## 3 Modelling Cumulative Methane Production
As a matter of example, a decrease in the value of the disintegration rate constant was reported by [PERSON] et al. [123] with the increase in solid content and particle size of the substrate. Any increase in temperature leads to a rise in the reaction rate, whereas reducing particle size increases the specific surface area making it more accessible to the enzyme attack. [PERSON] et al. [124] studied the effect of particle size using casein as model protein material. These authors found an increase in the hydrolysis constant from 0.034 to 0.298 1/d by decreasing this parameter. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Substrate** & \(k\) **(l/d)** & **Methane Yield (L/g VS)** & **Reference** \\ \hline \hline Cattle, pig manure & 0.106–0.149 & 0.217–0.287 & [125] \\ \hline Pig manure, swine & 0.213 & 0.202 & [86] \\ manure & 0.11 & 0.161 & [126] \\ \hline \hline  & 0.037–0.086 & 0.254–0.290 [1] & [127] \\  & 0.069–0.278 & - & [128] \\  & 0.082 & 0.239 & [129] \\  & 0.19 & 0.238 & [130] \\ \hline Chicken manure & 0.07–0.12 & 0.298–0.351 & [131] \\ \hline Grass & 0.107 & 0.400 & [129] \\ \hline Two-phase olive & & & \\ pomace & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of first-order decay constant available in the literature at mesophilic conditions. The composite structure of lignocellulosic materials translates into long degradation times, making inadequate the use of the first-order decay model to describe cumulative biogas evolution from batch tests. The slow degradation of cellulose and hemicellulose, contrary to what is observed in carbohydrates, makes the modified Gompertz model more suitable (Equation (2)). This model has demonstrated adequacy to evolved biogas data from substrates such as manures and agronomic wastes [138; 139]. \[B_{(t)}=B_{0}\cdot e^{-c\left(\frac{R_{max}\cdot e}{B_{0}}\lambda-t\right)+1} \tag{2}\]  
In this model, two additional parameters are introduced with regard to the first-order one. \(R_{max}\) represents the maximum methane production rate. \(\lambda\) represents the delay associated with the acclimation of the microflora to a different substrate and environment. The \(e\) number (2.718) is also used in this equation. The modified Gompertz model presents the advantage of fitting biogas evolution from complex substrates or those containing partial inhibitory compounds. Thus, \(\lambda\) values describe the delay in biogas production during the initial stage of the fermentation. Other models containing a lag phase have also been evaluated with good results, such as the transfer function model, logistic function, cone model and Richards' model [140; 141; 142]. Diauxic metabolism has also been considered appropriate for the description of cumulative biogas curves using the modified Gompertz and the logistic model for describing each sequential degradation stage [143; 144]. Table 3 shows a list of different values reported in the literature for \(\lambda\) and \(R_{max}\). Values of \(\lambda\) are usually in the range of 1.5-9.4 days [145; 146]. This parameter, as well as the maximum production rate, is affected by inhibitory substances either already present in the substrate or produced during the fermentation, as would be the accumulation of VFAs, long chain fatty acids and ammonia. [PERSON] et al. [147] showed that an increase in the lipid fraction during anaerobic co-digestion of slaughter wastewater caused an increase in the lag phase. Similar results were also reported by [PERSON] et al. [148] when evaluating the co-digestion of manure, slaughter wastes and glycerol. Increasing the content of glycerol in the mixture also caused a greater delay due to the fast initial conversion and accumulation of propionic acid.