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

• Published in: Mathematics (Basel) 2024-12, Vol.12 (24), p.4040
• Main Authors: Kovalenko, Savva, Kirillova, Evgenia, Chekanov, Vladimir, Uzdenova, Aminat, Urtenov, Mahamet
• Format: Article
• Language: English
• Subjects: asymptotic solution; Boundary conditions; Boundary value problems; Burgers equation; Charge exchange; Computer simulation; Computer-generated environments; Desalination; Differential equations; diffusion layer; Diffusion layers; Electric fields; Electrolytes; electromembrane system; Exact solutions; Ion exchange; Ion transport; ion-exchange membrane; Linear equations; Mathematical analysis; Mathematical models; Membranes; Nernst–Planck–Poisson equations; Nonlinear differential equations; Partial differential equations; Poisson equation; Singular perturbation; Space charge; space charge region
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12244040

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.