Mathematical model of salt ion stationary transport in the cross section of the channel at equilibrium

The equilibrium at the interphase boundaries largely determines the transfer processes and therefore studying it is an important task. The paper proposes a mathematical model of the problem of salt ion stationary transfer at the onset of equilibrium, namely at zero current, in the cross section of the desalination channel formed by anion exchange and cation exchange membrane in the form of a boundary value problem for systems of Nernst-Planck and Poisson equations in the potentiostatic mode. A numerical and asymptotic solution of this boundary value problem is obtained. The numerical and asymptotic solutions are compared, and their coincidences were shown with good accuracy. The acquired asymptotic solution allows for an exhaustive analysis of the equilibrium state depending on the initial concentration, potential jump, and properties of ion-exchange membranes and helps to establish the basic transfer patterns. It is shown that the stationary state of salt ion transfer process through the channel section coincides with the equilibrium state. The location and dimensions of the spatial charge and electroneutrality regions are established. The dependence of the electric field strength and concentration on the potential jump and the boundary values for cation and anion concentrations is obtained. The results of the research can be used to determine the optimal operating modes of electrodialysis water purification devices.

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