# Density and Viscosity in Biodiesel + Diesel Mixtures from Recycled Feedbacks
## 2 Materials and Methods
For density and viscosity, five measurements were performed for each point, respectively, which is the average of the measurements, while the reproducibility was considered as the standard deviation, obtaining 0.05 kg\(\cdot\)m\({}^{-3}\) for density and 0.02 mm\({}^{2}\cdot\)s\({}^{-1}\) for viscosity. All the characteristics of the equipment used in this work are included in Table 4.  
The excess molar volume (\(V^{\rm E}\)) was calculated by the difference between the real and the ideal molar volumes in terms of Equation (1):  
\[V^{\rm E}=\left(\frac{PM_{\rm M}}{\rho_{\rm M}}\right)-\left(\frac{x_{1}PM_{1} }{\rho_{1}}+\frac{x_{2}PM_{2}}{\rho_{2}}\right), \tag{1}\]  
where \(PM\) denotes the molecular weight, \(x\) is designated to the mole fraction, while the subscripts 1, 2 and M respectively correspond to biodiesel, diesel and mixture. The \(V^{\rm E}\) behavior can be fitted to the Redlich-Kister polynomial function expressed in Equation (2):  
\[V^{\rm E}=x_{1}(1-x_{1})\sum_{j=0}^{j=k}A_{j}(2x_{1}-1)^{j}, \tag{2}\]  
where \(k\) symbolizes the degree for the polynomial expression, and \(A_{j}\) corresponds to each parameter; Equation (2) was delimited to a third-degree function. The Extended Real Association Solution (ERAS) model is useful for predicting the excess molar volume of systems, which considers a physical and a chemical contribution for interactions [40; 41]. The application of this model involves the use of volumetric properties of pure fluids, namely thermal expansion coefficient (\(\alpha\)) and isothermal compressibility (\(\kappa\)). For the former, the values were determined from the data of this work; for the latter, it was necessary to find density data at different pressures from the literature. Diesel density was extracted from [42], density data of FAME were measured in our laboratory, and FABE density was taken from a previous work [7]. The ERAS model's aim is to correlate the excess molar volume of a mixture fitting three parameters, namely the association constant (\(\kappa_{\rm AB}\)), the cross-association volume (\(\Delta v_{\rm AB}^{*}\)) and the interaction parameter (\(X_{\rm AB}\)). The fitting was accomplished in a spreadsheet, with Gauss-Newton method. A brief description of the algorithm used for obtaining the parameters of ERAS model is as follows: reduction molar volume (\(v^{*}\)) is calculated with the temperature, molar volume and isobaric thermal expansion coefficient for each component. This reduction molar volume is used for calculating the stoichiometric hard-core volume fractions of the components (\(\Phi_{\rm i}\)), which also depends (separately) on the hard-core volume fraction of a component in the mixture (\(\phi_{\rm i}\)). A system of two equations with two unknowns (the volume fractions) is solved for the chemical contribution of the excess molar volume. Besides, the stoichiometric volume fractions lead to obtaining the surface fraction of each component (\(\Theta_{\rm i}\)), a part \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Model** & **Accuracy** & **Standard Uncertainty** \\ \hline \multicolumn{4}{c}{Synthesis of mixtures by successive loadings method} \\ \hline Balance & Precisa model EP 520A & \(\pm\)0.1 mg & 0.0004 in mole fraction \\ \hline \multicolumn{4}{c}{Density measurement} \\ \hline Vibrating tube densimeter & Anton Paar DMA 4500M & 0.01 kg\(\cdot\)m\({}^{-3}\) & 0.2 kg\(\cdot\)m\({}^{-3}\) \\ Thermometer & 100-\(\Omega\) platinum thermometer & 0.01 K & 0.02 K \\ \hline \multicolumn{4}{c}{Viscosity measurement} \\ \hline Capillarity viscometer & Cannon-Fenske & 0.01 mm\({}^{2}\cdot\)s\({}^{-1}\) & 0.14 mm\({}^{2}\cdot\)s\({}^{-1}\) \\ Thermometer & Polyscience, model PD07R & 0.02 K & 0.04 K \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Characteristics of the equipment used in this work. of the calculation of the reduction temperature and reduction pressure of the mixture, dependent on the reduction property of each component. For reduction pressure, the isothermal compressibilities are necessary.