Determination of Logistic Model Parameters From AM2 Model for Methane Production for Batch Bioreactors

of Logistic Model Parameters From AM2 Model for Methane Production for Batch Bioreactors. ABSTRACT Modeling the estimation of cumulated methane production generated from batch anaerobic bioreactors is of paramount importance. In this context, there are two main modeling approaches. The first approach is based on developing mathematical expressions representing the processes involved in the bioreactors. The AM1 model is known as the most complete one. However, it is a complicated model as it requires about 80 parameters to be tuned. A model named AM2 which is a simplified version of AM1 has also been developed. It is based on only two microbial growth processes and requires only 13 parameters. Nevertheless, both AM1 and AM2 models do not provide explicit mathematical expressions that enable to estimate the temporal evolution of methane production with respect to the parameters involved in the considered models. These models are much more useful for simulations and graphical visualizations of the dynamical behavior of the state variables including methane production. On the other hand, the second approach suggests semi-empirical or data-driven models which are based on simple explicit mathematical expressions that provide an estimate of the cumulated methane production (Logistic model, Gompertz model, etc.). But, this type of models require the identification of few parameters which are extracted from experimental data. Usually, they are simplistic and use only one growth process and thus cannot exhibit the influence of the many parameters involved in such complex dynamic biotechnological systems. In this paper, to overcome the complexity of the first type of models and to avoid the dependency on experimental data in the second type; an explicit analytical mathematical expression is proposed for estimating the cumulated methane production for batch anaerobic bioreactors. This analytical expression is derived via the adoption of some appropriate approximations performed on the set of differential equations characterizing the AM2 model. Therefore, the proposed analytical expression can be considered as an approximation of the AM2 model itself and this is the first contribution of this paper. Graphical profiles of the cumulated methane production are presented showing that of the proposed logistic expression and that of the AM2 model considered as a reference. To the best of the author’s knowledge, no such approach and result have been encountered in the literature. On the other hand, this expression resembles formally to the semi-empiric logistic model. However, the equivalent parameters of the proposed expression as self-defined by the parameters of the AM2 model and do not require experimental data to be identified as it is for the semi-empiric logistic model. Moreover, by comparing the proposed logistic model to the semi-empiric logistic model, an identification of the parameters of the semi-empiric model is linked to the parameters of the AM2 model, providing more insight into the methane production. This can be considered as a second contribution of this paper.

In this paper, to overcome the complexity of the first type of models and to avoid the dependency on experimental data in the second type; an explicit analytical mathematical expression is proposed for estimating the cumulated methane production for batch anaerobic bioreactors. This analytical expression is derived via the adoption of some appropriate approximations performed on the set of differential equations characterizing the AM2 model. Therefore, the proposed analytical expression can be considered as an approximation of the AM2 model itself and this is the first contribution of this paper. Graphical profiles of the cumulated methane production are presented showing that of the proposed logistic expression and that of the AM2 model considered as a reference. To the best of the author's knowledge, no such approach and result have been encountered in the literature.

