# Temperature Dependence of Density and Viscosity of Biobutanol-Gasoline Blends
## 3 Results and Discussion
### Mathematical Models
#### 3.4.4 Mathematical Multivariate Models  
The created three-dimensional mathematical multivariate polynomial models (see Equations (11) and (12)) have a significant application potential, because it is possible to accurately calculate the exact value of the volumetric mass density or the kinematic viscosity for the whole temperatures range from \(-10\) to 40 \({}^{\circ}\)C (\(\sim\)263-313 K) and with any possible concentration of the biobutanol in the gasoline (RON 95 and RON 98) with the obtained coefficients (Table 12). The volumetric mass density and kinematic viscosity values obtained by extrapolation from these multivariate models can also be used with a certain degree of accuracy.  
From the three-dimensional graphs using the colour scale, it can be clearly seen that the lowest values (blue colour) of the volumetric mass density (Figures 8 and 9) and kinematic viscosity (Figures 10 and 11) are reached in the biobutanol-gasoline blends with a concentration below about 40 vol.% and at temperatures higher than 0 \({}^{\circ}\)C (\(\sim\)273 K); on the contrary, the highest values (yellow colour) were reached by the biobutanol-gasoline blends above 80 vol.% of biobutanol and at low temperatures (<0 \({}^{\circ}\)C). The temperature dependence of the volumetric mass density or the kinematic viscosity and the effect of the biobutanol concentration in gasoline (RON 95 and RON 98) prove a nonlinear dependence like with the bioethanol-gasoline behaviour described in [19; 25].  
Figure 8: Three-dimensional graph created using multivariate polynomial model for volumetric mass density of biobutanol-gasoline (RON 95) blends. Experimental data are presented as the mean (\(N=5\)) for selected temperatures and concentrations.  
Figure 7: Vogel model graph for kinematic viscosity of biobutanol-gasoline (RON 98) blends.  
Figure 10: Three-dimensional graph created using multivariate polynomial model for kinematic viscosity of biobutanol-gasoline (RON 95) blends. Experimental data are presented as the mean (\(N=20\)) for selected temperatures and concentrations.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Volumetric Mass Density**} & \(\mathbf{x_{0}}\) & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & \(\mathbf{x_{4}}\) & \(\mathbf{R^{2}}\) & **SSE** \\  & (\(\mathbf{kg\cdot m^{-3}}\)) & (\(\mathbf{K^{-1}\cdot kg\cdot m^{-3}}\)) & (\(\mathbf{kg\cdot m^{-3}}\)) & (\(\mathbf{K^{-2}\cdot kg\cdot m^{-3}}\)) & (\(\mathbf{K^{-1}\cdot kg\cdot m^{-3}}\)) & & \\ \hline Bx-95 & 358.4 & 3.202 & 1.649 & \(-\)0.006287 & \(-\)0.003953 & 0.9944 & 178.1 \\ Bx-98 & 375.2 & 3.350 & 0.9477 & \(-\)0.006874 & \(-\)0.001859 & 0.9902 & 239.7 \\ \hline \multirow{2}{*}{**Kinematic Viscosity**} & \(\mathbf{y_{0}}\) & \(\mathbf{y_{1}}\) & \(\mathbf{y_{2}}\) & \(\mathbf{y_{3}}\) & \(\mathbf{y_{4}}\) & \(\mathbf{R^{2}}\) & **SSE** \\  & (\(\mathbf{mm^{2}\cdot s^{-1}}\)) & (\(\mathbf{K^{-1}\cdot mm^{2}\cdot s^{-1}}\)) & (\(\mathbf{mm^{2}\cdot s^{-1}}\)) & (\(\mathbf{K^{-2}\cdot mm^{2}\cdot s^{-1}}\)) & (\(\mathbf{K^{-1}\cdot mm^{2}\cdot s^{-1}}\)) & & \\ \hline Bx-95 & 52.91 & \(-\)0.3668 & 0.3809 & 0.0006412 & \(-\)0.001178 & 0.9715 & 7.021 \\ Bx-98 & 54.61 & \(-\)0.3801 & 0.4105 & 0.0006663 & \(-\)0.001265 & 0.9897 & 2.999 \\ \hline \end{tabular}
\end{table}
Table 12: Coefficients of polynomial multivariate models and statistical indicators (\(R^{2}\)—coefficient of determination, SSE—sum of squared estimate of errors).  
Figure 9: Three-dimensional graph created using multivariate polynomial model for volumetric mass density of biobutanol-gasoline (RON 98) blends. Experimental data are presented as the mean (\(N=5\)) for selected temperatures and concentrations.