Modeling of ion transport in a three-layer system with an ion-exchange membrane based on the Nernst-Planck and displacement current equations

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Abstract

Modeling of ion transport in a three-layer system containing an ion-exchange membrane and two adjacent diffusion layers makes it possible to describe the permselectivity of the membrane by determining its fixed charge density. For theoretical analysis of ion transport in such systems, the Nernst–Planck and Poisson equations are widely used. The article shows that in the galvanodynamic mode of operation of the membrane system, when the density of the flowing current is specified, the Poisson equation in the ion transport model can be replaced by the equation for the displacement current. A new model was constructed in the form of a boundary value problem for the system of the Nernst–Planck and displacement current equations. Based on this model, ion concentrations, electric field strength, space charge density and chronopotentiogram of the ion-exchange membrane and adjacent diffusion layers in direct current mode were numerically calculated. The calculation results of the proposed model are in good agreement with the modeling results based on the previously described approach using the Nernst–Planck and Poisson equations, as well as with the analytical assessment of the transition time. It is shown that in the case of the three-layer geometry of the problem, the required accuracy of numerical calculation using the proposed model is achieved with a smaller number of computational mesh elements and takes less (about 26.7 times for the considered system parameters) processor time compared to the model based on the Nernst–Planck and Poisson equations.

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About the authors

A. M. Uzdenova

Umar Aliev Karachai-Cherkess State University

Author for correspondence.
Email: uzd_am@mail.ru
Russian Federation, 369202, Karachaevsk, Lenina 29

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Supplementary files

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2. Fig. 1. Diagram of the membrane system and profiles of cation (c1, solid line) and anion (c2, dotted line) concentrations when a current of density i flows.

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3. Fig. 2. (a) Normalized concentration profiles of cations c1/c0 and anions c2/c0; (b) and (c) – enlargement of fragments of Fig. (a); (d) space charge density; (d) electric field strength. Results of calculations at times t = 0, 7.2, 100 s based on the NPP (solid lines) and NPS (dashed lines) approaches.

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4. Fig. 2. (a) Normalized concentration profiles of cations c1/c0 and anions c2/c0; (b) and (c) – enlargement of fragments of Fig. (a); (d) space charge density; (d) electric field strength. Results of calculations at times t = 0, 7.2, 100 s based on the NPP (solid lines) and NPS (dashed lines) approaches.

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5. Fig. 3. Chronpotentiograms calculated based on the NPP (solid line) and NPS (dashed line) approaches. The dotted line indicates the transition time τ = 7.2 s.

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6. Fig. A1. Values ​​of transport numbers of cations T1C (square markers) and anions T1C (round markers) in a cation exchange membrane at different values ​​of concentration of fixed charged groups χ. Calculation at current density i/ilim = 2.

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