Ion Transport in Electromembrane Systems under the Passage of Direct Current: 1D Modelling Approaches

Abstract

For a theoretical analysis of mass transfer processes in electromembrane systems, the Nernst–Planck and Poisson equations (NPP) are generally used. In the case of 1D direct-current-mode modelling, a fixed potential (for example, zero) is set on one of the boundaries of the considered region, and on the other—a condition connecting the spatial derivative of the potential and the given current density. Therefore, in the approach based on the system of NPP equations, the accuracy of the solution is significantly affected by the accuracy of calculating the concentration and potential fields at this boundary. This article proposes a new approach to the description of the direct current mode in electromembrane systems, which does not require boundary conditions on the derivative of the potential. The essence of the approach is to replace the Poisson equation in the NPP system with the equation for the displacement current (NPD). Based on the system of NPD equations, the concentration profiles and the electric field were calculated in the depleted diffusion layer near the ion-exchange membrane, as well as in the cross section of the desalination channel under the direct current passage. The NPD system, as well as NPP, allows one to describe the formation of an extended space charge region near the surface of the ion-exchange membrane, which is important for describing overlimiting current modes. Comparison of the direct-current-mode modelling approaches based on NPP and NPD showed that the calculation time is less for the NPP approach, but the calculation accuracy is higher for the NPD approach.

A Commemorative Special Issue in Honor of Professor Victor Nikonenko

“…Aminat Uzdenova [ 2 ] proposes a new approach to the theoretical description of the direct current mode in electromembrane systems, which contain ion exchange membranes as well as diluted and enriched diffusion layers. The new approach does not require boundary conditions on the derivative of the potential drop.…”

Time-Dependent Two-Dimensional Model of Overlimiting Mass Transfer in Electromembrane Systems Based on the Nernst–Planck, Displacement Current and Navier–Stokes Equations

“…In recent work [56], based on the mathematical model in the form of the boundary value problem for the system of NPD equations, concentration profiles and electric field strengths were calculated in a depleted diffusion layer near the surface of an ion-exchange membrane, as well as in the cross-section of a desalination channel under the action of an overlimiting DC current. It was shown that this approach allows one to describe the formation of the extended SCR, as well as the approach based on the NPP equations.…”

“…It was shown that this approach allows one to describe the formation of the extended SCR, as well as the approach based on the NPP equations. The equation for the displacement current contains only the time derivative and does not require the determination of boundary conditions [51,52,56]. In the equation for the displacement current, the small parameter stands at the time derivative of the electric field strengths.…”

“…In the equation for the displacement current, the small parameter stands at the time derivative of the electric field strengths. For this reason, numerical calculations based on the NPD equations differ from those based on the NPP equations by a longer calculation time and less error for the same values of the parameters [56]. In [51,52,56], 1D cases of model geometry are considered; the flow of an electrolyte solution is either not considered or is assumed to be laminar.…”