## 3 Results and Discussion
### Optimization of the Mean Rate of Acetic Acid Formation in the Two-Bioreactor System
However, any of the 550 combinations ensured that \((r_{A})_{global\,est}\) would fall in the range 0.116-0.135 g acetic acid-(100 mL-h)\({}^{-1}\) and thus be very close to the minimum of Table 4. Therefore, any combination would provide the lowest mean rate of acetic acid formation in the two-bioreactor system and lead to values of the operational variables falling in the ranges of Table 5.  
Applying the equations obtained in [27] to the other variables (Equations (8)-(16)) provided the estimated values of Table 6 for \(P_{m\,est}\), \(E_{u2\,est}\), \(V_{u2\,est}\), \(t_{cycle\,est}\), \(V_{m\,est}\), \(EtOH_{m1\,est}\), \(EtOH_{m2\,est}\), \(HAC_{m1\,est}\) and \(HAC_{m2\,est}\) under the operating conditions maximizing or minimizing \((r_{A})_{global\,est}\). The ranges of some variables reflect differences due to the influence of \(T_{2}\) --like \((r_{A})_{global\,est}\), other variables were independent of the temperature in the second reactor, however. \[ \begin{array}{rl}P_{m\,est}=&-243.705+18.324\cdot V_{u1}+72.736\cdot E_{l1}+ 21.525\cdot E_{u1}\\ &-9.708\cdot E_{l1}^{2}-1.102\cdot E_{u1}\cdot V_{u1}-0.534\cdot T_{1}\cdot V_{ u1}\\ &+0.742\cdot T_{1}\cdot E_{l1}-0.416\cdot E_{l2}\cdot E_{l1}+0.175\cdot T_{2} \cdot E_{l1}\\ &-0.12\cdot T_{1}\cdot E_{u1}-0.399\cdot T_{2}\cdot E_{u1}+0.101\cdot T_{2} \cdot E_{l2}\\ \end{array} , \tag{8}\]  
\[ \begin{array}{rl}E_{u2\,est}=&14.935-5.371\cdot E_{u1}+0.988\cdot E_{u1}^{2} -0.0592\cdot E_{l1}\cdot V_{u1}\\ &-0.456\cdot E_{u1}\cdot V_{u1}+0.0678\cdot E_{u1}\cdot E_{l1}\\ &+0.494\cdot E_{l2}\cdot E_{u1}-0.049\cdot T_{2}\cdot E_{l2}\\ \end{array} , \tag{9}\]  
\[ \begin{array}{rl}V_{u2\,est}=&4.921+1.399\cdot E_{l2}-0.59\cdot E_{u1}^{2} \\ &+0.665\cdot E_{u1}\cdot V_{u1}-0.324\cdot E_{l2}\cdot V_{u1}\\ &+0.172\cdot E_{l2}\cdot E_{u1}-0.0275\cdot T_{2}\cdot E_{u1}\\ \end{array} , \tag{10}\]  
\[ \begin{array}{rl}t_{cycle\,est}=&518.591-156.652\cdot V_{u1}\\ &-6.474\cdot T_{1}+13.556\cdot V_{u1}^{2}\\ &+1.076\cdot T_{1}\cdot V_{u1}-0.864\cdot E_{u1}\cdot E_{l1}\\ &-0.125\cdot E_{l2}\cdot E_{l1}+0.0326\cdot T_{2}\cdot E_{l1}\\ &-0.0735\cdot T_{2}\cdot E_{u1}+0.0358\cdot T_{2}\cdot E_{l2}\\ \end{array} , \tag{11}\]  
\[ \begin{array}{rl}V_{m\,est}=&4.306+4.739\cdot E_{u1}-0.841\cdot E_{u1}^{2} \\ &+0.336\cdot E_{u1}\cdot V_{u1}-0.0127\cdot T_{2}\cdot V_{u1}\\ &-0.125\cdot E_{l2}\cdot E_{l1}+0.0326\cdot T_{2}\cdot E_{l1}\\ &-0.0735\cdot T_{2}\cdot E_{u1}+0.0358\cdot T_{2}\cdot E_{l2}\\ \end{array} , \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**End** & \(V_{u1}\)**, **L** & \(E_{l1}\)**, **\% (v/v)** & \(E_{u1}\)**, **\% (v/v)** & \(T_{1}\)**, \({}^{\circ}\)C** & \(E_{l1}\)**, **\% (v/v)** & \(T_{2}\)**, \({}^{\circ}\)C** \\ \hline Lower & 5.45 & 4.0 & & & 28.0 & 2.5 Based on the previous results, the highest and lowest mean rate of acetic acid production were obtained by using extreme values of some operational variables. Thus, the minimum (\(r_{A}\))\({}_{global\,est}\) value (Table 4) was obtained with the lowest temperature in the first bioreactor (\(T_{1}=28\)\({}^{\circ}\)C), and the smallest values in the ranges of \(E_{l1}\) and \(E_{u1}\) [viz., 4% and 2% (\(v/v\)), respectively]. These results suggest that the smaller the amount of substrate available to AAB and the higher the acidity to which the bacteria are exposed during a fermentation cycle (see \(EtOH_{m1\,est}\) and \(HAc_{m1\,est}\) values in Table 6) are, the less suitable will be the medium for their metabolic action --and hence for acidification. Also, because the volume of medium unloaded from the first bioreactor was quite substantial (\(V_{u1}=5.75\) L), the amount of inoculum remaining in it for the next cycle was considerably reduced, and so was (\(r_{A}\))\({}_{global\,est}\) as a consequence. This result is consistent with those of modelling studies on a single bioreactor using both first-principles and black-box models [8; 9; 10; 22; 23; 31; 32].