## 2 Materials and Methods  
### Mathematical Model of the Two-Stage Anaerobic Digestion Process  
The two-stage anaerobic digestion process of corn steep liquor for sequential production of H\({}_{2}\) and CH\({}_{4}\) is carried out in two separate bioreactors [2]. During the first stage, relatively fast-growing acidogenic microorganisms engaged in the production of volatile fatty acids and H\({}_{2}\) are cultivated in the hydrogenic bioreactor BR\({}_{1}\). Slow-growing acetogenic and methanogenic bacteria are developed during the second stage in the methanogenic bioreactor BR\({}_{2}\). The volatile fatty acids are later transformed into CH\({}_{4}\) and CO\({}_{2}\) in BR\({}_{2}\).  
The process dynamics in the cascade BR\({}_{1}\) and BR\({}_{2}\) are presented, using mass balance, by a set of five Ordinary Differential Equations (ODEs) and two algebraic equations, as follows:  
\[\text{BR}_{1}\text{:}\]  
\[\frac{dS_{1}}{dt}=-Y_{1}\mu_{1}X_{1}+D_{1}(S_{1 in}-S_{1}), \tag{1}\]  
\[\frac{dX_{1}}{dt}=\mu_{1}X_{1}-D_{1}X_{1}, \tag{2}\]\[\frac{dAc_{1}}{dt}=Y_{2}\mu_{1}X_{1}-D_{1}Ac_{1}, \tag{3}\]  
\[Q_{H_{2}}=Y_{H_{2}}\mu_{1}X_{1}, \tag{4}\]  
\[\mu_{1}=\frac{\mu_{1 max}S_{1}}{K_{S_{1}}+S_{1}}. \tag{5}\]  
\(\text{BR}_{2}\):  
\[\frac{dX_{2}}{dt}=\mu_{2}X_{2}-D_{2}X_{2}, \tag{6}\]  
\[\frac{dAc_{2}}{dt}=-Y_{3}\mu_{2}X_{2}+D_{2}(Ac_{1}-Ac_{2}), \tag{7}\]  
\[Q_{CH_{4}}=Y_{CH_{4}}\mu_{2}X_{2}, \tag{8}\]  
\[\mu_{2}=\frac{\mu_{2 max}Ac_{2}}{K_{S_{2}}+Ac_{2}}. \tag{9}\]  
The system (1)-(5) describes the dynamics of the substrate concentration (\(S_{1}\)), [g/L]; the microbial biomass concentration (\(X_{1}\)), [g/L]; the product (acetate) formation (\(Ac_{1}\)) [g/L] in \(\text{BR}_{1}\). The algebraic equation describes the flow rate of the hydrogen (\(Q_{H_{2}}\)) [dm\({}^{3}\)/L\(\cdot\)h] in the gas phase of \(\text{BR}_{1}\). \(S_{1 im}\) [g/L] is the concentration of the input substrate. For the specific growth rate \(\mu_{1}\) [day\({}^{-1}\)] of hydrogen-producing microorganisms, a Monod-type function was adopted. \(D_{1}\) [day\({}^{-1}\)] is the dilution rate for the first bioreactor \(\text{BR}_{1}\); \(\mu_{1 max}\) [day\({}^{-1}\)] and \(K_{S_{1}}\) [g/L] are Monod kinetic coefficients; and \(Y_{1}\), \(Y_{2}\) and \(Y_{H_{2}}\) are yield coefficients, [g/g].  
The system (6)-(8) describes a one-step transformation of the inlet acetate \(Ac_{1}\) (coming from \(\text{BR}_{1}\)) into methane by methanogenic microorganisms. A Monod type function was also adopted for the specific growth rate of the methanogenic biomass. In the model, \(X_{2}\) is the microbial biomass concentration in \(\text{BR}_{2}\) [g/L], \(Ac_{2}\) is the acetate concentration in \(\text{BR}_{2}\) [g/L], \(Q_{CH_{4}}\) is the methane flow rate [dm\({}^{3}\)/L\(\cdot\)day], \(D_{2}\) [day\({}^{-1}\)] is the dilution rate for the second bioreactor \(\text{BR}_{2}\), \(\mu_{2}\) is the specific growth rate (Monod type) of methanogens [day\({}^{-1}\)], \(Y_{CH_{4}}\) and \(Y_{3}\) are yield coefficients, and \(\mu_{2 max}\) [day\({}^{-1}\)] and \(K_{S_{2}}\) [g/L] are kinetic coefficients.  
The so presented model structure is used for the first time for modelling the corn steep liquor TSAD process considered here based on real experimental data.