Geodesic
Mathematical model of the wastewater treatment process using biofilm
Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 2, pp. 25-36.

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The article proposes a mathematical model of wastewater treatment based on the use of biofilm; whose microorganisms destroy harmful impurities contained in water. For microorganisms, impurities are "food". A system of partial differential equations with boundary conditions is given. A system of partial differential equations with boundary conditions is given for one loading element, which is a cylindrical rod whose surface is covered with a biologically active film. This system includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on a part of the surface of this domain connected to each other through a boundary condition and a potential in a hyperbolic equation. Further, an asymptotic analysis of this system is carried out, which makes it possible to reduce the model of an individual element to the solution of a simple ordinary differential equation, and a strict mathematical justification of this method is given. In this case, a mathematical method is used to construct asymptotics in the so-called «thin regions». The proposed method is a simplification of a complex combined model based on the laws of hydrodynamics and diffusion. On this basis, a model of the operation of the entire wastewater treatment device containing a large (millions) of such elements is proposed.
Keywords: water treatment, biologically active layer, asymptotic analysis of solutions in a thin region, mathematical model of impurity treatment,systems of partial differential equations of mixed type. 
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T. N. Bobyleva; A. S. Shamaev; O. V. Yantsen. Mathematical model of the wastewater treatment process using biofilm. Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 2, pp. 25-36. http://geodesic.mathdoc.fr/item/SJIM_2023_26_2_a2/

[1] Khentse M., Armoes P., Lya-Kur-Yansen I., Arvan E., Ochistka stochnykh vod, Mir, M., 2006

[2] Bitton G., Wastewater Microbiology, Wiley-Interscience, N.Y., 2005

[3] B. DTAcunto, L. Frunzo, M. R. Mattei, “Continuum approach to mathematical modelling of multispecies biofilms”, Ricerche di Matematica, 66 (2017), 153–169 | DOI | MR

[4] J. P. Boltz, E. Mongenroth, D. Sen, “Mathematical modelling of biofilms and biofilm reactors for engineering design”, Water Sci. Technology, 62 (2010), 1821–1836 | DOI

[5] C. M. Guo, J. F. Chen, Z. Z. Zhang, L. J. Zhao, “Mathematical model of biofilm reactor treating industrial wastewater a review”, Adv. Materials Res., 356-360 (2011), 1739–1742 | DOI

[6] O. Wanner, R. Reichert, “Mathematical modelling of mixed-culture biofilms”, Biotechnol. Bioengrg., 48 (1996), 172–184 | 3.0.CO;2-N class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[7] E. Alpkvist, I. Yu. Klapper, “A multidimensional multispecies continuum model for heterogeneous biofilm development”, Bull. Math. Biol., 69:2 (2007), 765–789 | DOI | Zbl

[8] B. D'Acunto, L. Frunzo, M. R. Mattei, “Qualitative analysis of the moving boundary problem for a biofilm reactor model”, Math. Anal. Appl., 438 (2016), 474–491 | DOI | MR | Zbl

[9] A. Masic, H. J. Eberl, “A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor”, Bull. Math. Biol., 76 (2014), 27–58 | DOI | MR | Zbl

[10] Oleinik A.Ya., Vasilenko T.V., Rybachenko S.A., Khamad I.A., “Modelirovanie protsessov doochistki khozyaistvenno-bytovykh stochnykh vod na filtrakh”, Problemy vodopostachannya, vodovidvedennya ta gidravliki, 7 (2006), 85–97

[11] R. Christiansen, L. Hollesen, R. Harremoes, “Liquid film diffusion of reaction rate in submergen biofilters”, Water Res., 29:1 (1995), 947–952 | DOI

[12] S. V. Taylor, P. C.D. Milly, Jaffe P. R., “Biofilm growth and the related changes in the physical properties of a porous medium”, Water Resources Res., 26:9 (1990), 2161–2169 | DOI

[13] Vavilin V.A., Nelineinye modeli biologicheskoi ochistki i protsessov samoochischeniya v rekakh, Nauka, M., 1983

[14] Nazarov S.A., Asimptoticheskaya teoriya tonkii plastin i sterzhnei, Nauchn. kniga, Novosibirsk, 2002

[15] Oleinik O.A., Iosifyan G.A., Shamaev A.S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo MGU, M., 1990

[16] Agranovich M.S., “Smeshannye zadachi v lipshitsevoi oblasti dlya silno ellipticheskikh sistem 2-go poryadka”, Funkts. analiz i ego pril., 2011, no. 2, 1–22 | DOI