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 is empirically described by exponential saturation toward an asymptotic value over the sampled interval of 190–290 s, 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 non-conducting island length on vortex dynamics and desalination is examined; among the six lengths considered, 0.05 mm gives the highest computed degree of desalination (42.044%). Direct comparisons of longitudinal velocity give NMAE values of 2.50%, 14.73%, and 1.63% at three representative cross-sections, with the largest discrepancy near the vortex center.