2020, issue 2, p. 44-52

Received 11.06.2020; Revised 17.06.2020; Accepted 30.06.2020

Published 24.07.2020; First Online 27.07.2020


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UDC 519.6

Algorithms for Numerical Solution to the Problem of Diffusion in Nanoporous Media

N.A. Vareniuk 1,   N.I. Tukalevska 1 *

1 V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv

* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.


Introduction. Mathematical modeling of mass transfer in heterogeneous media of microporous structure and construction of solutions to the corresponding problems of mass transfer was considered by many authors [1–9, etc.]. In [6, 7] authors proposed a methodology for modeling mass transfer systems and parameter identification in nanoporous particle media (diffusion, adsorption, competitive diffusion of gases, filtration consolidation), which are described by non-classical boundary and initial-boundary value problems taking into account the mutual influence of micro- and macro-transfer flows, heteroporosity, the structure of microporous particles, multicomponent and other factors. In [8, 9] for a mathematical model of nonstationary diffusion of a single substance in a nanoporous medium described in [2] in the form of a multi-scale differential mathematical problem, the classical problems in the weak formulation were obtained. In this paper, algorithms for solving the above mathematical problems are constructed by using the finite element method. The results of the numerical solution of the test problem are presented. The results confirm the efficiency of the developed algorithms.

The purpose is to solve a problem of nonstationary diffusion of single substance in nanoporous medium by constructing discretization algorithms using FEM quadratic basis functions.

Results. Algorithms for the numerical solution of the problem of nonstationary diffusion of single substance in a nanoporous medium are proposed. Peculiarities of discretization of the region and construction of the matrix of masses, stiffness, and vector of right-hand sides when solving the problem by using FEM are described. The efficiency of the developed algorithms is confirmed by the results of solving a model example.


Keywords: mathematical modeling, numerical methods, nonstationary diffusion, nanoporous medium, finite element method.


Cite as: Vareniuk N.A., Tukalevska N.I. Algorithms for Numerical Solution to the Problem of Diffusion in Nanoporous Media. Cybernetics and Computer Technologies. 2020. 2. P. 44–52. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.2.5



           1.     Kärger J., Ruthven D. Diffusion and adsorption in porous solids. Handbook of porous solids Ed. by F. Shuth, K. W. Sing, J. Weitcamp. Wenheim: Wiely-VCH, 2002. P. 2089–2173. https://doi.org/10.1002/9783527618286.ch28

           2.     N’Gokoli-Kekele P. An analytical study of molecular transport in a zeolite crystallite bed. N’gokoli-Kekele P., Springuel-Huet, M.-A., Fraissard J. Adsorption. 2002. 8 (3). P. 35–44. https://doi.org/10.1023/A:1015266423017

           3.     Ruthven D. Principles of adsorption and adsorption processes. New York: Wiley-Interscience, 1984. 464 p.

           4.     Kärger J., Grinberg F., Heitjans P. Diffusion fundamentals. Leipzig: Leipziger Universitätsverlag, 2005. 615 p.

           5.     Petryk M., Leclerc S., Canet D., Fraissard J. Mathematical modeling and visualization of gas transport in a zeolite bed using a slice selection procedure. Diffusion Fundamentals. 2007. 4. P. 11.1–11.23.

           6.     Sergienko I.V., Petryk M.R., Khimich O.M., Kanet D. Mathematical modeling of masstransfer in media with particles of nanoporous structure. Kiev: NASU, 2014. 210 p. (in Ukrainian) http://elartu.tntu.edu.ua/handle/123456789/17812

           7.     Deineka V., Petryk M., Canet D., Fraissard J. Mathematical modeling and identification of mass transfer parameters in inhomogeneous and nanoporous media (adsorption, competitive diffusion). Kiev: NASU, 2014. 177 p. (in Ukrainian) http://elartu.tntu.edu.ua/bitstream/123456789/17811  

           8.     Deineka V., Sergienko I. Parameters identification by gradient methods of the problems of substance diffusion in nanoporous medium. J. of Automation and Information Sciences. 2010. Vol. 42, N 11. P. 117. https://doi.org/10.1615/JAutomatInfScien.v42.i11.10

           9.     Sergienko I., Deineka V. Identification of parameters of problems of diffusion of two-component substances in nanoporous media by gradient methods. Dopovidi NASU. 2010. № 12. P. 42–49. (in Russian) http://dspace.nbuv.gov.ua/handle/123456789/31095

       10.     Deineka V., Sergienko I., Skopetsky V. Models and methods for solving problem with conjugation conditions. Kiev: Naukova dumka, 1998. 615 p. (in Russian)

       11.     Computer software MIR-1 and MIR-2 (edited by Molchanov I.N.). Kiev: Naukova dumka, 1976. 280 p. (in Russian)



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