## 4 Conclusions  
The conducted tests confirmed the research hypothesis that the changes in lubricity, coefficient of friction, and percentage relative decrease in oil film thickness are similar for LO/DO mixtures prepared with fresh and ULO. The presented models and methods can be used in computational tribology as an element of the research process. The linear approximation of \(\mu\) and \(r\) trends is just a hypothesis that we made with a certain margin of error we have estimated. In the future, other, more complex approximation models (e.g., polynomial) might be made but with a higher value of the coefficient of determination.  
Dilution of LO with DO has an effect on lubricity, which was improved for the cases in question in the concentration range of DO in 0-2% m/m mixtures in comparison to the undiluted LO. On the other hand, at higher concentrations of DO in the mixtures (C > 2% m/m), lubricity of the mixture will deteriorate (the size of the wear scar increases up to a maximum value corresponding to pure DO).  
The lubricity of mixtures containing ULO exhibits an upward trend with a steeper slope in comparison to the mixtures based on the FLOs, which is attributed to the presence of oil aging products such as resins, organic acids, and relatively small amounts of asphaltenes in the mixture. As the DO content increases, the viscosity of lubricating LO/DO mixtures decreases simultaneously, leading to a deterioration in lubrication conditions. This results in an increase in the coefficient of friction and a decrease in oil film thickness. The regression model for the friction coefficient and oil film thickness did not show significant differences for the tested concentrations of DO in LO whether the mixtures were prepared using FLO or ULO.  
A qualitative visual assessment of the wear scar on the bead's surface confirmed an increase in the intensity of wear when conducting tests in a mixture with a higher concentration of diesel oil in the LO. At the same time, micro-milling marks were observed across the entire surface of the wear scar. In tests involving mixtures of DO with ULO, the occurrence of furrowing was observed. This may be attributed to the presence of metallic impurities in the LO, which are products of wear on engine components.  
Conceptualization, L.C. and M.S.; methodology, L.C. and M.S.; software, L.C. and M.S.; validation, L.C., M.S., R.S. and P.B.; formal analysis, L.C., M.S., R.S. and P.B.; investigation, L.C., M.S., R.S. and P.B.; resources, L.C., M.S., R.S. and P.B.; data curation, L.C.; writing--original draft preparation, L.C. and M.S.; writing--review and editing, L.C., M.S., R.S., and P.B.; visualization, L.C. and M.S.; supervision, L.C.; project administration, L.C.; funding acquisition, L.C. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by the Ministry of Science and Higher Education (MEiN) of Poland, grant number 1/S/KPBMiM/24.  
Not applicable.  
All data are available for this paper.  
The authors would like to thank [PERSON], [PERSON], and [PERSON] for technical support.  
The authors declare no conflicts of interest. The funders had no role in the design of the study, the collection, analysis, or interpretation of data, the writing of the manuscript, or the decision to publish the results.