Mathematical Modeling and Applied Calculation of Bioconveyer and Anaerobic Biofiltration

Abstract

Deep wastewater treatment is carried out by a bioconveyer technology and a direct-flow system of multistage biological treatment (DSMBT). To reduce excess biomass and energy costs, the first bioreactors (sections) of the DSMBT provide for partial removal of soluble organics in the absence of oxygen, the rest are intended for aerobic water treatment. Development of a method for engineering calculation of anaerobic biofiltration as applied to the first sections of DSMBT is the main aim of the study. Anaerobic substrate utilization is modeled at two levels using two stage biokinetics. The behavior of the substrate and its derivatives within a representative biofilm are analyzed, taking into account surface and molecular diffusion, limitation of the rates of the substrate and acid biodegradation, coexistence of two communities of microorganisms. The behavior of the original and newly formed organic substrates in the volume of a representative section is studied by analytical methods. A theoretical base and an engineering method for calculating anaerobic biofiltration are developed and illustrated, which can serve as the basis for applied optimization of the parameters of bioconveyer plants. It is justified to use the derived calculation dependencies for similar complex biological treatment plants with any filtering material.

1. Introduction

Deep treatment of wastewater containing a large amount of various organic contaminants has to be carried out in stages on special purification plants of complex design. Naturally, the technological process should provide at each stage a significant reduction in water pollution as a whole, but at the same time it can be aimed at the predominant removal of the dominant type of contamination. In such widespread situations, only rational technologies and structural decisions can achieve the desired result [1, 2, 3, 4, 5, 6, 7, 8]. The bioconveyer technology and the direct-flow system of multi-stage biological treatment (DSMBT) are their successful examples. Their successful examples are the bioconveyer technology and the direct-flow system of multi-stage biological treatment (DSMBT). The specified technology and complex plant were proposed, and then tested on an industrial scale by professor P.I. Gvozdyak (Institute of Colloid Chemistry and Water Chemistry of the National Academy of Sciences of Ukraine) [9, 10].

The original shape and high density of structural elements contribute to the high efficiency of the new generation technology. Thus, in fact the solid phase consists of a set of vertically and densely arranged (upper end is fixed) thin synthetic profiled fiber-nozzles (viy). Their surface has a complex configuration, which contributes to the strong attachment of microorganisms to them together with their metabolites. Due to the multiplicity and vertical orientation of the fibers, the space, that is filled by them in the sequence of bioreactors (sections) of DSMBT, has a high hydraulic resistance and anisotropy. Therefore, it is justified to interpret it as a specific (pseudo) porous medium. Simultaneously, the water flow between fibers is conditionally horizontal. This fact gives the right to use one-dimensional mathematical models of hydrodynamics in anisotropic porous medium, mass transfer and take into account changes in the permeability of the medium only in this direction.

Currently, there are acute problems of removing xenobiotics, disposal of waste from biological water treatment. It is possible to significantly reduce the amount of excess biomass, as well as to minimize energy costs, due to the decomposition of organic contaminants in the absence of oxygen. An appropriate technological process is implemented (in fact, low-molecular easily degradable compounds are processed) as a result of the successive and parallel course of a series of (bio)chemical reactions [11, 12, 13]. Detailed biokinetic model includes nine equations [14]. However, when developing engineering methods for calculating anaerobic biofiltration, it is sufficient to single out and formalize those processes and effects that determine the operation of the corresponding bioreactor-filter and, most importantly, the final result. The justification for the simplified consideration of the specified complex technological process as a two-stage process, which is just implemented below, is presented in the works [15, 16]. Taking into account the specifics of water purification by anaerobic microbiocenosis in a porous medium, its productive potential deserves special attention, in fact, the growth rate of anaerobes population under the most favorable conditions [17, 18, 19]. Actually, the vital activity of any microorganisms, including anaerobes, is significantly limited under the influence of many negative factors of a physical, chemical, biological nature [20, 21, 22, 23, 24]. Low content of the initial and especially intermediate (long-chain fatty acids) substrates may be the main limiting factor [25, 26, 27]. The adequacy of the initial model to the real conditions of biofiltration, its provision with reliable information are of great importance in mathematical modeling of the utilization of easily mineralizable organics in anaerobic bioreactor-filters. The source of such information are usually the articles that contain original data or data borrowed from experimental and theoretical studies, for example [28, 29, 30].

The principles of organizing the most complex technological process at DSMBT (deep biological treatment of highly toxic waters) are illustrated in Figure 1. It is essential that the bioconveyer technology is able to purify both relatively clean and extremely polluted water, while being waste-free. However, such a result can only be achieved with scientifically based debugging of DSMBT, namely, primarily due to the selection of populations of microorganisms that are most suitable for solving local applied problems. The presented three-stage layout of the technology of biological water treatment guarantees the production of high-quality drinking water with valuable biological additives. Developed on this base, the calculation methods make it possible to reliably assess the consequences of the action of DSMBT at the stages of microbiological treatment. The calculation method in relation to the second (aerobic) stage in the first version is set out in the articles [31, 32]. The method applied to the anaerobic stage is presented and discussed below. At the same time, these methods can serve as the basis for the applied optimization of the technological and design parameters of the first bioreactor banks (anaerobic I and aerobic II), and in the future, the entire purification plant. However, for full-scale calculations of DSMBT, it is also necessary to model the life activity of coexisting organisms that continue the trophic food chain - from protozoa to filter feeders and then predators. At present, modeling the removal of bacterial pollution, and at the same time, the waste products of the nanofauna, its mobilized individuals is problematic primarily due to the lack of suitable experimental information.

Figure 1.

Schematic diagram to calculate biological treatment at DSMBT: $S_0$ – Initial concentration of organic pollution; $S_L,S_R$ – Concentrations of its low-molecular and high-molecular components; $S_{B0}$ – Concentration of primary bacterial pollution;$S_B$ - concentration of total bacterial pollution.

The soluble organic component of pollution is finally eliminated and the content of high-molecular compounds drops to almost 0 in the bank II. In order to avoid organic shock load on it (can cause an emergency), an anaerobic reactor bank I is equipped (Figure 1). A comparative analysis of water treatment by aerobic and anaerobic microbiocenoses was carried out in the work [33]. At the same time, conditions are created in the bank III (usually includes two bioreactors) that favor the vital activity of organisms with more complex physiology (protozoa, filter feeders, predators). The first section (bioreactor) of the bank is inhabited by representatives of the nanofauna, while the second section is inhabited by the mesofauna. Of particular note is the ease of operation and flexibility in the management of DSMBT, its compactness (if we take into account the scale of the plant’s activity). Indeed, the service of the plant is comparatively easy. The plant is able to independently adapt to extreme conditions. If necessary, it is easy to forcibly adjust the composition of the microbiocenosis using specially selected microorganisms, to supply nutrients in the required amount.

