## 3 Results and Discussion
### Regression Model for Biodiesel Yield  
The BBD model was used that could correlate independent variables (M:O, CC, Rt, and RT) and dependent variables (biodiesel yield) based on experiments performed. The experimental input-output data that could fit models without any bias is presented in Table 4.  
Figure 4: The 3D surface plots of transesterification variables vs. biodiesel yield: (**a**) M:O and CC, (**b**) M:O and RT, (**c**) M:O and Rt, (**d**) CC and RT, (**e**) CC and Rt, and (**f**) Rt and RT. The model's statistical adequacy was evaluated based on regression coefficients. Note that the model with full quadratic terms (linear + interaction + square) resulted in a better regressor coefficient equal to 0.9869 (close to 1), which clearly indicates the best fit model. The adjusted R\({}^{2}\) (excluding insignificant terms: AC, CD, and C\({}^{2}\) from the model) value was found equal to 0.9737. The best fit quadratic model represents biodiesel yield as a mathematical function of transesterification variables (M:O, CC, Rt, and RT). \[ \begin{array}{ll}Yield\ (\%)=&+54.44389+1.24074\ A+7.61333\ B+0.388222\ C+0.330889D\\ &+0.666667\ AB+0.011111 AC+0.003333 AD+0.16 BC-0.0916670 BD\\ &-0.001444 CD-0.16963A^{2}-5.54B^{2}-0.003956C^{2}-0.001136D^{2}\end{array}  \tag{5}\]  
The experimental data were analyzed for the developed quadratic model that could estimate linear, square, and interaction factor effects. All linear terms (M:O, CC, RT, and Rt) were found to have significant contributions (as their \(p\)-values \(<\) 0.05). The effect of the reaction temperature was found to be significant as their effect (measured with F-value) was found to be comparatively lower than M:O, CC, and Rt. This can be clearly seen in the main effect plot result shown in Figure 3. The square terms of M:O, CC, and Rt were found to be significant as their \(p\)-values are \(<\)0.05, which clearly indicates their relationship with biodiesel yield is non-linear. The squared term of reaction temperature was found to be insignificant (\(p\)-value \(>\) 0.05), and hence the relationship with biodiesel yield was found to be linear. The curvature effects are in good agreement with statistical \(p\)-values of squared terms of transesterification variables (see Table 5). Although all individual terms were found to be significant, the interaction terms (M:O and RT, and RT and Rt) were found to be insignificant with negligible influence on biodiesel yield. Excluding insignificant terms (M:O \(\times\) RT, RT \(\times\) Rt, and RT\({}^{2}\)) from the model-derived regression equation creates imprecise input-output relationships and reduces the prediction accuracy; therefore, the model was tested for prediction accuracy with experimental values with inclusion of all terms in the equation resulted in the best fit data, close to the trend line (see Figure 5). The better prediction ensures the regression equations derived for predicting the biodiesel yield are accurate and can be used to perform optimization. \begin{table}
\begin{tabular}{c c c c} \hline
**Model** & **Regression Equation** & **Regression Coefficient** \\ \hline Linear & Yield = 83.78 + 0.302 A + 2.483 B + 0.047 C - 0.027 D & R\({}^{2}\) = 0.3141, and \\  & Yield = 89.297 - 1.304 A - 3.467 B - 0.067 C + 0.1264 D + & Adj. R\({}^{2}\) = 0.1998 \\ Linear + Interaction & 0.667 AB + 0.011 AC + 0.0033 AD + 0.16 BC - 0.0917 BD & R\({}^{2}\) = 0.4437, and \\  & – 0.0014 CD & Adj. R\({}^{2}\) = 0.1347 \\  & Yield = 54.44389 + 1.24074 A + 7.61333 B + 0.388222 C + & \\  & 0.330889 D + 0.66667 BB + 0.011111 AC + 0.003333 AD + & R\({}^{2}\) = 0.9869, and \\  & 0.16 BC - 0.0916670 BD - 0.00144 CD - 0.16963 A\({}^{2}\) 5.54 & Adj. R\({}^{2}\) = 0.9737 \\  & \({}^{2}\) - 0.003956 C\({}^{2}\) - 0.001136 D\({}^{2}\) & \\ \hline \end{tabular}
\end{table} 
Table 4: Fitted models correspond to experimental input–output data.