INTRODUCTION
Since the beginning of the 21st century, sustainable development is considered as more and more affected by the climate change. One severe factor that influences negatively the environment and the climate change is the misuse of energy generation policy which is responsible of the emission of large amounts of greenhouse gas caused by the combustion of conventional fossil fuels. Nowadays, it is firmly admitted that renewable energy sources have the potential to reduce emissions of greenhouse gas when replacing fossil fuels and thereby to mitigate climate change. In this context, bioenergy systems can contribute to climate change mitigation if they replace traditional fossil fuel use (Pawlita-Posmik et al., 2018;Sawin and Sverrisson, 2016).
Biogas can be produced by anaerobic digestion of almost every wet organic feedstock such as animal waste, crop residues, domestic food, industrial wastewater, municipal sewage sludge, etc. Biogas production has been growing steadily in recent years and has made its contribution to renewable energy generation. As a consequence, it reduced negative impacts on the environment, both in the form of greenhouse gas emissions and the pollution of soil and water courses (Wellinger et al., 2013). Anaerobic digestion and biogas technologies contribute also through the reduction of harmful methane emissions from food and farming wastes, providing energy and food security, improving waste management and sanitation, and reducing poverty and hunger (Jain, 2016;Baeyens, Appels, Peng and Dewil, 2016;Wall, Dumont and Murphy, 2018). Moreover, methane gas is also an option for storage of renewable power. It is much easier to handle and safer than hydrogen gas. Many techniques and optimization methods are used to improve the efficiency of anaerobic bioreactors in regard to the expected outputs: gaz production, purification of water and digestates (Mutz, Hengevoss, Hugi and Gross, 2017). The use of membrane in bioreactors for wastewater treatment is a technology that is recently gaining more importance (Sepehri and Sarrafzadeh, 2018).
Thereby, it is worth to have means for predicting and evaluating the methane production produced by anaerobic bioreactors. In this field of research, there are two main approaches (Noll and Henke, 2020).
The first approach is based on mathematical expressions representing the processes involved in the bioreactors. The earlier mathematical models of anaerobic bioreactors have been proposed in the 1970s (Noykova et al., 2002;Yu et al., 2013;Zaher et al., 2009). Depending on the number of biochemical processes considered, there are more or less complex models that have been proposed. The ADM1 model (Anaerobic Digestion Model No. 1) is the most complete model to simulate anaerobic reactors which takes into account most processes involved in bioreactors (Ozgun, 2019;Vavilin et al., 2000). However, this model is very complex because it includes biochemical and physicochemical processes and requires more than 80 parameters to be tuned. The need for more practical models has led to develop the AM2 model which can be considered as a light version of ADM1. It corresponds to the process of anaerobic digestion involving only two phases. Many authors have used AM2 model to study batch and continuous anaerobic digesters with different substrates in various conditions (Noykova et al., 2002;Vavilin, 2000;Zaher et al., 2009). From a mathematical point of view, the AM2 model corresponds to a set of coupled ordinary differential equations of the first order with nonlinear left-hand sides. Because of this nonlinearity, direct analytical solutions about state variables are not easy to determine. Therefore, the solutions are mainly obtained numerically for simulating and visualizing graphically the evolution of the state variables.
Nevertheless, both AM1 and AM2 models do not provide explicit mathematical expressions that enable to estimate the temporal evolution of methane production with respect to the parameters and initial conditions of the considered processes. These models are much more useful for simulations and visualizations of the dynamical behavior of the state variables. In addition, the prediction of methane production in anaerobic reactors by means of mathematical models such as AM1 and AM2 can have difficulties to fit the experimental measures because of process complexity and sensitivity to the parameters and conditions of experiments.
For these reasons, many empirical or data-driven models have been developed constituting a second approach. They attempt to approximate the methane production by proposing simple formulas to capture the dynamical profile of the methane production. They have adopted various methods: enzymatic, chemical, kinetics, statistical and microbiological growth (Ali et al., 2018;Dewil et al., 2011;Dittmer et al., 2021;Simonov, 2009;Zhang, 2020;Zwienstituting et al., 1990). These models attempt to provide an accurate estimate of the cumulated methane production. One can cite the Transfer Function, Logistic function, Gompertz function, etc which contain some parameters that require identification and adjustment based on experimental data analysis (Parra-Orobio et al., 2017). Usually, this type of models use only one equivalent growth process which is too simplistic and cannot reflect the influence of the many parameters involved in such complex dynamical systems. Among many models, the logistic one is largely used for methane production in anaerobic bioreactors. It provides a mathematical expression that is used for cumulated methane prediction. It requires identification by experimental data analysis of three parameters which have not necessarily biological meanings. The logistic expression has been artificially modified in order to give biological meanings to its three parameters (Crescenzo and Paraggio, 2019;Gerber and Span, 2008;Lemon et al., 2006;Pererva et al., 2020;Van et al., 2018).
This paper intends to present essentially two main contributions.
The first one consists of proposing an explicit analytical mathematical expression for estimating the dynamical cumulated methane production for batch anaerobic bioreactors. This expression is derived via some appropriate approximations applied on the set of differential equations which are characterizing the AM2 model. This explicit function is in the form of a logistic one and depends on the parameters of the AM2 model. Thus, it can be considered as an approximation of the AM2 model for estimating the cumulated methane production. A graphical illustration of both the cumulated methane production obtained by simulation of the AM2 model and of the proposed explicit logistic function is presented enabling to evaluate qualitatively and quantitatively the approximations adopted. The result obtained by simulation is considered as a reference curve.
The second contribution comes from the fact that the proposed function resembles formally to the empiric logistic model which requires the experimental identification of 3 parameters. Thus, by comparing the parameters of the empiric logistic model and its modified version to the proposed one, a link is established between their parameters. Thus, the parameters which have to be identified via experimental data can be expressed in terms of the AM2 model parameters. This approach bridges the AM2 model with both the empirical and the theoretical logistic function ones providing more insight to the methane production.
To the best of the author's knowledge, no such approach or similar results have been encountered in literature concerning the estimation of cumulated methane production of batch anaerobic bioreactors. As a consequence, this proposed logistic function enables to overcome the complexity of the first type of models as well as the dependency on experimental data in the second type of models.
The paper is organized as follows: First section gives an introduction about the issue under consideration. The second section presents the AM2 model. The third section determines the concentration of the acidogenic bacteria population X1(t) which is required in order to determine the concentration of the methanogenic bacterial population X2(t). Section four establishes the general solution of X2(t) and its particular solution given an exponential approximation of X1(t). Section five reveals the formal resemblance between the logistic function and the inferred expression X2(t) that leads to predict the cumulated methane production. Last section concludes the paper.

