Analytical Solutions and Computer Modeling of a Boundary Value Problem for a Nonstationary System of Nernst–Planck–Poisson Equations in a Diffusion Layer

Savva Kovalenko, Evgenia Kirillova (), Vladimir Chekanov, Aminat Uzdenova and Mahamet Urtenov
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Savva Kovalenko: Faculty of Computer Technologies and Applied Mathematics, Kuban State University, 350040 Krasnodar, Russia
Evgenia Kirillova: Faculty of Architecture and Civil Engineering, RheinMain University of Applied Sciences, 65197 Wiesbaden, Germany
Vladimir Chekanov: Department of Digital, Robotic Systems and Electronics, North-Caucasus Federal University, 355017 Stavropol, Russia
Aminat Uzdenova: Department of Computer Science and Computational Mathematics, Umar Aliev Karachay-Cherkess State University, 369202 Karachayevsk, Russia
Mahamet Urtenov: Faculty of Computer Technologies and Applied Mathematics, Kuban State University, 350040 Krasnodar, Russia

Mathematics, 2024, vol. 12, issue 24, 1-22

Abstract: This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems.

Keywords: electromembrane system; diffusion layer; ion-exchange membrane; space charge region; Nernst–Planck–Poisson equations; asymptotic solution; singularly perturbed boundary value problems; galvanodynamic mode (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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