Modeling the action of anaerobic biofilm

Modeling the action of anaerobic biofilm

A mathematical problem of the action of a representative biofilm in the absence of oxygen is formulated. The anaerobic process of decomposition of a dissolved organic matter is considered as a two-stage process, proceeding due to the vital activity of two groups of microorganisms. An approximate a...

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Дата:2021
Автор: Poliakov, V.L.
Формат: Стаття
Мова:English
Опубліковано: Видавничий дім "Академперіодика" НАН України 2021
Назва видання:Доповіді НАН України
Теми:
Механіка
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Цитувати:Modeling the action of anaerobic biofilm / V.L. Poliakov // Доповіді Національної академії наук України. — 2021. — № 6. — С. 52-58. — Бібліогр.: 15 назв. — англ.

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spelling irk-123456789-1848172022-07-18T01:26:22Z Modeling the action of anaerobic biofilm Poliakov, V.L. Механіка A mathematical problem of the action of a representative biofilm in the absence of oxygen is formulated. The anaerobic process of decomposition of a dissolved organic matter is considered as a two-stage process, proceeding due to the vital activity of two groups of microorganisms. An approximate analytic solution allowing one to calculate the concentration and consumption of primary and secondary organic substrates with minimal errors has been obtained. On test examples, their rates of transfer through the biofilm surface are determined, and the possibility of the movement of volatile fatty acids in both directions is discussed. Сформульовано математичну задачу дії репрезентативної біоплівки за відсутності кисню. Анаеробний процес розкладу розчиненої органіки розглядається як двостадійний, який протікає завдяки життєдіяльності двох груп мікроорганізмів. Одержано наближений аналітичний розв’язок, що дозволяє з мінімальними похибками розраховувати концентрації і витрати первинного і вторинного органічних субстратів. На тестових прикладах визначено їх витрати через поверхню біоплівки і демонструється реальність руху летючих кислот в обох напрямках. 2021 Article Modeling the action of anaerobic biofilm / V.L. Poliakov // Доповіді Національної академії наук України. — 2021. — № 6. — С. 52-58. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2021.06.052 http://dspace.nbuv.gov.ua/handle/123456789/184817 532.546:628.16 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Механіка
Механіка
spellingShingle Механіка
Механіка
Poliakov, V.L.
Modeling the action of anaerobic biofilm
Доповіді НАН України
description A mathematical problem of the action of a representative biofilm in the absence of oxygen is formulated. The anaerobic process of decomposition of a dissolved organic matter is considered as a two-stage process, proceeding due to the vital activity of two groups of microorganisms. An approximate analytic solution allowing one to calculate the concentration and consumption of primary and secondary organic substrates with minimal errors has been obtained. On test examples, their rates of transfer through the biofilm surface are determined, and the possibility of the movement of volatile fatty acids in both directions is discussed.
format Article
author Poliakov, V.L.
author_facet Poliakov, V.L.
author_sort Poliakov, V.L.
title Modeling the action of anaerobic biofilm
title_short Modeling the action of anaerobic biofilm
title_full Modeling the action of anaerobic biofilm
title_fullStr Modeling the action of anaerobic biofilm
title_full_unstemmed Modeling the action of anaerobic biofilm
title_sort modeling the action of anaerobic biofilm
publisher Видавничий дім "Академперіодика" НАН України
publishDate 2021
topic_facet Механіка
url http://dspace.nbuv.gov.ua/handle/123456789/184817
citation_txt Modeling the action of anaerobic biofilm / V.L. Poliakov // Доповіді Національної академії наук України. — 2021. — № 6. — С. 52-58. — Бібліогр.: 15 назв. — англ.
