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

Electromembrane processes underlie the functioning of electrodialysis devices and nano- and microfluidic devices, the scope of which is steadily expanding. One of the main aspects that determine the effectiveness of membrane systems is the choice of the optimal electrical mode. The solution of this problem, along with experimental studies, requires tools for the theoretical analysis of ion-transport processes in various electrical modes. The system of Nernst–Planck–Poisson and Navier–Stokes (NPP–NS) equations is widely used to describe the overlimiting mass transfer associated with the development of electroconvection. This paper proposes a new approach to describe the electrical mode in a membrane system using the displacement current equation. The equation for the displacement current makes it possible to simulate the galvanodynamic mode, in which the electric field is determined by the given current density. On the basis of the system of Nernst–Planck, displacement current and Navier–Stokes (NPD–NS) equations, a model of the electroconvective overlimiting mass transfer in the diffusion layer at the surface of the ion-exchange membrane in the DC current mode was constructed. Mathematical models based on the NPP–NS and NPD–NS equations, formulated to describe the same physical situation of mass transfer in the membrane system, differ in the peculiarities of numerical solution. At overlimiting currents, the required accuracy of the numerical solution is achieved in the approach based on the NPP–NS equations with a smaller time step than the NPD–NS equation approach. The accuracy of calculating the current density at the boundaries parallel to the membrane surface is higher for the model based on the NPD–NS equations compared to the model based on the NPP–NS equations.