## 5 Results and Discussion
### Density Evaluation for Biodiesel/Diesel Blends at Varying Temperatures Using Empirical Equation  
The empirical Equation (9) was used to predict the effect of temperature and diesel volume percentage on blend densities. The empirical parameters of Equation (9) were determined using all the experimental data of blends for both vegetable and animal fat biodiesel in the fitting process of Aquasim. The result of parameter estimation for a and b values gave the values of a = 0.126 and b = 0.795. The experimental and theoretical results are presented in Figure 6a for diesel/vegetable biodiesel volume percentage blends and in Figure 6b for diesel/animal fat biodiesel volume percentage blends. It was observed that the theoretical values were close to the experimental values, achieving high values of E and low values of MSE. Specifically, for vegetable biodiesel mixtures, the values were E = 0.87 and MSE = 0.3-0.4%, while for animal fat biodiesel mixtures, the values were E = 0.8784 and MSE = 0.2-0.3%. It is worth noting that empirical equations are widely used in engineering process design as a prior tool for decision-making. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **D100\%-BD0\%**, & **D85\%-BD15\%**, & **D70\%-BD30\%**, & **D55\%-BD50\%**, & **D35\%-BD50\%**, & **D15\%-BD5\%**, \\  & **D95\%-BD5\%**, & **D80\%-BD20\%**, & **D65\%-BD35\%**, & **D50\%-BD50\%**, & **D30\%-BD70\%**, & **D10\%-BD90\%**, \\  & **D90\%-BD10\%** * & **D75\%-BD25\%** & **D60\%-BD40\%** & **D40\%-BD50\%** & **D25\%-BD75\%**, & **D5\%-BD95\%**, \\ \hline
**Parameters** & **Group 1** & **Group 2** & **Group 3** & **Group 4** & **Group 5** & **Group 6** \\ \hline a\({}_{0}\) (gr-cm\({}^{-3}\)) & \(-0.5089\) & \(-0.3161\) & \(1.7982\) & \(0.7752\) & \(-2.8150\) & \(-2.9471\) \\ \hline a\({}_{1}\) (gr-cm\({}^{-3}\).K\({}^{-1}\)) & \(0.0093\) & \(0.0084\) & \(-0.0059\) & \(0.0015\) & \(0.0264\) & \(0.0274\) \\ \hline a\({}_{2}\) (gr-cm\({}^{-3}\).K\({}^{-2}\)) & \(-1.62\times 10^{-5}\) & \(-1.52\times 10^{-5}\) & \(9.07\times 10^{-6}\) & \(-4.13\times 10^{-6}\) & \(-4.73\times 10^{-5}\) & \(-4.92\times 10^{-5}\) \\ \hline b\({}_{0}\) (MPa) & \(400\) & \(400\) & \(400\) & \(400\) & \(400\) & \(400\) \\ \hline b\({}_{1}\) (MPa-K\({}^{-1}\)) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) \\ \hline b\({}_{2}\) (MPa-K\({}^{-2}\)) & \(0.001\) & \(0.001\) & \(0.001\) & \(0.001\) & \(0.001\) & \(0.001\) \\ \hline C & \(0.0831\) & \(0.0831\) & \(0.0831\) & \(0.0831\) & \(0.0831\) & \(0.0831\) \\ \hline \end{tabular}  
* Data groups are given in order to relate the varying density of blends with the temperature given in Tables S1 and S2 in the Supplementary Materials.  
\end{table} 
Table 3: Results of the parameter estimation using Tammann–Tait equation (Equations (3)–(5)) for volume percentage vegetable animal fat biodiesel/diesel blends at varying temperatures. Estimation of Specific Gravity for Biodiesel/Diesel Blends at Varying Temperatures Using Empirical Equation  
Specific gravity of diesel and its blends with biodiesel is affected by temperature. In order to use a mathematical expression able to predict the specific gravity, API values were used as obtained from the experimental data (Tables S3 and S4), which were carried out with diesel and biodiesel blends. The values of specific gravity of each condition tested were calculated by using the API value (Equation (10)). [PERSON] and [PERSON] [28] present a correlation between specific gravity and temperature (Equation (11)). The parameter estimation of constants c and d that are present in Equation (11) resulted in the values shown in Tables 4 and 5. The predicted theoretical and experimental values of the specific gravity Sg using Equation (11) are presented in Figure 7. As shown in Figure 7, the theoretical calculations are in good agreement with the experimental values, which is confirmed by the results of the evaluation of the blends of diesel with vegetable biodiesel (E = 0.5375-0.9120, MSE = 0.1%) and with animal fat biodiesel (E = 0.5182-0.9988, MSE = 0.1%).