Makhamet Urtenov

To enhance desalination efficiency and reduce experimental costs, the development of advanced mathematical models for EMS is essential. In this study, we propose a novel hybrid approach that integrates neural networks with high-accuracy numerical simulations of electroconvection. Based on dimensionless similarity criteria (Reynolds, Péclet numbers, etc.), we establish functional relationships between critical parameters, such as the dimensionless electroconvective vortex diameter and the plateau length of current-voltage curves. Training datasets were generated through extensive numerical experiments using our in-house developed mathematical model, while multilayer feedforward neural networks with backpropagation optimization were employed for regression tasks. The resulting AI (artificial intelligence) -driven hybrid models enable rapid prediction and optimization of EMS design and operating parameters, reducing computational and experimental costs. This research is situated at the intersection of membrane science, artificial intelligence, and computational modeling, forming part of a broader foresight agenda aimed at developing next-generation intelligent membranes and adaptive control strategies for sustainable water treatment. The proposed methodology offers a scalable framework for integrating physics-informed modeling and machine learning into the design of high-performance electromembrane systems.

This paper presents the first theoretical investigation of the effect of a variable equilibrium coefficient on the steady-state transport of a binary electrolyte in a desalination channel cross-section of the electrodialyzer. To address this problem, we developed a new mathematical model in the form of a boundary value problem for an extended system of stationary Nernst–Planck–Poisson equations. We obtained a numerical solution to this problem using the finite element method. Analysis of this solution revealed that the channel cross-section has a complex structure: it is divided into seven regions dominated by different processes, and, consequently, the solution to the boundary value problem behaves differently in each of them. Existing models of the diffusion layer or channel cross-section typically assume a constant equilibrium coefficient. In this paper, we demonstrated that in the channel cross-section, the velocity change corresponding to the equilibrium constant is related not only to the field strength but also to the magnitude of the space charge. In the space-charge region, in the boundary layers near the ion-exchange membranes, intense dissociation of water molecules occurs, and the higher the equilibrium coefficient, the more intense this dissociation is. We have shown that an internal boundary layer (recombination region) arises deep within the solution, associated with the recombination reaction of H+ and OH− ions. In this study, we found that with increasing equilibrium coefficient, fluxes increase, while with increasing fluxes, the electric field strength decreases proportionally, and equilibrium is reached. We demonstrate that by calibrating a single fitting parameter in the model, the simulation results can be matched to experimental data with high accuracy. Thus, our proposed model and its numerical solution provide a completely new understanding of the ion transport process in electromembrane systems, taking into account the influence of the dissociation/recombination reaction of water molecules.

This paper theoretically investigates the effect of changes in the equilibrium coefficient on the steady-state transport of a binary electrolyte in a desalination channel cross-section of the electrodialysis apparatus using a mathematical model in the form of a bounda-ry-value problem for an extended system of stationary Nernst-Planck-Poisson equations. A numerical solution is obtained using the finite element method and demonstrates that the channel cross-section has a complex structure, specifically, it is divided into seven segments where different processes dominate, and therefore the solution to the bounda-ry-value problem behaves differently in each of them. This paper demonstrates that, un-like existing approaches, the change in the equilibrium coefficient rate is associated not only with the field strength but also with the magnitude of the space charge. In the space-charge region, in the boundary layers near the ion-exchange membranes, intense dissociation of water molecules occurs, and the higher the equilibrium coefficient, the more intense this dissociation is. It is shown that an internal boundary layer (recombina-tion region) arises deep within the solution, associated with the recombination reaction of H+ and OH− ions. It is shown that with an increase in the equilibrium coefficient, the flows increase, and with an increase in flows, the electric field strength decreases propor-tionally, and equilibrium arises. It is shown that by choosing the fitting parameter of the mathematical model it is possible to align the results obtained using the mathematical model and experimental data with arbitrary accuracy. Thus, the proposed model and its numerical solution provide a new understanding of the process of ion transport in electromembrane systems, taking into account the influence of the dissociation/recombination reaction of water molecules.

This article proposes a 3D mathematical model of the influence of electrical heterogeneity of the ion exchange membrane surface on the processes of salt ion transfer in membrane systems with axial symmetry; in particular, we investigate an annular membrane disk in the form of a coupled system of Nernst–Planck–Poisson and Navier–Stokes equations in a cylindrical coordinate system. A hybrid numerical–analytical method for solving the boundary value problem is proposed, and a comparison of the results for the annular disk model obtained by the hybrid method and the independent finite element method is carried out. The areas of applicability of each of these methods are determined. The proposed model of an annular disk takes into account electroconvection, which is understood as the movement of an electrolyte solution under the action of an external electric field on an extended region of space charge formed at the solution–membrane boundary under the action of the same electric field. The main regularities and features of the occurrence and development of electroconvection associated with the electrical heterogeneity of the surface of the membrane disk of the annular membrane disk are determined; namely, it is shown that electroconvective vortices arise at the junction of the conductivity and non-conductivity regions at a certain ratio of the potential jump and angular velocity and flow down in the radial direction to the edge of the annular membrane. At a fixed potential jump greater than the limiting one, the formed electroconvective vortices gradually decrease with an increase in the angular velocity of rotation until they disappear. Conversely, at a fixed value of the angular velocity of rotation, electroconvective vortices arise at a certain potential jump, and with its subsequent increase gradually increase in size.

