## 3 Results and Discussion
### Application of Kinetic Modelling  
The accumulated methane production curve was simulated using four models; first-order kinetics, modified Gompertz model, logistic function model, and transference function model to analyze the best combination reactor and S/I that has potential for maximum biomethane production. Parameters as hydrolysis rate constant, maximum methane production rate, lag phase, biogas yield potential was studied; the fitness of these models depends on bio-kinetic parameters (\(\lambda\), \(R^{2}\), \(RMSE\), \(Y\), and \(Rm\)) which are presented in Table 4. The simulation of experimental values and predicted values were plotted in Figures 4 and 5. Among all applied kinetic models, the best-fitted model was the logistic function model followed by the modified Gompertz model. The transference function model could not replicate the experimental data for all reactors, the coefficient of determination for all reactors have low values, whereas \(RMSE\) values are high in the TF model and the % differences were very high therefore this model could not be the best-fitted model. The logistic model was best-fitted for both Experiments. The \(R^{2}\) values show that the predicted model was best fitted with experimental data, and the variance between predicted and experimental BMP's was less than 10%, this low value demonstrates that the logistic model predicts the performance of reactors accurately. Furthermore, the lag phase suggests the time required by methanogenic bacteria to produce methane. However, according to our results of all reactors, the lag phase increases with an increase in organic loading ratio due to higher substrate concentration (solid substrate) which prolongs the hydrolysis process and increases the period of acclimation for microorganisms [5,46]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reactors** & **pH** & **VS Reduction (\%)** \\ \hline
**Control 1** & 7.0 & 88.9 \\
**WSRS** & 7.0 & 98.8 \\
**BARS** & 6.9 & 98.5 \\
**BAWS** & 7.8 & 98.1 \\
**BAWSRS** & 7.0 & 99.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: pH and VS Reduction % after Co-digestion of S/I Ratio 1.5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reactors** & **pH** & **VS Reduction (\%)** \\ \hline
**Control 2** & 7.1 & 87.0 \\
**WSRS** & 7.1 & 99.2 \\
**BARS** & 5.5 & 99.4 \\
**BAWS** & 6.5 & 97.7 \\
**BAWSRS** & 7.0 & 96.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: pH and VS Reduction % after Co-digestion of S/I Ratio 2.5. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Parameters} & \multicolumn{4}{c}{**S11.5**} & \multicolumn{4}{c}{**S12.5**} \\ \cline{2-9}  & **WSSS** & **RAWS** & **BAWS** & **BAWSS** & **WSSS** & **RAWS** & **BAWS** & **BAWS** & **BAWSS** \\ \hline \hline \multicolumn{9}{c}{**Modified Competics Model**} \\ \hline R\({}^{2}\) & 0.972 & 0.977 & 0.977 & 0.981 & 0.981 & 0.983 & 0.975 & 0.984 \\ \hline RMSE & 0.981 & 0.299 & 0.638 & 0.951 & 0.984 & 0.662 & 0.688 & 0.636 \\ \hline \(\lambda\) days & 10 & 18 & 22 & 17 & 18 & 25 & 22 & 21 \\ \hline Rm mECH/VS-4 & 4.123 & 0.320 & 0.296 & 0.581 & 0.298 & 0.091 & 0.0181 & 0.1574 \\ \hline Predicted Median Field NetCLtigWS & 338.088 & 134.788 & 180.000 & 259.990 & 255.493 & 37.723 & 13.860 & 133.586 \\ \hline Experimental Median NetCLtigWS & 303.088 & 177.965 & 188.299 & 337.900 & 244.785 & 65.793 & 15.748 & 151.356 \\ \hline Difference* (\(\mathbf{Y}\)) & 6.8 & 7.21 & 4.84 & 6.23 & 5.86 & 5.63 & 7.44 & 5.82 \\ \hline \multicolumn{9}{c}{**Logistic Function Model**} \\ \hline R\({}^{2}\) & 0.990 & 0.992 & 0.990 & 0.990 & 0.994 & 0.992 & 0.990 & 0.994 \\ \hline RMSE & 0.138 & 0.23 & 0.115 & 0.136 & 0.335 & 0.662 & 0.666 & 0.200 \\ \hline \(\lambda\) days & 10 & 17 & 21 & 15 & 19 & 36 & 33 & 25 \\ \hline Rm mECH/VS-4 & 2.566 & 1.477 & 2.368 & 2.773 & 13.80 & 0.670 & 0.191 & 2.002 \\ \hline Predicted Median Field NetCLtigWS & 335.716 & 161.425 & 170.274 & 66.334 & 224.178 & 98.955 & 14.511 & 38.107 \\ \hline Experimental Median Yield SentCLtigWS & 390.088 & 177.965 & 188.299 & 337.900 & 244.