## 3 Results and Discussion
### R/S Analysis  
\(R/S\) analysis is applied to the pH series of the four experimental conditions. The time scale interval is given by \(N_{S,\mathrm{min}}\) = 20 data (20 s) and \(N_{S,\mathrm{max}}\) = 3600 data (1 h), which according to [PERSON] et al. [37], is the time scale where the characteristic phenomena corresponding to each of the stages of the anaerobic digestion of crude whey can be identified. Figure 4 shows the statistic \((R/S)\)2 as a function of the time scale Ns, observing that the power-law (\((R/S)\) = \(aNs^{H}\)) exhibits three changes in slope (i.e., three different values of the Hurst exponent, H\({}_{1}\), H\({}_{2}\), and H\({}_{3}\)), named as Zones 1, 2, and 3. The presence of different values of the Hurst exponent suggests that the \(R/S\) analysis can identify three characteristic phenomena immersed in the fluctuations of the pH series. These zones correspond to those reported in the anaerobic digestion of whey [37] and tequila vinasse [35; 36], where each region was attributed to a different anaerobic digestion stage. Based on the above observations, Zone 1 is related to hydrolysis, Zone 2 corresponds with the acidogenesis and acetogenesis stages, and Zone 3 is related to methanogenesis. It should be noted that in conventional anaerobic digestion processes, the experimental conditions favor methane-rich biogas production. In contrast, in this study, the operating conditions favor the VFA production, so the methanogenic stage is not significant. Figure 3: The dynamic profile of COD consumption and distribution of VFA produced. Table 2 shows the values of the Hurst exponent for each zone, where it is observed that the H\({}_{1}\) values are around 1.0, indicating that in this time scale, there are characteristic phenomena with similar behavior for all the operating conditions evaluated. On the other hand, the Hurst exponent values in Zones 2 and 3 show more significant variability between operating conditions, which indicates that the phenomena on these characteristic scales are more susceptible to changes in pH and temperature. According to the above, Zone 1 could be correlated with the hydrolytic stage. In all experiments, the COD consumption is mainly to carbohydrate degradation, which is consumed up to 95% in the first two days. Zones 2 and 3 could be correlated with the acidogenic-acetogenic stages, which show the formation of different fatty acids depending on the operating conditions evaluated. For example, at pH = 5.5, higher production of butyric and caproic acids is obtained, while at pH = 7.5, higher production of acetic and propionic acids is obtained. Note that the H\({}_{3}\) value is higher when there is a more significant formation of butyric and caproic acids. In order to identify correlations between the key variables and the Hurst exponents, the \(R/S\) analysis is implemented considering 6 h windows, which are shifted every hour, covering the whole series. This allows one to obtain the Hurst exponent values as a function of scale and time. For the four experimental conditions, Figure 5 shows the dynamic \(R/S\) analysis, where the three zones described in Figure 4 can be identified, noting that the H values show temporal variations in each zone, which can be correlated with those of the process stages. In order to avoid different interpretations of the dynamic Hurst exponent, the average is carried out in each zone, which leads to an H dynamic profile for each zone (i.e., Hd\({}_{1}\), Hd\({}_{2}\), and Hd\({}_{3}\)). These profiles are compared with the experimental measurements of COD and VFA. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Zone/Test**} & **T = 35 \({}^{\circ}\)C** & **T = 35 \({}^{\circ}\)C** & **T = 40 \({}^{\circ}\)C** & **T = 40 \({}^{\circ}\)C** \\  & **pH = 5.5** & **pH = 7.5** & **pH = 5.5** & **pH = 7.5** \\ \hline Zone 1 & 0.85819 & 0.99983 & 1.01371 & 0.9804 \\ Zone 2 & 1.12514 & 0.73615 & 0.51536 & 0.79128 \\ Zone 3 & 1.05433 & 0.89266 & 1.28973 & 1.05855 \\ \hline \hline \end{tabular}
\end{table}