series Доповіді НАН України
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fulltext 52 МЕХАНІКА MECHANICS ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6: 52—58 Ц и т у в а н н я: Poliakov V.L. Modeling the action of anaerobic biofilm. Допов. Нац. акад. наук Укр. 2021. № 6. С. 52—58. https://doi.org/10.15407/dopovidi2021.06.052 Biofiltration of water with a high content of dissolved organic substances has a number of sig- nificant advantages and disadvantages in comparison with other methods of biological treat- ment. A characteristic feature of the aerobic biofiltration is a consistently large amount of the active biomass fixed on the elements of the solid phase, which ensures the rapid removal of or- ganic pollution with the formation of a significant amount of by-products and secondary pollu- tants of the treated water. At the same time, the free space is seriously reduced, and the hydraulic resistance becomes more significant. It is possible to avoid the noted negative consequences of the intense biooxidation due to the treatment of polluted waters in the absence of oxygen [1, 2]. It is an important feature of the anaerobic wastewater treatment that, as a rule, an increase in the biomass and the release of exudates are considerably reduced, the permeability of the partially clogged filter medium increases, and, finally, the energy is consumed more economically [3—5]. Dissolved organic matter is directly utilized in the biological phase of a biofilter which is formed by numerous biofilms. The rate of degradation of the matter and the composition of its products are closely related to the oxygen content. In its absence, the slow metabolism is a char- acteristic of vital activities of anaerobic microorganisms [6—8]. As a result, the corresponding biological phase economically consumes the energy for the biosynthesis and respiration and thus https://doi.org/10.15407/dopovidi2021.06.052 UDC 532.546:628.16 V.L. Poliakov Institute of Hydromechanics of the NAS of Ukraine, Kyiv E-mail: v.poliakov.ihm@gmail.com Modeling the action of anaerobic biofilm Presented by Corresponding Member of the NAS of Ukraine O.Ya. Oliinyk A mathematical problem of the action of a representative biofilm in the absence of oxygen is formulated. The anaerobic process of decomposition of a dissolved organic matter is considered as a two-stage process, proceeding due to the vital activity of two groups of microorganisms. An approximate analytic solution allowing one to calcu- late the concentration and consumption of primary and secondary organic substrates with minimal errors has been obtained. On test examples, their rates of transfer through the biofilm surface are determined, and the possibility of the movement of volatile fatty acids in both directions is discussed. Keywords: anaerobic biofilm, organic substrate, decomposition, volatile fatty acids, analytic solution, concen tration, consumption. 53ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6 Modeling the action of anaerobic biofilm allows performing a primary wastewater treatment. Therefore, it is natural that the anaerobic bio- filtration as a technological process and the action of a representative biofilm under such con- ditions as its basis are of a great practical interest. The decomposition of organic matter within an anaerobic biofilm is considered below by analytic methods. Modern ideas of the anaerobic process and its formal description [9] are interpreted with in- significant simplifications. First of all, it is considered as a two-stage process [10]. In the first stage, the organic substrate is transformed into volatile fatty acids and carbon dioxide. In the second stage, the volatile fatty acids eventually decompose to carbon dioxide and methane. Thus, no intermediate oxidation step is distinguished. Therefore, we will analyze the behavior of the substrate and its by-products (volatile fatty acids, 2 4CO CH ) within an arbitrary flat biofilm fl in thickness. The mathematical problem is formulated with respect to the corresponding mass concentrations ( 1, 2, 3, 4)is i  assuming only the diffusion (surface and molecular) transfer, significant limitation of the rates of bio- degradation of the substrate and acids, functioning and coexistence within a single biofilm of two groups of microorganisms (acidogenetic and methanogenetic). The model includes, first of all, the following system of equations: 2 1 1 1 1 1 2 1 1 1 1 m B e s d s s D Y K sdx     , (1) 2 1 2 1 1 12 2 2 2 2 2 2 2 2 1 1 1 m B s Bm B e s s Y sd s s D Y K s K sdx        , (2) 3 1 3 2 2 1 1 1 2 2 23 3 2 1 1 2 2 m B s B m B s B e s s Y s Y sd s D K s K sdx         , (3) 4 2 2 2 2 24 4 2 2 2 m B s B e s Y sd s D K sdx     . (4) Here, eiD is the effective diffusion coefficients of the i -th substrate, 1, 2Y are the effec tive eco- nomic coefficients characterizing the decomposition of the initial substrate and volatile fatty ac- ids by the corresponding groups of microorganisms 1 1 2 1 3 1 1 1 s B s B s B Y Y Y Y    ,    2 2 3 2 4 2 2 1 s B s B s B Y Y Y Y ; (5) i js BY are the conversion factors (equal to the mass of the i -th substrate used per unit of the j -th variety of a biomass; and mj and Bj are the specific growth rate and density of the j -th bio- mass. This system is complemented by the usual boundary conditions for a biofilm on a solid ele- ment in the liquid medium: 0, 0ids x dx   ; , ( )i f ei Li i i ds x l D k S s dx    , (6) 54 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6 V. L. Poliakov where kLi is the coefficient of transfer of the i-th substrate through the liquid film, Si is the con- centration of the i -th substrate outside both films Dimensionless variables and parameters are introduced using, as scales, 0S (reference va- lue for the concentration of the primary substrate, for example, the input value for a fixed-bed reactor), R (a characteristic microsize; for example, radius of a grain or a thread), and 1eD as follows: 0 0 , ,i i i i S s S s S S   2 2 0 1 0 , , , , ,mj Bj e si Li j si Li j ej e ei R D K Rkx x D K k R Y D S D S D         f f l l R  . If a theoretical analysis of the anaerobic biofiltration is carried out solely for the purpose of monitoring the quality of the biological purification of water, and if the associated combustible gas is not of practical interest, then it is sufficient to restrict ourselves to a truncated system of equations for 1s , 2s , namely: 2 1 1 1 2 1 1s d s s K sdx    , (7) 2 1 2 12 2 2 1 1 2 2 2 1 1 s B s s d s s s K s D K sdx        (8) with the boundary conditions 0x  , 1, 2 0 d s dx  ; (9) fx l , 1, 2 1, 2 1, 2 1, 2( )L L d s k S s d x   . (10) The solution of problem (7) — (10) can be significantly simplified due to the usually large initial content of an organic substrate in wastewater. In such situations, it is reasonable to believe that the primary substrate decomposes at the maximum rate. It should be noted that this assump- tion cannot be applied to volatile fatty acids, whose decomposition is limited due to their low content, and, at the same time, an inhibitory effect is possible [11—13]. Therefore, the indicated maximum rate will be 1 , and Eq. (8) takes the form 2 2 2 2 12 2 2s d s s K sdx      , (11) where 2 11 1 1s BY D    . Then the distribution of the primary substrate across the biofilm is represented by the expression 2 21 1 1 1 1 1 ( ; , ) ( ) 2 f f f L l s x l S S x l k         . (12) 55ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6 Modeling the action of anaerobic biofilm The approximate problem (9) — (11) is solved by averaging the right-hand side of Eq, (11). Previously, this technique was used in the absence of the internal source of a degradable substrate [14]. Then, following the previous procedure, the equation was derived for the mean value avu on the interval [0, ]X   2 2 20 ( )1 ( ) ( ) X av s s x dx u X X K s x , (13) which has the form 2 1 2 arctg 1 ( ) ( ) ( ) s av s av f av f av K x u u u x u       . (14) Here, 2 22 2 2 1 2 2( ) 2s f f f av L K S Dl l u k        . Now, the function arctg f x  is expanded in a series in the argument, and only its first term is used. As a result, the