concentrations of glucose and oil, respectively. The parameters Y1 and Y2 correspond to the yield coefficients of glucose and oil in BS, respectively. The ODE was solved with the Berkeley – Madonna program, integrating with the Runge–Kutta method to solve the ODE, and also comparing the experimental values with those calculated by the model. Parameter estimation was carried out using the deepest descent methodology proposed by Marquardt in 1963 [102]; this technique minimizes the difference of the sum of squares between the experimental data and those calculated by the model. Table 2.4 shows the parameters estimated by the program. The estimation of the initial and final values of the three variables are similar to experimental data. However, the conversion yield estimation (Y1 and Y2) does not reflect the global yield of raw materials.
Figure 2.10 presents the results of the model simulation (lines) compared to the experimental data. Calculated values for substrate consumption fit well to those obtained experimentally. Glucose and oil consumption have a classic descending behavior of first‐order kinetics. However, the pattern of the BS formation, the most important variable of the process, does not fit correctly to experimental data, particularly during the first 3 days of culture. The curve calculated by the model is not sigmoidal; this latter is a typical pattern for microbial fermentations. Levenspiel [103] pointed out that the kinetics of the formation of the product of interest (sophorolipid in this case) is the basis for obtaining the design equations of ideal reactors. This is done by plotting the inverse of the product formation rate against the sophorolipid concentration (Figure 2.11).
Using this plot, the size of the batch reactor will be proportional to the area under the curve of the inverse of the reaction rate, evaluated between the limits of product formation (0–100 g/kgdry mass). On the other hand, in the case of continuous culture of SSF, the size of the bioreactor will be proportional to a rectangle whose base is the increase in product formation (0–100 g/kgdry mass) and the height of the rectangle is the value of the inverse of the reaction rate (1/rSL) evaluated at the point of discharge (~ 100 g/kgdry mass). A comparison of these two types of bioreactor systems indicates that the batch reactor is much smaller than the continuous culture reactor; therefore, up to this level of analysis, the batch reactor is the most suitable type of reactor to produce the sophorolipid by SSF. However, it is necessary to deepen the analysis of the concavity of the sophorolipid production curve to have a model that better represents the behavior of the production of BS in SSF.
Table 2.4 Estimated parameters to characterize the kinetics of the production of sophorolipids by SSF. The Runge–Kutta method of 4 order was used, in a time interval of 0–360 hours with an increase in time step of 0.05 hours.
| Parameters | Units | Parameter value | |
|---|---|---|---|
| First‐order constant rate | k 1 | h−1 | 0.01784 |
| First‐order constant rate | k 2 | h−1 | 0.00771 |
| Yield of glucose to sophorolipids | Y 1 | g SL/g Glucose | 3.99 × 10−5 |
| Yield of oil to sophorolipids | Y 2 | g SL/g oil | 0.69998 |
| Residual glucose constant | B | g/kg dry mass | 0.06616 |
| Residual oil | C | g/kg dry mass | 0.15279 |
| Initial glucose | G 0 | g/kg dry mass | 251.20 |
| Initial oil | Oil 0 | g/kg dry mass | 153.20 |
| Initial sophorolipid | SL 0 | g/kg dry mass | 5.92 × 10−14 |
Figure 2.11 Plot of the equation design of ideal bioreactors for the production of sophorolipids using a first‐order reaction rate.
To analyze the change in the concavity of the sophorolipid production curve already indicated, the Gompertz model was used. This model is a sigmoidal equation that has the characteristic of not being symmetric at the inflexion point; as is the case with the logistic equation, it is a very flexible model to simulate behavior such as that described for the production of BS in SSF. Equations (2.4) and (2.5) show the differential and integral form of the model. Equation (2.6) shows the solution of Equation (2.5) for the initial conditions of the fermentation:
Where P is the product concentration (sophorolipid), Pmax and P0, are the maximum (88.9) and initial (1.61) concentrations of BS in g/kgdry mass; k is a specific first‐order constant (0.032 h−1), and b (8.898) is a parameter associated with the initial concentration of BS. The mentioned parameters were estimated using Excel’s Solver subroutine. Figure 2.12 shows the result of the simulation of the experimental data by the Gompertz model. It is worth noticing the use of the Gompertz model, allows predicting the change in the concavity of the production rate of BS in SSF. The correlation coefficient between experimental and calculated data was 0.98.