EQUATIONS GOVERNING THE AM2 MODEL
The mathematical AM2 model is based on the laws of growth (Simonov et al., 2009). It involves the following dynamic variables: X1 is the concentration of the acidogenic bacteria population; X2 is the concentration of the methanogenic bacterial population; S1 is the concentration of the substrate of carbonaceous material and S2 is the substrate concentration of volatile fatty acids. For batch systems, the mathematical model is expressed in the form of a coupled differential equations of the first order system: For the growth process, the function of Monod µ1 for acidogens bacteria and the function of Haldane µ2 for methanogens bacteria are respectively: with 1 the maximal growth rate and KS1 the constant of half-saturation, and with 2 the maximal growth rate, KS2 the constant of saturation and K12, the constant of inhibition.
The flow of methane which is the end product depends directly on the growth of methanogenic bacteria population X2, according to the relation: The cumulative quantity C(t) of the produced methane can be estimated by: For the implementation, this model can be written in a state space form such as: with the state space vector: )) and initial conditions: There are nine parameters involved in this model (µ1m, KS1, µ2m, KS2, KI2, k1, k2, k3, k4). To solve mathematically this system of differential equations, we must also provide four initial conditions: S1(0), S2(0), X1(0) and X2(0).
This ODE system has been implemented by means of Euler's integration method and also has been solved by using the ODE function of Scilab software. The graphical results obtained by solving this system correspond to a simulation carried out with the following parameter values of the two phases processes ( Table 1). Table 2. ANALYTICAL APPROACH FOR S1(t) AND X1(t) Temporal Evolution of Substrate S1(t) To analyze the system of differential equations of AM2 model, consider the differential equations X1(t) et S1(t) corresponding to equations (1) and (3). It can be noticed that they are in fact decoupled from equations X2(t) and S2(t). Therefore, by combining (1) and (3), one can write the following equalities:

The initial condition values are shown in
by integrating (10), one gets the following linear relationship between S1(t) and X1(t) that is: where X10 and S10 are respectively the initial values of X1(t) and S1(t).
To explicit X1(t), let's replace µ1 in (10) by its expression in (1) so that one gets the relation:  Table 2. Initial conditions values S1(0) S2(0) X1(0) X2(0) 10 g/l 2 g/l 0.4 g/l 0.01 g/l replacing X1(t) by its expression obtained in (11), one gets the following first order non linear differential equation in terms of S1(t) with respect to the time independent variable: with: = 10 + 1 . 10 By separating the variables from both sides in (13), it can be rewritten in the following form: By introducing the initial conditions, one obtains the solution of (14) in an analytical inverse form t=f(S1): It is not possible to express S1(t) in a standard form. So, from (15), S1(t) can be easily computed and graphically represented by only generating values to S1 and computing consequently the corresponding time t.

Temporal Evolution of Bacteria X1(t)
The dynamic evolution of bacteria X1(t) can be derived from relation (11) as follows: Similarity to S1(t), X1(t) cannot be expressed in a direct analytical standard form but can be expressed in the inverse form: t=g(X1) as follows: The analytical expressions (15) and (17) include the model parameters (Ks1, u1m, k1) and the initial values (S10, X10). They enable to analyze the evolution of S1(t) and X1(t) as well as their sensitivity to the related model parameters. By varying the values of X1(t), the values of the corresponding time are computed and the function X1=g-1 (t) can be drawn. Graphics of X1(t) obtained by numerical simulation from the AM2 model (blue segments) and also from the analytical expression (17) (red circles) as implemented under Scilab software are presented in Figure 1. They appear superimposed and thus are similar.
Moreover, in order to obtain an approximate analytical expression of X1(t) derived from (17), one needs to examine this expression which is composed of three parts. A graphical representation of the parts composing (17) can reveal the combination that can provide the best approximation of X1(t). Let's consider the first part of (17) that is constituted of the following expression: This expression (17-a) is represented graphically by the blue + in Figure 1. This curve is very close to the theoretical curve drawn from (17) with red circles. The profiles of these two curves are almost exponential, therefore, (17-a) can be Figure 1. Analytical expression of 1 ( ) and its approximation considered as an approximate expression of (17). The second part of expression (17) that is left is: It corresponds to the curve that is approximately a line (in blue star) situated at the left side of Figure 1 and which can be neglected.

