## 3 Results and Discussion
### Optimization of \(P_{m\,est}\)  
The variable (\(r_{A}\))\({}_{global\,est}\) possesses a high industrial interest. In practice, however, the aim of a vinegar producing plant is to obtain as much acetic acid per hour of operation (i.e., to maximize the overall production of acid, \(P_{m}\)). This variable can be estimated with an error of 0.7 g acetic acid\(\cdot\)h\({}^{-1}\) from Equation (8). In this Equation, only three of the operational variables (\(E_{l1}\), \(E_{u1}\) and \(V_{u1}\)) have a direct influence on \(P_{m\,est}\), but as explained before in the case of Equation (5), not all the terms in Equation (8) have the same influence on \(P_{m\,est}\) (see Table 7); the higher the value of the statistic \(F\), the more significant the term is for \(P_{m\,est}\); it can be seen from Table 7 that the most influential term is that of \(E_{l1}\). \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Variable**} & \multicolumn{1}{c}{**Conditions Minimizing (\(r_{A}\))\({}_{global\,est}\) (0.11 g**} & \multicolumn{1}{c}{**Conditions Maximizing (\(r_{A}\))\({}_{global\,est}\) (0.27 g**} \\  & **Acetic Acid (100 mL\(\cdot\)h)\({}^{-1}\))** & **Acetic Acid (100 mL\(\cdot\)h)\({}^{-1}\))** \\ \hline \(P_{m\,est}\), g acetic acid\(\cdot\)h\({}^{-1}\) & 20.6 \(\pm\) 0.7 \(\leftrightarrow\) 22.0 \(\pm\) 0.7 & 27.5 \(\pm\) 0.7 \(\leftrightarrow\) 28.6 \(\pm\) 0.7 \\ \(E_{u2\,est}\), \% (\(v/v\)) & 0 \(\pm\) 0.3 \(\leftrightarrow\) 0.4 \(\leftrightarrow\) 0.3 & 1.4 \(\pm\) 0.3 \(\leftrightarrow\) 2.2 \(\pm\) 0.3 \\ \(V_{u2\,est}\), L & 7.77 \(\pm\) 0.22 \(\rightarrow\) 7.99 \(\pm\) 0.22 & 7.63 \(\pm\) 0.22 \(\leftrightarrow\) 7.97 \(\pm\) 0.22 \\ \(t_{cycle\,est}\), h & 51.1 \(\pm\) 2.5 & 20.8 \(\pm\) 2.5 \\ \(V_{m\,est}\), L & 14.04 \(\pm\) 0.33 \(\leftrightarrow\) 14.32 \(\pm\) 0.33 & 14.06 \(\pm\) 0.33 \(\leftrightarrow\) 14.36 \(\pm\) 0.33 \\ \(EtOH_{m1\,est}\), \% (\(v/v\)) & 3.8 \(\pm\) 0.2 & 5.2 \(\pm\) 0.2 \\ \(EtOH_{m2\,est}\), \% (\(v/v\)) & 2.4 \(\pm\) 0.4 \(\leftrightarrow\) 2.7 \(\pm\) 0.4 & 3.6 \(\pm\) 0.4 \(\leftrightarrow\) 3.9 \(\pm\) 0.4 \\ \(HAc_{m1\,est}\), \% (\(w/v\)) & 7.7 \(\pm\) 0.2 & 6.3 \(\pm\) 0.2 \\ \(HAc_{m2\,est}\), \% (\(w/v\)) & 8.8 \(\pm\) 0.4 \(\leftrightarrow\) 9.1 \(\pm\) 0.4 & 7.5 \(\pm\) 0.4 \(\leftrightarrow\) 7.9 \(\pm\) 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of the state variables under the operating conditions maximizing and minimizing (\(r_{A}\))\({}_{global\,est}\). Below is described the procedure followed to identify the operating conditions maximizing (the industrial target) or minimizing (an undesirable outcome) \(P_{m\,est}\). The procedure of Section 2.3 to develop a MATLAB script to identify the optimum (maximum or minimum) \(P_{m\,est}\) value was used (see file \"Optimal_Prod.m\" in Supplementary Materials). The conditions maximizing \(P_{m\,est}\) can be identified by solving Equations (17) and (18), the latter containing the constraints imposed by the values of the experimental variables under the experimental conditions used. \[ \begin{array}{cc}\max&P_{m\,est}\\ \text{s.t.}&,\end{array}  \tag{17}\]  
\[ \begin{array}{cc}4.25\leq V_{u1}\leq 5.75\\ 4\leq E_{l1}\leq 6\\ 2\leq E_{u1}\leq 4\\ 28\leq T_{1}\leq 32\\ 2.5\leq E_{l2}\leq 4.5\\ 28\leq T_{2}\leq 32\end{array}  \tag{18}\]  
where \(T_{2}\) is the temperature in the second bioreactor. Table 8 shows the value of each operational variable maximizing \(P_{m\,est}\) and hence the solution to the optimization problem. In a similar way to the previous discussion on \((r_{A})_{global\,est}\), plotting the response surfaces of \(P_{m\,est}\) varying just a pair of the operational variables at once, it is possible to show that significant changes can only be found when the variable \(E_{l1}\) is included. By way of example, response surfaces varying \(E_{l1}\) and \(E_{u1}\) on one side and \(E_{u1}\) and \(E_{l2}\) on the other are shown in Figure 4a,b, respectively. Values from Table 8 are used for the remaining variables.