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.

At present, it is customary to consider the overlimit operating modes of electromembrane systems to be effective, and electroconvection as the main mechanism of overlimiting transfer. The breakdown of the space charge is a negative, “destructive” phenomenon, since after the breakdown the size and number of electroconvective vortices are significantly reduced, which leads to a decrease in mass transfer. Therefore, electromembrane desalination processes must be carried out before space charge breakdown occurs. Thus, the actual problem arises of determining at which potential jumps a breakdown of the space charge occurs at a given concentration of the solution. Electromembrane systems are used for desalination at electrolyte solution concentrations ranging from 1 to 100 mol/m3. In a theoretical study of increasing the efficiency of the desalination process, mathematical modeling is used in the form of a boundary value problem for the system of Nernst–Planck and Poisson (NPP) equations, which refers to “hard” problems that are difficult to solve numerically. This is caused by the appearance of a small parameter at the derivative in the Poisson equation in a dimensionless form, and, correspondingly, a boundary layer at ion-exchange membranes, where concentrations and other characteristics of the desalination process change exponentially. It is for this reason that the numerical study of the boundary value problem is currently obtained for initial concentrations of the order of 0.01 mol/m3. The paper proposes a new numerical–analytical method for solving boundary value problems for the system of Nernst–Planck and Poisson equations for real initial concentrations, using which the phenomenon of space charge breakdown (SCB) in the cross section of the desalination channel in potentiostatic and potentiodynamic modes is studied. The main regularities of the appearance and interaction of charge waves, up to their destruction (breakdown), are established. A simple formula is proposed for engineering calculations of the potential jump depending on the concentration of the solution, at which the breakdown of the space charge begins.

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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.