Mathematical Modeling of the Influence of Non-Uniform Electrical Conductivity of an Ion-Exchange Membrane on Salt Ion Transport in a Desalination Channel with Forced Convection

In this work, the influence of the electrical heterogeneity of the surface of an ion-exchange membrane on salt ion transport in a desalination channel with forced convection in the potentiodynamic mode is studied for the first time by the successful combined application of various mathematical methods: mathematical modeling based on a boundary value problem for the coupled system of the Navier–Stokes, Nernst–Planck, and Poisson equations; the finite element method for its numerical solution; an approximate analytical solution of the Stokes equations constructed by the method of images, in which the flow at the junction of the conducting and non-conducting regions is described as a flow induced by a point torque (rotlet); and a first-order relaxation model describing the growth of the vortex size. Since such problems have not been considered before, a new mathematical apparatus for their solution has been developed. It is established that the presence of electrical heterogeneity on the membrane surface leads to the formation of a heteroelectroconvective vortex at the conducting/non-conducting boundary, which arises as a result of space-charge accumulation at the junction point and the emergence of a significant non-potential electric force that swirls the fluid. It is shown that the radius of the heteroelectroconvective vortex grows exponentially up to a stationary value determined by the geometry of the non-conducting island, that its stabilization occurs before the appearance of ordinary electroconvective vortices, and that it is described quite accurately by the first-order relaxation model. With a further increase in the potential drop and the development of ordinary electroconvection, cyclic destruction and restoration of the heteroelectroconvective vortex occur. The influence of the length of the non-conducting island on the vortex dynamics and the degree of desalination is investigated, and a ratio between the sizes of the conducting and non-conducting regions that is optimal with respect to the degree of desalination is found.