## 5 Results and Discussion
### Density Evaluation for Biodiesel/Diesel Blends at Varying Temperatures Using Tammann-Tait Equation  
Figures 4 and 5 present the experimental and the calculated dependence of density on temperature for different diesel/biodiesel blends, according to Equations (3)-(5). Figure 4a,b illustrate examples of the experimental data of diesel/vegetable biodiesel blends together with the estimated values using Equations (4) and (5) (see Section 4.2). In addition, Figure 5a,b present examples of the experimental and calculated values of diesel/animal fat biodiesel. The values of the parameters of the Tammann-Tait equation (b\({}_{0}\) (MPa), b\({}_{1}\) (MPa\(\cdot\)K\({}^{-1}\)), b\({}_{2}\) (MPa\(\cdot\)K\({}^{-2}\)) and C) were adopted from [PERSON] et al. [19]. The parameters a\({}_{0}\) (gr\(\cdot\)cm\({}^{-3}\)), a\({}_{1}\) (gr\(\cdot\)cm\({}^{-3}\).K\({}^{-1}\)), and a\({}_{2}\) (gr\(\cdot\)cm\({}^{-3}\).K\({}^{-2}\)) were calculated using  
Figure 3: (**a**) Experimental and theoretical calculations of density (\(\rho\), g/cm\({}^{3}\)) dependence on diesel volume percentage (v%) mixed with animal fat biodiesel at a constant temperature of 288 K; (**b**) residuals of density versus volume percentage. Figure 2: (**a**) Experimental and theoretical calculations of density (\(\rho\), g/cm\({}^{3}\)) dependence on diesel volume percentage (v%) mixed with vegetable biodiesel at a constant temperature of 288 K; (**b**) residuals of density versus volume percentage. Aquasim and their values are shown in Table 2 (vegetable biodiesel) and Table 3 (animal fat biodiesel) as determined by the parameter estimation process, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **D100\%-BD0\%,** & **D85\%-BD15\%,** & **D70\%-BD30\%,** & **D55\%-BD45\%,** & **D35\%-BD65\%,** & **D15\%-BD85\%,** & **D80\%-BD50\%,** & **D30\%-BD70\%,** & **D10\%-BD90\%,** \\  & **D95\%-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\({}^{-3}\)) & \(-0.1543\) & \(2.4919\) & \(1.8164\) & \(2.3069\) & \(4.5449\) & \(5.2909\) \\ \hline a\({}_{1}\) (gr-cm\({}^{-3}\).K\({}^{-1}\)) & \(0.0067\) & \(-0.0114\) & \(-0.0064\) & \(-0.0098\) & \(-0.0253\) & \(-0.0303\) \\ \hline a\({}_{2}\) (gr-cm\({}^{-3}\).K\({}^{-2}\)) & \(-1.14\times 10^{-5}\) & \(1.95\times 10^{-5}\) & \(1.05\times 10^{-5}\) & \(1.64\times 10^{-5}\) & \(4.34\times 10^{-5}\) & \(5.19\times 10^{-5}\) \\ \hline b\({}_{0}\) (MPa) & \(400\) & \(400\) & \(400\) & \(400\) & \(400\) & \(400\) \\ \hline b\({}_{1}\) (MPa\(\cdot\)K\({}^{-1}\)) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) & \(-1.31\) \\ \hline b\({}_{2}\) (MPa\(\cdot\)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 \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 2: Results of the parameter estimation using Tammann–Tait equation (Equations (3)–(5)) for volume percentage vegetable biodiesel/ diesel blends at varying temperatures. Figure 4: Experimental and theoretical calculations of volume percentage diesel/vegetable biodiesel blends in relation to density (\(\rho\), g/cm\({}^{3}\)) and temperature (T, K): (**a**) D0%, D25%, D50%, D75%, and D100%, (**b**) D5%, D20%, D45%, D70%, and D95%. Figure 5: Experimental and theoretical calculations of volume percentage diesel/animal fat biodiesel blends in relation to density (\(\rho\), g/cm\({}^{3}\)) and temperature (T, K): (**a**) D0%, D25%, D50%, D75%, and D100%, (**b**) D5%, D20%, D45%, D70%, and D95%. Equations (3)-(5) adequately describe the temperature dependence of density for the various mixing fractions of diesel and biodiesel, as shown in Figures 4 and 5.