Hybrid numerical-analytical method for solving the problems of salt ion transport in membrane systems with axial symmetry

The aim of this study is to develop a new hybrid numerical-analytical method for solving boundary value problems with axial symmetry, for example, with a rotating membrane disk, based on matching the asymptotic solution near the cation-exchange membrane (CEM) with the numerical solution in the rest of the region. For this, the following method is used:
1) a basic mathematical model for the transfer of salt ions in an electrochemical cell with a rotating cation-exchange membrane disk is proposed based on the general conservation laws represented by the Nernst-Planck-Poisson and Navier-Stokes equations with natural boundary and initial conditions. This model contains no fitting parameters or simplifying assumptions. However, the numerical solution of the corresponding boundary value problem presents significant computational difficulties for real solution concentrations and large jumps in the potential and angular velocity of the membrane disk rotation, associated with large concentration and potential gradients near the CEM in the quasi-equilibrium space charge region (SCR);
2) the solution region is divided into two parts, one of which is a small cation increase region (CIR) located near the CEM, and the remaining main part of the region (MPOR);
3) in the CIR, an analytical solution is found by the method of matching asymptotic solutions;
4) a simplified mathematical model is constructed in the MPOR, which differs from the basic mathematical model in such a boundary condition at the boundary with the CIR, which then allows us the solution of the corresponding boundary value problem to be matched with the solution in the CIR.
The main result is a hybrid numerical-analytical method that allows one to carry out a numerical analysis of the transfer of salt ions at real concentrations of a binary salt electrolyte solution in a wide range of changes in the potential jump and the angular velocity of the membrane disk.
Based on the results of the work, the following conclusion can be drawn, that the combination of the analytical (asymptotic) method of solving in the region of the boundary layer and the numerical solution in the rest of the region, with the exception of the boundary layer, with their subsequent splicing, makes it possible to construct an effective hybrid numerical-analytical method for solving the problems of salt ion transport in membrane systems with axial symmetry.