To describe the oscillatory behavior of continuous ethanolic fermentation using yeasts or bacteria, the mathematical model employed should contain the necessary expressions showing the variation of the state variables with time. Some models have been proposed with this aim and contrasted with the experimental data in order to validate their suitability. Among these models, the mathematical descrip­tion of Jarz^bski (1992) should be highlighted. This model is based on stochastic assumptions taking into account the distribution of cell population and corresponds to the continuous fermentation in a stirred tank using the yeast Saccharomyces cerevisiae. This model divides the cell biomass into three groups: viable biomass (cells able to reproduce and biosynthesize ethanol), nonviable biomass (cells able to biosynthesize ethanol, but not to reproduce), and dead biomass (cells able nei­ther to reproduce nor biosynthesize ethanol). Moreover, the kinetic model includes terms describing the maintenance of viable and nonviable cells. The results derived from this model were compared to the experimental data obtained by Perego et al. (1985). The equations of the model are as follows:

rate (in h-1), p is the specific growth rate (in h-1). The subindexes v, nv, and d refer

to viable, nonviable, and dead cells, respectively. The biomass growth (death) rate expressions are as follows:

(7.9)

where pmax, K1, K2 , and Pc are kinetic constants, which can be found in the original work of Jarz^bski (1992).

In a previous work (Restrepo et al., 2007), the simulation of the continuous system using the available information and parameters reported by Jarz^bski (1992) was performed, but the results were not satisfactory because there was no appropriate correspondence with the experimental data reported by Perego et al. (1985) for continuous fermentation using sugarcane molasses as a feedstock. Using the software Matlab™ (Mathworks, Inc., USA), the curves corresponding to the dynamic behavior of such fermentation system were obtained (Figure 7.11). Nondynamic analysis (bifurcation analysis) is a powerful tool to study the oscil­latory behavior of continuous ethanolic fermentations. This tool was used in the previous mentioned work in order to assess whether the model can describe the dynamic behavior of the system. For the analysis of the fermentation system stud­ied, the software MatCont developed in the University of Gent (Belgium) was employed as a toolbox in Matlab package. The results of bifurcation analysis for the Jarz^bski’s model are shown in Figure 7.12. From this diagram in the zone hav­ing a physical sense (positive dilution rates), it can be observed that the dynamic system behaves in an ordinary way and presents no sustained oscillations (Hopf bifurcations). In the zone of negative dilution rates, limit point (LP or node-saddle bifurcation) and Hopf bifurcation are present indicating that the model does have the possibility to represent the oscillations.

For representing the sustained oscillations and the instability of the contin­uous fermentation, the Jarz^bski’s model was modified and adjusted in such a way that the model represents not only the oscillatory fermentation reported by Perego, Jr. et al. (1985) appropriately, but also the data for continuous nonoscil­latory fermentation obtained by these same authors. The modification consisted in the simplification of the nonviable biomass growth rate (pnv) considering that the inhibition of cell biomass is mainly due to the high ethanol concentration. In addition, the expression describing the biomass death rate (pnv) was changed in such a way that this rate was proportional to a fraction d of the viable biomass growth rate (pnv). Therefore, the model modified comprises the system (7.8) and the following rate

Once the model was modified, an optimization applying a simple multiobjec­tive strategy was performed with the aim of adjusting the model parameters by minimizing the nonlinear least squares. Using the Nelder-Mead algorithm, a point is searched without calculating the derivatives. This facilitates the formulation of the two objective functions used (one for adjusting the biomass concentration, and another one for adjusting the substrate concentration). These functions numerically integrate the system using the numeric differentiation formulae (NDF) and an algo­rithm for controlling the integration step size with a defined absolute tolerance. Then, the experimental data reported for both steady-state fermentation and oscil­latory fermentation were employed to calculate the sum of the squares of the residu­als, which were subsequently minimized by using the mentioned algorithm. The outcomes obtained are presented in Figure 7.13. As can be observed, the adjust­ment of the model to the experimental data was appropriate and allowed describing the behavior of the two types of fermentations studied. Unlike the nonmodified Jarz^bski model, all the concentrations had physical sense. In the previous work reported (Restrepo et al., 2007), the values of the kinetic parameters were calcu­lated. These values are presented in Table 7.5.

D (1/h)

D (1/h)

FIGuRE 7.14 Response diagram of the continuous ethanolic fermentation process for viable cell biomass in dependence on the dilution rate D.

When the bifurcation analysis has been performed based on the modified model, a Hopf bifurcation can be noted in the response diagram near D = 0.089 h-1 for the viable biomass in dependence on the dilution rate (Figure 7.14) indicating the appearance of the oscillatory fermentation. Similarly, the point with the maximum concentration of viable cells was also established. This point matches with the max­imum ethanol concentration and corresponds to an inlet substrate concentration of

137.5 kg/m3. Finally, through bifurcation analysis, it is possible to delimit the zones of the operating diagram S0 versus D (Figure 7.15). In particular, the parameter S0 has demonstrated a strong influence on the fermentation behavior. Small changes in the nutrient composition of the culture broth entering the continuous fermenter can provoke significant changes in the response variables (concentrations of biomass, ethanol, and residual substrate). Knowing these zones, the task of operating the fermenter in the stable zones becomes easier. Moreover, the design of the control

FIGURE 7.15 Operating diagram of the continuous ethanolic fermentation for inlet sub­strate concentration (S0) and dilution rate (D).

structure can be based on these nonlinear analysis tools. Therefore, the modified model can predict when the instability of the fermentation will occur. At this point, the instability of the system can be considered during the early step of conceptual design, which permits a better design of the automatic control system. This issue is of great importance during the operation of industrial ethanolic fermentations.