## 2 Materials and Methods
### Validation of Numerical Model
#### 2.2.2 Validation Results  
In this section, the computer code is verified by comparing the computational outcomes obtained from the present simulations, analytical results of Cengel [56], and numerical results of Huminic and Huminic [57], considering similar geometrical dimensions and boundary conditions. Huminic and Huminic [57] investigated the consequence of HNFs on the performance of a smooth elliptical tube using a single-phase model numerically. Figure 4 compares the analytical and numerical results. As can be seen from Figure 4, there is an excellent agreement between the analytical and numerical data [56; 57] and the numerical data obtained from the present study. The maximum deviation is 1.14% in terms of the average friction factor for different Reynolds numbers.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{4}{c}{**Value**} \\ \cline{2-4}  & **Ethylene Glycol (EG)** & **Multiwalled Carbon Nanotubes** & **Magnesium Oxide Nanoparticles** \\ \hline
**Chemical Formula/Symbol** & **C\({}_{2}\)H\({}_{6}\)O\({}_{2}\)** & **COOH-Functionalized MWCNTs** & **MgO** \\ \hline \multirow{2}{*}{\begin{tabular}{c} Molar mass \\ Appearance \\ Qdor \\ Density \\ \end{tabular} } & 62.07 (g/mol) & \multirow{2}{*}{\begin{tabular}{c} 40.30 (g/mol) \\ - \\ 2400 (kg/m\({}^{2}\)) \\ \end{tabular} } \\  & Clear, colorless liquid & Black & White \\ \multirow{2}{*}{\begin{tabular}{c} Qdor \\ Density \\ \end{tabular} } & 1113.2 (g/m\({}^{2}\)) & \(\sim\)2100 (kg/m\({}^{2}\)) & 3580 (kg/m\({}^{2}\)) \\ \multirow{2}{*}{\begin{tabular}{c} Specific heat capacity \\ Melting point \\ \end{tabular} } & 2470 (/kg/K) & 630 (W/m\(\cdot\)K) & 964 (W/m\(\cdot\)K) \\  & \(-\)12.9 (\%) & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } \\  & Boiling point & 197.3 (\%) & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } \\ Thermal conductivity (@20 \({}^{\circ}\)C) & 0.244 (W/m\(\cdot\)K) & 50 (W/m\(\cdot\)K) & 45 (W/m\(\cdot\)K) \\ Viscosity (@20 \({}^{\circ}\)C) & 0.016 (Pa.s) & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } \\  & - & \(>\)97\% & \(>\)99\% \\ \multirow{2}{*}{\begin{tabular}{c} Content of -COOH \\ Outer diameter \\ \end{tabular} } & - & 2.56 (wt\%) & \multirow{2}{*}{\begin{tabular}{c} - \\ \end{tabular} } \\  & - & 5-15 (nm) & 40 (nm) \\ \multirow{2}{*}{\
\begin{tabular}{c} Inner diameter \\ Length \\ SSA \\ \end{tabular} } & - & \(\sim\)50 (nm) & - \\ \multirow{2}{*}{
\begin{tabular}{c} Morphology \\ \end{tabular} } & - & \(>\)233 (m\({}^{2}\)/g) & \(\sim\)25 (m\({}^{2}\)/g) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Material properties [56].  
Figure 2: Schematic of computational domain and grid used in the simulations.  
Figure 4: Average friction factor versus Reynolds number: comparison between the present results and those reported by [PERSON] (2016) and Huminic and Huminic (2017).  
Figure 3: Grid independence test in terms of (**a**) local velocity and (**b**) local temperature.