## 4 Conclusions  
The two serially arranged bioreactors system for the acidification process has been optimized in various respects using previous quadratic polynomial models. The values of the operational variables maximizing or minimizing the mean rate of acetic acid production were obtained: a maximum value of \((r_{A})_{global\,est}\) = 0.27 g acetic acid-(100 mL-h)\({}^{-1}\) was achieved when \(V_{u1}\) = 4.25 L, \(E_{l1}\) = 6% (\(v/v\)), \(E_{u1}\) = 3.07% (\(v/v\)), \(T_{1}\) = 32 \({}^{\circ}\)C, \(E_{l2}\) = 4.5% (\(v/v\)) and a minimum value of \((r_{A})_{global\,est}\) = 0.11 g acetic acid-(100 mL-h)\({}^{-1}\) was achieved when \(V_{u1}\) = 5.75 L, \(E_{l1}\) = 4% (\(v/v\)), \(E_{u1}\) = 2% (\(v/v\)), \(T_{1}\) = 28 \({}^{\circ}\)C, \(E_{l2}\) = 4.5% (\(v/v\)).  
In addition, the conditions leading to the maximum and minimum productivity were: \(P_{m\,est}\) = 36.6 g acetic acid-h\({}^{-1}\) was achieved when \(V_{u1}\) = 4.25 L, \(E_{l1}\) = 5.1% (\(v/v\)), \(E_{u1}\) = 4% (\(v/v\)), \(T_{1}\) = 32 \({}^{\circ}\)C, \(E_{l2}\) = 4.5% (\(v/v\)), \(T_{1}\) = 32 \({}^{\circ}\)C, \(E_{l2}\) = 4.5% (\(v/v\)), \(T_{2}\) = 32 \({}^{\circ}\)C to achieve 34.6 \(\leq P_{m}\leq\) 35.5 g acetic acid-h\({}^{-1}\) with \(0.2\leq E_{u2}\leq\) 1.5% (\(v/v\)). Such conditions allowed near-maximum productivity to be obtained.  
One other aim fulfilled by modelling the system and using results of previous research work was to compare the performance of the system with the two reactors operating in series and in parallel. \(P_{m}\) can be maximized with no restrictions on \(E_{u2}\) by using either configuration. However, if the substrate is to be depleted [e.g., \(E_{u2}\) = 0.5% (\(v/v\))], then the serial configuration will be more productive.  
The following are available online at [[http://www.mdpi.com/xxx/s1](http://www.mdpi.com/xxx/s1)]([http://www.mdpi.com/xxx/s1](http://www.mdpi.com/xxx/s1)). \"Optimal_rA.m\" and \"Optimal_Prod.m\": MATLAB scripts for optimizing \((r_{A})_{global\,est}\) and \(P_{m\,est}\), respectively. File \"S1.pdf\": Combinations of operational variables to analyze the maximum and minimum of \((r_{A})_{global\,est}\) and \(P_{m\,est}\).  
Conceptualization, I.G.-G.; methodology, C.M.A.-C., I.M.S.-D. and I.G.-G.; formal analysis, C.M.A.-C., I.M.S.-D., J.E.J.-H. and [PERSON]-G.; data curation, C.M.A.-C. and I.M.S.-D.; writing--original draft preparation, J.E.J.-H. and I.G.-G.; writing--review and editing, J.E.J.-H., I.G.-G. and I.M.S.-D.; funding acquisition, I.G.-G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by \"XXIII Programa Propio de Fomento de la Investigacion 2018\" (MOD 4.2. SINERGIAS, Ref XXIII. PP Mod 4.2) from University of Cordoba (Spain) and by \"Programa PAIDI\" from Junta de Andalucia (RNM-271).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Optimizing Production**} & \multicolumn{3}{c}{**Optimizing Production with a Specific Final \(E_{u2}\) Value**} \\ \hline Working mode & Parallel & Series & Parallel & Series & \\  & (31 \({}^{\circ}\)C) & \(T_{1}\) = 31 \({}^{\circ}\)C & \(T_{1}\) = 32 \({}^{\circ}\)C & (31 \({}^{\circ}\)C) & \(T_{1}\) = 31 \({}^{\circ}\)C & \(T_{1}\) = 32 \({}^{\circ}\)C \\  & \(T_{2}\) = 31 \({}^{\circ}\)C & \(T_{2}\) = 28 \({}^{\circ}\)C & & \(T_{2}\) = 31 \({}^{\circ}\)C & \(T_{2}\) = 32 \({}^{\circ}\)C \\ \(p_{m}\) & & & & & & \\ g acetic acid-h\({}^{-1}\) & 35.2 \(\pm\) 0.5 & 34.8 \(\pm\) 0.7 & 35.9 \(\pm\) 0.7 & 29.6 \(\pm\) 0.5 & 33.2 \(\pm\) 0.7 & 34.2 \(\pm\) 0.7 \\ \(E_{u2}\) & & & & & & \\ \% (\(v/v\)) & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 14: Comparison of acetic acid production and final ethanol concentration at the time of unloading with two bioreactors operating in series and in parallel.