Mathematical description of ion transport in membrane systems

A review of works resulting in development of a hierarchic system of mathematical models is presented. Different scales of ion transport are considered: 1) in the membrane, where two approaches are used, the irreversible thermodynamics and modelling of the membrane material, the last permits to link physicochemical properties with structure-kinetic parameters of the membrane; 2) in a three-layer system composed of a membrane with two adjoining diffusion layers; and 3) in an electrodialysis 2D cell, where the differential equation of convective diffusion is used.

A three-scale system of mathematical models is developed to describe electrodialysis process: • the microheterogeneous model permits to calculate the membrane transport characteristics such as the diffusion permeability, transport number and conductivity as well as the fluxes through the membrane from the properties of the phases constituting the membrane; the transport equations following from the irreversible thermodynamics; • the three-layer models use the microheterogeneous model to calculate the effective transport numbers and yield these values as functions of the current density and diffusion layer thickness; • the 2D convective-diffusion model uses the effective transport numbers in its boundary conditions and permits to calculate the distribution of concentrations, current density and diffusion layer thickness in an ED compartment as well as current-voltage characteristics. Softwares realising the models of each scale are developed. One of these, “ED Manager” contains a data base on electrodialysis apparatus and cells properties as well as tools for treating these data and modelling apparatus behaviour in a non-studied range of parameters.