# Density and Viscosity in Biodiesel + Diesel Mixtures from Recycled Feedbacks
## 3 Results and Discussions
### Volumetric Properties
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(A_{0}\)1  & \(A_{1}\)1  & \(A_{2}\)1  & \(A_{3}\)1  & \(s\)1  & \(AAD\) (\%) \\ \hline \multicolumn{6}{c}{**waste cooking oil methyl ester biodiesel (1) + diesel (2)**} \\
293.15 & 0.1123 & \(-1.4763\) & \(-0.6015\) & 0.8471 & 0.0188 & 13.94 \\
303.15 & 0.1618 & \(-1.5012\) & \(-0.7173\) & 0.6607 & 0.0222 & 12.69 \\
313.15 & 0.1653 & \(-1.6925\) & \(-0.7872\) & 0.7788 & 0.0249 & 11.60 \\
323.15 & 0.0841 & \(-1.8618\) & \(-0.6734\) & 0.8722 & 0.0420 & 94.60 \\
333.15 & 0.0892 & \(-1.9400\) & \(-0.5437\) & 0.8304 & 0.0332 & 31.50 \\
343.15 & 0.3930 & \(-2.2957\) & \(-1.1090\) & 1.2698 & 0.0372 & 14.37 \\
353.15 & 0.3798 & \(-2.4213\) & \(-1.0682\) & 1.3524 & 0.0420 & 13.72 \\
363.15 & 0.4071 & \(-2.5143\) & \(-1.2786\) & 1.1875 & 0.0420 & 11.75 \\ \hline \multicolumn{6}{c}{**waste beef tallow butyl ester biodiesel (1) + diesel (2)**} \\
293.15 & \(-0.3844\) & \(-1.7615\) & \(-0.1971\) & 0.4791 & 0.0183 & 11.39 \\
303.15 & \(-0.3726\) & \(-1.7793\) & \(-0.1586\) & 0.4493 & 0.0207 & 12.81 \\
313.15 & \(-0.3623\) & \(-1.8012\) & \(-0.0256\) & 0.3727 & 0.0199 & 10.80 \\
323.15 & \(-0.3722\) & \(-1.8347\) & 0.1367 & 0.5746 & 0.0163 & 8.98 \\
333.15 & \(-0.3235\) & \(-1.9433\) & 0.2209 & 0.6131 & 0.0116 & 5.96 \\
343.15 & \(-0.2798\) & \(-1.9425\) & 0.2941 & 0.4672 & 0.0016 & 0.68 \\
353.15 & \(-0.1983\) & \(-1.9498\) & 0.2679 & 0.3144 & 0.0078 & 4.30 \\
363.15 & \(-0.1390\) & \(-2.1558\) & 0.7994 & \(-0.1146\) & 0.0182 & 7.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Redlich–Kister parameters for the excess molar volume correlation. Positive excess molar volumes for biodiesel + diesel systems have been reported [24; 26; 53]. The binary systems based on pure chemicals agreed with the preceding information: alkane + methyl ester [54; 55], alkyl ester + 1-alkanol [56; 57; 58], alkanol + cycloalkane [55]. Conversely, positive and negative excess molar volumes have been reported along the composition range for coconut biodiesel + diesel [25], 2-octanol + octane [59] and methyl ester biodiesel (from beef tallow) + alcohol [52; 60] mixtures, where the negativity was found at low alcohol compositions for the last ones. There is no doubt that interaction molecular effects between chemical groups can alter the excess properties, for instance, non-polar (hydrocarbons from diesel), esters, alcohols, water and glycerol; all of this was confirmed in the study of \(V^{\rm E}\) for ternary systems (b biodiesel + diesel + alcohol or their corresponding base components) [52; 53; 55] where regions with negative \(V^{\rm E}\) values were observed. Regarding the ERAS model, the results are presented in Table 6. Deviation was evaluated in terms of \(AAD\) and standard deviation \(s\left(\text{m}^{3}\cdot\text{mol}^{-1}\right)\), \(\sqrt{\sum_{i=1}^{N}\left(V_{\rm exp}^{\rm E}-V_{\rm cal}^{\rm E}\right)^{2}/ \left(N-m\right)}\), where \(m\) refers to the number of parameters. The cross-association constant (\(\kappa_{AB}\)) had some dependence in temperature for FABE + diesel system. Its values span from \(-0.4293\) to \(0.3198\), which are characteristic of non-associating compounds [61]. The cross-association volume (\(\Delta v_{AB}^{*}\)) for the FAME + diesel system was positive at any temperature, reflecting the weak intermolecular interactions. For the system FABE + diesel, the cross-association volume was negative, which indicated the interaction of diesel with FABE was stronger than its interaction with FAME. However, the values were much lower than those of associating compounds, even if they exhibit small association degree [61]. The interaction parameter (\(X_{AB}\)) reflects the difference of dispersive interactions and presents higher values than the ones used for associating compounds (<12 MPa) [62].