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Mathematical models not only ensure information about kinetic-metabolic nature of fermentations, but also facilitate their control and optimization. In the study, flexible ten models were evaluated and employed to describe the ethanol fermentation in a biofilm reactor with a carob extract medium (CEM). Findings indicated that W model well fitted the experimental data of cell growth (root mean square error (RMSE) = 0.289 g/L, mean absolute error (MAE) = 0.237 g/L, regression coefficient (R2) = 0.9944, bias factor (BF) = 1.021, and accuracy factor (AF) = 1.047), ethanol production (RMSE = 1.609 g/L, MAE = 1.277 g/L, R2= 0.9774, BF = 1.178, and AF = 1.283), and substrate consumption (RMSE = 2.493 g/L, MAE = 1.546 g/L, R2= 0.9931, BF = 1.001 and AF = 1.053). In the prediction of kinetic parameters, W model also gave better and well-directed results compared with the other mathematical models used in the study. When an independent set of experimental data was used in the validation of mathematical models, similar validation results were obtained and W model was also successful. Consequently, W model could be used for more progress of fermentation process in biofilm reactor with CEM, which can serve as a universal equation.

The medium composition and fermentation parameters play a critical role in the fermentation process. Therefore, in the present study, the effects of fermentation parameters (initial glucose concentration (IGC), inoculation ratio (IR), initial pH value (IPV), stirring rate (SR), and temperature (T)) and medium composition (yeast extract (YE) and peptone (PEP)) on bioethanol production from glucose by using Scheffersomyces stipitisin shake flask fermentation were studied, respectively. According to results, the highest ethanol yield (YP/S) was achieved to be 47.63% when IGC, IR, IPV, SR, T, and nitrogen concentrations were 15 g/L, 5% (v/v), 5.5, 150 rpm, 30 °C, 5 g/L YE, and 20 g/L PEP, respectively. Nevertheless, the best fermentation parameters were 15 g/L IGC, 5% (v/v) IR, 5.5 IPV, 150 rpm SR, 30 °C T, no YE, and 5 g/L of PEP, respectively. At these conditions, kinetic results including bioethanol production, YP/S, ethanol yield per biomass, maximum production rate, sugar utilization yield, and theoretical YP/Swere 8.29 g/L, 47.26%, 5.08 g ethanol/g biomass, 0.41 g/L/h, 95.86%, and 92.49%, respectively. In conclusion, these results indicated that although the concentrations of ingredients (YE and PEP) used in the fermentation medium were considerably decreased from 10 and 20 g/L to 0 and 5 g/L, the kinetic results of the fermentation were preserved compared to the initial fermentation studies. Nevertheless, further improvement of the determined fermentation parameters could be performed in the bioreactor.

This study aimed to develop a model to estimate experimental substrate specificity (inulin and sucrose) data for Aspergillus nigerinulinase. For this purpose, ten mathematical models were used. Root-mean-square-error (RMSE), mean-absolute-error (MAE), mean standard deviation (MSD), determination coefficient (R2), slope (m), Akaike information criterion (AIC), bias factor (BF), and accuracy factor (AF) were used to be able to evaluate the models. Besides, the experimental enzyme kinetics were also compared with those calculated from the models. Also, the statistical significance of the models was determined by f-testing. Furthermore, the sensitivity analysis of the selected model was performed by making a 5% increase and decrease in each model parameter. Based on the results, the von Bertalanffy model was the best (RMSE = 28.48 and 27.34 U/mL, MAE = 21.53 and 25.14 U/mL, MSD = 15.23 and 17.77 U/mL, R2= 0.9908 and 0.9962, m= 1.02 and 1.00, AIC = 81.93 and 84.79, BF= 1.00 and 1.00, and AF= 1.04 and 1.06 for inulinase and invertase-type activity, respectively). The predicted inhibition constant and maximum enzyme velocity for inulin and sucrose, which were highly good agreement with their experimental values, were calculated as 17.43 mg/mL and 1101 U/mL and 51 mg/mL and 1625 U/mL, respectively. According to the f-testing results, only the Mitscherlich model was significant for invertase-type activity while other all models except for the Stannard model were statistically significant for inulinase activity (p< 0.05). In the case of a deviation of 5% in the experimental values, extremely satisfactory results were obtained by the sensitivity analysis. Consequently, this is the first study to develop a model for predicting the kinetic parameters of the inulinase enzyme. The suggested model can be used in the industry to readily calculate the parameters regarding enzyme kinetics.

In this study, various mathematical functions were utilized to fit the observed values and the kinetic parameters of fermentation in immobilized-cell stirred tank reactor (ICSTR) with carob extract medium (CEM). The best model was selected based on the residual sum of squares (RSS), root-mean-square-error (RMSE), mean-absolute-error (MAE), determination coefficient (R2), slope (m), bias factor (BF), accuracy factor (AF), the objective function (Φ-factor), and f-testing. Therefore, the Stannard (ST) model was good agreement with the actual biomass production (X), ethanol production (P), and sugar consumption (S) data (RSS = 0.22, 6.23, and 45.70 g/L; RMSE = 0.17, 0.88, and 2.39 g/L; MAE = 0.14, 0.64, and 1.72 g/L; R2= 0.9948, 0.9977, and 0.9972; m= 1.03, 1.02, and 1.02; BF= 1.35, 1.08, and 0.97; AF= 1.41, 1.14, and 1.04; and Φ-factor = 0.008, 0.004, and 0.005 (∑Φ-factor = 0.017) for X, P, and S, respectively). In the prediction of kinetic parameters belonging to the experimental fermentation, the ST model also gave better, satisfactory, and well-directed results than the other mathematical models. Besides, the ST model was also accomplishedly validated using two independent sets of the experimental data yielded from the different carbon sources in various concentrations. Consequently, the models proposed for X, P, and Scan serve as universal functions to predict the real values and the kinetic parameters of the ethanol fermentation. Besides, these models can be applied to improve the ethanol fermentation in the ICSTR with CEM.