## 2 Material and Methods
### Kinetic Modeling  
Biogas production kinetics was modeled by using three models, (a) modified Gompertz model, (b) logistic function model, (c) transference function model to estimate performance parameters, and (d) first-order kinetics model was used for determination of bio-kinetic parameter. Simulation of experimental data for the determination of the best fit model was performed by using SPSS (IBM SPSS statistics 20), and graphical representation was made by using Origin 8.0 software.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Wheat Straw** & **Rice Straw** & **Bagasse** & **Inoculum** \\ \hline
**Total Solids \%** & 99 & 98.5 & 98 & 85 \\
**Volatile Solids \%** & 83.33 & 62.42 & 80 & 58.6 \\
**Total Organic Carbon (TOC) \%** & 46.29 & 34.46 & 44.44 & 32.555 \\
**Moisture Content \%** & 1 & 1.5 & 2 & 90.6 \\
**pH** & - & - & - & 6.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Proximate Analysis of Substrates.  
1. Logistic Function Model \[\text{M}=\frac{Mo}{1+exp\Big{(}\frac{4 Rm(\lambda-t)}{Mo}+2\Big{)}}\] (2) 2. Modified Gompertz Model \[\text{M}=Mo.\exp\big{[}-\exp(\frac{Rm.e}{Mo}(\lambda-t)+1]\] (3) 3. Transference Function Model \[\text{M}=Mo\bigg{[}1-\exp\bigg{(}\frac{-Rm(t-\lambda)}{Mo}\bigg{)}\bigg{]}\] (4) 4. First Order Kinetics \[\text{M}=\text{Mo (1}-\exp\big{(}-k_{\text{h}}\text{ *t}\big{)}\] (5) \[\text{RMSE}=\big{[}\frac{1}{\text{m}}\sum_{j=1}^{m}\bigg{(}\begin{array}{ c}dj\\ Yj\end{array}\bigg{)}^{2}\big{]}^{\frac{1}{2}}\] (6)  
where M (mL/gVS) is cumulative methane production, \(Mo\) (mL/gVS) refers to methane yield potential, \(t\) is to be time (days) is the \(\lambda\) lag phase (day) i.e., minimum time required for methane production, and \(Rm\) (mL/g VS. d) is the maximum methane production rate. The Modified Gompertz Model and Logistic Function Model are sigmoidal function with (S-shape) curves and are usually compared to illustrate exponential bacterial growth. With these non-linear regression models' methane production rate, lag phase and methane production potential were determined by using experimental values of batch assays. Though both models appear to be similar, the major difference among the two models is that the curve of the modified Gompertz model is symmetrical, and that of the logistic growth model is asymmetric [29]. The logistic growth model fits for methane production and assumes that is the rate of methane production is directly proportional to the amount of gas produced and to the maximum capacity of methane production, moreover, estimation of growth rate for the population of cells is mostly done by using logistic growth model (LF) [30; 31]. In this study, the modified version of the logistic model is used, as shown in Equation (2) [32]. The first-order kinetics model (Equation (5)) determines the cumulative methane production and hydrolysis rate constant (\(k_{\text{h}}\)), it is the crucial kinetic parameter, and in the anaerobic digestion process, the hydrolysis is considered as the rate-limiting reaction which governs the process. Simulation of the experimental data for the best fit model was performed, and selection of the best-fitted model was based on the kinetic parameters, which are R2 coefficient of determination, RMSE Root mean square error, \(Rm\), \(\lambda\) and \(\%\) difference (\(\Upsilon\)) between the predicted and experimental values RMSE was calculated using the Equation (6).