## 3 Results and Discussion
### Optimizing \(P_{m\,est}\) While Ensuring Enough Substrate Depletion  
For economy and legal reasons, the amount of ethanol present in industrially produced vinegar must not exceed prescribed levels depending on each specific type of vinegar. It is therefore important to identify the operating conditions optimizing acetic acid production (\(P_{m\,est}\) in Equation (8)) under a constraint on the ethanol concentration at the time the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**End** & \(V_{u1}\)**, L** & \(E_{11}\)**, \% (\(v/v\))** & \(E_{u1}\)**, \% (\(v/v\))** & \(T_{1}\)**, \({}^{\circ}\)**C** & \(E_{12}\)**, \% (\(v/v\))** & \(T_{2}\)**, \({}^{\circ}\)**C** \\ \hline Lower & 5.55 & & 3.6 & 31.2 & 2.5 & 31.3 \\ \(\ddagger\) & \(\ddagger\) & 4.0 & \(\ddagger\) & \(\ddagger\) & \(\ddagger\) & \(\ddagger\) \\ Upper & 5.75 & & 4.0 & 32.0 & 2.9 & 32.0 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Upper and lower end of the ranges over which the operational variables minimized \(P_{m\,est}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \
\begin{tabular}{c} **Conditions Minimizing \(P_{m\,est}\)** \\ **(14.7 g Acetic Acid-h-1)** \\ \end{tabular} &
\begin{tabular}{c} **Conditions Maximizing \(P_{m\,est}\)** \\ **(36.6 g Acetic Acid-h-1)** \\ \end{tabular} \\ \hline \((r_{A})_{global\,est}\), g acetic acid-(100 mL-h)\({}^{-1}\) & \(0.18\pm 0.01\) & \(0.21\pm 0.01\) \\ \(E_{u2\,est}\), \% \((v/v)\) & \(0.0\pm 0.3\) & \(4.3\pm 0.3\) \\ \(V_{u2\,est}\), L & \(7.56\pm 0.22\) & \(6.71\pm 0.22\) \\ \(t_{cycle}\), h & \(43.0\pm 2.5\) & \(19.2\pm 2.5\) \\ \(V_{m\,est}\), L & \(11.58\pm 0.33\) & \(12.07\pm 0.33\) \\ \(EtOH_{m1\,est}\), \% \((v/v)\) & \(4.1\pm 0.2\) & \(4.8\pm 0.2\) \\ \(EtOH_{m2\,est}\), \% \((v/v)\) & \(1.8\pm 0.4\) & \(4.5\pm 0.4\) \\ \(HAC_{m1\,est}\), \% \((w/v)\) & \(7.4\pm 0.2\) & \(6.7\pm 0.2\) \\ \(HAC_{m2\,est}\), \% \((w/v)\) & \(9.7\pm 0.4\) & \(7.0\pm 0.4\) \\ \hline \hline \end{tabular}
\end{table}
Table 12: Values of the state variables under the operating conditions maximizing and minimizing \(P_{m\,est}\).  
second bioreactor is unloaded (\(E_{u2\,est}\) in Equation (9)). The optimization problem is stated in Equations (20) and (21).  
\[\max\;\;P_{m\,est}\;\;, \tag{20}\]  
s.t. \[4.25\leq V_{u1}\leq 5.75\] \[4\leq E_{l1}\leq 6\] \[2\leq E_{u1}\leq 4\] \[28\leq T_{1}\leq 32\] \[2.5\leq E_{l2}\leq 4.5\] (21)  
\[28\leq T_{2}\leq 32\] \[E_{u2\,est}=\text{constant}\]  
where _constant_ is the ethanol concentration at unloading time in the second reactor to be imposed.  
Table 13 shows the values of the operational variables providing the solution to the previous problem with \(E_{u2\,est}\) values from 0.2% to 1.5% (\(\
icefrac{{\
u}}{{\
u}}\)).  
The solution was obtained by using the MATLAB script of Section 3.2 (see file \"Optimal_Prod.m\" in Supplementary Materials) as expanded with the constraint on \(E_{u2\,est}\). The previous range of \(E_{u2\,est}\) was used to consider virtually complete and less marked depletion of the substrate because some vinegars (particularly those under a designation of origin or aged in casks) are allowed to contain higher levels of residual ethanol.  
As expected, the greater \(E_{u2\,est}\) was (i.e., the less markedly the substrate was depleted in the second bioreactor), the more favored was AAB activity and the higher was acetic acid productivity as a result.