## Materials and Methods
### Mathematical Methods
#### 2.6.1 Polynomial Models and Fitting Methods
\(E_{\mathit{u1}}\) must be greater than 1% (_v/v_) in order to prevent ethanol depletion from happening too early in the first bioreactor--the second bioreactor was used for this purpose. Also, too low a value would slow down acidification and decrease productivity as a result [15]. 2. \(E_{\mathit{u1}}\) must not be too high. Otherwise, ethanol not consumed in the first bioreactor would have to be used in the second and the probability of complete depletion before unloading would be diminished. Also, ethanol concentrations higher than 5-6% (_v/v_) could detract from microbial activity [15; 45]. Therefore, \(E_{\mathit{u1}}\) must not exceed 5% (_v/v_). 3. \(E_{\mathit{u1}}<(E_{\mathit{1}})_{\mathit{max}}\) (Equation (2)). Constraints on \(V_{\mathit{u1}}\):  
This variable must range from 1 to 7 L if the self-aspirating turbine with which the bioreactors were equipped is to operate properly. The turbine was used to feed oxygen to the microorganisms and help homogenize the culture medium. Constraints on \(T_{\mathit{1}}\) and \(T_{\mathit{2}}\):  
The values of these variables must be compatible with the activity of acetic acid bacteria (AAB), which are the microorganisms affecting the process. The optimum temperature for AAB to oxidize ethanol into acetic acid is 25-35 \({}^{\circ}\)C. Because AAB activity drops outside this range [46; 47], temperatures from 26 to 34 \({}^{\circ}\)C have been used here. Constraints on \(E_{\mathit{2}}\):  
1. The ethanol concentration in the feed wine is typically 12% (_v/v_). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Experiment No. ** & \(V_{\mathit{u1}}\) & \(E_{\mathit{u1}}\) & \(E_{\mathit{1}}\) & \(E_{\mathit{12}}\) & \(T_{\mathit{1}}\) & \(T_{\mathit{2}}\) \\ \hline  
2. \(E_{l2}\) must be less than the maximum ethanol concentration in the medium of the second bioreactor at any time. Such a concentration depends on \(E_{u1}\), \(V_{u1}\) and the ethanol concentration of feed wine, and can be easily obtained from a simple mass balance: \[(E_{l2})_{max}=\frac{V_{u1}\cdot E_{u1}+(8-V_{u1})\cdot 12}{8},\] (3)  
Based on the previous constraints, \(E_{u1}\), \(V_{u1}\), \(E_{l1}\) and \(E_{l2}\) are mutually related; also, feasible \(E_{u1}\) and \(V_{u1}\) values span the range 1-5% (\(v\)/\(v\)) and 1-7 L, respectively. However, not all potential combinations of the values of the operational variables can be used in practice. Thus, the initial volume of the second bioreactor--unloaded from the first--\(V_{u1}\), together with \(E_{u1}\), imposes some additional constraints on \(E_{l1}\) and \(E_{l2}\). As a result, the experimental design is subjected to the following constraints:  
* Based on constraint 2 on \(E_{l1}\) and the experiments of Table 1, \(E_{l1}\) level (\(-1\)) must not be less than \(E_{u1}  
considered for level (0) were 1-5% (_v/v_) for \(E_{u1}\), \(E_{\Pi 1}\) and \(E_{\Pi 2}\); and 1-7 L for \(V_{u1}\), all in 0.5 steps. A 0.5 step over the range 0.5-5 was used for iteration of levels (\(-2\)) and (\(+2\)) in each case. The script initially identified 148611 feasible combinations. Such a large number required additional constraints to be imposed based on the following practical arguments:  
A1. There must be a simultaneous maximum difference between levels (\(-1\)) and (\(+1\)) in \(E_{u1}\) and \(V_{u1}\) (see Table 1). This constraint expanded the feasible range of operational conditions and allowed the influence of each variable and mutual relations to be more accurately detected and examined as a result. A2. Because in the semi-continuous operation mode it is a common practice to unload at least half of the working volume from the first bioreactor, \(V_{u1}\) level (\(-1\)) must be greater than or equal to 4 L.  
A3. Because the initial ethanol concentration in the second bioreactor must not be too high in order to facilitate appropriate performance (see constraint 2 on \(E_{u1}\)), \(E_{u1}\) level (\(+2\)) must be less than 5% (_v/v_).