## 3 Results and Discussions
### Kinetic Constants  
In Table 3, the kinetic constants \(\mu_{max}\), K\({}_{s}\), and X\({}_{m}\) of the Monod, Logistic, and Contois models are reported. Without pH control, the specific experimental growth rate was more appropriately described by the Contois model (R\({}^{2}\) = 0.990). The specific growth rate decrease was not only linked to the glucose consumption, as expressed by the Monod model (R\({}^{2}\) = 0.742), or to the biomass increase in the fermenting medium, as assumed in the Logistic model (R\({}^{2}\) = 0.874), but both contributions could also be considered relevant for describing the microbial growth kinetics. The maximum specific growth rate \(\mu_{max}\) and the semi-saturation constant K\({}_{s}\) calculated by the Contois model were 0.993 h\({}^{-1}\) and 124.961 g/g, respectively, indicating that only during the first stages of the exponential phase, when the glucose to biomass concentration S/X ratio was higher or of the same order of magnitude than the constant K\({}_{s}\), the specific growth rate was approximately higher or equal to half of the maximum growth rate (Figure 3). Subsequently, the growth rate dropped to lower values as it became more influenced by both the biomass increase and the substrate decrease that occur during the fermentation process. \begin{table}
\begin{tabular}{c c c c} \hline
**Process Conditions** & **t\({}_{d}\) (h)** & **k (h\({}^{-1}\))** & **n** \\ \hline Without pH control & 1.52 & 0.66 & 6.58 \\ With pH control & 1.21 & 0.83 & 9.92 \\ \hline \end{tabular}
\end{table} 
Table 2: Fermentation parameters evaluated during the exponential growth phase of fermentation with and without pH control: doubling time (t\({}_{d}\)); constant growth rate (k); generation number (n). When the fermentation process was carried out with pH control, both the Monod and Contois models satisfactorily described the specific growth rate trend, with R\({}^{2}\) values of 0.987 and 0.985, respectively. The Monod model provided \(\upmu_{\text{max}}\) and K\({}_{\text{s}}\) values of 0.734 h\({}^{-1}\) and 4.880 g/L, respectively, indicating that the glucose concentration in the fermenting medium always ensured a higher specific growth rate than half of its maximum value and satisfied the nutritional requirement of the microorganism, as shown in Figure 4a. In Figure 4b, the specific growth rate as a function of the ratio of S/X is reported. The ratio of S/X was always higher than the constant Ks value determined for the Contois model, ensuring a high specific growth rate during the entire process, without any limitations provided by the decreasing glucose content and/or the increasing biomass concentration. Moreover, the logistic model provided a high value for the maximum biomass concentration \(X_{\text{m}}=28.665\) g/L, confirming that the biomass density in the fermenting suspension never limited the cell growth rate. In [PERSON] et al. (2014) and [PERSON] et al. (2017), the growth rate of five Lactobacillus (_Lactobacillus casei_, _Lactobacillus delbruuckii_, _Lactobacillus fermentum_, and _Lactobacillus lactis_) in a batch submerged culture of whey with added lactose was modelled using Monod, Logistic, and Contois models, respectively. As well as for LP CBA L74 used in this work without pH control, Contois resulted to fit the experimental data obtained using _Lactobacillus bulgarcius_ with higher accuracy (R\({}^{2}\) = 0.95) than that resulting from the Monod and Logistic models (R\({}^{2}\) = 0.890 and R\({}^{2}\) = 0.897, respectively) (R\({}^{2}\)). Moreover, both Monod and Contois models provided an acceptable fitting of the growth rate when _Lactobacillus casei_ was used during the fermentation process (R\({}^{2}\) = 0.965 and R\({}^{2}\) = 0.955, respectively) while a lower accuracy (R\({}^{2}\) = 0.892) was observed when the Logistic equation was applied (R\({}^{2}\),R\({}^{2}\)). Figure 3: Specific growth rate \(\upmu\) as function of glucose to biomass concentration S/X ratio during fermentation without pH control.