DOI: https://doi.org/10.15802/stp2020/199710

### MATHEMATICAL MODELING OF WATER PURIFICATION WITH FILTER

V. D. Petrenko, M. I. Netesa, O. L. Tiutkin, O. V. Gromova, V. А. Kozachyna

#### Abstract

Purpose. To analyze the effectiveness of water purification in water treatment systems, an important task is the development of mathematical models that allow determining the degree of water purification at the design stage. The main purpose of the work is to construct numerical models for calculating the filtration process and mass transfer in the filter. Methodology. The calculation of the filtering process of contaminated water in the filter is carried out in two stages. At the first stage, the flow rate field in the filter is calculated. To solve this problem, the classical filtration equations are used. At the second stage of the calculation, the flow of contaminated water in the filter is simulated. To solve this problem, the mass transfer equation is used, which expresses the law of mass conservation. This equation takes into account the transfer of impurities by the filtration flow, the transfer of impurities due to dispersion and the sorption of impurities in the filter material. The solution of the filtration equation is carried out using the alternating triangular method of A. A. Samarskyi. The unknown pressure value based on this method is determined by the explicit formula of point-to-point computation. For numerical integration of the mass transfer equation in the filter, a difference splitting scheme is used. Findings. The current trend in the field of water supply and sanitation is the creation of multidimensional and multifactor mathematical models. Such models make it possible to replace a physical experiment with a computational one. The complex of water treatment facilities necessarily includes water purification filters. The filter efficiency affects the efficiency of other treatment facilities of the technological treatment scheme. A mathematical model has been developed that allows analyzing the water purification process in the filter. Based on the developed numerical model, a package of application programs has been developed for computer simulation of the filter water purification process. The results of a computational experiment on modeling the filtering process of contaminated water in a filter are presented. Originality. The paper proposes a numerical two-dimensional filter model based on the filtration equation and the mass transfer equation. A feature of the developed mathematical models is the possibility of modeling the velocity field and the process of impurity transfer taking into account the geometric shape of the filter. Practical value. The calculation time for one variant of the task based on the developed numerical model is several seconds, which is important for conducting serial calculations in practice. Models can be used as an alternative to laboratory experiments.

#### Keywords

water purification; mathematical modeling; filter; numerical model

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#### References

Airapetian, T. S., & Karahiaur, A. S. (2018). Theoretical studies of biological purification in combined aerotanks with fixed biocenosis. Scientific Bulletin of Civil Engineering, 91(1), 200-205. DOI: https://doi.org/10.29295/2311-7257-2018-91-1-200-205 (in Russian).

Biliaiev, N. N., & Kozachina, V. A. (2015). Modelirovaniye massoperenosa v gorizontalnykh otstoynikakh: monografiya. Dnepropetrovsk: Aktsent PP. (in Russian)

Biliaiev, N. N., & Nagornaya, E. K. (2012). Matematicheskoye modelirovaniye massoperenosa v otstoynikakh sistem vodootvedeniya: monografiya. Dnepropetrovsk: Novaya ideologiya. (in Russian)

Kanalizatsiia. Zovnishni merezhi ta sporudy. Osnovni polozhennia proektuvannia, 128 DBN V.2.5-75-2013 (2013). (in Ukrainian)

Kozachyna, V. A., Shynkarenko, V. I., Bondarenko, I. O., Gabrinets, V. A., & Horiachkin, V. M. (2019). Water Cleaning Modeling in a Horizontal Settler. Science and Transport Progress, 5(83), 36-42. DOI: https://doi.org/10.15802/stp2019/184467 (in Ukrainian)

Oleynik, A. Y., & Airapetyan, T. S. (2015). The modeling of the clearance of waste waters from organic pollutions in bioreactors-aerotanks with suspended (free flow) and fixed biocenoses. Reports of the National Academy of Sciences of Ukraine, 5, 55-60. DOI: https://doi.org/10.15407/dopovidi2015.05.055 (in Ukrainian)

Polyakov, V. L. (2018). On the modeling of a dissolved iron removal from underground water by filtration. Reports of the National Academy of Sciences of Ukraine, 12, 37-45. DOI: https://doi.org/10.15407/dopovidi2018.12.037 (in Russian)

Polyakov, V. L., & Martynov, S. Yu. (2019). Calculation of the removal of iron from groundwater in a fast filter. Reports of the National Academy of Sciences of Ukraine, 3, 35-45. DOI: https://doi.org/10.15407/dopovidi2019.03.035 (in Russian)

He, Y., Uehara, S., Takana, H., & Nishiyama, H. (2016). Numerical Modelling and Simulation of Chemical Reactions in a Nano-Pulse Discharged Bubble for Water Treatment. Plasma Science and Technology, 18(9), 924-932. DOI: https://doi.org/10.1088/1009-0630/18/9/09 (in English)

