## 3 Results and Discussion
### Thermal Properties of Electro-Insulating Liquids Modified by Fullerene C\({}_{60}\)
There was no influence of modification of base liquids with TiO\({}_{2}\) nanoparticles on their density and thermal expansion. Table 7 presents the calculated values based on Equations (4), (6), and (7) of the criterion numbers of the analyzed liquids. As can be seen from the calculations, for temperatures in the range of 40-80 \({}^{\circ}\)C, for both mineral oils modified with surfactant and mineral oils modified with surfactant and titanium dioxide TiO\({}_{2}\) nanoparticles, turbulent flows are possible. In turn, for all ester-derived nanofluids analyzed (both \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Kind of Liquid** & **Temperature** [\({}^{\circ}\)**C] & **Gr** & **Pr** & **Nu** \\ \hline \multirow{4}{*}{MO} & 25 & \(7.9\times 10^{8}\) & 212 & 736 \\  & 40 & \(21.9\times 10^{8}\) & 125 & 869 \\  & 60 & \(61.6\times 10^{8}\) & 74 & 1028 \\  & 80 & \(137.4\times 10^{8}\) & 50 & 1178 \\ \hline \multirow{4}{*}{MO + Span\({}^{\circledR}\) 20} & 25 & \(7.4\times 10^{8}\) & 225 & 736 \\  & 40 & \(20.7\times 10^{8}\) & 132 & 868 \\  & 60 & \(53.3\times 10^{8}\) & 82 & 1017 \\  & 80 & \(126.9\times 10^{8}\) & 54 & 1179 \\ \hline \multirow{4}{*}{MO + Span\({}^{\circledR}\) 20 + TiO\({}_{2}\)} & 25 & \(7.1\times 10^{8}\) & 209 & 708 \\  & 40 & \(19.5\times 10^{8}\) & 127 & 840 \\  & 60 & \(49.3\times 10^{8}\) & 81 & 985 \\  & 80 & \(119.9\times 10^{8}\) & 51 & 1137 \\ \hline \multirow{4}{*}{SE} & 25 & \(0.9\times 10^{8}\) & 641 & 520 \\  & 40 & \(3.0\times 10^{8}\) & 339 & 623 \\  & 60 & \(10.3\times 10^{8}\) & 177 & 757 \\  & 80 & \(26.9\times 10^{8}\) & 107 & 884 \\ \hline \multirow{4}{*}{SE + Span\({}^{\circledR}\) 20} & 25 & \(0.9\times 10^{8}\) & 658 & 520 \\  & 40 & \(2.9\times 10^{8}\) & 348 & 625 \\  & 60 & \(9.4\times 10^{8}\) & 188 & 751 \\  & 80 & \(25.7\times 10^{8}\) & 110 & 878 \\ \hline \multirow{4}{*}{SE + Span\({}^{\circledR}\) 20 + TiO\({}_{2}\)} & 25 & \(0.9\times 10^{8}\) & 654 & 513 \\  & 40 & \(2.8\times 10^{8}\) & 351 & 614 \\  & 60 & \(9.0\times 10^{8}\) & 189 & 742 \\  & 80 & \(24.9\times 10^{8}\) & 111 & 873 \\ \hline \multirow{4}{*}{NE} & 25 & \(0.8\times 10^{8}\) & 575 & 480 \\  & 40 & \(2.1\times 10^{8}\) & 343 & 559 \\  & 60 & \(6.0\times 10^{8}\) & 199 & 657 \\  & 80 & \(13.7\times 10^{8}\) & 131 & 755 \\ \hline \multirow{4}{*}{NE + Span\({}^{\circledR}\) 20} & 25 & \(0.8\times 10^{8}\) & 565 & 479 \\  & 40 & \(2.2\times 10^{8}\) & 336 & 557 \\  & 60 & \(6.0\times 10^{8}\) & 194 & 654 \\  & 80 & \(13.8\times 10^{8}\) & 128 & 751 \\ \hline \multirow{4}{*}{NE + Span\({}^{\circledR}\) 20 + TiO\({}_{2}\)} & 25 & \(0.8\times 10^{8}\) & 551 & 473 \\  & 40 & \(2.2\times 10^{8}\) & 328 & 553 \\ \cline{1-1}  & 60 & \(6.0\times 10^{8}\) & 191 & 649 \\ \cline{1-1}  & 80 & \(13.6\times 10^{8}\) & 125 & 742 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Calculation results of Grashof, Prandtl, and Nusselt numbers of pure electro-insulating liquids and nanofluids based on surfactant Span\({}^{\circledR}\) 20 and titanium dioxide (TiO\({}_{2}\)). synthetic and natural), turbulent flow may occur at a temperature of 80 \({}^{\circ}\)C. As in the previous case, the turbulent nature of the flow can positively influence the dispersion of titanium dioxide TiO\({}_{2}\) nanoparticles in the analyzed nanofluids. Modification of electro-insulating liquids with surfactant Span(r) 20 and titanium dioxide TiO\({}_{2}\) nanoparticles does not significantly change the value of the Grashof number. In the case of the analyzed liquids, it can be noticed that modifying the mineral oil and the synthetic ester with the surfactant Span(r) 20 causes a minimal increase in the Prandtl number. In turn, the addition of titanium dioxide TiO\({}_{2}\) nanoparticles to the resulting liquids decreases the value of this criterion number. In the case of a natural ester, it can be seen that modifying with dispersant causes a reduction in the value of the Prandtl number. The addition of titanium dioxide TiO\({}_{2}\) nanoparticles further enhances this change.