# Density and Viscosity in Biodiesel + Diesel Mixtures from Recycled Feedbacks
## 4 Conclusions  
This work was a first effort to study the volumetric and transport properties for diesel + waste material biodiesel systems at 0.078 MPa and different temperatures and compositions. The density of waste cooking oil biodiesel ranged from 837.9 to 888.3 kg\(\cdot\)m\({}^{-3}\); for waste beef tallow biodiesel, it ranged from 782.0 to 832.2 kg\(\cdot\)m\({}^{-3}\), values that were similar to those found in the literature. Thermodynamic and transport-derived properties were analyzed to understand the molecular interactions of waste beef tallow butyl ester biodiesel or waste cooking oil ester biodiesel with fossil diesel. The calculated properties were the excess molar volume, partial molar volume, isobaric expansion coefficient and viscosity deviation. The excess molar volume performed a sigmoidal shape that tended to change from positive to negative values as the composition of biodiesel increased, considering that such behavior was interpreted as a contribution of opposing effects. The weak interactions induced by repulsion forces were the most relevant on the former region, and the stronger interactions predominated over the effect of steric hindrance of esters on the latter. The slight variations on the partial molar volumes confirmed the mentioned effects on volumetric properties. The negative values on the viscosity deviations were ascribed to minimum molecular interactions accompanied by the insertion of the smaller molecules into the structure of the bigger ones. Finally, the ERAS model was applied successfully to the excess molar volumes of this work, while the McAllister four-body model depicted the excess viscosity with deviation lower than 25.92%. The application of the ERAS model required some thermophysical properties, hence the necessity for experimental data of volumetric properties for the development and application of theoretical models. The measurement was limited at atmospheric pressure. In future work, it will be desirable to extend the pressure interval for a better understanding of the behavior of the mixtures and a wider availability of data for the use and testing of models. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15073812/s1](https://www.mdpi.com/article/10.3390/app15073812/s1)]([https://www.mdpi.com/article/10.3390/app15073812/s1](https://www.mdpi.com/article/10.3390/app15073812/s1)), Table S1: Density for biodiesel + diesel systems from 293.15 to 363.15 K at 0.078 MPa as a function of mole fraction of biodiesel (\(x_{1}\)); Table S2: Excess molar volume for biodiesel + diesel systems from 293.15 to 363.15 K as a function of mole fraction of biodiesel (\(x_{1}\)); Table S3: Partial molar volumes for biodiesel (1) + diesel (2) systems as a function of mole fraction of biodiesel (\(x_{1}\)). Table S4: Isobaric expansion coefficient for biodiesel (1) + diesel (2) systems as a function of mole fraction of biodiesel (\(x_{1}\)). Table S5: Kinematic viscosity for biodiesel (1) + diesel (2) systems as a function of mole fraction of biodiesel (\(x_{1}\)). **Author Contributions:** Conceptualization, A.Z.-M. and J.D.-G.; methodology, J.D.-G. and G.S.-R.; software, F.J.V.-S. and H.I.P.-L.; validation, G.S.-R. and J.D.-G.; formal analysis, F.J.V.-S. and G.S.-R.; investigation, G.S.-R. and H.I.P.-L.; resources, G.S.-R. and F.J.V.-S.; data curation, F.J.V.-S. and H.I.P.-L.; writing--original draft preparation, A.Z.-M. and O.E.-S.; writing--review and editing, F.J.V.-S. and O.E.-S.; visualization, J.D.-G. and H.I.P.-L.; supervision, A.Z.-M.; project administration, A.Z.-M. and O.E.-S.; funding acquisition, A.Z.-M. and O.E.-S. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by CONAHCyT, grant number CF-2023-I-311 and INSTITUTO POLITECNICO NACIONAL, grant number 20250588. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.