## 2 Modeling and Computational Details
### Slip Flow Treatment  
The traditional Navier-Stokes equations have been used comfortably to describe macro-scale flows. However, they lack accuracy at local rarefied gas flow areas. To bridge this gap, a slip boundary condition at the walls has been developed and imposed. At low Kn values (Kn \(<\) 0.1), [PERSON] and [PERSON] [39] worked on the developing the regional heat transfer of a circular duct under temperature jump and slip velocity conditions with constant wall temperature. They reported that the second-order slip flow model was not significant. Therefore, the first-order slip condition proved to be valid [26]. The rarefaction of gas flow was measured using the Kn, in accordance with Equation (10), as follows:  
\[\rm{Kn}=\frac{\lambda}{D_{h}} \tag{10}\]  
where Dh is the pipe diameter with a constant value of 0.05 m. Moreover, \(\lambda\) is the local mean free path for a solid spherical molecule. It is defined as the average distance a molecule moves before it collides with another molecule, as defined in Equation (11). The value of the Lennard-Jones characteristic length of air (\(\rm{\upsigma}\)) was manipulated based on the operated pressure and temperature of Ansys-Fluent to force the value of Kn to be (0.01, 0.03, 0.05, 0.07, or 0.1) at a constant Boltzmann constant (\(\rm{k_{B}}\)) value of 1.38066 \(\times\) 10\({}^{-23}\) J/K. In this way, the effects of the absolute pressure value on the \(\rm{\upsigma}\) value became insignificant, and the derivation of pressure with respect to the traveled distance remains important to study. This approach is called the low-pressure slip flow method for laminar flow in Ansys-Fluent, where \(\rm{\upsigma}\) is a constant value input into Equation (11), and the temperature and pressure are extracted from the CFD simulation in order to estimate \(\lambda\). \[\lambda=\frac{16\mu}{5\rho}\sqrt{\frac{m}{2\pi kT}}=\frac{Tk_{B}}{\sqrt{2}\pi \sigma^{2}P} \tag{11}\] \begin{table}
\begin{tabular}{l c c} \hline \hline
**Property** & **Air** & **Al2O3** \\ \hline Specific Heat: C\({}_{\rm p}\) (J/kg\(\cdot\)K) & 1006 & 765 \\ Density: \(\rm{\uprho}\) (kg/m\({}^{3}\)) & 1 & 3970 \\ Thermal Conductivity: k (W/m\({}^{2}\cdot\)K) & 0.025 & 40 \\ Thermal Diffusivity: \(\rm{\upalpha}\) (m\({}^{2}\)/s) & \(19.00\times 10^{-6}\) & \(13.17\times 10^{-6}\) \\ Thermal Expansion Coefficient: \(\rm{\upbeta}\) (1/K) & \(3.33\times 10^{-3}\) & \(8.50\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the utilized base fluid and nanoparticles. In a pure convection problem, heat transfer through the wall of a tube is characterized by a suitable thermal boundary condition, which is directly or indirectly specified at the interface of the wall-fluid. The set of conditions known as thermal boundary conditions describe the temperature and/or heat flow at the tube's inner wall. Since the velocity and temperature are only determined in the fluid region, it is not necessary, in such problems, to provide a solution to the temperature problem for the solid wall. Consequently, the adopted slip and temperature jump boundary conditions followed the first-order [PERSON]'s model [2] with wall perpendicular velocity (\(\mathrm{u_{g}}=0\)), as follows:  
\[u_{w}-u_{g}=\left(\frac{2-\sigma_{v}}{\sigma_{v}}\right)Kn\;D_{h}\;\frac{ \partial u}{\partial n}\approx\left(\frac{2-\sigma_{v}}{\sigma_{v}}\right)\; \frac{\lambda}{\delta}\;\left(u_{g}-u_{c}\right) \tag{12}\]  
\[T_{w}-T_{g}=2\left(\frac{2-\sigma_{T}}{\sigma_{T}}\right)\;\frac{2\gamma}{ \gamma+1}\;\frac{k}{\mu C_{v}}\;Kn\;D_{h}\;\frac{\partial T}{\partial n} \approx 2\left(\frac{2-\sigma_{T}}{\sigma_{T}}\right)\;\frac{2\gamma}{ \gamma+1}\;\frac{k}{\mu C_{v}}\;\frac{\lambda}{\delta}\;\left(T_{g}-T_{c}\right) \tag{13}\]  
where \(\delta\) is the distance from cell center to the wall, \(\mathrm{\sigma_{v}}\) and \(\mathrm{\sigma_{T}}\) represent the momentum and thermal accommodation coefficients.