Substrate biodegradation in different sections of the same bank is described by models of the same type, similar coefficients of which may differ significantly. Since the diffusion mechanism of mass transfer on the scale of each section can be neglected [34], the relationship between neighboring sections and banks (by analogy with a layered medium of a rapid filter, multi-stage filters) is almost one-sided. As a result, the values of pollution concentrations at the outlet of each section, excluding the last section, can be considered as initial values when modeling biofiltration. Hereinafter, the term biofiltration will be systematically used, although it does not quite correspond to the real conditions in the specific medium under consideration.

Thus, the algorithm for calculating the action of DSMBT is actually reduced to the sequential determination of the output concentrations of substrates for the sections of the first, second and, as a result, the third bioreactor bank. Therefore, in order to develop a method for calculating DSMBT, it is first necessary to analyze, using analytical methods, the utilization of the organic substrate in representative anaerobic and aerobic banks (sections) under nonuniform boundary conditions. Similar studies in relation to aerobic conditions were carried out earlier in the above-mentioned works of the author.

It should be emphasized that the anaerobic biofiltration, organized within one or two sections of the first bank of DSMBT, as well as the previously studied the aerobic biofiltration, are modeled in a similar way, namely, at two levels. A key role in the anaerobic utilization of dissolved organic matter is also played by biofilms which differ significantly in size and properties (composition, adaptability, activity, strength, etc.) from aerobic analogues. And above all, due to the relatively slow growth of anaerobic microorganisms, the current $\left(l_f\right)$ and limiting $\left(l_{\mathit{fm}}\right)$ thicknesses of the biofilm formed by them, as a rule, are clearly less than those of aerobic biofilms. Therefore, when choosing the radius of the fiber $\left(R_t\right)$ as a linear scale, the following relation ${\overline{l}}_{\mathit{fm}}=l_{\mathit{fm}}/R_t<<1$ also holds.

It is appropriate to note that this development can be useful in the case of an in-depth study of the action of biofilms with poor aeration of polluted waters and, as a result, the localization of oxygen in their outer part (aerobic zone). However, the internal part of the biofilm (anaerobic zone) can also make a noticeable contribution to the processing of the organic substrate. Of course, ignoring such a contribution is uncritical in view of its usually relative smallness. However, for the operation of the first bank of the DSMBT, the vital activity of anaerobes is of decisive importance for the operation of the first bank of the DSMBT. Therefore, the attention is focused below precisely on the steady-state (due to the stable long-term operation of this plant) anaerobic biofiltration [35, 36].

2. Theoretical analysis

2.1 Statement and solution of mathematical problem

Thus, the behavior of the substrate and its derivatives within an arbitrary flat biofilm with a thickness $l_f$ (in view of $l_f<<R_t$) is analyzed by analytical methods at the first stage of studies of stationary anaerobic biofiltration in a (pseudo) porous medium. It is here that it transforms in several stages, so that the end products are volatile acids (secondary substrate), carbon dioxide and combustible biogas, mainly methane. In fact, two stages dominate, at which these products are formed. The internal mathematical problem in this case is formulated with respect to the corresponding mass concentrations $s_i\left(i=1,2,3,4\right)$ assuming only diffusion (surface + molecular diffusion) mass transfer, significant limitation of the biodegradation rates of the substrate and acids, coexistence within a single biofilm of two communities of microorganisms (acid-producing and methane-producing). The problem includes, first of all, the following system of equations

$$D_{e1}\frac{d^2{s}_1}{{\mathit{dx}}^2}=\frac{1}{Y_1}\frac{{\mu }_{m1}{\rho }_{B1}{s}_1}{K_{s1}+s_1},$$E1

$$D_{e2}\frac{d^2{s}_2}{{\mathit{dx}}^2}=\frac{1}{Y_2}\frac{{\mu }_{m2}{\rho }_{B2}{s}_2}{K_{s2}+s_2}-\frac{{\mu }_{m1}{\rho }_{B1}{Y}_{s_2/B_1}{s}_1}{K_{s1}+s_1},$$E2

$$D_{e3}\frac{d^2{s}_3}{{\mathit{dx}}^2}=\frac{{\mu }_{m1}{\rho }_{B1}{Y}_{s_3/B_1}{s}_1}{K_{s1}+s_1}+\frac{{\mu }_{m2}{\rho }_{B2}{Y}_{s_3/B_2}{s}_2}{K_{s2}+s_2},$$E3

$$D_{e4}\frac{d^2{s}_4}{{\mathit{dx}}^2}=\frac{{\mu }_{m2}{\rho }_{B2}{Y}_{s_4/B_2}{s}_2}{K_{s2}+s_2}.$$E4

Here, $D_{\mathit{ei}}$ is the effective diffusion coefficient of the i-th substance; $Y_{1,2}$ are the effective economic coefficients that characterize the decomposition of the initial substrate and volatile acids by the corresponding groups of microorganisms in general. Also.

$$Y_1=\frac{1}{Y_{s_1/B_1}+Y_{s_2/B_1}+Y_{s_3/B_1}},Y_2=\frac{1}{Y_{s_{21}/B_2}+Y_{s_3/B_2}+Y_{s_4/B_2}};$$E5

$Y_{s_i/B_j}$ are the conversion coefficients equal to the mass of the formed or consumed substance, which falls on the biomass unit of the $j$-th variety; ${\mu }_{\mathit{mj}}$, ${\rho }_{\mathit{Bj}}$ are the specific growth rate and density of the $j$-th biomass. This system is supplemented with standard boundary conditions for biofilms (on the surfaces separating the biofilm from the liquid film and fiber-nozzle).

$$x=0,\frac{{\mathit{ds}}_i}{\mathit{dx}}=0;x=l_f,D_{\mathit{ei}}\frac{{\mathit{ds}}_i}{\mathit{dx}}=k_{\mathit{Li}}\left(S_i-s_i\right);$$E6

where $k_{\mathit{Li}}$ is the transfer coefficient of the $i$-th substance through the liquid film; $S_i$ is the concentration of the $i$-th substance in the liquid medium outside both films.