quadratic equation for avu was obtained:             2 2 2 2 2 2 2 1 1 2 2( ) [ ( ) ( )] ( ) 0f f av f f s f f av f fl u l K S l u l S , (15) where 2 2 2 2( ) ( 2 ) 2f f L f f Ll k l Dl k    . Only one root has the physical meaning, namely, 2 2 2 2 21 1 2 1 2 2 2 2 1 ( , ) 1 1 4 2 ( ) ( ) ( ) s s av f f f f f f f K S K S S u l S l l l                                    . (16) By simplifying Eq. (11) with regard for (13) and integrating it twice, we get the follo wing ex- pression for the concentration 2s : 2 2 2 2 2 2 2 1 2 ( ; , ) [ ( , ) ] 2 2 f f f av f L l l x s x l S S u l S k              . (17) Thus, the relative value of 2s at the biofilm surface is               2 2 2 2 2 1 2 2 2 2 2 1 ( , ) [ ( , ) ] 2 f f f av f s L L f l s l S S u l S S K S k k l D 2 1 1 2 2 2 2 1 2 2 2 ( ) 1 ( ) 1 ( ) 4 ( )f f f f s f f f f f l l K S l S l                                       . (18) Using the representations for 1s (12) and 2s (17) and doubly integration Eqs. (3) and (4) under the appropriate conditions (6), it is easy to find also the concentrations 3s and 4s of the final decomposition products  2 4CO , CH . 56 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6 V. L. Poliakov The actual efficiency of a biofilm in relation to both components of the organic pollution can be assessed by calculating their relative flow rates 1fi , 2fi across the border between both films: 1 1( ) ,f f fi l l  (19) 2 2 2 1( ) [ ( , ) ].f f f av fi l l u l S   (20) The approximate solution to the problem of the steady-state action of an anaerobic biofilm obtained above is illustrated by test examples. The possible inaccuracies in calculating the mic ro- characteristics ( , )i fis i due to the use of an adaptive averaging of the local organic substrate uti- lization function (applied before to an aerobic biofilm) were discussed in [14]. It was found that they do not exceed a few percent and are less than the errors due to the experimental determi- nation of the model coefficients. The subject of many calculations was the relative flow rates of both substrates through the surface of the active biofilm under consideration 1 2( , )f fi i . It is they that determine the effi- ciency of the elements of the biological phase (biofilms of any form) in relation to the organic pollution and underlie the modeling of the anaerobic biofiltration. Calculations were performed using formulas (19) and (20) with a continuous change in the relative thickness fl from 0 to 0.2. Thus, the range of its real values was covered with a large margin. The initial content of volatile fatty acids was also discretely varied from 0 to 0.5. In the accepted model, the stable presence of volatile fatty acids is allowed already at the inlet to the filter 2( 0)S  . In practice, a similar si- tuation is typical of the successively acting second and subsequent anaerobic filters [15]. The ini- tial information included the following fixed relative values of the coefficients: 2 0.25,sK  2 10Lk  , 1,B D   10.4, 10    . Two characteristic values of (10 and 20) were also selected Fig. 1. Dependence 2( )f fi l : 1 — 20 0S  ; 2 — 20 0.1S  ; 3 — 20 0.25S  ; 4 — 20 0.5S  . Fig. 2. Dependences 1( )f fi l , 2( )f fi l : 1 — 1fi , 2—5 — 2fi ; 2 — 20 0.5S  ; 3 — 20 0.25S  ; 4 — 20 0.1S  ; 5 — 20 0S  . 57ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6 Modeling the action of anaerobic biofilm for 2 . Graphs presenting the dependence 2( )f fi l for 2 10  are shown in Fig. 1 and, for 2 20  , in Fig. 2. There, the single graph due to the constancy of 1 is given for 1( )f fi l as well. When establishing the quantities 1 2,f fi i , their sign is of fundamental importance, since it deter mines the direction of the transfer of the corresponding substrate. The “+” sign means that the im- purity moves into the biofilm, and the “–” sign — in the opposite direction. It is obvious that the primary substrate is only consumed by the biofilm. Therefore, 1fi is necessarily positive. The directionality of the secondary substrate is dictated by the ratio between its concentrations out- side both