# Mathematical modeling of the effect of spacers on mass transfer in electromembrane systems

## Abstract

The transfer of ions near ion-exchange membranes causes concentration polarization, which significantly complicates mass transfer in electromembrane systems. Spacers are used to neutralize the effect of concentration polarization and increase mass transfer. Spacers reduce the thickness of the boundary layer by increasing the mixing depth of the solution and creating a normal component of convective transport; ions can reach membranes faster, and the current increases, from a hydrodynamic point of view. However, spacers significantly increase the hydrodynamic resistance and consequently the cost of pumping the solution.

For the first time, the main regularities of the transfer of salt ions in the desalination channel of an electrodialysis apparatus with spacers of various shapes and arrangements are determined, taking into account electroconvection, in overlimiting current modes. Namely, it is shown, using the current-voltage characteristic, that spacers of different shapes and locations are optimal at different stages of the desalination process.

The paper presents the results of mathematical and simulation modeling of the salt ion transport process in electromembrane systems with spacers in overlimiting current modes. 2D direct numerical simulation was carried out for the coupled system of the Nernst–Planck–Poisson and Navier–Stokes equations without fitting parameters. The finite element method was used in combination with the method of successive approximations and segregation to solve boundary value problems for systems of nonlinear differential equations with partial derivatives. The novelty of the method lies in the fact that after discretization in time, the problem on each time layer is split into hydrodynamic and electrochemical problems, each of which is solved by the method of successive approximations until a complete mutual agreement.

## About the authors

### Anna V. Kovalenko

Kuban State University

Author for correspondence.
Email: savanna-05@mail.ru
ORCID iD: 0000-0002-3991-3953
SPIN-code: 3693-4813
Scopus Author ID: 55328224000
http://www.mathnet.ru/person112835

Dr. Tech. Sci., Associate Professor; Head of Department; Dept. of Data Analysis and Artificial Intelligence

149, Stavropolskaya st., Krasnodar, 350040, Russian Federation

### Aminat M. Uzdenova

Umar Aliev Karachay–Cherkess State University

Email: uzd_am@mail.ru
ORCID iD: 0000-0001-5951-9876
SPIN-code: 2810-2267
Scopus Author ID: 55821149800
http://www.mathnet.ru/person188273

Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Informatics and Computational Mathematics

29, Lenina st., Karachayevsk, 369202, Russian Federation

### Anna Ovsyannikova

Financial University under the Government of the Russian Federation

Email: anna_ovsyannikov@bk.ru
ORCID iD: 0000-0002-1716-3100
SPIN-code: 7540-6190
Scopus Author ID: 57925463800
http://www.mathnet.ru/person145605

Cand. Pedagogic. Sci.; Associate Professor; Dept. of Mathematics

29, Leningradskii pr., Moscow, 125167, Russian Federation

### Makhamet Kh. Urtenov

Kuban State University

Email: urtenovmax@mail.ru
ORCID iD: 0000-0002-0252-6247
SPIN-code: 7189-0748
Scopus Author ID: 6603363090
http://www.mathnet.ru/person119069

Dr. Phys. & Math. Sci., Professor; Head of Department; Dept. of Applied Mathematics

149, Stavropolskaya st., Krasnodar, 350040, Russian Federation

### Ramazan A. Bostanov

Umar Aliev Karachay–Cherkess State University

Email: bost-rasul@yandex.ru
ORCID iD: 0000-0002-8502-7653
SPIN-code: 6093-2105
http://www.mathnet.ru/person30830

Cand. Phys. & Math. Sci.; Dean of Faculty of Physics and Mathematics

Russian Federation, 29, Lenina st., Karachayevsk, 369202, Russian Federation

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