Dimensionless variables and parameters are introduced using as scales $S_{10}$ (concentration of the primary substrate at the inlet to the section under consideration), $R_t$ (characteristic microsize), $D_{e1}$ as follows:

${\overline{S}}_i,{\overline{s}}_i=\frac{S_i,s_i}{S_{10}}$, $\overline{x}=\frac{x}{R_t}$, ${\overline{\lambda }}_j=\frac{{\mu }_{\mathit{mj}}{\rho }_{\mathit{Bj}}{\lambda }_j}{{\mathrm{{\rm Y}}}_j{D}_{e1}{S}_{10}}$, ${\overline{D}}_e=\frac{D_{e2}}{D_{e1}}$, ${\overline{K}}_{\mathit{si}}=\frac{K_{\mathit{si}}}{S_{10}}$, ${\overline{k}}_{\mathit{Li}}=\frac{R_t{k}_{\mathit{Li}}}{D_{\mathit{ei}}}$, ${\overline{l}}_f=\frac{l_f}{R_t}$. If the theoretical analysis of anaerobic biofiltration is performed solely for the purpose of monitoring the quality of biological water treatment, and the combustible gas released along the way is of no practical interest, then it is enough to restrict ourselves to a truncated system of the equations for ${\overline{s}}_1$, ${\overline{s}}_2$, namely,

$$\frac{d^2{\overline{s}}_1}{d{\overline{x}}^2}=\frac{{\overline{\lambda }}_1{\overline{s}}_1}{{\overline{K}}_{s1}+{\overline{s}}_1},$$E7

$$\frac{d^2{\overline{s}}_2}{d{\overline{x}}^2}=\frac{{\overline{\lambda }}_2{\overline{s}}_2}{{\overline{K}}_{s2}+{\overline{s}}_2}-\frac{{\mathrm{{\rm Y}}}_1{\mathrm{{\rm Y}}}_{s2/B_1}}{\overline{D}}\frac{{\overline{\lambda }}_1{\overline{s}}_1}{{\overline{K}}_{s1}+{\overline{s}}_1}$$E8

and the boundary conditions.

$$\overline{x}=0,\frac{d{\overline{s}}_{1,2}}{s\overline{x}}=0_{;}$$E9

$$\overline{x}={\overline{l}}_f,\frac{d{\overline{s}}_{1,2}}{d\overline{x}}={\overline{k}}_{L1,L2}\left({\overline{S}}_{1,2}-{\overline{s}}_{1,2}\right).$$E10

The solution of problem (7)–(10) can be significantly simplified due to the usually large initial content of the substrate in wastewater. In such situations, which are just characteristic of bioconveyer technologies, it is reasonable to assume that the primary substrate decomposes at a maximum rate. It should be noted that this assumption does not apply to volatile acids, the decomposition of which is not only limited due to their low content. At the same time, an inhibitory effect is also possible. Therefore, the indicated maximum rate will be ${\overline{\lambda }}_1$ and Eq. (8) is reduced to the form

$$\frac{d^2{\overline{s}}_2}{d{\overline{x}}^2}=\frac{{\overline{\lambda }}_2{\overline{s}}_2}{{\overline{K}}_{s2}+{\overline{s}}_2}-{\tilde{\lambda }}_1,$$E11

where ${\tilde{\lambda }}_1=Y_1{\mathrm{{\rm Y}}}_{s_2/B_1}{\overline{\lambda }}_1/\overline{D}$. In that case the distribution of the primary substrate across the biofilm is represented by the following expression

$${\overline{s}}_1\left(\overline{x}\right)={\overline{S}}_1-\frac{{\tilde{\lambda }}_1{\overline{l}}_f}{{\overline{k}}_{L1}}+\frac{{\tilde{\lambda }}_1}{2}\left({\overline{x}}^2-{\overline{l}}_f^2\right).$$E12

Problem (9)–(11) is approximately solved by averaging the right side of Eq. (11). Previously, this technique was used in the absence of an internal source of degradable substrate $\left({\tilde{\lambda }}_1=0\right)$ and then substantiated on the test examples [31]. Therefore, following the previous procedure and keeping the notation, we derive an equation for the average value $u_{\mathit{av}}$ on the interval $\left[0,X\right]$

$$u_{\mathit{av}}\left(\mathrm{X}\right)=\frac{1}{\mathrm{X}}\underset{0}{\overset{\mathrm{X}}{\int }}\frac{{\overline{s}}_2\left(\overline{x}\right)d\overline{x}}{{\overline{K}}_{s2}+{\overline{s}}_2\left(\overline{x}\right)},$$E13

which looks like

$$u_{\mathit{av}}+\frac{2{\overline{K}}_{s2}}{\left({\overline{\lambda }}_s{u}_{\mathit{av}}-{\tilde{\lambda }}_1\right)\cdot {\mathrm{\Psi }}_f\left(u_{\mathit{av}}\right)\overline{x}}\mathit{\text{arctg}}\frac{\overline{x}}{{\mathrm{\Psi }}_f\left(u_{\mathit{av}}\right)}=1.$$E14

Here ${\mathrm{\Psi }}_f^2=\frac{2\left({\overline{K}}_{s2}+{\overline{S}}_2\right)}{{\overline{\lambda }}_2{u}_{\mathit{av}}-{\mathrm{X}}_1}-\frac{2{\overline{D}\overline{l}}_f}{{\overline{k}}_{L2}}-{\overline{l}}_f^2$. Now the function $\mathit{\text{arctg}}\frac{\overline{x}}{{\mathrm{\Psi }}_f}$ is expanded into a series in terms of the argument and only its first term is preserved. The result is a quadratic equation for $u_{\mathit{av}}$

$${\overline{\lambda }}_2{\varphi }_f\left({\overline{l}}_f\right)u_{\mathit{av}}^2-\left[{\overline{\lambda }}_2{\varphi }_f\left({\overline{l}}_f\right)+{\overline{\lambda }}_2{\overline{K}}_{s2}+{\overline{\lambda }}_2{\overline{S}}_2+{\overline{\lambda }}_1{\varphi }_f\left({\overline{l}}_f\right)\right]u_{a2}+{\overline{\lambda }}_1{\varphi }_f\left({\overline{l}}_f\right)+{\overline{\lambda }}_2{\overline{S}}_2=0,$$E15

where ${\varphi }_f\left({\overline{l}}_f\right)={\overline{\lambda }}_2\left({\overline{k}}_{L2}{\overline{l}}_f^2+2{\overline{D}\overline{l}}_f\right)/2{\overline{k}}_{L2}$. Physical meaning has only one root, namely,

$$u_{\mathit{av}}\left({\overline{l}}_f,{\overline{S}}_2\right)=\frac{1}{2}\left\{\frac{{\overline{K}}_{s2}+{\overline{S}}_2}{{\varphi }_f\left({\overline{l}}_f\right)}+\frac{{\tilde{\lambda }}_1}{{\overline{\lambda }}_2}+1-\sqrt{{\left(\frac{{\overline{K}}_{s2}+{\overline{S}}_2}{{\varphi }_f\left({\overline{l}}_f\right)}+\frac{{\tilde{\lambda }}_1}{{\overline{\lambda }}_2}+1\right)}^2-\frac{4{\overline{S}}_2}{{\varphi }_f\left({\overline{l}}_f\right)}-\frac{4{\tilde{\lambda }}_1}{{\overline{\lambda }}_2}}\right\}.$$E16

