DOI: https://doi.org/10.15587/2312-8372.2017.109309

### Mathematical modeling of the process of fluid filtration through a multi-layer filtering element

Anastasia Vecherkovskaya, Svitlana Popereshnyak

#### Abstract

The paper presents a mathematical model of the fluid filtration process in a porous medium. The mathematical model of fluid filtration in a porous medium is based on the equation of continuity and the Darcy law for a hollow porous cylinder. The porous cylinder consists of three materials having different mechanical and filtration properties, made of polypropylene fibers by extrusion, consisting of seven layers.

Filtration is described by different types of research laws that establish the relationship between a velocity vector of the fluid filtration and the field of pressure. There are several ways to describe mathematically the process of filter colmatation. Present work considers a single-component model of the flow of suspension through a porous medium. An analysis of this model revealed that it can be implemented in the presence of empirical coefficients that take into account porosity of the sediment deposited on the walls of a porous space, linear functions that describe adsorption and desorption of precipitation on the pore walls, as well as functional dependences for the permeability of filter and the viscosity of suspension.

An efficient technique to calculate complex models is to apply parallel programming methods, which make it possible to split calculations into streams performed on the basis of supercomputers, clusters, and other high-performance computing systems.

The proposed method for organizing distributed calculations for a finite-element filter model could be used to construct a hydrodynamic model of the filtration process, as well as its program implementation, which is planned in the future.

#### Keywords

mathematical model of the liquid filtration process; porous medium; Darcy law; distributed calculations

#### Full Text:

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#### References

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Darcy, H. (1856). Les fontaines publiques de la ville de Dijon. Paris, 647.

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#### GOST Style Citations

Parmakli, I. I. Rasprostranenie volny zagriazneniia pri fil'tratsii suspenzii v poristoi srede [Text] / I. I. Parmakli, S. A. Poslavskii // Materialy mezhdunarodnoi konferentsii «Sovremennye problemy matematiki i eio prilozheniia v estestvennyh naukah i informatsionnyh tehnologiiah». – Kharkiv, 2012. – P. 84.

Tieffilterpatrone und methode und vorrichtung zu deren herstellung [Electronic resource]: Patent DE 69331102 T2 / Szczepanski C., Aune M., Schneider J.; assignee: Osmonics Inc. – Appl. No. DE1993631102; Filed August 19, 1993; Publ. August 22, 2002. – Available at: \www/URL: http://www.google.com.pg/patents/DE69331102T2

Koaleszenzelement [Electronic resource]: Patent DE 69723714 D1 / Whitney A., Williamson M., Clendenning A., Hibbard R., Griffin M.; assignee: Pall Corp. – Appl. No. DE1997623714; Filed September 30, 1997; Publ. August 28, 2003. – Available at: \www/URL: http://www.google.com.pg/patents/DE69331102T2

Construction of filter (versions) and method of filtration [Electronic resource]: Patent RU2182509C2 / Calbau B., Dadri D. J.; assignee: Donaldson Company, Inc. – Appl. No. RU2000109355A; Filed September 9, 1997; Publ. May 20, 2002. – Available at: \www/URL: https://patents.google.com/patent/RU2182509C2/en

Fil'troval'nyi element dlia zhidkih i gazovyh sred [Text]: Patent RU 2326716; MPK VO1D 39/16 (2006.01) / Troian D. A.; assignee: Scientific and Productive Firm «Rubicon», Private Enterprise «Unifilter». – Appl. No. 2006124622/15; Filed January 20, 2008; Publ. June 20, 2008, Bull. No. 17. – 5 p.

Vecherkovskaya, A. Comparative analysis of mathematical models forming filter elements [Text] / A. Vecherkovskaya, S. Popereshnyak // 2017 XIIIth International Conference on Perspective Technologies and Methods in MEMS Design (MEMSTECH). – IEEE, 2017. doi:10.1109/memstech.2017.7937545

Basniev, K. S. Neftegazovaia gidromehanika [Text]: Handbook / K. S. Basniev, N. M. Dmitriev, G. D. Rozenberg. – Moscow-Izhevsk: Institute for Computer Research, 2005. – 544 p.

Selivanov, V. V. Mehanika razrusheniia deformiruemogo tela [Text]: Handbook / V. V. Selivanov // Prikladnaia mehanika sploshnyh sred. – Moscow: MSTU n. a. N. E. Bauman, 1999. – Vol. 2. – 420 p.

Hasanov, M. M. Opredelenie optimal'nogo perioda otrabotki nagnetatel'noi skvazhiny na neft' [Text] / M. M. Hasanov, V. A. Krasnov, V. A. Korotovskih // Nauchno-tehnicheskii Vestnik OAO «NK «Ros-neft». – 2007. – No. 5. – P. 19–22.

Darcy, H. Les fontaines publiques de la ville de Dijon [Text] / H. Darcy. – Paris, 1856. – 647 p.

Charnyi, I. A. Podzemnaia gidrogazodinamika [Text] / I. A. Charnyi. – Moscow-Izhevsk: SIC «Regular and chaotic dynamics», Institute for Computer Research, 2006. – 436 p.

Andrianov, I. V. Asimptoticheskaia matematika i sinergetika: put' k tselostnoi prostote [Text] / I. V. Andrianov, R. G. Barantsev, L. I. Manevich. – Moscow: Editorial URSS, 2004. – 304 p.

Wriggers, P. Nonlinear Finite Element Methods [Text] / P. Wriggers. – Springer Berlin Heidelberg, 2008. – 560 p. doi:10.1007/978-3-540-71001-1

Sementsov, H. N. Development of informative support for automatic antisurge protection system and regulation of gas pumping plant [Text] / H. N. Sementsov, L. I. Davydenko // Eastern-European Journal of Enterprise Technologies. – 2014. – Vol. 4, No. 11 (70). – P. 20–24. doi:10.15587/1729-4061.2014.26311

Vambol, S. Assessment of improvement of ecological safety of power plants by arranging the system of pollutant neutralization [Text] / S. Vambol, V. Vambol, O. Kondratenko, Y. Suchikova, O. Hurenko // Eastern-European Journal of Enterprise Technologies. – 2017. – Vol. 3, No. 10 (87). – P. 63–73. doi:10.15587/1729-4061.2017.102314

Copyright (c) 2017 Anastasia Vecherkovskaya, Anastasia Vecherkovskaya