films 2S and at their common boundary fs . Thus, the volatile fatty acids will diffuse from the outside at 2fs S (Fig. 1 and curve 2 in Fig. 2), and 2fi will be negative at 2fs S (curves 3—5 in Fig. 2). Therefore, the theoretical basis has been developed for a further research of the operation of an anaerobic fixed-bed bioreactor by analytic methods, in essence, thanks to the solution of the problem of the action of a representative anaerobic biofilm. The derived dependences can be used to specify the model coefficients at the structural level. REFERENCES 1. Andrus, D.F. (1981). Development of a dynamic model and control strategy for the anaerobic decomposition process. In: A. James. (ed.) Mathematical models of water pollution, Moscow, Mir, pp. 321-345 (in Russian). 2. Dmitrenko, G.N. (2005). Oxygen-free microbial processes in water purification. J. Water Chemistry and Technology, 27, № 1, pp. 85-103. (in Russian). 3. Buffiere, P., Steyer, J.P., Fonade, C. & Moletta, R. (1998). Modeling and experiments on the influence of bio- film size and mass transfer in a fluidized bed reactor for anaerobic digestion. Water Res., 32, № 3, pp. 657-668. 4. Cakir, F.Y. & Stenstrom, M.K. (2005). Greenhouse gas production: a comparison between aerobic and an- aerobic wastewater treatment technology. Water Res., 39, pp.4197-4203. 5. Knobel, A.N. & Lewis, A.E. (2002). A mathematical model of a high sulphate wastewater anaerobic treat- ment system. Water Res., 36, pp. 257-265. 6. Aspe, E., Marti, M.C. & Roeckel, M. (1997). Anaerobic treatment of fishery wastewater using a marine se- diment inoculum. Water Res., 31, № 9, pp. 2147-2160. 7. Merkel, W., Manz, W., Szewzyk, U. & Krauth, K. (1999). Population dynamics in anaerobic wastewater re- actors: modelling and in situ characterization. Water Res., 33, № 10, pp. 2392-2402. 8. Ribes, J., Keesman, K. & Spanjers, H. (2004). Modelling anaerobic biomass growth kinetics with a substrate threshold concentration. Water Res., 38, pp. 4502-4510. 9. Huang, J.-S. & Jih, C.-G. (1997). Deep-biofilm kinetics of substrate utilization in anaerobic filters. Water Res., 31, № 9, pp. 2309-2317. 10. Escudie, R., Conte, T., Steyer, J.P. & Delgenes, J.P. (2005). Hydrodynamic and biokinetic models of an anaerobic fixed-bed reactor. Process Biochemistry, 40, pp. 2311-2323. 11. Aguilar, A., Casus, C. & Lema, J.M. (1995). Degradation of volatile fatty acids by differently enriched me- thanogenic cultures: kinetics and inhibition. Water Res., 29, № 2, pp. 505-509. 12. Kus, F. & Wiesmann, U. (1995). Degradation kinetics of acetate and propionate by immobilized anaerobic mixed cultures. Water Res., 29, № 6, pp. 1437-1443. 13. Zonta, Z., Alves, M.M., Flotas, X. & Palats, J. (2013). Modelling inhibitory effects of long chain fatty acids in the anaerobic digestion process. Water Res., 47, № 3, pp. 623-636. 14. Poliakov, V.L. (2011). Modeling the biofiltration of water with limited organic substrate content. Aerobic biofilm. Dopov. Nac. akad. nauk Ukr., № 5, pp. 72-77 (in Russian). 15. Gvozdiak, P.I. (2019). Biochemistry of water. Biotechnology of water. Kiev-Mohyla Academy. 228 p. (in Ukrainian). Received 03.10.2021 58 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6 V. L. Poliakov В.Л. Поляков Інститут гідромеханіки НАН України, Київ E-mail: v.poliakov.ihm@gmail.com МОДЕЛЮВАННЯ ДІЇ АНАЕРОБНОЇ БІОПЛІВКИ Сформульовано математичну задачу дії репрезентативної біоплівки за відсутності кисню. Анаеробний процес розкладу розчиненої органіки розглядається як двостадійний, який протікає завдяки життєдіяль- ності двох груп мікроорганізмів. Одержано наближений аналітичний розв’язок, що дозволяє з мінімаль- ними похибками розраховувати концентрації і витрати первинного і вторинного органічних субстратів. На тестових прикладах визначено їх витрати через поверхню біоплівки і демонструється реальність руху летючих кислот в обох напрямках. Ключові слова: анаеробна біоплівка, органічний субстрат, розклад, летючі кислоти, аналітичний розв’я- зок, концентрація, витрата.

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