# Density and Viscosity in Biodiesel + Diesel Mixtures from Recycled Feedbacks
## 3 Results and Discussions
### Viscosity  
The kinematic viscosity (\(v\)) obtained via the Cannon-Fenske viscometers for the two binary systems is presented in Table S5 and Figure 8.  
Figure 8: Kinematic viscosity for biodiesel (1) + diesel (2) systems as a function of temperature. Symbols indicate different mole fractions of biodiesel (\(x_{1}\)). Red and blue symbols denote waste cooking oil methyl ester biodiesel and waste beef tallow butyl ester biodiesel, respectively. Figure 7: Isobaric expansion coefficient for biodiesel (1) + diesel (2) systems as a function of mole fraction of biodiesel (\(x_{1}\)). Red and blue lines denote waste cooking oil methyl ester biodiesel and waste beef tallow butyl ester biodiesel, respectively. The kinematic viscosity for biodiesel samples (FAME and FABE) was greater than the one for diesel at a fixed temperature; consequently, \(v\) for binary mixtures decreased as the composition of diesel incremented. Besides, the samples tended to be less viscous with temperature increasing; this behavior was attributed to the influence of the ester carbon chain length, the number of double and triple bonds and the conformation of the fatty acids used as feedstock. Viscosities for waste cooking oil methyl ester biodiesel were compared with some selected data from the literature [23; 64; 65; 66; 67; 23], ranging deviations of 2.28 and 0.55 mm\(\cdot\)s\({}^{-1}\) at 303.15 and 343.15 K, respectively; even at 313.15 K, the deviations reach 1.83 mm\(\cdot\)s\({}^{-1}\) as maximum. All the data sets correspond to waste cooking oil methyl ester biodiesel; the difference in viscosity values confirms that the ester profile for each sample is not the unique one that alters viscosity. Impurities such as glycerides and water content, the unreacted material indicated in the reaction yield and the presence of polyunsaturated alkyl esters can change the viscosity of biodiesel [69]. Moreover, [PERSON] and [PERSON] [68] reported different viscosity data for waste cooking oil biodiesel by distinct reaction and purification methods that varied from 4.63 to 5.82 mm\(\cdot\)s\({}^{-1}\). The comparison is displayed in Figure 9.  
The calculated dynamic viscosity (\(\eta=\
u\cdot\rho\)) is contained in Table 7.  
For excess viscosity, the isothermal values are plotted in Figure 10 as a function of composition along with \(\eta\) calculated by the McAllister four-body model, whose parameters are declared in Table 8.  
Figure 10: Viscosity deviation for biodiesel (1) + diesel (2) systems as a function of mole fraction of biodiesel (\(x_{1}\)). Red and blue symbols and lines denote waste cooking oil methyl ester biodiesel and waste beef allow butyl ester biodiesel, respectively. Figure 9: Kinematic viscosity for waste cooking oil methyl ester biodiesel in comparison with other literature data: \(\bullet\), this work; \(\odot\), [PERSON] and [PERSON] 2015 [64]; \(\square\), [PERSON] et al. 2013 [23]; \(\blacksquare\), [PERSON] and [PERSON], 2017 [65]; \(\blacklozenge\), [PERSON] and [PERSON] 2015 [66]; \(\circ\), [PERSON] and [PERSON] 2013 [67]; \(\triangle\), [PERSON] and [PERSON] 2010 [68]. Dynamic viscosity presented the same behavior as kinematic viscosity. The value of this property decreases with temperature rising; the viscosity increases as the value of \(x_{1}\) increases, since diesel is less viscous than biodiesel. Therefore, a mixture with a higher biofuel content has a greater resistance to deformation, implying greater effort for its transport in the engine injection. The molecular interactions can be analyzed by means of viscosity deviation where molecules with different shapes and size coexisted in both systems.