After the simplification of Eq. (11) considering Eq. (13) and then its double integration, the expression for the concentration ${\overline{s}}_2$ is derived

$${\overline{s}}_2\left(\overline{x};{\overline{l}}_f,{\overline{S}}_2\right)={\overline{S}}_2-\left[{\overline{\lambda }}_2{u}_{\mathit{av}}\left({\overline{l}}_f,{\overline{S}}_2\right)-{\tilde{\lambda }}_1\right]\left(\frac{{\overline{l}}_f}{{\overline{k}}_{L2}}+\frac{{\overline{l}}_f^2}{2}-\frac{{\overline{x}}^2}{2}\right).$$E17

Thus, the relative value ${\overline{s}}_2$ on the outer surface of the biofilm will be

$${\overline{s}}_{2f}\left({\overline{l}}_f,{\overline{S}}_2\right)={\overline{S}}_2-\left[{\overline{\lambda }}_2{u}_{\mathit{av}}\left({\overline{l}}_f,{\overline{S}}_2\right)-{\tilde{\lambda }}_1\right]\frac{{\overline{l}}_f}{{\overline{k}}_{L2}}.$$E18

Using the representations for ${\overline{s}}_1$ Eq. (12) and ${\overline{s}}_2$ Eq. (17) with the help of double integrating Eqs. (3) and (4) under the appropriate conditions Eq. (6) it is also easy to find the concentrations ${\overline{s}}_3$, ${\overline{s}}_4$ of the main end products of decomposition $\left({\mathit{CO}}_2,{\mathit{CH}}_4\right)$.

To assess the actual active capacity of the biofilm in relation to both components of organic pollution allows the calculation of their relative rates${\overline{i}}_{f1}$, ${\overline{i}}_{f2}$ across the boundary between both films

$${\overline{i}}_{f1}\left({\overline{l}}_f\right)={\tilde{\lambda }}_1{\overline{l}}_f,$$E19

$${\overline{i}}_{f2}\left({\overline{l}}_f,{\overline{S}}_2\right)={\overline{l}}_f\left[{\overline{\lambda }}_2{u}_{\mathit{av}}\left({\overline{l}}_f,{\overline{S}}_2\right)-{\tilde{\lambda }}_1\right].$$E20

At the second stage of theoretical studies of steady-state anaerobic biofiltration, attention is focused on the behavior of the initial and newly formed (secondary) organic substrates on the scale of the section selected for consideration. The basis of the corresponding mathematical model (the biofiltration compartment in the general model) is the system of equations for the convective transfer of both substrates within the given section. Neglecting the diffusion (dispersion) mechanism, the system of macrotransfer equations can be written in the dimensional form

$$V\frac{{\mathit{dS}}_1}{\mathit{dy}}+I_{B1}\left(l_f\right)=0,$$E21

$$V\frac{{\mathit{dS}}_2}{\mathit{dy}}+I_{B2}\left(l_f,S_2\right)=0,$$E22

where $y$ is the coordinate in the direction of water movement. Here, the functions of utilization of primary $I_{B1}\left(l_f\right)$ and secondary $I_{B1}\left(l_f\right)$ substrates at the specific surface area of the biological phase ${\mathrm{\Omega }}_B$ are presented, for example, in the following form.

$$I_j={\mathrm{\Omega }}_B{D}_{\mathit{ej}}\frac{{\mathit{ds}}_j}{\mathit{dx}}\left|{}_{x=l_j};\right.j=1,2$$E23

In the analyzed case of bioconveyer technology and the anaerobic section with biomass at the fibers of a radius $R_t$, the area ${\mathrm{\Omega }}_B$ is expressed in terms of the fraction of space free from them $n_0$ (analogous to porosity for porous media) as follows

$${\mathrm{\Omega }}_B=2\left(1-n_0\right)/R_t.$$E24

A similar expression as applied to a medium of grains with a radius $R_g$ will be

$${\mathrm{\Omega }}_B=3\left(1-n_0\right)/R_g.$$E25

Equations (21) and (22) are supplemented with the boundary conditions.

$$y=0,S_j=S_{j0}.$$E26

To establish a relationship between the relative thickness ${\overline{l}}_f$ and concentration of the total biomass $B$ as the sum of $B_1$ (acid-producing bacteria and their metabolites) and $B_2$ (methane-producing bacteria and their metabolites), the generalized balance equation between its increase and decrease is used

$${\mathrm{{\rm Y}}}_{B_1/s_1}{I}_1+{\mathrm{{\rm Y}}}_{B_2/s_2}{I}_2=k_d\left(B_1+B_2\right)=k_{d}B.$$E27

Equation (27) are transformed after a series of transformations to this equality in the dimensionless form

$$u_{\mathit{av}}\left({\overline{l}}_f,{\overline{S}}_2\right)={\psi }_B=\frac{1}{{\overline{\lambda }}_2}\left[{\overline{k}}_{d2}\left({\overline{\rho }}_B+1\right)-\frac{{\overline{\lambda }}_1}{\overline{D}}\left(\frac{{\mathrm{{\rm Y}}}_{B_1/s_1}}{{\mathrm{{\rm Y}}}_{B_2/s_2}}-{\mathrm{{\rm Y}}}_1{\mathrm{{\rm Y}}}_{s_2/B_1}\right)\right],$$E28

where ${\overline{k}}_{d2}=k_d{\rho }_{B2}{\mathrm{{\rm Y}}}_{s_2/B_2}{R}_t^2/\left(D_{e2}{S}_{10}\right)$, ${\overline{\rho }}_B={\rho }_{B1}/{\rho }_{B2}$. An expression is derived from Eq. (28) for the concentration ${\overline{S}}_2$ as a function of ${\overline{l}}_f$ taking into account Eq. (16)

$${\overline{S}}_2\left({\overline{l}}_f\right)=\frac{{\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1}{{\overline{\lambda }}_2}{\varphi }_f\left({\overline{l}}_f\right)+\frac{{\psi }_B{\overline{K}}_{s2}}{1-{\psi }_B}.$$E29

