## 3 Methodology  
### Selection of Suitable Nanoparticles--Therrophysical Properties of Nanofluid  
In the present study, the thermophysical properties of different nanoparticles, such as Al\({}_{2}\)O\({}_{3}\), TiO\({}_{2}\), Cu, MWCNT, and SWCNT, are considered with mineral oil as base fluid and are shown in Tables 1-4. To obtain the effective thermophysical properties of the hybrid nanofluid, the following equations [35] are given:  
The effective density of the hybrid nanofluid is denoted by \(\rho_{m}\) and is expressed as  
\[\rho_{lnf}=\rho_{f}(1-\phi_{2})\Bigg{[}(1-\phi_{1})+\phi_{1}\Bigg{(}\frac{\rho _{s}}{\rho_{f}}\Bigg{)}\ \Bigg{]}+\phi_{2}\rho_{s2} \tag{8}\]where \(\phi\) is the solid volume fraction of the nanoparticles. The heat capacitance of the hybrid nanofluid is denoted by \(\big{(}\rho c_{p}\big{)}_{lnf}\) and it can be expressed as:  
\[\big{(}\rho c_{p}\big{)}_{lnf}=\big{(}\rho c_{p}\big{)}_{f}(1-\phi_{2})\bigg{[}( 1-\phi_{1})+\phi_{1}\bigg{(}\frac{\big{(}\rho c_{p}\big{)}_{s1}}{\big{(}\rho c_{ p}\big{)}_{f}}\bigg{)}\bigg{]}+\phi_{2}\big{(}\rho c_{p}\big{)}_{s2} \tag{9}\]  
In addition, \(\mu_{lnf}\) is the effective dynamic viscosity of the hybrid nanofluid, which is determined by:  
\[\mu_{lnf}=\frac{\mu_{f}}{\big{(}1-\phi_{1}\big{)}^{2.5}(1-\phi_{2})^{2.5}} \tag{10}\]  
\[\frac{k_{lnf}}{k_{bf}}=\frac{k_{s2}+(s-1)k_{bf}-(s-1)\phi_{2}\Big{(}k_{bf}-k_{ s2}\Big{)}}{k_{s2}+(s-1)k_{bf}+\phi_{2}\Big{(}k_{bf}-k_{s2}\Big{)}} \tag{11}\]  
where,  
\[\frac{k_{bf}}{k_{f}}=\frac{k_{s1}+(s-1)k_{f}-(s-1)\phi_{1}\Big{(}k_{f}-k_{s1} \Big{)}}{k_{s1}+(s-1)k_{f}+\phi_{1}\Big{(}k_{f}-k_{s1}\Big{)}} \tag{12}\]  
In these equations, indexes \(s\) and \(f\) represent the solid particles and fluid, respectively. Finally, the thermophysical properties of the nanoparticles are shown in Table 1. The thermophysical properties of the nanofluid mixture tabulated in Tables 2-5 are determined using the empirical correlations represented in Equations (8)-(12). The volume fractions of the hybrid nanoparticles and the thermophysical properties such as density, viscosity, specific heat, and thermal conductivity of base fluid and nanoparticles are considered in determining the mixture properties. The nanoparticles are assumed as monodispersed spherical particles of uniform size.  
In Table 3, numerical values of thermophysical properties for the Al\({}_{2}\)O\({}_{3}\)-Cu hybrid nanofluid concentration range are listed and were used in these simulations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nanoparticles** & **Thermal Conductivity (W/m-K)** & **Specific Heat (J/kg-K)** & **Density (kg/m\({}^{3}\))** \\ \hline Al\({}_{2}\)O\({}_{3}\) & 40 & 765 & 3970 \\ Cu & 401 & 385 & 8933 \\ SWCNT & 6600 & 425 & 2600 \\ MWCNT & 3000 & 796 & 1600 \\ TiO\({}_{2}\) & 8.5 & 703 & 4160 \\ \hline \hline \end{tabular} For the range of Al\({}_{2}\)O\({}_{3}\)–TiO\({}_{2}\) hybrid nanofluid concentrations, the effective thermophysical properties listed in Table 2 were used in these simulations.  
\end{table}
Table 1: Thermophysical and mechanical properties of nanoparticles [40; 41; 42; 43].  
In Table 4, numerical values of thermophysical properties for the Al\({}_{2}\)O\({}_{3}\)-SWCNT concentration range are listed and were used in these simulations.  
In Table 5, numerical values of thermophysical properties for the Al\({}_{2}\)O\({}_{3}\)-MWCNT concentration range are listed and were used in these simulations.