## 3 Results and Discussion  
### Optimization of the Mean Rate of Acetic Acid Formation in the Two-Bioreactor System  
One potentially useful variable for assessing the \"health\" of a bioprocess and its productivity is the rate of biotransformation. Because the bioreactors operated in a non-steady state here, the target variable was the mean reaction rate, which can be estimated from the rate of ethanol consumption or that of acetic acid formation [6]. When volatile losses are negligible, as it is the case [27], both rates are identical. We focused on the mean rate of acetic acid formation, \((r_{A})_{global}\), because it is easier to estimate. In [27], the polynomial model for the mean rate of acetic acid formation in the two- bioreactor system was obtained (Equation (5)). The model allowed \((r_{A})_{global\;est}\) to be estimated with an error of 0.01 g acetic acid\(\cdot\)(100 mL\(\cdot\)h)\({}^{-1}\). \[ \begin{array}{ll}(r_{A})_{global\;est}=&-0.51+0.43\cdot E_{u1}-0.0654\cdot E_{u1 }^{2}\\ &-0.00456\cdot E_{l2}\cdot V_{u1}-0.00468\cdot E_{u1}\cdot E_{l1}\\ &+0.000672\cdot T_{1}\cdot E_{l1}+0.000839\cdot E_{l2}\cdot T_{1}\end{array} , \tag{5}\]  
As Equation (5) shows, not all the operational variables have a direct influence on \((r_{A})_{global\;est}\); additionally, as explained in [27], particularly in its Supplementary Material (\"S3.docx\" file), not all the terms in Equation (5) have the same influence on \((r_{A})_{global\;est}\); the higher the value of the statistic \(F\) (see Table 1), the more significant the term is for \((r_{A})_{global\;est}\); it can be seen from Table 1 that the most influential term is that of \(E_{u1}\). This model is of especial practical interest as it allows the operating conditions leading to maximum--desirable--or minimum levels of the industrially useful variables to be identified. We used the procedure described in Section 2.3 to develop a MATLAB script [30] that allowed the maximum or minimum value of \((r_{A})_{global\;est}\) to be calculated (see file \"Optimal_rA.m\" in Supplementary Materials). The optimization problem addressed to maximize \((r_{A})_{global\;est}\) is represented by Equations (6) and (7). The latter sets the constraints imposed by the experimental conditions on the operational variables. \[ \begin{array}{ll}\max&(r_{A})_{global\;est}\\ &\text{s.t.}\end{array} , \tag{6}\]  
\[ \begin{array}{ll}4.25\leq V_{u1}\leq 5.75\\ 4\leq E_{l1}\leq 6\\ 2\leq E_{u1}\leq 4\\ 28\leq T_{1}\leq 32\\ 2.5\leq E_{l2}\leq 4.5\end{array} , \tag{7}\]  
where \(V_{u1}\) is the volume of medium unloaded from the first bioreactor into the second, \(E_{l1}\) the ethanol concentration during loading of the first bioreactor, \(E_{u1}\) the ethanol concentration at the time the first bioreactor is unloaded, \(T_{1}\) the temperature in the first bioreactor and \(E_{l2}\) the ethanol concentration in the second bioreactor during loading. Solving the previous optimization problem provided the operating conditions of Table 2. \begin{table}
\begin{tabular}{c c c} \hline
**Terms** & **F** & **Coefficient** \\ \hline Constant & & \(-0.51\) \\ \(E_{u1}^{2}\) & 342.526 & \(-0.0654\) \\ \(E_{u1}\) & 392.086 & \(0.43\) \\ \(T_{1}E_{l1}\) & 56.948 & \(0.000672\) \\ \(E_{u1}E_{l1}\) & 28.095 & \(-0.00468\) \\ \(E_{l2}T_{1}\) & 119.001 & \(0.000839\) \\ \(E_{l2}V_{u1}\) & 122.922 & \(-0.00456\) \\ \hline \end{tabular}
\end{table} 
Table 1: Significance of the terms of Equation (5). (Adapted from Table S3.14 in [27], Supplementary Material (“S3.docx” file)). \begin{table}
\begin{tabular}{c c c c c c} \hline \(V_{u1}\), & \(E_{l1}\), & \(E_{u1}\), & \(T_{1}\), & \(E_{l2}\), & \(T_{2}\), & \((r_{A})_{global\;est}\) \\ L & **\% (\%/\(v\))** & **\% (\%/\(v\))** & \(\circ\)C & **\% (\(v\)/\(v\))** & \(\circ\)C & **g Acetic Acid\(\cdot\)(100 mL\(\cdot\)h)\({}^{-1}\)** \\ \hline
4.25 & 6.0 & 3.07 & 32.0 & 4.5 & – & 0.27 \\ \hline \end{tabular}
\end{table} 
Table 2: Values of the operational variables needed to maximize \((r_{A})_{global\;est}\).