Bilayer ion-exchange membranes consisting of a thick heterogeneous membrane layer and a thin homogeneous membrane layer are studied theoretically and experimentally for their electrochemical properties such as overall and partial current–voltage curves, external and internal diffusion-limited currents, effective transference numbers, and specific selectivity for counterions and coions to be separated. For a diffusion layer (I)/modifying layer (II)/substrate membrane (III)/diffusion layer (IV) system in ternary solutions of electrolytes, we develop and verify a four-layer mathematical model with quasi-equilibrium boundary conditions. For the conditions of small current densities and limiting currents, we obtain analytical expressions for calculations of the limiting current, effective transference numbers, and specific selectivity coefficients. A numerical approach to calculating the specific selectivity coefficients is proposed. The mathematical model is verified for two electromembrane systems: one system is an anisotropic bilayer membrane with its MK-41 and MF-4SK polymer matrices having the same type of charge in a mixed CaCl₂ + NaCl solution, and in another one its MA-41 and MF-4SK matrices have opposite types of charge in a mixed Na₂SO₄ + NaNO₃ solution. The transport parameters of both the modifying film (MF-4SK) and initial membranes (MK-40 and MA-41) are determined in single-component and mixed electrolyte solutions in separate experiments.

The paper shows the possibility of using a microheterogeneous model to estimate the transport numbers of counterions through ion-exchange membranes. It is possible to calculate the open-circuit potential and power density of the reverse electrodialyzer using the data obtained. Eight samples of heterogeneous ion-exchange membranes were studied, two samples for each of the following types of membranes: Ralex CM, Ralex AMH, MK-40, and MA-41. Samples in each pair differed in the year of production and storage conditions. In the work, these samples were named “batch 1” and “batch 2”. According to the microheterogeneous model, to calculate the transport numbers of counterions, it is necessary to use the concentration dependence of the electrical conductivity and diffusion permeability. The electrolyte used was a sodium chloride solution with a concentration range corresponding to the conditional composition of river water and the salinity of the Black Sea. During the research, it was found that samples of Ralex membranes of different batches have similar characteristics over the entire range of investigated concentrations. The calculated values of the transfer numbers for membranes of different batches differ insignificantly: ±0.01 for Ralex AMH in 1 M NaCl. For MK-40 and MA-41 membranes, a significant scatter of characteristics was found, especially in concentrated solutions. As a result, in 1 M NaCl, the transport numbers differ by ±0.05 for MK-40 and ±0.1 for MA-41. The value of the open circuit potential for the Ralex membrane pair showed that the experimental values of the potential are slightly lower than the theoretical ones. At the same time, the maximum calculated power density is higher than the experimental values. The maximum power density achieved in the experiment on reverse electrodialysis was 0.22 W/m², which is in good agreement with the known literature data for heterogeneous membranes. The discrepancy between the experimental and theoretical data may be the difference in the characteristics of the membranes used in the reverse electrodialysis process from the tested samples and does not consider the shadow effect of the spacer in the channels of the electrodialyzer.