## 4 Conclusions  
Despite the broad available experience and technical knowledge available on the biotransformation of ethanol into acetic acid in the vinegar production process, a number of essential questions remain unanswered. Such is the case, for example, with the nature of the microbiota that effects the process and with its complex metabolic interactions. In practice, the process continues to require more or less extensive modelling in order to relate operational variables to specific objective functions. Black-box models based on generalized polynomials have proved especially suitable for representing the behaviour of acidification systems in the form of response surfaces. In this work, an effective experimental design based on useful data for the unequivocal calculation of the coefficients of the polynomial equations has been developed. Once the main operational variables have been identified, their admissible ranges have been established. That information has been used into an experimental design in order to identify the combinations of values of the operational variables that would allow the number of experiments maximizing the predictive ability of a model to be minimized. Another aim carried out in this work was to model key variables related to the acidification process performed with two serial bioreactors working in the semi-continuous mode by using quadratic polynomial equations. The operational variables considered were the ethanol concentration at the time of unloading (\(E_{t1}\)), unloaded volume (\(V_{u1}\)), ethanol concentration during loading (\(E_{11}\)) and operating temperature (\(T_{1}\)) in the first reactor, and the ethanol concentration during loading (\(E_{2}\)) and operating temperature (\(T_{2}\)) in the second. As it has been shown, the variation ranges for these variables are subject not only to physico-chemical constraints, but also to others arising from the fact that the two reactors operate in a serial mode and from the limited number of combinations allowed for by the experimental design. As a result, a fractional factorial design with 30 experiments (see Tables 5-7) considering the previous constraints has been obtained, with it being necessary to perform only 18 of them (see Tables 5 and 6). With the gathered experimental data, second-order polynomials for several target variables of interest in terms of the considered operational variables involved in the industrial production process were developed. Specifically, models for the mean overall rate of acetic acid formation in the two-bioreactor system [\((r_{A})_{global}\)], total acetic acid production in the system (\(P_{m}\)), ethanol concentration at the time the second bioreactor was unloaded (\(E_{u2}\)), the volume unloaded from the second bioreactor (\(V_{u2}\)), cycle duration (\(t_{cycle}\)), the mean overall volume in the two bioreactors during a cycle (\(V_{m}\)), the mean ethanol concentration in the first bioreactor during a cycle (\(EtOH_{m1}\)), the mean ethanol concentration in the second bioreactor during a cycle (\(EtOH_{m2}\)), the mean acetic acid concentration in the first reactor during a cycle (\(HA_{cm1}\)) and the mean acetic acid concentration in the second reactor during a cycle (\(HA_{cm2}\)) were obtained. The experimental results and their estimates were correlated via Forward Stepwise Regression, which allowed models with high predictive ability and minimal errors to be obtained. The resulting goodness of fit allowed a set of polynomials to be established that accurately reproduced the experimental results with only 18 experiments rather than the 30 needed in theory. Such models should allow us to carry out further optimization studies. With all models, the variables associated to the first bioreactor were the more influential on the process.