The study of ion transport in membrane systems in overlimiting current modes is an important problem of physical chemistry and has an important application value. The influence of the space charge on the transport of salt ions under overlimiting current modes was first studied in the work of Rubinstein and Shtilman and later in the works of many authors. The purpose of this research is to study, using the method of mathematical modeling, the reasons of formation and properties of the local maximum (minimum) space charge in membrane systems under overlimiting current conditions. It is shown that, in the diffusion layer of the cation-exchange membrane (CEM), the local maximum of the space charge appears due to the limited capacity (exchange capacity) of the membrane at a given potential jump, i.e., the local maximum of space charge appears due to the presence of a local minimum of space charge at the surface of the CEM. The local maximum of the space charge moves as a single soliton-like wave into the depth of the solution. Unlike real solitons, this charged wave changes its size and shape, albeit quite slowly. In the section of the desalination channel, the situation is completely different. First, the space charge of the anion-exchange membrane (AEM) has a negative value, so we should be talking about the local minimum (or the maximum of the absolute value of the charge). However, this is an insignificant clarification. Secondly, the space charge waves of different signs begin to interact, which leads to a new effect, namely the effect of the breakdown of the space charge. The dependence of the local maximum on the input parameters—the cation diffusion coefficient, the growth rate of the potential jump, and the initial and boundary concentrations—is studied.

This work is devoted to problems of numerical calculation of the theoretical current-voltage characteristics (CVC) and galvanography of electromembrane systems in the potentiometric mode. The purpose of this work is the derivation of the formula for calculating the CVC and galvanography of electromembrane systems in the potentiometric mode, which allows calculating stable relative to random errors and rounding errors in over-limiting current densities. To use the formulas, it is necessary to create a mathematical model for calculating the local current density. In this paper, as this model model is used transport of ions in binary salt in the desalination channel, taking into account electroconvection. The electromembrane systems have been used for water medium preparing in mobile mini systems for generating heat and electrical energy as well as vapour producing. The quality of clean water influences on reliability and life time of these systems, including vapour generator and dam-vapour turbine. The obtained formula in the form of a double integral of the local current density is derived, which allows calculating them steadily with respect to random errors and rounding errors. The formulas allowing considering CVC stably with respect to random errors and rounding errors are obtained, and the physical meaning of these formulas is clarified. The calculation of the theoretical CVC with the use of a mathematical model of ion transfer binary salts given electroconvection and shows that it qualitatively coincides with the experimental current-voltage curves. The quantitative difference can be explained by the fact that the mathematical model does not take into account the reaction of water dissociation / recombination, gravitational convection and other transport mechanisms.

In electromembrane systems, the theoretical study of salt ion transport usually uses mathematical models of salt ion transport in the depleted diffusion layer of ion-exchange membranes. This study uses a one-dimensional mathematical model of salt ion transport in a cross-section of a desalination channel formed by anion-exchange and cation-exchange membranes, taking into account an effect of a dissociation/recombination reaction of water molecules. The reaction on the one hand leads to an overlimiting mass transfer due to the effect of exaltation of the limiting current. On the other hand, an appearance of new electric charge carriers (hydrogen and hydroxyl ions) can reduce the space charge that occurs in membranes and suppress an electroconvective mechanism of overlimiting transport. Thus, there is a problem of studying these phenomena together, taking into account their mutual influence, and this article is devoted to the solution of this problem. Theoretically, using a method of mathematical modeling and numerical research, main regularities are established; in particular, it is shown that the dissociation/recombination reaction of water molecules does not lead to the destruction of the double electric layer at the membranes, but also creates a new double electric layer in the middle of the desalination channel. Thus, the space charge and the dissociation/recombination reaction significantly affect each other and simultaneously the transport of salt ions.

In practice, the current-voltage characteristic (CVC) is the most important characteristic of transport in electromembrane systems, since it is using CVC that the concept of limiting current is introduced, various modes of operation of electromembrane systems are analyzed, and their efficiency is evaluated. At present, experimental CVC methods of Fourier analysis, wavelet analysis, and dynamical systems are well studied. At the same time, the study of theoretical CVC is not sufficiently developed. Previously, we derived a formula for calculating the CVC of a non-stationary 1:1 transfer of an electrolyte in the cross-section of the desalination channel, which includes an anion-exchange (AEM) and cation-exchange (CEM) membranes, and establishing the fundamental laws of changes in CVC over time. The simulation is based on the NernstPlanck-Poisson equations. In this paper, we analyze this formula and identify the fundamental laws of the CVC of non-stationary 1:1 transfer of the electrolyte in the cross-section of the desalination channel. It is shown that in the prelimiting mode, the migration current and the diffusion current give approximately the same contribution to the total current, and in the overlimiting mode, the main contribution is given by the migration current, the value of the displacement current does not depend on time and is proportional to the sweep speed. It is found that the average conduction current is many times greater than the displacement current, starting from a few seconds. The results obtained allow to construct and analyze the CVC for the cross-section of the desalination channel.

In electromembrane systems, a theoretical study of salt ion transfer usually uses mathematical models of salt ion transfer in the depleted diffusion layer of ion-exchange membranes. In this paper, a new mathematical model of ion transport in the cross-section of the desalination channel formed by two ion-exchange membranes – anion-exchange (AEM) and cation-exchange (CEM), taking into account the non-catalytic dissociation/recombination reaction of water molecules. The model is a boundary value problem for a non-stationary system of Nernst-Planck and Poisson equations. A numerical analysis of the boundary value problem is performed and the main regularities of the 1:1 salt ion transfer process are established, in particular, the occurrence and development of space charge breakdown is shown. The interaction of the space charge and the noncatalytic dissociation/recombination reaction of water molecules are theoretically investigated.