## 4 Conclusions  
The aim of the present work was to perform a kinetic analysis of the results reported by [PERSON] et al. [11] for rice flour fermentation performed using _Lactobacillus paracasi_ CBA L74 with and without pH control. [PERSON] et al. [11] had already demonstrated the possibility of improving the functional properties of the fermented rice product by controlling pH at 5.8, but a comparison between the kinetic performances, corresponding to the two different process conditions, had not been conducted yet. When pH control was performed, the microorganism showed a better duplication capacity during the exponential growth phase, \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\alpha\) (g/g) & \(\beta\) (g/g h) & l \(\alpha\)/\(\beta\) l & **RSS** & \(\mathbf{R^{2}}\) \\ \hline Without pH control & 0.001 & 0.489 & 0.002 & 0.0013 & 0.958 \\ With pH control & 19.539 & \(-\)8.637 & 2.262 & 0.3305 & 0.762 \\ \hline \end{tabular}
\end{table} 
Table 4: The kinetic constants of the Luedeking–Piret model (\(\alpha\): growth-linked constant; \(\beta\): non-growth-linked constant), RSS, and the correlation coefficient R\({}^{2}\), calculated for fermentations with and without pH control. Figure 4: Specific growth rate \(\mu\) as function of the glucose concentration S (**a**) and the glucose to biomass weight S/X ratio, expressed in logarithmic scale (**b**) during the fermentation process performed with pH control. providing higher values of the generation number n and the constant growth rate k (9.92 and 0.83 h\({}^{-1}\), respectively) and a lower doubling time t\({}_{\text{d}}\) of 1.21 h in comparison with the values obtained in the absence of pH control. Three kinetic models were proposed to describe the bacterial growth rate in both conditions: the Monod, Logistic, and Contois models. When fermentation was carried out without pH control, the best mathematical model able to describe the experimental data was the Contois model, showing that the specific growth rate was affected both by the glucose reduction and the biomass production that occurred during the fermentation process. When pH control was applied, both Monod and Contois models satisfactorily described the specific growth rate trend. Therefore, the constants Ks and \(\mu_{\text{max}}\) confirmed that the glucose content in the fermenting suspension satisfied the nutritional requirements of microorganisms and that the growth kinetic was not affected by the decreasing glucose content and/or the increasing biomass concentration. Moreover, the Luedeking-Piret model was applied for modeling the lactic acid production rate, which appeared mainly non-growth linked, as confirmed by the \(\alpha/\beta\) ratio of 0.002, calculated in the absence of pH control. The Luedeking-Piret model was not appropriate for describing the experimental lactic acid rate when pH was controlled. In conclusion, the kinetic study reported in this work allowed us to validate the results discussed by [PERSON] et al. [11], showing the convenience of controlling pH during the rice flour fermentation process with LP CBA L74 and also in terms of the microbial growth rate. Therefore, optimal growth conditions in terms of the pH value of the fermenting medium allowed the microorganism to also improve its fermentation performance from a kinetic point of view, providing high specific growth rates that were not significantly influenced by the increasing biomass amount or the substrate consumption in the fermenting medium. Conceptualization, R.C.C., M.G., F.P. and R.N. ; methodology, R.C.C., M.G., F.N., F.P. and R.N. ; validation, R.C.C., M.G. and R.N. ; Resources, A.B. ; formal analysis, R.C.C., M.G., F.N. and F.P. ; data curation, R.C.C., F.N. and F.P. ; writing--original draft, R.C.C. ; writing--review and editing, R.C.C. ; supervision, M.G. and R.N. ; project administration, M.G. and R.N. All authors have read and agreed to the published version of the manuscript.