## 3 Results
### Modeling of Reactors  
The Stover-Kincannon model was used to fit the five different combinations of data from the reactors. The averaged values were employed and the size of a combination was up to 18 points. The results are summarized in Table 7. The fit for R2 was very good and the best among all datasets. The datasets for R1 and R3 had the worst fit, which suggested that the largest difference in the kinetic behavior among all possible combinations existed in these reactors. The RMSE values for R1 and R2 were also rather large, which meant that there were significant differences in the behaviors of these reactors. The goodness of fit in R1 and R3 were equal; however, R3 could produce a maximum amount of methane that was more than double the daily amount of methane from R1. R3 was also able to handle a larger OLR, which was related to the intrinsic properties of the substrate mixtures. In addition, the model predicts that R2 was far from reaching its maximum in terms of the production that can be handled by its largest OLR. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Influent** & **Reactor Tube** & **Separation Zone** \\ \hline Re & 295.42 & 295.42 & 295.42 \\ Peaxial & 1.10 & 9.33 & 38.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Hydraulic parameters of the reactors. The datasets for R2 and R3 were of special interest. Since the mixtures in these reactors showed synergistic effects and they seemed to behave similarly, we examined if the mixtures followed similar kinetics. We found that the difference among them was moderate (R2, R3); however, the differences between R1 and R2 (R1, R2) or R1 and R3 (R1, R3) were larger. The fits and measured data are shown in Figure 6. By following both estimated models (red and violet lines), it was noticed that significant differences existed at low OLRs, with these being much smaller at higher OLRs. The fluctuations in R3 between 4.5 and 6.5 gCOD/L were most likely the main cause of the misfit, which was observed in the green but not in the red line. The performance of R3 (green line) was closely related to mixture preparation and the degree of mixing in the feeding tank since mixture 3 had the highest DM content, contrary to the smooth behavior of R2 (light blue line). Therefore, we decided to work on both mixtures individually, given that both models were not able to converge in most of the working intervals. Since working at low HRTs usually reduces the MY [44], we took into account the other response variables in order to establish an optimal operational OLR. Hence, empirical models of MY and \({}^{\eta}\)\({}_{\text{COD}}\) versus OLR were also fitted. The fits for reactors 2 and 3 are shown in Figure 7a,b, respectively. \begin{table}
\begin{tabular}{c c c c} \hline
**Data Set** & **RMSE (L\({}_{\text{CH}}\)/L\({}_{\text{react}}\)/d)(-)** & **M\({}_{\text{max}}\) (L\({}_{\text{CH}}\)/L\({}_{\text{react}}\)/d)** & **M\({}_{\text{B}}\) (gCOD/L/d)** \\ \hline (R1, R2, R3) & 0.214 & 1.29 & 6.08 \\ (R1, R2) & 0.188 & 1.25 & 8.21 \\ (R1, R3) & 0.245 & 1.07 & 4.20 \\ (R2, R3) & 0.118 & 1.48 & 5.07 \\ (R1) & 0.132 & 0.68 & 3.59 \\ (R2) & 0.031 & 1.76 & 9.99 \\ (R3) & 0.132 & 1.52 & 5.12 \\ \hline \end{tabular}
\end{table} 
Table 7: Fit analysis using the Stover–Kincannon model. Figure 6: Stover–Kincannon model fitting for the datasets of R1, R2, and R3. No significant fit was found for \({}^{\eta}\)\({}_{\rm COD}\), as per the pre-established criterion set for R\({}^{2}\). The R\({}^{2}\) values for R2 and R3 were 0.649 and 0.635, respectively. Therefore, no strong dependency on OLR existed. The fits should be described by more complex models that take into account mass transfer relationships. The MY was satisfactorily described by its inverse relationship with OLR for R2. The R\({}^{2}\) was 0.881 with a D-W of 3.6 (\(p\)-value = 0.990). While the fit for R3 was smaller, the R\({}^{2}\) of 0.782 with D-W of 2.22 (\(p\)-value = 0.4709) was still significant.