Modeling of Ion Transport in a Three-Layer System with an Ion-Exchange Membrane Based on the Nernst–Planck and Displacement Current Equations

Modeling of ion transport in a three-layer system containing an ion-exchange membrane and two adjacent diffusion layers makes it possible to describe the permselectivity of the membrane by determining its fixed charge density. For theoretical analysis of ion transport in such systems, the Nernst–Planck and Poisson equations are widely used. The article shows that, in the galvanodynamic mode of operation of the membrane system when the density of the flowing current is specified, the Poisson equation in the ion transport model can be replaced by the equation for the displacement current. A new model is constructed in the form of a boundary value problem for the system of the Nernst–Planck and displacement current equations, based on which the concentrations of ions, electric field strength, space charge density, and chronopotentiogram of the ion-exchange membrane and adjacent diffusion layers in a direct current mode are numerically calculated. The calculation results of the proposed model are in a good agreement with the results of the modeling based on the previously described approach using the Nernst–Planck and Poisson equations as well as with the analytical assessment of the transition time. It is shown that, in the case of the three-layer geometry of the problem, the required accuracy of the numerical calculation using the proposed model is achieved with a smaller number of computational mesh elements and takes less (about 26.7-fold for the system parameters under consideration) processor time in comparison with the model based on the Nernst–Planck and Poisson equations.