialysis Conditions

Reverse electrodialysis (RED) is a relatively recent technology for renewable energy harvesting from the interaction of river and seawater. This paper revisits the thermodynamic equilibrium governing the ionic transport processes through ion-exchange membranes (IEMs) under RED conditions and theoretically derives approximate analytical expressions for the ionic concentrations at the inner boundaries of a permeable membrane with well-stirred baths. The equation for the Donnan ion partitioning at the membrane–solution interface, which is based on the equality of the electrochemical potential in the two phases, is analysed for binary salts with symmetric (1:1) and asymmetric (2:1) electrolytes, by considering bathing solutions with the equivalent concentrations 0.02 M in the dilute bath, and 0.5, 1, and 1.5 M in the concentrate one. Simple approximate analytical expressions exhibiting the evolution with the membrane fixed-charge concentration of the counter-ionic concentrations at the inner boundaries of the membrane, the concentration gradients inside the membrane, the total Donnan electric potential, and the ionic partitioning coefficients have been derived. The approximate generalised expressions for a general z1:z2 binary electrolyte are also presented for the first time.