Nadezhdin, I., Papasidero, D., Goryunov, A., & Manenti, F. (2016). Optimisation of EDM process for water purification. Chemical Engineering Transactions, 52, 325-330. DOI: https://doi.org/10.3303/CET1652055 (in English)

Oleynik, O., Airapetian, T., & Kurganska, S. (2019). Evaluation of the performance of aerotanks due to add-on attached biocenosis. Science and Transport Progress, 4(82), 37-46. DOI: https://doi.org/10.15802/stp2019/175883 (in English)

Rezakazemi, M., Ghafarinazari, A., Shirazian, S., & Khoshsima, A. (2012). Numerical modeling and optimization of wastewater treatment using porous polymeric membranes. Polymer Engineering & Science, 53(6), 1272-1278. DOI: https://doi.org/10.1002/pen.23375 (in English)

Giacomo, V., & Evangelista, S. (2018). Experimental and numerical analysis of the hydraulic performance of filtering cartridges for water treatment. EPiC Series in Engineering, 3, 2187-2195. DOI: https://doi.org/10.29007/b26c (in English)

#### GOST Style Citations

1. Айрапетян Т. С., Карагяур А. С. Теоретические исследования биологической очистки в комбинированных аэротенках с прикрепленным биоценозом. Науковий Вісник Будівництва. 2018. T. 91. No. 1. С. 200–205. DOI: https://doi.org/10.29295/2311-7257-2018-91-1-200-205.1
2. Беляев Н. Н., Козачина В. А. Математическое моделирование массопереноса в горизонтальных отстойниках: монография. Днипро : Акцент ПП, 2015. 115 с.
3. Беляев Н. Н., Нагорная Е. К. Математическое моделирование массопереноса в отстойниках систем водоотведения: монография. Днипро: Новая идеология, 2012. 112 с.
4. ДБН В.2.5-75:2013. Каналізація. Зовнішні мережі та споруди. Основні положення проектування. [Чинний від 2014-01-01]. Київ : Мінрегіон України, 2013. 128 с.
5. Козачина В. А., Шинкаренко В. І., Бондаренко І. О., Габрінець В. О, Горячкін В. М. Моделювання очищення води у горизонтальному відстійнику. Наука та прогрес транспорту. 2019. № 5 (83). P. 36–42. DOI: https://doi.org/10.15802/stp2019/184467
6. Олійник О. Я, Айрапетян Т. С. Моделювання очисних стічних вод від органічних забруднень в біо-реакторах-аеротенках зі зваженим (вільно плаваючим) і закріпленим біоценозом. Доповідь НАН України. 2015. № 5. С. 55–60. DOI: https://doi.org/10/15407/dopovidi2015.05.055
7. Поляков В. Л. О моделировании удаления фильт рованием растворенного железа из подземных вод. Доповідь НАН України. 2018. № 12. С. 37–45. DOI: https://doi.org/10.15407/dopovidi2018.12.037
8. Поляков В. Л., Мартынов С. Ю. Расчет обезжелезивания подземных вод на скором фильтре. Доповідь НАН України. 2019. № 3. С. 35–45. DOI: https://doi.org/10.15407/dopovidi2019.03.035
9. He, Y., Uehara S., Takana H., Hideya Nishiyama. Numerical Modelling and Simulation of Chemical Reactions in a Nano-Pulse Discharged Bubble for Water Treatment. Plasma Science and Technology. 2016. Vol. 18. No 9. P. 924–932. DOI: https://doi.org/10.1088/1009-0630/18/9/09
10. Nadezhdin I., Papasidero D., Goryunov A., Manenti F. Optimisation of EDM process for water purification. Chemical Engineering Transactions. 2016. Vol 52. P. 325–330. DOI: https://doi.org/https://doi.org/10.3303/CET1652055
11. Oleynik O., Airapetian T., Kurganska S. Evaluation of the performance of aerotanks due to add-on attached biocenosis. Science and Transport Progress. 2019. № 4 (82). P. 37–46. DOI: https://doi.org/10.15802/stp2019/175883
12. 12. Rezakazemi M., Ghafarinazari A., Shirazian S., Khoshsima A. Numerical Modeling and Optimization of Wastewater Treatment Using Porous Polymeric Membranes. Polymer Engineering and Science. 2013. Vol. 53. Iss. 6. P. 1272–1278. DOI: https://doi.org/10.1002/pen.23375
13. Viccione G., Evangelista S. Experimental and numerical analysis of the hydraulic performance of filtering cartridges for water treatment. EPiC Series in Engineering. 2018. Vol. 3. P. 2187–2195. DOI: https://doi.org/10.29007/b26c