Behavior evolution of S1(t) and X1(t)
Although the equation of X1(t) and S1(t) are non-linear, there profiles present behaviors like systems whose dynamics are characterized by a transient response followed by a steady state response. So, to determine the steady state asymptotic value of X1(t) which is reached for large time values, it requires combining (3), (5), (11) and equaling to zero the first derivative of X1(t): This leads to the condition: − 1 . 1 + = 0 , which consequently leads to the asymptotic steady state value of acidogenic bacteria: X1(t) = 1 = 1 = 10 + 10 1 ≅ 1.17 / .
Similarly, from equation (11), we can determine the asymptotic steady state value for S1(t) which tends to zero. This means that the concentration of the substrate of carbonaceous material will be almost completely consumed at relatively large time durations.

Settling time for X1(t) and S1(t)
In such systems, it is important to determine the settling time tset which is defined as the time to reach the steady state value up to 5%. For X1(t), it corresponds to: X1(tset)=X1set=0.95*X1S = 1.11 g/l.
By replacing this settling value X1set in (12), the settling time ts is about 45 days. From equation (11), we can estimate the settling value for S1(t) which is about 50 days. The comparison of these results are in full agreement with those given by simulation of the system of differential equations composing AM2 model.

TEMPORAL EVOLUTION OF X2(t) Establishment of Differential Equations for X2(t)
From the previous equations (2) and (4), we can write: On the other hand, by using (1) and (2), equation (4) can be written in the following form: by integrating this last equation (19), we obtain: which can also be written as: with: ( ) = 2 . 1 ( ) + and = 20 − 2 10 + 3 . 20 By replacing S2(t) from (21) into the (18), we can explicit a nonlinear differential equation of the first order on X2(t) which depends on X1(t) via the term b(t): Under this form, the differential equation (22) cannot be solved to obtain an analytical expression. This is only possible in the case where X1(t) is independent from time which is not the case in our general problem since we know already the profile of X1(t) as given by (17).

Approximation of the Growth Process of Bacteria µ2(S2)
To obtain analytical solutions for X2(t) and S2(t), it is necessary to make some approximations that simplify the differential equations (21) and (22). That is the only way to make them amenable to expressions that are analytically integrable. To be solvable analytically, it needs again to be simplified to the level that the grown law of the methanogenic process of the bacterial population becomes linear of the form: By adopting these simplifications, (22) can be written as follows: Now, the differential equation (24) corresponds to Bernoulli's differential equation [Parker, 2020]. It can be rewritten to match the standard form of Bernoulli's differential equation for the particular case where the coefficient n=2; that is: Therefore, equation (24) structured in the standard Bernoulli's form becomes: And To solve (25), it is converted into another simpler differential equation of the first order that has a general solution (Parker, 2020): To explicit the general solution in the form of (29), it requires the provision of P(t) which means to provide the profile of X1(t). Here again, the problem of integrating this function is not obvious in standard analytical expressions. So, in this paper, we will use an exponential approximation for X1(t) as announced earlier.

Exponential Approximation of X1(t)
We approximate X1(t) by the exponential part expressed by (17-a) that is: The exponential approximation of X1(t) can be written in a form that corresponds to a Logistic Function as follows:

Determination of X2(t) with Exponential Approximation of X1(t)
To explicit X2(t) from (29) given the exponential approximation of X1(t), we need to define the functions Pe(t) and Qe(t) respectively from (26) and (27). In expression (26), we replace X1(t) par the exponential approximation as follows: it can be written in the form: Now, we can determine X2(t) from expression (29). The integral at the numerator can be computed: With some basic transformations, this expression can be written : and: ( ) = − 1. . ( 1 + 1 ) + ( 0 + 1 ).
The expression of the numerator is: The denominator can be expressed as follows: By taking into account the initial value of X2(0) = X20; it gives: c0=1/x20. Finally, the expression X2e(t) can be written as follows: In this last expression (34), the parameters (poe, p1e, q0, B) depend from the parameters of the AM2 model. Then, X2e(t) will depend on the values of these parameters. However, the integral on the denominator cannot be written in standard forms but can only be expressed with special functions such as the error function erf(x), the incomplete gamma function ɣ(x, z) and exponential integral function Ei(x). It is thus difficult to analyze X2e(t) including these special functions.

