## 3 Results and Discussion  
### Kinetic Modeling of Cell Growth  
In order to create a model for describing the growth of microorganisms, it is necessary to measure growth curves. A kinetic model can be used to define the growth curve. To achieve this, the logistic model was implemented and evaluated for biomass production to comply with the experimental data [23]. Table 1 shows the calculated parameters of the logistic equation. In Equation (8), \(X_{0}\) and \(\mu_{max}\) were found by plotting the values of \(ln\left(\frac{X}{X_{\text{max}}-X}\right)\) against \(t\). As shown in Figure 1A,D,G,J,M, the slope of the plots gives \(\mu_{max}\), which were 0.4076, 0.4002, 0.296, 0.3618, and 0.3283 h\({}^{-1}\) for Media A, B, C, D, and E, respectively (Table 1). Please note that Xo values for each medium in Table 1 are slightly different due to the nature of repeated biofilm fermentation without any new inoculation. When yeast extract was not used in the medium (Medium C), \(\mu_{max}\) considerably decreased. On the other hand, when ammonium sulfate was not used and yeast extract was used in the environment, \(\mu_{max}\) slightly reduced. Additionally, when mineral salts were only utilized in the medium and no nutrients were employed in the medium, \(\mu_{max}\) also reduced. It is thought that decrease of \(\mu_{max}\) in Medium C and relative increase of \(\mu_{max}\) in Media D and E compared to Medium C is probably related to the release of the nutrients in PCS materials used in the biofilm reactor [44]. Besides, \(X_{0}\) is calculated from the \(y\)-shift of the plots in Equation (8) (Table 1). It was found that the calculated \(X_{0}\) values were all lower than the experimental \(X_{0}\) values. The experimental and predicted cell growth data were also plotted versus time (Figure 3). It can be said that the logistic equation underestimated the actual cell growth at lag phase while it over-predicted the experimental cell growth data at log phase (Figure 3). Besides, the logistic equation only predicted the observed cell growth data at acceleration, deceleration, and stationary phases (Figure 3). Additionally, the cell growth in the biofilm reactor including Media A-E was described by the logistic function based on the experimental data, by applying the model parameters given in Table 1. The model results indicated that the lowest values of RMSE and MAE were achieved when Medium E was used in the biofilm reactor. On the contrary, the maximum levels were determined when using Medium A in the biofilm reactor. In addition, the resulting \(R^{2}\) and slope values ranged between 0.96 and 0.99 and between 1.06 and 1.19. Similarly, Medium A gave the lowest value of \(R^{2}\) while Medium E yielded its highest value. Therefore, it can be concluded that the estimated results gave a relatively good agreement with the experimental data, with \(R^{2}\) values above 0.9 [36]. This indicates that the suggested model was sufficient to define _S. cerevisiae_ growth in batch cultivation. Similar to \(R^{2}\), Medium A yielded the highest slope value; however, Medium E gave the lowest slope value. The slope is a number that indicates how much and which direction a line of regression is fitting. It was used when estimating the variable \(Y\) (dependent), where \(Y\) represents the predicted values. Therefore, as long as the slope was close to 1, the estimated values were closer to the experimental values [45]. On the other hand, Table 1 shows the BF and AF values. Results indicated that the logistic equation was \"acceptable\" due to BF values (1.11 \(<\) BF \(<\) 1.43). According to the AF value, the logistic model was \"acceptable\" for Media C, D and E (1.20 \(\leq\) AF \(\leq\) 1.30), except for Media A and B (AF \(>\) 1.30) (Table 1). In conclusion, the logistic model adequately described the growth of the yeast _S. cerevisiae_ in the biofilm reactor including different nutritional compositions.