## 2 Modeling and Computational Details
### Slip Flow Treatment
The value of \(\mathrm{\sigma_{v}}\) and \(\mathrm{\sigma_{T}}\) for many engineering applications of gas-surface material is between 0.8 and 1.0. For air, it is often assumed that \(\mathrm{\sigma_{v}}=\mathrm{\sigma_{T}}\approx 1\)[28]. For all currently simulated Re values, the resulting error in estimating the value of Nu was less than 7% at \(\mathrm{\sigma_{v}}=\mathrm{\sigma_{T}}\) values between 0.9 and 1.0. Consequently, the value of \(\mathrm{\sigma_{v}}=\mathrm{\sigma_{T}}=1.0\) was adopted throughout the current study. Throughout the current study, the average Nu was calculated based on Equation (14) by integrating the local Nu in the flow direction. The average Nu depends on many parameters, as follows: the applied heat flux on the pipe walls (\(\mathrm{q^{\prime\prime}}=5\;W\)), \(\mathrm{D_{h}}=0.05\;\mathrm{m}\), \(\mathrm{k_{nf}}\), the fluid temperature at the wall (\(\mathrm{T_{w}(x)}\)), and the fluid mean temperature at a given location in the flow direction (\(\mathrm{T_{m}(x)}\)). Under the constant properties assumption, \(\rho\), \(\mathrm{C_{p}}\), and the horizontal axial velocity (\(\mathrm{U}\)) were taken at the inlet of the pipe with the constant pipe radius (\(\mathrm{R}\)). \(\mathrm{T_{w}(x)}\) values were extracted from the CFD model for each different simulation, while \(\mathrm{T_{m}(x)}\) values were calculated based on Equation (15). \[Nu=\int_{0}^{L}Nu(x)dx=\int_{0}^{L}\frac{h(x)D_{h}}{k_{nf}}dx=\int_{0}^{L} \frac{q^{\prime\prime}D_{h}}{k_{nf}(T_{w}(x)-T_{m}(x))}dx \tag{14}\]  
\[T_{m}(x)=T_{m,\;intel}+\frac{2q^{\prime\  
the effects of viscous dissipation, natural convection, radiation heat transfer, the mass-weighted average bulk fluid temperature, and the first-order temperature jump boundary condition.  
The value of R was constant at 2.5 cm, and the values of Kn, the specific heat ratio (\(\gamma\)), and Pr were variables based on the specific simulated case and nanoparticle volume fraction, as listed in Table 1. Furthermore, the value of \(\mathrm{Kr_{T}}\) was connected with the value of Kn based on the assumption of \(\sigma_{\mathrm{v}}=\sigma_{\mathrm{T}}=1.0\).  
\[\frac{U}{U_{m}}=\frac{2\left[1-\left(r/R\right)^{2}\right]+Kn}{1+8 Kn} \tag{16}\]  
\[ReC_{f}=\frac{16}{8 Kn+1} \tag{17}\]  
\[Nu=\frac{1}{\frac{1}{48(8 Kn+1)^{2}}+\frac{1}{12(8 Kn+1)}+\frac{1}{8}+\frac{2 \gamma Kn_{T}}{(1+\gamma)P_{T}}} \tag{18}\]  
Structured mesh elements were dependent on the density of elements in the perpendicular direction, with respect to the pipe wall. Five different mesh sizes, as listed in Table 2, were utilized for the mesh sensitivity analysis at six values of Kn, at Re = 1750, which lies within the laminar flow region inside the pipe, and at Pr = 0.765. The number of mesh elements varied between 3000 (AS0) and 450,000 (AS4). They corresponded to perpendicular distances of 1.042 mm (AS0) and 83.3 \(\upmu\)m (AS4) between the first cell center in touch with the pipe wall and the cell face center of the wall. These distances are important in solving gas slip velocities and temperature jump values at cells in touch with constant heat flux pipe walls. It is worth mentioning that Nu is independent of the value of wall heat flux when a no-slip wall boundary condition is utilized. All simulated mesh sizes showed similar behavior of decreasing values with the increase in Kn values, as shown in Figure 2. For (U/U\({}_{\mathrm{m}}\)) and (ReC\({}_{\mathrm{f}}\)) simulations, as shown in Figure 2a,b, the maximum error in (U/U\({}_{\mathrm{m}}\)) corresponded to AS0 mesh, with values between 0.6 and 0.7%, while the minimum error in (U/U\({}_{\mathrm{m}}\)) corresponded to AS3 and AS4 meshes, with values between 0 and 0.4% for Kn values between 0.0 and 0.1, respectively. The difference in the reported errors between AS0 and AS3 was 0.6% at Kn = 0.0 and reduced to 0.3% at Kn = 0.1 in favor of mesh AS3 and AS4. Similar results were reported of the error values for (ReC\({}_{\mathrm{f}}\)) simulations, with values between 1 and 1.5% at AS0 mesh and 0.0 and 1.2% at AS3 and AS4 meshes.