## Materials and Methods
### Determination of Heat Transfer in the Analyzed Nanofluids  
As previously mentioned, effective cooling is one of the basic functions that must be provided by the electro-insulating liquid. This function is performed by thermal conductivity and convection, which depends on such liquid properties as specific heat, viscosity, density, and thermal expansion. In order to determine the heat transfer in the analyzed electrical insulating nanofluids, the Newton equation should be used:  
\[q=\alpha\cdot\Delta T, \tag{1}\]  
Figure 4: Comparison of prepared samples of Span® 20 and titanium dioxide TiO\({}_{2}\)-based nanofluids with samples of base liquids; from the left: MO—mineral oil, MO + TiO\({}_{2}\)—mineral oil doped with TiO\({}_{2}\), SE—synthetic ester, SE + TiO\({}_{2}\)—synthetic ester doped with TiO\({}_{2}\), NE—natural ester, NE + TiO\({}_{2}\)—natural ester doped with TiO\({}_{2}\). where \(q\) is heat flux density [W\(\cdot\)m\({}^{-2}\)], \(\alpha\) is the heat transfer coefficient [W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-1}\)], and \(\Delta T\) is the temperature drop between the solid surface and liquid [K]. After transformation, the dependence is obtained by:  
\[\Delta T=\frac{q}{\alpha} \tag{2}\]  
In order to determine the heat transfer coefficient, the theory of similarity and dimensional analysis is used, which is based on the convergence of physical phenomena with empirical studies. It allows reducing a significant number of properties and parameters determining the heat transfer coefficient. The heat transfer process by liquid can be determined using the similarity criteria (dimensionless modules). However, one must distinguish between natural and forced convection, since in both cases, the method of determining the heat transfer coefficient varies. In the analyzed case, the considerations were limited to the case of heat transfer with free flow of liquids (natural convection), which is characteristic for most transformers. In the convective heat transfer analysis, the heat transfer coefficient is determined on the basis of the Nusselt criterion. This criterion combines the movement of heat in the fluid stream with the penetration of heat through the boundary surface. It defines the ratio of the heat transfer rate through convection to the heat transfer rate due to thermal conductivity. This criterion is defined by the dependence:  
\[Nu=\frac{\alpha\cdot\delta}{\lambda}, \tag{3}\]  
where \(\delta\) is the characteristic dimension [m] and \(\lambda\) is the thermal conductivity coefficient [W\(\cdot\)m\({}^{-1}\)\(\cdot\)K\({}^{-1}\)]. Before determining the similarity criteria, it is necessary to determine the quantities that affect the course of free convection. In the case of transformers, heat transport will depend on the temperature difference (\(\Delta T\)) between the heat transfer surface (winding, core) and the temperature of the electro-insulating liquid. In addition, convection is caused by buoyancy forces, which depend on the difference in density, \(\rho\). In turn, the difference in density results from the temperature and the thermal expansion coefficient \(\beta\). The gravity acceleration, \(g\) also plays an important role, which, together with the difference in density, affects the specific gravity. In the case of transformers, the characteristic height, \(\delta\) is the height of the heating surface. In addition, when considering the boundary layer, it should be noted that the factor causing its increase is the kinematic viscosity, \(v\). In the case of heat transfer by convection, the viscosity of the liquid causes the movement of particles in the boundary layer to slow down.