Of fundamental importance for modeling biofiltration is the assumption of a weak dependence of the rate of biomass loss on the concentration of dissolved organic matter, which allows us to assume $k_d=k_d\left(l_f\right)$. From a formal point of view, it is much easier when describing the utilization of substrates in the bioreactor medium to operate instead of $I_j$using the equivalent expressions from the right side of Eq. (27). Then system Eqs. (21) and (22) takes the following form

$$\frac{d{\overline{S}}_1}{d\overline{y}}+{\chi }_t{\overline{k}}_{d1}{\overline{l}}_f=0,$$E30

$$\frac{d{\overline{S}}_2}{d\overline{y}}+{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_f=0,$$E31

where ${\chi }_t=2\left(1-n_0\right)\frac{D_{e1}L}{{\mathit{VR}}_t^2}$ (for fiber-nozzles), ${\overline{k}}_{d1}=\frac{k_d{\rho }_{B1}{R}_t^2}{{\mathrm{{\rm Y}}}_{s_1/B_1}{D}_{e1}{S}_{10}}$.

The subsequent analysis by analytical methods of the general stabilized situation in the reactor medium and the choice of an appropriate calculation scheme are determined by the degree of saturation with biomass of the space between the fibers. Since the behavior of the secondary substrate is much more difficult to formalize, therefore, the attention will be given specially to it.

The determination of the relative thickness of biofilms in the inlet section of the medium $l_{f0}$ is of key importance for concretizing the situation. Here it is possible to express ${\overline{l}}_f$ through ${\overline{S}}_{20}$ and finally derive the formula

$${\overline{l}}_{f0}=\sqrt{\frac{{\overline{D}}^2}{{\overline{k}}_L^2}+\frac{2}{{\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1}\left({\overline{S}}_{20}-\frac{{\psi }_B{\overline{K}}_{s2}}{1-{\psi }_B}\right)}-\frac{\overline{D}}{{\overline{k}}_L}.$$E32

Then ${\overline{l}}_{f0}$ is correlated with the limit value ${\overline{l}}_{\mathit{fm}}$. The first two more realistic situations turn out if $l_{f0}>l_{\mathit{fm}}$. Equations (30) and (31) can be easily integrated in this case taking into consideration the boundary conditions.

$$\overline{y}=0,{\overline{S}}_1=1;{\overline{S}}_2={\overline{S}}_{20}$$E33

As a result, the following linear representations are obtained for the desired concentrations

$${\overline{S}}_1\left(\overline{y}\right)=1-{\chi }_t{\overline{k}}_{d1}{\overline{l}}_{\mathit{fm}}\overline{y},$$E34

$${\overline{S}}_2\left(\overline{y}\right)={\overline{S}}_{20}-{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_{\mathit{fm}}\overline{y}.$$E35

The next computational step is to calculate the value ${\overline{S}}_{2m}$(using Eq. (29) for given values ${\overline{l}}_{\mathit{fm}}$, ${\overline{S}}_{20}$) related to

$${\overline{S}}_{2m}=\left({\psi }_B-\frac{{\tilde{\lambda }}_1}{{\overline{\lambda }}_2}\right){\varphi }_f\left({\overline{l}}_{\mathit{fm}}\right)+\frac{{\psi }_B{\overline{K}}_{s2}}{1-{\psi }_B}.$$E36

Then the coordinate ${\overline{y}}_m$ of the section, in which ${\overline{S}}_2={\overline{S}}_{2m}$, is calculated by the formula following from Eq. (35),

$${\overline{y}}_m=\frac{{\overline{S}}_{20}-{\overline{S}}_{2m}}{{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_{\mathit{fm}}}.$$E37

Two situations are possible depending on the ratios ${\overline{y}}_m$≷1. First, at ${\overline{y}}_m>1$ (the entire medium contains the maximum amount of biomass) the most important for practice output concentrations ${\overline{S}}_{\mathit{ie}}$ are simply calculated from (34) and (35).

$${\overline{S}}_{\mathit{ie}}=1-{\chi }_t{\overline{k}}_{d1}{\overline{l}}_{\mathit{fm}},{\overline{S}}_{2e}={\overline{S}}_{20}-{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_{\mathit{fm}}$$E38

Calculations of anaerobic biofiltration are much more difficult if $1>{\overline{y}}_m>0$. Then two characteristic zones are formed in the medium, where, respectively, $l_f=l_{\mathit{fm}}$ and $l_f<l_{\mathit{fm}}$. Within the first zone, as before, linear distributions Eqs. (34) and (35) are valid up to the section $y=y_m$. In the second zone$\left(1\ge y\ge y_m\right)$, these distributions are already non-linear due to decreasing $l_f$ along the flow. The corresponding distribution ${\overline{l}}_f\left(\overline{y}\right)$ is found from Eq. (31), which is transformed to this form

$$\frac{d}{d{\overline{l}}_f}{\overline{S}}_2\left({\overline{l}}_f\right)\cdot \frac{d{\overline{l}}_f}{d\overline{y}}+{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_f=0$$E39

and solved under the condition.

$$\overline{y}={\overline{y}}_m,{\overline{l}}_f={\overline{l}}_{\mathit{fm}}.$$E40

Based on Eq. (31), an expression for $d{\overline{S}}_2/d{\overline{l}}_f$ is derived and then a solution to problem Eqs. (39) and (40) is obtained in the form of an inverse function

$$\overline{y}-{\overline{y}}_m=\frac{1}{{\chi }_t{\overline{D}\overline{k}}_{d2}}\left[\frac{{\overline{S}}_2\left({\overline{l}}_{\mathit{fm}}\right)}{{\overline{l}}_{\mathit{fm}}}-\frac{{\overline{S}}_2\left({\overline{l}}_f\right)}{{\overline{l}}_f}+\underset{{\overline{l}}_f}{\overset{{\overline{l}}_{\mathit{fm}}}{\int }}\frac{{\overline{S}}_2\left(\zeta \right)}{{\zeta }^2}\mathit{d\zeta }\right].$$E41

Expression Eq. (41) can be simplified after integration taking into account Eq. (29) and some transformations, so that

$$\overline{y}-{\overline{y}}_m=\frac{1}{{\chi }_t{\overline{D}\overline{k}}_{d2}}\left[\left({\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1\right)\left({\overline{l}}_{\mathit{fm}}-{\overline{l}}_f\right)+\frac{\overline{D}}{{\overline{k}}_{L2}}ln\frac{{\overline{l}}_{\mathit{fm}}}{{\overline{l}}_f}\right].$$E42

