## 2 Results and Discussion
### Statistical Analysis  
#### 2.3.1 Model Fitting and Analysis of Variance (ANOVA)  
Table 4 presents the conditions proposed by the experimental design to determine the temperature, time, molar ratio (FFAs/alcohol) and biocatalyst content for the production of biodiesel using the free fatty acids from residual babassu oil, ethanol and Novozym(r) 435 as biocatalyst. In total, 27 experimental tests were performed (24 tests at factorial points, 3 with repetitions at the central point). According to the author [PERSON] et al. 2020 [71], repetitions at the central point are necessary, as they allow the estimation of pure experimental errors and verify the fit of the model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & **Temperature** & **Time** & **Molar Ratio** & **Bicatalyst Content** & **Conversion** \\  & **(\({}^{\circ}\)C)** & **(hours)** & **(FFAs/alcohol)** & **(grams)** & **(\%)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: Complete experimental design with 18 factorial points, 6 axial points (2 axial points on the axis of design variable) and 3 replications at the central point, totaling 27 experiments, with the result for the conversion into esters for the esterification of residual babassu oil. Based on Table 4, it is possible to observe that the highest conversion to fatty acid ethyl esters was found in test 16 with a conversion of 91.12%, at a temperature of 50 \({}^{\circ}\)C, for 6 h, in the molar ratio of 1:15 (FFAs/alcohol) and in the biocatalyst content of 0.15 g. Through the software Statistica(r) 10 software (Statsoft, Tulsa, OK, USA), it was possible to determine the optimized conditions for the synthesis of biodiesel (0.14 g of biocatalyst, molar ratio of 1:18 (FFAs/alcohol), temperature of 48 \({}^{\circ}\)C in 4 h), obtaining a yield theoretical of 95.87%. The validation of the proposed model was performed by chromatographic analysis of ethyl esters, following the standard EN 14,103 [72] with some modifications, the value obtained was 96.84%. The Resolution 51/2016 of the Brazilian National Petroleum Agency states that to be considered biodiesel the conversion to esters must present a minimum value of 96.5%. Based on the chromatographic analysis, biodiesel was produced in the present study. In Table 5, the ANOVA results are presented to adjust the response surface model of second order, using the mean square method. According to equation 2, the coefficients of the response surface model were analyzed. To test the statistical significance of the model, the F-test for ANOVA was used [73]. A very low P (<0.0001) of the model in the F-test proves that the regression is statistically significant at the 95% confidence level. The calculated F of 58.98 was considerably higher than the tabulated F of 2.82, which confirms the significance of the model. To determine the quality of the model, the coefficient of determination (R\({}^{2}\)) was evaluated. The obtained R\({}^{2}\) was 0.99, which indicates that 99% of the experimental data are compatible with the data provided by the model. The adjusted R\({}^{2}\) was 0.97, a high value, stating the significance of the model. The model presented an R\({}^{2}\) value above 99%, confirming an excellent relationship between the theoretical values predicted by the model and the results obtained experimentally. This correlation is given by the second order polynomial regression equation between the conversion of esters and the process variables: temperature (\({}^{\circ}\)C), molar ratio (FFAs: alcohol), biocatalyst content and time, which is given below:  
\[ \begin{array}{rcl}\text{Y}&=&32.3289+1.8085\text{X}_{1}-0.0221\text{X}_{1}^{2 }+4.7503\text{X}_{2}-0.3768\text{X}_{2}^{2}-0.3234\text{X}_{1}\text{X}_{2}\\ &&+0.0066\text{X}_{1}\text{X}_{3}+0.0029\text{X}_{1}\text{X}_{4}-0.0047\text{X} _{2}\text{X}_{3}-0.0163\text{X}_{2}\text{X}_{4}+9.1230\end{array}