Mathematical Modeling of a Cation-Exchange Membrane Containing Two Cations

Transport phenomena in an ion-exchange membrane containing both H⁺ and K⁺ are described using the multicomponent diffusion (extended Stefan–Maxwell) equations. Expressions for macroscopic transport parameters, i.e., conductivity, proton transference number, water electro-osmotic coefficient, and transport parameters characterizing diffusion at zero current, are derived as a function of the binary interaction parameters, D_ij, used in the multicomponent transport equations. As experimental data for only four transport properties are available in the literature, the six D_ij values cannot be determined in an unequivocal manner. It is in harmony with the data that D_H+,K+ is large, and linear variations of ln(D_ij) with y_HM are assumed for the other D_ij coefficients. Values for the slopes of those linear variations are refined by nonlinear least-square regression on the four experimental transport properties. General governing equations to describe complete transport in the membrane with H⁺ and K⁺ are presented, and the model is used with particular boundary conditions to describe the behavior of a membrane used in a CO₂–H₂O electrolyzer. This provides some insights on macroscopic quantities such as the ohmic drop and water transport that are relevant for cell operation.