## 6 Conclusions  
The goal of this study was to apply suitable mathematical equations to predict the physicochemical properties of mixtures under varying conditions of temperature and mixing ratios. To describe the relationship between the density (\(\rho\), kg/m\({}^{3}\)) of the mixtures, the volume percentage (v%) of biodiesel mixed with diesel, and temperature (T, K), [PERSON]'s mixing equation, the Tammann-Tait equation, and empirical formulas were employed. Mathematical models were also used to predict the specific gravity (Sg) of the mixtures. The information provided by the mathematical correlation of the physicochemical properties of diesel/biodiesel is a useful tool that can be used to select the appropriate blending rates of the different fuels based on the desired properties of the final product. Theoretical calculations were able to accurately predict (E up to 0.9988 and MSE up to 0.4%) the experimental findings. Therefore, based on [PERSON]'s and Tammann-Tait equation estimations and the experimental findings, the density of the blends decreased as the percentage of diesel in the mixture increased and as the temperature in the mixture increased. Moreover, the density and specific gravity of the blends increased as the volume percentage of biodiesel increased and temperature decreased, as estimated by the [PERSON] and [PERSON] empirical equation. In any case, based on experimental data and theoretical calculations, blend densities (vegetable, 0.822-0.8823 gr/cm\({}^{3}\); animal fat, 0.22-0.882 gr/cm\({}^{3}\)) were within the specifications given for biodiesel (0.86-0.9 gr/cm\({}^{3}\)) as its percentage in the blend increased and, respectively, for diesel (0.82-0.86 gr/cm\({}^{3}\)) as its percentage in the blend increased. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Par.**} & **D100\%-** & **D95\%-** & **D90\%-** & **D85\%-** & **D80\%-** & **D75\%-** & **D70\%-** & **D65\%-** & **D60\%-** & **D55\%-** \\  & **BD0\%** & **BD5\%** & **BD10\%** & **BD15\%** & **BD20\%** & **BD25\%** & **BD30\%** & **BD35\%** & **BD40\%** & **BD45\%** \\ \hline
**c** & 0.8199 & 0.8258 & 0.8292 & 0.8358 & 0.8404 & 0.8456 & 0.8526 & 0.8562 & 0.8562 & 0.8605 \\ \hline
**d** & 0.0001 & \(-\)0.0001 & \(-\)0.0001 & \(-\)0.0003 & \(-\)0.0004 & \(-\)0.0006 & \(-\)0.0007 & \(-\)0.0007 & \(-\)0.0008 & \(-\)0.0009 \\ \hline \multirow{2}{*}{**Par.**} & **D50\%-** & **D45\%-** & **D40\%-** & **D35\%-** & **D30\%-** & **D25\%-** & **D20\%-** & **D15\%-** & **D10\%-** & **D5\%-** \\  & **BD50\%** & **BD55\%** & **BD60\%** & **BD65\%** & **BD70\%** & **BD75\%** & **BD80\%** & **BD85\%** & **BD90\%** & **BD95\%** \\ \hline
**c** & 0.8632 & 0.8668 & 0.8722 & 0.8756 & 0.8765 & 0.8789 & 0.8818 & 0.8845 & 0.8881 & 0.8901 \\ \hline
**d** & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.0010 & \(-\)0.0010 & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.0008 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter values used in Equation (11) of diesel volume percentage mixed with animal fat biodiesel. Figure 7: Experimental and theoretical calculations of (**a**) diesel/vegetable biodiesel and (**b**) diesel/animal fat biodiesel volume percentage blends in relation to specific gravity Sg and temperature (T, K).