## 3 Results and Discussion
### Multifractal Analysis  
The multifractal analysis is a useful tool to identify information at different time scales associated with the inherent complexity in bioprocesses. The anaerobic treatment of raw cheese whey is a highly complex process involving transport phenomena and bioreactions, affecting overall process stability. This interaction depends on the evaluated operating conditions and is reflected in the type and quantity of products generated.  
In Figure 8, the application of the multifractal \(R/S\) analysis is presented, considering \(q\in\) [0.2, 20]. For the experimental test at T = 40 \({}^{\circ}\)C and initial pH = 5.5, Figure 8a shows the statistic \((R/S)q\), noting that the slopes (\(H_{q}\)) of the three zones show variations at different values of q-norm. Figure 8b shows that the \(H_{q}\) values of the three zones present nonlinear variations with respect to the q-norm that exhibit significant differences in each zone, suggesting the presence of a multifractal behavior in the global process, as well as in each of the stages that make up the process.  
Figure 7: Correlations between \(\mathrm{Hd}_{3}\) and VFA produced measurements.  
The degree of multifractality of each process stage is evaluated with the multifractal index (IM). The higher the IM value, the greater the multifractal behavior. Table 3 shows the multifractal index of the three zones for the four experimental tests. The lowest values of the multifractal index correspond to Zone 1, associated with the hydrolytic stage. Regardless of the operating conditions evaluated, a similar behavior is observed in all experiments. Whereas Zone 3, associated with the acidic stage, shows the highest values of the multifractal index, which correspond to the generation of products obtained in this stage. It should be noted that different types of VFA are produced and the quantity produced depends on the operating conditions.  
Globally, the multifractal index identifies which stage of the process presents more significant interactions between the underlying phenomena. In order to determine the changes in multifractality as the process is carried out, the multifractal index is calculated considering 6 h moving windows. Figure 9 shows the scale-temporal changes of the multifractal index, where it is observed that the dynamic changes of the IM correspond to the variations observed in the consumption/production rates of the key variables. In Zone 1, it is observed that the most important changes in IM occur in the first 30 h of the process, where the maximum consumption of carbohydrates is carried out. In Zone 3, more significant dynamic changes are observed associated with the changes in the VFA production rate. Increases in the IM correspond to the increases in the VFA production rate.  
Figure 8: Multifractal analysis for pH time series to pH = 7.5 and T = 40 \({}^{\circ}\)C, (**a**) \(R/S\) analysis applied to different q-norm and (**b**) \(H_{q}\) as a function of q-norm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**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 & 0.0725 & 0.0919 & 0.0469 & 0.1107 \\ Zone 2 & 0.028 & 0.1929 & 0.0679 & 0.1414 \\ Zone 3 & 0.0115 & 0.1691 & 0.1756 & 0.1039 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Multifractal index by zone for the four experimental tests.