Now, in order to determine the value ${\overline{S}}_2$ at any value $\overline{y}$ within the second zone, it is enough to attach Eqs. (29)–(42). Therefore, there is a parametric representation for ${\overline{S}}_2\left(\overline{y}\right)$ and the thickness ${\overline{l}}_f$is here the parameter, which decreases from ${\overline{l}}_{\mathit{fm}}$ to ${\overline{l}}_{\mathit{fe}}$. Calculations of ${\overline{S}}_2\left(\overline{y}\right)$ can be simplified by getting rid of ${\overline{l}}_f$ due to the use of the dependence ${\overline{l}}_f\left({\overline{S}}_2\right)$. This dependence has the form Eq. (32), where ${\overline{l}}_{f0}$ and ${\overline{S}}_{20}$ are formally replaced by ${\overline{l}}_f$ and ${\overline{S}}_2$. As a result, the desired distribution ${\overline{S}}_1\left(\overline{y}\right)$ is described by the inverse function $\overline{y}\left({\overline{S}}_2\right)$.

To ascertain the distribution ${\overline{S}}_1\left(\overline{y}\right)$ in the same zone, first of all, Eq. (30) is presented as follows

$$\frac{d{\overline{S}}_1}{d{\overline{l}}_f}+{\chi }_t{\overline{k}}_{d1}{\overline{l}}_f\frac{d}{d{\overline{l}}_f}\overline{y}\left({\overline{l}}_f\right)=0.$$E43

The corresponding boundary condition will be.

$${\overline{l}}_f={\overline{l}}_{\mathit{fm}},{\overline{S}}_1={\overline{S}}_{1m}=1-{\chi }_t{\overline{k}}_{d1}{\overline{l}}_{\mathit{fm}}{\overline{y}}_m.$$E44

Since according to Eq. (39)

$$\frac{d\overline{y}}{d{\overline{l}}_f}=-\frac{1}{{\chi }_t{\overline{D}\overline{k}}_{d2}{\overline{l}}_f}\frac{d{\overline{S}}_2}{d{\overline{l}}_f},$$E45

then integration of Eq. (43) taking into account Eqs. (44) and (45) gives

$${\overline{S}}_1\left({\overline{l}}_f\right)={\overline{S}}_{1m}+\frac{{\overline{k}}_{d1}}{{\overline{D}\overline{k}}_{d2}}\left[{\overline{S}}_2\left({\overline{l}}_f\right)-{\overline{S}}_{2m}\right].$$E46

Consequently, the relative concentration of dissolved organic matter at the outlet of the biofilter ${\overline{S}}_e$ will be

$${\overline{S}}_e={\overline{S}}_{1e}+{\overline{S}}_{2e}={\overline{S}}_{1m}-\frac{{\overline{k}}_{d1}}{{\overline{D}\overline{k}}_{d2}}{\overline{S}}_{2m}+\left(1+\frac{{\overline{k}}_{d1}}{{\overline{D}\overline{k}}_{d2}}\right){\overline{S}}_{2e},$$E47

where ${\overline{S}}_{2e}$is calculated by Eq. (29) with the value ${\overline{l}}_{\mathit{fe}}$ previously found by selection from the equation

$$\left({\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1\right)\left({\overline{l}}_{\mathit{fm}}-{\overline{l}}_{\mathit{fe}}\right)+\frac{\overline{D}}{{\overline{k}}_{L2}}ln\frac{{\overline{l}}_{\mathit{fm}}}{{\overline{l}}_{\mathit{fe}}}={\chi }_t{\overline{D}\overline{k}}_{d2}\left(1-{\overline{y}}_m\right).$$E48

The third situation with its characteristic relation ${\overline{l}}_{f0}\le {\overline{l}}_{\mathit{fm}}$ is largely similar to the second situation $\left(1>y_m>0\right)$. Its calculation with minimal differences duplicates the computational procedure described above. Since here there is no zone of maximum saturation with biomass at all, then ${\overline{l}}_f$ starting with the value ${\overline{l}}_{f0}$ becomes smaller according to Eq. (32) as the distance from the inlet section increases. With known ${\overline{l}}_{f0}$ are sequentially calculated:

biofilm thickness at the biofilter outlet from the equation

$$\left({\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1\right)\left({\overline{l}}_{f0}-{\overline{l}}_{\mathit{fe}}\right)+\frac{\overline{D}}{{\overline{k}}_{L2}}ln\frac{{\overline{l}}_{f0}}{{\overline{l}}_{\mathit{fe}}}={\chi }_t{\overline{D}\overline{k}}_{d2},$$E49

the output concentration of the secondary substrate

$${\overline{S}}_{2e}=\left({\overline{\lambda }}_2{\psi }_B-{\tilde{\lambda }}_1\right)\left(\frac{{\overline{l}}_{\mathit{fe}}^2}{2}+\frac{\overline{D}}{{\overline{k}}_{L2}}{\overline{l}}_{\mathit{fe}}\right)+\frac{{\psi }_B{\overline{K}}_{s2}}{1-{\psi }_B},$$E50

the output concentration of the primary substrate

$${\overline{S}}_{1e}=1-\frac{{\overline{k}}_{d1}}{{\overline{D}\overline{k}}_{d2}}\left({\overline{S}}_{20}-{\overline{S}}_{2e}\right).$$E51

Obviously, when polluted water passes through the section of DSMBT under study, it is realistic to reduce the concentration of dissolved organic matter by a relative value

$$\mathrm{\Delta }\overline{S}=1+{\overline{S}}_{20}-{\overline{S}}_{1e}-{\overline{S}}_{2e}.$$E52

