## 3 Results and Discussion
### Kinetic Modeling of Cell Growth
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter Estimation** & **Medium A** & **Medium B** & **Medium C** & **Medium D** & **Medium E** \\ \hline  & & \multicolumn{3}{c}{Cell growth} & & \\ \(\mu_{max}\) (h\({}^{-1}\)) & 0.4076 & 0.4002 & 0.2960 & 0.3618 & 0.3283 \\ \(X_{0}\) (g/L) & 0.80 & 0.53 & 0.82 & 0.73 & 0.62 \\ \(X_{max}\) (g/L) & 9.98 & 9.73 & 9.73 & 9.02 & 8.96 \\ RMSE (g/L) & 1.04 & 0.71 & 0.61 & 0.57 & 0.41 \\ MAE (g/L) & 0.80 & 0.57 & 0.49 & 0.46 & 0.31 \\ \(R^{2}\) & 0.9561 & 0.9750 & 0.9791 & 0.9804 & 0.9914 \\ Slope & 1.19 & 1.10 & 1.08 & 1.09 & 1.06 \\ BF & 1.29 & 1.28 & 1.19 & 1.18 & 1.19 \\ AF & 1.38 & 1.37 & 1.26 & 1.25 & 1.23 \\  & Ethanol production & & & & & \\ \(\beta\) (g /g X.h) & 0.0011 & 0.0241 & 0.0329 & 0.0203 & 0.0571 \\ \(a\) (g /g X) & 2.6724 & 2.2205 & 1.6429 & 2.2757 & 1.3559 \\ RMSE (g/L) & 3.46 & 1.71 & 1.84 & 1.98 & 1.43 \\ MAE (g/L) & 2.11 & 1.22 & 1.40 & 1.54 & 0.88 \\ \(R^{2}\) & 0.9217 & 0.9794 & 0.9533 & 0.9644 & 0.9870 \\ Slope & 0.92 & 0.98 & 0.97 & 0.97 & 1.07 \\ BF & 0.80 & 0.86 & 0.91 & 0.85 & 0.85 \\ AF & 1.26 & 1.18 & 1.16 & 1.21 & 1.17 \\  & & & & & & \\ Z (g /g X.h) & 0.0012 & 0.0284 & 0.0106 & 0.0120 & 0.0048 \\ \(\gamma\) (g /g X) & 5.3443 & 5.3409 & 5.5775 & 5.4862 & 5.723 \\ RMSE (g/L) & 1.51 & 1.40 & 0.38 & 2.19 & 1.01 \\ MAE (g/L) & 0.87 & 0.65 & 0.20 & 1.34 & 0.63 \\ \(R^{2}\) & 0.9961 & 0.9921 & 0.9997 & 0.9936 & 0.9949 \\ Slope & 1.00 & 1.00 & 1.00 & 0.99 & 0.99 \\ BF & 1.04 & 1.01 & 1.02 & 0.86 & 0.96 \\ AF & 1.04 & 1.10 & 1.02 & 1.16 & 1.07 \\ \hline \(\Phi\)-value & 0.212 & 0.078 & 0.075 & 0.096 & 0.055 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The calculated model parameters for logistic, Luedeking-Piret (LP), and Modified Luedeking-Piret (MLP) equations. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Medium** & **A** & **B** & **C** & **D** & **E** & **A** & **B** & **C** & **D** & **E** \\ \hline \(X\) kinetics & \multicolumn{6}{c}{Experimental} & & \multicolumn{6}{c}{Logistic model} \\ \(\Delta X\) (g/L) & 7.28 & 8.01 & 7.78 & 7.47 & 7.56 & 9.19 & 9.20 & 8.89 & 8.28 & 8.33 \\ \(Y_{XS}\) (\%) & 14.83 & 14.98 & 15.17 & 14.82 & 15.14 & 18.60 & 16.60 & 17.16 & 17.28 & 17.14 \\ \(Q_{X}\) (g/L) & 0.71 & 0.77 & 0.64 & 0.65 & 0.64 & 0.77 & 0.79 & 0.65 & 0.67 & 0.65 \\ \(\lambda\) (h) & 4.83 & 5.18 & 5.71 & 4.31 & 4.50 & 0.70 & 1.81 & 1.78 & 1.24 & 1.94 \\ \(\mu_{max}\) (h\({}^{-1}\)) & 0.41 & 0.40 & 0.30 & 0.36 & 0.33 & 0.41 & 0.40 & 0.30 & 0.36 & 0.33 \\ \(P\) kinetics & \multicolumn{6}{c}{Experimental} & & \multicolumn{6}{c}{LP model} \\ \(\Delta B\) (g/L) & 24.51 & 24.12 & 21.80 & 23.50 & 19.24 & 24.82 & 25.75 & 21.52 & 23.07 & 22.48 \\ \(Y_{PS}\) (\%) & 49.94 & 45.07 & 42.48 & 46.58 & 38.55 & 50.26 & 46.49 & 41.55 & 48.12 & 46.22 \\ \(Y_{pX}\) (g/g) & 3.37 & 3.01 & 2.80 & 3.14 & 2.55 & 2.70 & 2.80 & 2.42 & 2.78 & 2.70 \\ \(Q_{P}\) (g/L/h) & 2.56 & 2.19 & 1.40 & 1.90 & 1.16 & 2.05 & 1.90 & 1.22 & 1.64 & 1.13 \\ \(\eta\) (\%) & 97.73 & 88.20 & 83.14 & 91.16 & 75.43 & 98.36 & 90.97 & 81.31 & 94.17 & 90.46 \\ \(S\) kinetics & \multicolumn{6}{c}{Experimental} & & \multicolumn{6}{c}{MLP model} \\ \(\Delta S\) (g/L) & 49.09 & 53.51 & 51.30 & 50.44 & 49.91 & 49.38 & 55.40 & 51.81 & 47.94 & 48.63 \\ \(Y_{XS}\) (g/g) & 6.74 & 6.68 & 6.59 & 6.75 & 6.60 & 5.38 & 6.02 & 5.83 & 5.79 & 5.84 \\ \(Q_{S}\) (g/L/h) & 4.31 & 5.00 & 3.69 & 3.80 & 3.47 & 4.10 & 4.40 & 3.65 & 3.77 & 3.71 \\ \(SUY\) (\%) & 87.02 & 86.87 & 85.77 & 88.06 & 86.14 & 87.54 & 89.95 & 86.61 & 83.70 & 83.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The calculated kinetics from the experimental and the predicted data related to cell growth, ethanol production and substrate consumption.