## 2 Materials and Methods  
### Numerical Model Methodology  
Figure 1a shows a schematic of a HE with elliptical corrugated tubes and its geometrical parameters. Figure 1b illustrates a schematic of the elliptical tube considered for the present study. All boundary conditions are illustrated in Figure 1b. Figure 1c presents a detailed image of the HE with elliptical corrugated tubes studied in this paper. The following assumptions are considered to perform the required simulations:  
* The flow is in the steady state [49]. * The ethylene glycol/magnesium oxide-multiwalled carbon nanotube (EG/MgO-MWCNT) hybrid nanofluid is simulated as a single-phase fluid. * Laminar flow regime (\(50\leq\text{Re}\leq 1000\)), \(T_{\text{{null}}}=400\) K, and \(T_{\text{{in}}}=313\) K and \(323\) K.
* The convergence criterion is considered to be less than \(10^{-6}\) for all the effective variables. * Gravity effects and kinetic losses are considered. Figure 1: Schematic of (**a**) heat exchanger with elliptical corrugated tubes, (**b**) elliptical tube and boundary conditions, and (**c**) elliptical corrugated tube. * Velocity inlet and constant inlet temperature boundary conditions are used for the inlet. * The pressure outlet boundary condition is employed for the heat exchanger exit. * The constant temperature boundary condition is used on the side walls. * The diffusion terms in the energy and momentum equations are approximated using second-order central difference. * For the convective terms, the second-order upwind scheme is employed. The governing equations for two-phase NF flow and HT in the microchannel are as follows [50; 51; 52; 53; 54]:  
Continuity equation:  
\[\
abla\cdot\left(\rho_{nf}\ V_{m}\right)=0 \tag{1}\]  
where \(\rho_{nf}\) is the NF density and \(V_{m}\) is the mean velocity. Momentum equation:  
\[\
abla\cdot\left(\rho_{nf}\ V_{m}V_{m}\right)=-\
abla P+\
abla\cdot\left(\mu_{ nf}\
abla V_{m}\right)+\left(M\cdot\
abla\right)\cdot B \tag{2}\]  
where \(\mu_{nf}\) is the NF viscosity. Energy equation:  
\[\
abla\cdot\left(\rho_{nf}\ c\ V_{m}\ T\right)=\
abla\cdot\left(k_{nf}\ \
abla\ T\right)+\mu_{0}(M\cdot\
abla)\cdot H \tag{3}\]  
Table 1 shows the geometrical parameters of the heat exchanger with elliptical corrugated tubes implemented in the present investigation. The average Nu and thermal resistance coefficient for elliptical and circular tubes are presented in Table 2. Table 3 presents the required coefficients to calculate the equations of Table 1 in terms of the Nusselt number at different weight concentrations. Table 4 shows the required coefficients to calculate the equations of Table 1 in terms of the thermal resistance coefficient at different weight concentrations. The thermophysical properties of the EG/MgO-MWCNT hybrid nanofluid are presented in Table 5. In Table 6, thermal and hydrodynamic equations are given to evaluate the different effective parameters. Table 7 presents material properties, and Table 8 illustrates the thermophysical objects of the NF considered in this paper. \begin{table}
\begin{tabular}{c c} \hline \hline
**Equation** & **Eq. No.** \\ \hline \(\dot{Q}=mc_{p}(T_{in}-T_{out})\) & (13) \\ \(h=\frac{\dot{Q}}{A_{e}A\overline{I}_{m}}=\frac{\dot{m}\sigma_{p}(T_{in}-T_{out}) }{A_{e}\left(\frac{(T_{in}-T_{in})-(T_{in}-T_{out})}{\ln\frac{\dot{m}\sigma_{p} }{T_{out}}}\right)}\) & (14) \\ \(A_{e}=2\pi L\sqrt{\frac{\dot{q}^{2}+\dot{p}^{2}}{2}}\) & (15) \\ \(Nu=\frac{\dot{h}D_{e}}{\dot{k}}\) & (16) \\ \(\dot{W}=\frac{\dot{m}\Delta P}{\dot{\rho}}\) & (17) \\ \(f=\frac{\Delta P\dot{\rho}D_{e}}{\dot{\rho}L^{\dagger}}\) & (18) \\ \(\dot{S}_{gont}=\dot{S}_{gont},T+\dot{S}_{gont},F\) & (19) \\ \(\dot{S}_{gont},T=\frac{\dot{Q}}{NukT_{in}L}\) & (20) \\ \(\dot{S}_{gont},F=\frac{\dot{g}\pi^{2}L^{2}}{\rho^{2}\Pi^{2}\Pi^{2}\Pi^{2}(T_{ out}-T_{in})}\ln\frac{T_{out}}{T_{in}}\) & (21) \\ \(\dot{\psi}=\frac{S_{gont}}{m\sigma_{p}}=\frac{\Delta S_{gont}}{Q}\) & (22) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Implemented thermal hydrodynamic equations in the model [54,55].