The article presents a 3D mathematical modeling of the influence of electrical inhomogeneity of the ion-exchange membrane surface on the processes of salt ion transfer in membrane systems with axial symmetry, in particular, an annular membrane disk. This model is presented as a boundary value problem for the coupled system of Nernst – Planck – Poisson and Navier – Stokes equations in a cylindrical coordinate system. A hybrid numerical-analytical method for solving the boundary value problem is proposed, and the results for the annular disk model obtained by the hybrid method and the independent finite element method are compared. The applicability areas of each of these methods are determined. The main patterns and features of the occurrence and development of electroconvection associated with the electrical inhomogeneity of the membrane disk surface are determined using the example of an annular membrane disk. In particular, it is shown that electroconvective vortices arise at the junction of conductivity and non-conductivity regions at a certain ratio of the potential jump and angular velocity. Electroconvective vortices flow down in the radial direction to the edge of the annular membrane. At a fixed jump in potential, greater than the limiting value, the formed electroconvective vortices gradually decrease with increasing angular velocity of rotation until they disappear. And vice versa, at a fixed value of angular velocity of rotation, electroconvective vortices arise at a certain jump in potential and with its subsequent increase gradually increase in size.

This article is devoted to numerically modeling the effect of the geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMSs) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The main patterns of salt ion transport are established taking into account diffusion, electromigration, forced convection, electroconvection, and the geometric modification of the surface of ion-exchange membranes. It is shown that the geometric modification of the surface of ion-exchange membranes significantly changes both the formation and development of electroconvection. A significant combined effect of electroconvection and geometric modification of the surface of ion-exchange membranes in the desalination channel on the salt ion transport is shown, as well as a complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization in the desalination channel.

В статье проведено математическое моделирование автоволн в слабопроводящей жидкости (ферроколлоиде) и в проводящей жидкости (растворе соли) с учетом перезарядки ионов в областях пространственного заряда в виде краевой задачи для нестационарной системы уравнений Нернста-Планка-Пуассона. С использованием предложенной математической модели, проведено теоретическое исследование возникновения процесса самоорганизации автоволн в электромембранной системе и в ферроколлоиде (магнитной жидкости). Основным механизмов в возникновении автоволн является перезарядка магнитных частиц (в случае ферроколлоида) и перезарядка ионов (катионов и анионов) в случае электромембранной системы. Примесные ионы участвую только в процессе переноса заряда. Проведен численный анализ краевой задачи математической модели и установлены основные закономерности явления перезарядки на перенос магнитных частиц. Показано, существование солитоноподобных автоволновых решений для магнитных частиц и ионов соли. Суммарная концентрация основных ионов практически не изменяется, в отличии от этого суммарная концентрация примесных ионов уменьшается. В статье предлагается новый математический метод приближенного аналитического решения краевой задачи, основанный на выводе нелинейного дифференциального уравнения с частными производными для потенциала в области одиночной волны. Показано, что это уравнение можно свести с использованием ряда преобразований, включая и преобразование Хопфа-Коула, к каноническому уравнению параболического типа, и в этом смысле найдено точное аналитическое решение. Кроме того, найдено простое аналитическое приближение для одиночной волны, проведено сравнение с численным решением и показано их качественное и количественное совпадение (с точностью ∼ 3%). The article presents mathematical modeling of autowaves in a weakly conductive liquid (ferrocolloid) and in a conductive liquid (salt solution), taking into account ion recharge in the regions of spatial charge in the form of a boundary value problem for a nonstationary system of Nernst-Planck-Poisson equations. Using the proposed mathematical model, a theoretical study of the occurrence of the self-organization process – autowaves in an electromembrane system and in a ferrocolloid (magnetic liquid) was carried out. The main mechanisms in the occurrence of autowaves are the recharging of magnetic particles (in the case of a ferrocolloid) and the recharging of ions (cations and anions) in the case of an electromembrane system. Impurity ions are involved only in the charge transfer process. A numerical analysis of the boundary value problem of the mathematical model is carried out and the main regularities of the phenomenon of recharging on the transfer of magnetic particles are established. The existence of soliton-like autowave solutions for magnetic particles and salt ions is shown. The total concentration of basic ions practically does not change, in contrast, the total concentration of impurity ions decreases. The article proposes a new mathematical method for the approximate analytical solution of a boundary value problem based on the derivation of a nonlinear partial differential equation for a potential in the region of a single wave. It is shown that this equation can be reduced using a number of transformations, including the Hopf–Cole transformation, to a canonical parabolic equation, and in this sense an exact analytical solution is found. In addition, a simple analytical approximation for a single wave was found, a comparison with the numerical solution was carried out and their qualitative and quantitative coincidence was shown (with an accuracy of ∼ 3%)

This article investigates the boundary value problem for an extended stationary system of Nernst–Planck–Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10⁻¹² and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry.

The article is devoted to the theoretical analysis of the effect of geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMS) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on boundary value problems for the coupled system of Nernst-Planck-Poisson and Navier-Stokes equations. The main patterns of salt ion transport are established taking into account diffusion, electromigration, forced convection, electroconvection, and geometric modification of the surface of ion-exchange membranes. It is shown that geometric modification of the surface of ion-exchange membranes significantly changes both the formation and development of electroconvection. A significant combined effect of electroconvection and geometric modification of the surface of ion-exchange membranes in the desalination channel on the salt ion transport is shown, as well as a complex, nonlinear and non-stationary interaction of all the main effects of concentration polarization in the desalination channel.

The article investigates the boundary value problem for an extended stationary system of Nernst-Planck-Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10−12 and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry.