Analysis of the theoretical CVC of electromembrane systems

In a number of recent studies, the authors derived and substantiated a formula for calculating the theoretical current voltage curve (CVC) of electromembrane systems (EMS). This formula was created for the flow cell of desalination of the electrodiaysis apparatus formed by an anion exchange (AEM) and a cation exchange membranes (CEM) in the potentiodynamic mode based on the charge conservation law. In this paper, the formula for the CVC (I –V characteristic) is investigated, its physical meaning, and the contribution of various factors to the CVC are revealed. A new simplified formula is proposed for calculating CVC, stable with respect to rounding errors. The critical values of the current density were determined and the current voltage curve was divided into separate sections. In the article, we showed that for characteristic values of the average flow rate of the electrolyte solution, the initial concentration in all sections of the CVC, the contribution of the convective current is small. The main role belongs to the electromigration (ohmic) current, especially in the overlimiting sections of the current – voltage curve. The contribution of the diffusion current in limiting and underlimiting sections is quite significant, although less than the ohmic current

Stationary model of salt ion transfer in two-dimensional electrodialysis desalting channel in galvanostatic mode

Introduction. The theoretical description of the ion transport in membrane systems in the galvanostatic mode is presented. A desalting channel of the electrodialysis apparatus is considered as a membrane system. The work objectives are the development and verification of a two-dimensional mathematical model of the stationary transport of salt ions in the desalting channel of the electrodialysis apparatus for the galvanostatic mode.Materials and Methods. A new model of ion transfer is proposed. It is based on the Nernst –Planck – Poisson equations for the electric potential and on the equation for the electric current stream function. A numerical solution to the boundary value model problem by the finite element method is obtained using the Comsol Multiphysics software package.Research Results. The developed mathematical model enables to describe the stationary transfer of binary salt ions in the desalting channel of the electrodialysis apparatus. Herewith, the violation of the solution electroneutrality and the formation of the dilated domain of space charge at overlimiting currents in the galvanostatic mode are considered. A good agreement between the physicochemical characteristics of the transfer calculated by the models for the galvanostatic and potentiostatic modes implies adequacy of the constructed model.Discussion and Conclusions. The developed model can interpret the experimental study results of ion transfer in membrane systems if this process takes place in the galvanostatic mode. Some electrokinetic processes are associated with the appearance of a dilated domain of space charge at overlimiting currents. When describing the formation of this domain, it is possible to find out how the processes dependent on it affect the ion transfer in the galvanostatic mode

Mathematical modelling of space charge breakdown in membrane systems taking into account the non-catalytic dissociation/ recombination reaction of water molecules

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

Analysis of the theoretical current-voltage characteristic of non-stationary transport in the cross-section of the desalination channel

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

Numerical and Asymptotic Study of Non-Stationary Mass Transport of Binary Salt Ions in the Diffusion Layer near the Cation Exchange Membrane at Prelimiting Currents

In this paper, we consider a depleted stationary diffusion layer adjacent to the ion-exchange membrane. The main goal is to study the structure of the diffusion layer over time. A one-dimensional non-stationary mathematical model of the transport of a binary electrolyte in a diffusion layer in a potentiostatic mode is investigated using the Nernst-Planck and Poisson equations. For the first time, it is shown that the left boundary of the space charge region is established quickly, approaching a certain straight line xc =const asymptotically. Using this fact, a new asymptotic solution is constructed. The original feature of the proposed asymptotic method is that it is based not only on asymptotic simplifications in the equations, but also on replacing the exact description of the structure of the diffusion layer with an approximate one

Modeling and numerical analysis of the effect of dissociation/recombination of water molecules on the transport of salt ions in diffusion layer

Introduction. The paper presents a theoretical study on binary salt ion transport considering the water dissociation/recombination reaction. The work objectives are as follows: to build a mathematical model; to develop an algorithm for the numerical solution to the boundary value problem corresponding to the mathematical model; to work out the similarity theory including the transition to a dimensionless form using characteristic quantities; to determine a physical meaning of trivial similarity criteria; to find nontrivial similarity criteria; to build and analyze the volt-ampere characteristic (VAC).Materials and Methods. The theoretical study and numerical analysis of the transport of binary salt ions consider the dissociation/recombination reaction of water. In this case, the heat transfer equation and the mathematical model of electrodiffusion of four types of ions simultaneously (two salt ions, as well as ????+ and ????????−ions) in the diffusion layer of electromembrane systems with a perfectly selective membrane are used. For the first-order differential equations, a singularly perturbed boundary-value problem is set. In the equation for the electric field, the right side is independent of the intensity. In the numerical solution to the digitized system of equations by the Newton-Kantorovich method, this causes the stability of the method. In this regard, the boundary-value problem is reduced for numerical solution: a transition to a system of the second-order equations is provided, and the missing boundary conditions for the electric field strength are calculated.Research Results. A new mathematical model, a numerical algorithm to solve a boundary value problem, and software are developed. A numerical analysis is carried out, and fundamental laws of the transport of salt ions are determined considering the dissociation/recombination reaction of water molecules, temperature effects, and Joule heating. The VAC is built and analyzed.Discussion and Conclusions. The transport of binary salt ions through a diffusion layer near a cation exchange membrane is considered. A mathematical model of this process is proposed. It takes into account the temperature effects due to dissociation/recombination reactions of water molecules and Joule heating in a solution. The basic laws of the transport of salt ions are established considering the dissociation/recombination reaction of water molecules and temperature effects. The temperature effects of the dissociation/recombination reaction and the Joule heating in the electroneutrality region (ENR) are almost imperceptible (with the exception of the recombination region, RR). The Joule heating in the space-charge region (SCR) is by two orders of magnitude larger than the cooling effect of the water dissociation reaction. Upon recombination, approximately the same heat is released in the RR as during Joule heating in the expanded SCR. However, due to the small size of the RR, the effect of this heat is imperceptible. Therefore, we can assume that there is only one heat source at the interface in the SCR, which, due to its noticeable size, causes a significant increase in temperature in the entire diffusion layer. It follows that the emergence and development of gravitational convection is possible. General conclusions, following from the results obtained, open up the possibility of intensifying the process of transport of salt ions in the electrodialysis machines

Coupled transport phenomena in overlimiting current electrodialysis

The long-term goals of this work is to develop a better understanding of the coupled transport phenomena role in enhancement of salt ions transfer at electrodialysis of moderate-dilute and dilute electrolyte solutions in membrane modules of various constructions. Dependences of partial salt ions and H+(OH¿) fluxes through an anion-exchange membrane as a function of voltage applied to a membrane or a membrane channel were measured. On the base of the data obtained, reasons of the growth of salt ions `overlimiting' mass transfer in electrodialysis desalination channels with intermembrane distance and different NaCl solution concentration were analysed. It is shown that the role of gravitational convection may be important in desalination channels with a quite great intermembrane distance and electrolyte solution concentration. In this case a salt ions concentration at a membrane¿solution interface does not decrease to values sufficient for promoting the H+ and OH¿ generation and electroconvection. In desalination channels with small intermembrane distances, when treating dilute solutions, the salt concentration at the membrane surface reaches very small values. It promotes the H+ and OH¿ generation and a space charge region arising. An additional salt ions transfer occurs due to the exaltation effect as well as to the electroconvection. In this case the gravitational convection contribution is negligible