## 3 Results and Discussion
### Model Fitting for Fermentation Temperature as an Input Variable  
Figure 2 shows kinetic parameters (ultimate methane yield, maximum daily methane yield and lag period) of husks, pulp and mucilage determined from non-linear regression of the LOG and GOM models operated at 21 \({}^{\circ}\)C, 30 \({}^{\circ}\)C and 37 \({}^{\circ}\)C.  
Fitting equations were established for the three parameters against the respective fermentation temperature T:  
\[S=c_{1}+c_{2}\cdot T \tag{7}\]  
\[R_{m}=c_{3}+c_{4}\cdot e^{\frac{T-21}{c_{5}}} \tag{8}\]  
\[\lambda=c_{6}+c_{7}\cdot T \tag{9}\]  
The ultimate methane yield (S) and the lag period (\(\lambda\)) fit well with a linear equation while the maximum daily methane yield (\(R_{m}\)) fits with the exponential function. The coefficients c\({}_{1}\)-c\({}_{7}\) are shown in Table 3.  
Figure 2: Lag period \(\lambda\) (A), maximum daily methane yield \(R_{m}\) (B) and ultimate methane yield S (C), from the modified Logistic model (LOG) and modified Gompertz model (GOM) vs. fermentation temperature T.  
The lag periods (\(\lambda\)) tend to decrease with increasing temperatures from 21 \({}^{\circ}\)C to 37 \({}^{\circ}\)C (Figure 2 A). In studies by [PERSON], [PERSON] and [PERSON] (2019), the lag period was represented by a hyperbolic type function, while growth of bacteria was examined in temperature ranges between 6 \({}^{\circ}\)C and 43 \({}^{\circ}\)C. The linear fit in this study (21 \({}^{\circ}\)C-37 \({}^{\circ}\)C) is similar to the results fromYing [PERSON], [PERSON] and [PERSON] (2019) at the same temperature range. The methane yield rate (R\({}_{m}\)) increased with increasing temperatures for all substrates and there was only a small difference between the substrates, regardless of the model type (Figure 2B). The fitting equation indicated strong dependence of the methane yield rate on temperature. The highest yield rate from each fermentation temperature was recorded from mucilage followed by pulp and husks. The Ratkowsky, Hinshelwood and Schoolfield growth models estimate that the growth yield increases from a minimum temperature to the optimum temperature and then starts to decrease again (2019). In this study, the fermentation temperature did not exceed the optimum temperature of 37 \({}^{\circ}\)C and hence, did not show a decline. [PERSON], et al. (2019) reported that the maximum daily methane yield rate rose exponentially within a range from low to optimal fermentation temperatures, in full-scale continuous feed biogas digesters fed with pig and cow manure. The ultimate methane yield (S) prediction from the modified LOG model shows an increasing trend with increasing temperatures for all substrates; however, the GOM model shows a decreasing trend with increasing temperatures for pulp and mucilage (Figure 2C). Introducing Equations (7)-(9) into Equation (1) (GOM model) and Equation (2) (LOG model) leads to the generalized modified Logistic model (gLOG) and the generalized modified Gompertz model (gGOM) with fermentation temperature T as an input variable. Thus, parameter values (S, \(R_{m}\) and \(\lambda\)) were estimated for fermentation temperatures of 21 \({}^{\circ}\)C, 30 \({}^{\circ}\)C and 37 \({}^{\circ}\)C to generalize the models (Table 4). The performance of the generalized models was validated with the measured SMY and showed high goodness of fit (R\({}^{2}\) > 0.959). Figure 3 shows the measured experimental data fitted with the generalized models. The generalized models described the experimental data very well. However, both the gLOG and the gGOM model underestimated the SMY from husks within the first weeks of fermentation at 21 \({}^{\circ}\)C. At the same temperature, the SMY from mucilage was slightly underestimated by the generalized LOG model and overestimated by GOM models. The deviation of the estimates values from the measured values could be attributed to the performance of fitting equations to the parameter values prior to application in the generalized model.