## 3 Results and Discussion
### Optimization of the Mean Rate of Acetic Acid Formation in the Two-Bioreactor System
Because of the number of involved operational variables, it is not straightforward to graphically show the optimum obtained with the conditions of Table 2. Nevertheless, considering the previous comments on the significance of the terms in Equation (5), when plotting the response surfaces of \((r_{A})_{global\ est}\) 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 \(E_{u1}\) is included. By way of example, response surfaces varying \(E_{u1}\) and \(E_{l2}\) on one side and \(E_{l1}\) and \(E_{l2}\) on the other are shown in Figure 3a,b, respectively. Values from Table 2 are used for the remaining variables. Since \((r_{A})_{global\ est}\) could be subject to an error of up to 0.01 g acetic acid-(100 mL-h)\({}^{-1}\), we checked whether other combinations of values of the operational variables falling within the ranges imposed by their errors would lead to maximum values over the range \(0.26\leq(r_{A})_{global\ est}\leq 0.27\) g acetic acid-(100 mL-h)\({}^{-1}\). A systematic analysis provided the ranges shown in Table 3. With a 0.2 interval for each variable, such ranges led to a total of 162 combinations of which only 109 (see Table S1 in file \"S1.docx\" in Supplementary Materials) fell within the previous range of \((r_{A})_{global\ est}\). However, using all 162 combinations ensured that this variable would be very close to its maximum value: 0.25-0.27 g acetic acid-(100 mL-h)\({}^{-1}\). Therefore, any of the 162 combinations would, in theory, lead to the highest possible mean rate of acetic acid formation in the two-bioreactor system--and hence to values of each operational variable falling in the ranges of Table 3.  
Conversely, minimizing \((r_{A})_{global\ est}\) led to the results of Table 4 (see file \"Optimal_rA.m\" in Supplementary Materials). The problem was like that stated in Equations (6) and (7), except that the aim was to minimize \((r_{A})_{global\ est}\) rather than maximize it. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**End** & \(\mathbf{V_{u1}}\)**, L** & \(\mathbf{E_{l1}}\)**, \% (\%/\(\mathbf{v}\))** & \(\mathbf{E_{u1}}\)**, \% (\%/\(\mathbf{v}\))** & \(\mathbf{T_{1}}\)**, \({}^{\circ}\)**C** & \(\mathbf{E_{l2}}\)**, \% (\(\mathbf{v}\)/\(\mathbf{v}\))** & \(\mathbf{T_{1}}\)**, \({}^{\circ}\)**C** \\ \hline Lower & 4.25 & 5.6 & 2.8 & 31.6 & 4.1 & \\ \(\updownarrow\) & \(\updownarrow\) & \(\updownarrow\) & \(\updownarrow\) & \(\updownarrow\) & \(\updownarrow\) & – \\ Upper & 4.50 & 6.0 & 3.2 & 32.0 & 4.5 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Upper and lower end of the ranges over which the operational variables maximized \((r_{A})_{global\ est}\). Figure 3: Response surfaces of \((r_{A})_{global\ est}\) varying \(E_{u1}\) and \(E_{l2}\) (**a**) as well as \(E_{l1}\) and \(E_{l2}\) (**b**), using values from Table 2 for the remaining operational variables. As in the previous case, the error made in modelling \((r_{A})_{global\,est}\) was used to examine the combinations of values of the operational variables, with provision for their errors (see Table 5), leading to minimum values over the range \(0.11\leq(r_{A})_{global\,est}\leq 0.12\) g acetic acid-(100 mL-h)\({}^{-1}\). The ranges of Table 5 for the operational variables, with 0.2 unit intervals, led to a total of 550 combinations only 29 of which provided an \((r_{A})_{global\,est}\) value falling in the previous range of minimum values (see Table S2 in file \"S1.docx\" in Supplementary Materials).