2.2 Calculation of test examples and discussion of results

The approximate solutions obtained above for stationary internal (biofilm) and external (bioreactor medium) problems are illustrated by the test examples. Possible inaccuracies in the calculation of micro characteristics due to the use of adaptive averaging of the local function of the organic substrate utilization in relation to the aerobic biofilm were evaluated. It was found that they do not exceed a few percent and, as a rule, are noticeably smaller than the errors that occurred due to the experimental determination of the model coefficients. The relative flow rates of both substrates through the surface of the representative flat biofilm $\left(i_{f1},i_{f2}\right)$ were the subject of numerous calculations. They determine the active capacity of the elements of the biological phase (biofilms of any shape) in relation to organic pollution and underlie the modeling of anaerobic biofiltration. Calculations were performed using Eqs. (19) and (20) with a continuous change in the relative thickness ${\overline{l}}_f$ from 0 to 0.2. Thus, the range of its real values was covered with a large margin. The initial content of volatile acids was also discretely varied from 0 to 0.5. In the adopted model, a stable presence of volatile acids is actually allowed already at the inlet to the bioreactor $\left(S_{20}>0\right)$. In practice, such a situation is typical, along with DSMBT also for sequentially operating second and subsequent anaerobic bioreactors (bank I). The initial information included the following fixed relative values of the coefficients: ${\overline{K}}_{s2}=0.25,{\overline{k}}_{L2}=10,{\overline{\rho }}_B=\overline{D}=1,\chi =0.4,{\overline{\lambda }}_1=10$. Also, two characteristic values (10 and 20) were chosen for ${\tilde{\lambda }}_1$. Graphs of the dependence ${\overline{i}}_{f2}\left({\overline{l}}_f\right)$ for ${\tilde{\lambda }}_1=10$ are presented in Figure 2 and for ${\tilde{\lambda }}_1=20$ - in Figure 3. Here, the only graph for ${\overline{i}}_{f1}\left({\overline{l}}_f\right)$ is given due to the constancy of ${\tilde{\lambda }}_1$.

Figure 2.

Dependence ${\overline{i}}_{f2}\left({\overline{l}}_f\right)$: 1 – ${\overline{S}}_{20}=0$, 2 – ${\overline{S}}_{20}=0.1$, 3 – ${\overline{S}}_{20}=0.25$, 4 – ${\overline{S}}_{20}=0.5$.

Figure 3.

Dependences: ${\overline{i}}_{f1}\left({\overline{l}}_f\right)$, ${\overline{i}}_{f2}\left({\overline{l}}_f\right)$: 1 – ${\overline{i}}_{f1}$, 2-5 – ${\overline{i}}_{f2}$; 2 – ${\overline{S}}_{20}=0.5$, 3 – ${\overline{S}}_{20}=0.25$, 4 – ${\overline{S}}_{20}=0.1$, 5 – ${\overline{S}}_{20}=0$.

When determining the values ${\overline{i}}_{f1},{\overline{i}}_{f2}$, their sign is of fundamental importance, since it governs the direction of transfer of the corresponding substrate. The “+” sign means that the impurity moves inside the biofilm, and the “-” sign means - in the opposite direction. Obviously, the primary substrate is only consumed by the biofilm, and therefore the consumption rate ${\overline{i}}_{f1}$ is necessarily positive. The orientation of the secondary substrate is dictated by the ratio between its concentrations outside both films $S_2$ and at their common boundary $s_{2f}$. Thus, volatile acids will diffuse from the outside at $s_{2f}<S_2$ (Figure 2 and curve 2 in Figure 3), and ${\overline{i}}_{f2}$ will already be negative at $s_f>S_2$ (curves 3-5 in Figure 3).

Therefore, thanks to the solution to the problem of the action of a representative anaerobic biofilm, in essence, a theoretical basis has been developed for subsequent studies using analytical methods for the operation of purification plants for biological treatment under anaerobic conditions. Also, the derived dependencies can be used to specify the model coefficients at the microlevel.

In the second series of examples, the subject of calculations were relative macrocharacteristics – biofilm thickness, which can be interpreted as a reduced biomass, concentrations of primary and secondary substrates. A larger and less significant part of the initial information for the analysis of the technological process was common to all the examples mentioned and accepted only in a dimensionless form. The indicated information included: ${\overline{K}}_{s2}=0.1,$ ${\overline{k}}_{L2}=10,$ ${\overline{\rho }}_B=\overline{D}=1,{\chi }_t=0.4$. The coefficients ${\overline{\lambda }}_i,{\overline{k}}_{\mathit{di}}\left(i=1,2\right)$, controlling the amount of the biological phase in the bioreactor medium, varied continuously $\left({\overline{k}}_{\mathit{di}}\right)$ or discretely. In order to reduce the amount of calculations, it was assumed that ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$, ${\overline{k}}_{d1}$ and ${\overline{k}}_{d2}$ are identical. In parallel, two options for saturation of this medium with biomass were considered. Formally, they differ in the ratio between the maximum (at the inlet to the medium) and the ultimate thicknesses. The ratio $l_{\mathit{fm}}>l_{f0}$ is true for the first and here the main option. Therefore, it is real to increase the biomass when creating more comfortable conditions for it everywhere in the medium, with the possible exception of the inlet cross-section. In the second option, a strict restriction is imposed on the growth of biomass, expressed by the ratio $l_f\le l_{\mathit{fm}}=0.1$, at the inlet cross-section of the medium. It is obvious that the determination of the value $l_{f0}$ according to Eq. (32) becomes decisive for the choice of the calculation procedure for a known value $l_{\mathit{fm}}$. Figure 4 presents the results of the corresponding calculations for four characteristic values ${\overline{\lambda }}_i$. Here, a change of $l_{f0}$ with a large margin from 0 to 0.4 was allowed. In fact, the value 0.4 is comparable to the porosity, for example, of the granular media. Thus, the value ${\overline{l}}_{f0}$ cannot in principle exceed the value of ${\overline{l}}_{\mathit{fm}}$, which is usually significantly less than 0.4. Nevertheless, as will be shown below, the data on ${\overline{l}}_{f0}$ can be useful not only for choosing a calculation variant, but also for technological calculations. A high sensitivity of ${\overline{l}}_{f0}$ with respect to biokinetic parameters, but especially to ${\overline{k}}_{\mathit{di}}$, is obvious from Figure 4. In fact, small changes in ${\overline{k}}_{\mathit{di}}$ cause incomparably large changes in ${\overline{l}}_{f0}$. It is natural that an increase in the rate of biomass loss leads to a thinning of the biofilm. Based on Eq. (32), it is easy to indicate such a relationship between the model coefficients, in which the biomass is not able to accumulate at all. It is described by the equation ${\psi }_B={\overline{S}}_{20}/\left({\overline{S}}_{20}+{\overline{K}}_{s2}\right)$. On the contrary, with a decrease in ${\overline{k}}_{d2}$ and, thus, tending of ${\overline{\lambda }}_2{\psi }_B$ to ${\tilde{\lambda }}_1$ ${\overline{l}}_{f0}$ will grow indefinitely.

Figure 4.

Dependence ${\overline{l}}_{f0}\left({\overline{k}}_{\mathit{di}}\right)$: 1– ${\overline{\lambda }}_i=10$, 2 – ${\overline{\lambda }}_i=20$, 3 – ${\overline{\lambda }}_i=30$, 4 – ${\overline{\lambda }}_i=40$.

