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.
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