## 3 Results and Discussion
### \(2^{3}\) Factorial Design  
Figure 5 shows the lactic acid and 1,3-propanediol obtained in the experiments using an M9 medium supplemented with 10% glycerol. The three variables were chosen as observed and experimentally. Agitation was maintained at 50 rpm, as the bacterium _Klebsiella_ sp. 2 GPP is facultatively anaerobic and has obtained better results for lactic acid production with smaller amplitudes, leading to less oxygenation [15; 18]. The significance of the effects on temperature, inoculum concentration, and pH was evaluated for experiments using glycerol as a carbon source, based on analysis of variance (ANOVA) using the Statistica program as a tool. Based on the Pareto diagram for standardized effect estimates concerning lactic acid production, considering the equivalent of 48 h of fermentation in glycerol experiments, only temperature and the nonlinear effect of curvature (\(d_{curv}\), corresponds to a quadratic contribution of experimental runs performed at the central point of the experimental planning) are significant for lactic acid concentration. Figure 4: HPLC chromatogram of (**A**) standard samples of L-(+)-Lactic acid and D-(\(-\))-Lactic acid enantiomers, and (**B**) fermentation medium by using _Klebsiella_ sp. 2 GPP with glycerol as a carbon source. The ANOVA test indicated that the confidence level for these analyzes was 95% (\(p=0.05\)), as depicted in Figure 6. The correlation coefficient (R2) was equal to 96.4%. As the curvature corresponds to a nonlinear behavior between factor(s) and the response variable(s), for any 2k experimental design, a check for curvature should be considered a _sine qua non_ to evaluate the linearity between the variables, because it provides significant insights into the process features, allowing for an appropriate choice of new experimental planning strategies that allow for the use of second-order models. \(\delta_{curv}\) is generally expressed as a quadratic contribution at the midpoint of the experimental design, given as [50; 51; 52]:  
\[\delta_{curv}=\left\{ \begin{array}{ll}1,\;\text{if the point is a center point}\\ 0,\;\text{otherwise}\end{array} \right. \tag{3}\]  
By considering only the significant variables, the fermentation process model with glycerol can be represented according to Equation (4). \[\text{Y}=1.75+0.83\cdot\text{X}_{1}+1.13\cdot\delta_{curv} \tag{4}\]  
Equation (4) clearly shows how important the curvature effect is, and unequivocally indicates the presence of critical, statistically significant nonlinear effects, considering the order of magnitude of the estimated parameters. Figure 5: Production of lactic acid and 1,3-propanediol obtained in M9 medium with 10% glycerol for 48 h for the conditions related to the factorial design experiments. The central point corresponds to Experiments 9, 10, and 11, respectively, described in Table 3.  
By analyzing the level curves (Figure 7), it is possible to verify that there is a tendency towards higher lactic acid production when the temperature (X\({}_{1}\)) and pH increases (X\({}_{3}\)) may represent higher lactic acid production. Inoculum concentration was not significant. [PERSON] et al. [53] obtained excellent results with fermentations at 37 \({}^{\circ}\)C for _K. pneumoniae_ and verified that, when pH was alkaline, the predominant form in the fermentation broth is the lactic acid conjugate base (lactate), and lactate is less toxic to the bacterial cell. They also concluded that an intrinsic mechanism of bacteria self-protection should cause the exchange of metabolic pathways to adapt to the environmental variations in the pH of _K. pneumoniae._  
Figure 6: (**A**) Pareto chart for standardized effects estimative of the temperature (X\({}_{1}\)), Inoculum concentration (X\({}_{2}\)), and pH (X\({}_{3}\)) and (**B**) prediction of the regression model for experimental data for the production of lactic acid (N.B. linear fit: \(y=0.0741+0.964\)-\(x\)). The dashed lines stand for the regression bands built with a 95% confidence level. According to research by [PERSON] et al.