It is important to note that the biomass is distributed extremely unevenly along the section with the exception of the ultimate saturation zone $\left(1\ge \overline{y}\ge {\overline{y}}_m\right)$ when the selected initial data is used. This, in particular, is evidenced by Figure 5, which shows the profiles calculated by (42) for ${\overline{\lambda }}_i=30,k_{\mathit{di}}=17$ and 18. The calculations were performed in parallel for the two above-mentioned options. It is natural that the nature of the biomass distribution for them differed significantly. In the case of restrained biomass growth, a significant amount of it is at a greater distance from the inlet section than in the case of unlimited growth (combined graphs 3, 4 and 3, 5). The utilization of dissolved organics should inevitably occur with less intensity within the ultimate saturation zone, but more intense outside. In order to assess the consequences of severe limitation of biomass growth for the efficiency of biofiltration process, the distributed concentrations of the initial and newly formed substrates were calculated. First of all, the distribution functions ${\overline{S}}_i\left(\overline{y}\right)$ were found for the case $l_{\mathit{fvm}}>l_{f0}$ with ${\overline{\lambda }}_i=20$ and three values of ${\overline{k}}_{\mathit{di}}$. The corresponding curves calculated by Eqs. (29) and (46) are shown in Figure 6. As noted earlier, a decrease in the rate of biomass loss contributes to its greater concentration, so that both substrates degrade more strongly throughout the entire bioreactor medium with the stable functioning of the microbiocenosis.

Figure 5.

Change in relative thickness of biofilms along bioreactor medium: 1, 2 – ${\overline{l}}_{\mathit{fm}}>l_{f0}$; 3-5 – ${\overline{l}}_{f0}>l_{\mathit{fm}}$; 1, 3 и 5 – ${\overline{k}}_{\mathit{di}}=17$; 2, 3 и 4 – ${\overline{k}}_{\mathit{di}}=18$.

Figure 6.

Decrease in concentrations of primary and secondary substrates along bioreactor: 1-3 – ${\overline{S}}_1$, 4-6 – ${\overline{S}}_2$; 1, 4 – ${\overline{k}}_{\mathit{di}}=13$; 2, 5 – ${\overline{k}}_{\mathit{di}}=12$; 3, 6 – ${\overline{k}}_{\mathit{di}}=11$.

It is obvious that any redistribution of biomass along the bioreactor should be reflected in the appropriate way in the shape of the profiles ${\overline{S}}_i\left(\overline{y}\right)$. Figure 7 indicates the significance of such changes in the two options under consideration, but only within the section (bioreactor). Indeed, the pairs of the calculated curves (1 and 4, 2 and 3) corresponding to ${\overline{\lambda }}_i=40$ and two values of ${\overline{k}}_{\mathit{di}}$ diverge significantly in the area $1\ge \overline{y}\ge {\overline{y}}_m$, but they quickly converge with a subsequent increase in $\overline{y}$. Moreover, the calculation of the content of both the primary and the secondary substrates in the filtrate gives almost the same results with and without taking into account the restriction on biomass growth. Thus, it is permissible to determine the output concentrations of organic pollution components, ignoring the inaccessibility of a part of the pore space for the biological phase.

Figure 7.

Decrease in concentration of primary substrate along bioreactor medium: 1, 2 – ${\overline{l}}_{f0}>l_{\mathit{fm}}$; 3, 4 – ${\overline{l}}_{f0}<l_{\mathit{fm}}$; 1, 4 – ${\overline{k}}_{\mathit{di}}=21$; 2, 3 – ${\overline{k}}_{\mathit{di}}=23$.

In conclusion, changes in the residual content of dissolved organic matter due to variation in the loss of biomass on account of the detachment, respiration and grazing were analyzed. The relative value $\mathrm{\Delta }\overline{S}$ was finally calculated by Eq. (52) for three values of ${\overline{\lambda }}_i$. In this case, the range of the calculations for ${\overline{k}}_{\mathit{di}}$ was from 10 to 30. The corresponding set of graphs of the dependence $\mathrm{\Delta }\overline{S}\left({\overline{k}}_{\mathit{di}}\right)$ is shown in Figure 8. Naturally, the calculated curves are limited from above by the total initial value 1.35. Once $\mathrm{\Delta }\overline{S}$ becomes formal equal to this value it means the absence of active biomass in the section of DSMBT under consideration and, as a result, the complete incapacity of the latter. With a decrease in ${\overline{k}}_{\mathit{di}}$, the thickness ${\overline{l}}_f$ increases, verging towards ${\overline{l}}_{\mathit{fm}}$. Evidently, the best quality of water treatment here is achieved if ${\overline{l}}_f$ becomes equal to ${\overline{l}}_{\mathit{fm}}$ or, in other words, the entire medium will contain the maximum possible amount of active biomass. Then the corresponding maximum values of the output concentrations ${\overline{S}}_{1e}$ and ${\overline{S}}_{2e}$ are simply calculated by Eqs. (14) and (15) and as a result

Figure 8.

Dependence $\mathrm{\Delta }\overline{S}\left({\overline{k}}_{\mathit{di}}\right)$: 1 – ${\overline{\lambda }}_i=20$, 2 – ${\overline{\lambda }}_i=30$, 3 – ${\overline{\lambda }}_i=40$.

$$\mathrm{\Delta }{\overline{S}}_{\max }={\chi }_t{\overline{l}}_{\mathit{fm}}\left({\overline{k}}_{d1}+{\overline{D}\overline{k}}_{d2}\right).$$E53

3. Conclusions

Summing up, it should be stated that a method for engineering calculation of anaerobic biofiltration has been developed. It can be successfully applied in the case of any porous media with a regular structure. Here it is specified in relation to a biological treatment plant of a special design with a pseudo-porous medium. The method makes it possible to reliably predict the stable effect of any anaerobic biofilm, the concentration of biomass in the bioreactor medium, its permeability, and, most importantly, the concentration of organic pollution in the filtrate. It can serve as a basis for substantiating rational technological and design parameters of DSMBT. It is also easy to use the method for granular media with minimal adjustments to the calculated dependencies. This development is based on a number of assumptions that correspond to real situations with biofiltration under anaerobic conditions. Further improvement of this method is possible taking into consideration in addition:

• limitation of the decomposition rate of the primary substrate,

• inhibition of the decomposition of the secondary substrate,

• the influence of the acid–base status of the aquatic environment,

• isolation and special consideration of the stage of biooxidation,

• influence of biomass content on the rate of its loss,

• complex composition of the primary substrate.