Determination of X2(t) with Exponential Approximation of X1(t)
It is difficult to analyze X2e(t) with these special functions. For this reason, in order to overcome these difficulties, we will consider an approximation that simplifies the expression (34). To this end, we will neglect the parts that vanishes for relatively large values of t that correspond to the steady state response.
so, consider the numerator (32), for relatively large t, the term p1.B.exp (-t/B) tends to vanish in the numerator and in the denominator. Therefore, the expression Ne(t) becomes: with this approximation De(t) becomes integrable and is: replacing this new results in (29) or (34), we obtain the final expression: which shows that the growth of methanogens bacteria under our approximations can be written in the form of a logistic function with parameters related to the AM2 model: with:

Determination of the Parameters of the Logistic Function
To evaluate the cumulated methane production from the AM2 model, we may use the expression (7) and (8). The cumulated quantity of methane over a given period t is proportional to X2(t).
If we consider our approximation, it can be written as: where k4 is the production parameter of methane for the AM2 model.
It can be rewritten after some elementary operations and arranged in the form of a logistic function: The empirical logistic function that is used for predicting the cumulative methane production is usually expressed as follows (Crescenzo and Paraggio., 2019; Gerber and Span., 2018;Lemon et al., 2006).
where V(t) is the methane volume in time t and (a,b,c) are three parameters that have to be determined via an analysis of a set of experimental data.
The expressions V(t) in (40) and V2e(t) in (41) become comparable. By comparing them, we can identify the parameters in both models. The empirical parameters expressed in terms of the approximate AM2 adopted model can be written in the form: in order to give a meaning to the parameters of the empirical logistic function, it is useful to use its modified version known as the modified logistic function. By some elementary mathematical transformations, the logistic function with (a, b, c) parameters can be transformed into the modified logistic function and expressed as: In (42), Vm(t) is the cumulated methane volume at time t. It requires the determination of three parameters via an analysis of the experimental data. They are: the maximum methane production (A) (mL), the duration of the lag phase (λ) (hours), and the maximum rate of methane production (vmax) (mL·h-1).
To make the approximate AM2 model V2e(t) comparable to the modified logistic function Vm(t), we transform again the expression (40) to resemble (42) as follows: By comparing (42) with (43), the parameters of the modified logistic function can be expressed in terms of the approximate AM2 model parameters: We notice that the parameters of the logistic function depend from the parameters of the AM2 model as well as from the initial conditions. Thus, the proposed approach relates the parameters of the logistic function with those of the AM2 model giving a biotechnological sense to them.
The lag time has to be positive, we can impose a condition that is: (1 + 1 ) − 1.

Comparison of Profiles AM2 Model to the Proposed Logistic Function
To analyze what the approximations adopted over the AM2 in order to get an analytical expression for the cumulated methane production, we draw the curve obtained by solving numerically the system of AM2 equations. It is the blue curve represented in Figure 2. The red curve corresponds to the approximate analytical expression V2e(t) based on AM2 model.

CONCLUSION
A mathematical expression formally comparable to the empirical logistic function is proposed for estimating the cumulative methane production for batch reactors. It is derived through some appropriate approximations performed on the set of differential equations characterizing the AM2 model.
The comparison of the empirical logistic function with the proposed expression enabled to express the three parameters of the empiric logistic function as well as its modified version in terms of the AM2 parameters.
The relations between the sets of both parameters reveals the complexity of the methane production processes. This approach bridges the mathematical AM2 model with the empirical one providing more insight in the methane production.
As perspectives, a study has to be carried out in order to analyze the influence of the parameters of the AM2 on the proposed logistic function. This study has also to compare the parameters of the empiric logistic function which are identified experimentally with those obtained by the proposed logistic function.

Funding:
No external funding is received for this article.

Declaration of interest: Author declares no competing interests.
Ethics approval and consent to participate: Not applicable.
Availability of data and materials: All data generated or analyzed during this study are available for sharing when appropriate request is directed to the author.