## 2 Materials and Methods
### Numerical Model--Mesh Sensitivity Analysis and Model Validation  
In order to demonstrate the model reliability and solution independence from the grid, a mesh sensitivity analysis was conducted. Among the five configurations investigated by [PERSON] et al. (2017), the present authors decided to validate the developed numerical model on the results for the E1-2 configuration with 0\({}^{\circ}\) fiber inclination angle and cylindrical shape. For what concerns the solution independence from the grid, the E1-2 was adopted as reference case. The comparison was made in terms of friction factor \(f\) and thermal power \(Q\) for all the five velocities analyzed, as defined below [27]:  
\[f=\frac{\Delta P}{L}\frac{D}{2\rho_{f}\Omega_{in}^{2}} \tag{4}\]  
\[Q=\dot{m}c_{p,f}(T_{out}-T_{in}) \tag{5}\]  
where \(\frac{\Delta P}{L}\) is the pressure gradient (Pa m\({}^{-1}\)), \(D\) the hydraulic diameter (m), and \(\dot{m}\) the mass flow rate (kg s\({}^{-1}\)). The subscript f refers to the fluid, i.e., air.  
Three different meshes were simulated. The coarsest one, denoted as \"Mesh 1\" in Figure 4, consisted of some 6.2 M elements. \"Mesh 2\" 10.4 M and \"Mesh 3\" 18.3 M elements, respectively. As it can be observed from Figure 4, the coarsest mesh resulted in friction factor and thermal power values that deviated from [PERSON] et al. (2017) results by more than \(-10\%\) (see the lower dotted line). Mesh 2 and Mesh 3, instead, showed \(f\) and \(Q\) values that differed less than \(10\%\) with respect to [27]. From what was calculated, the present authors decided to consider Mesh 2 as the best trade-off between solution accuracy and computational efforts. Therefore, the other meshes for all the other cases were obtained by  
Figure 3: Geometry of the simulated system.  
adopting the same characteristics of Mesh 2. In Table 3, the number of elements for all the cases is reported.  
Figure 5 clearly demonstrates the validity of the developed numerical model. As shown in Figure 5a, the friction factor deviates less than 10% (represented by the dotted lines) from [PERSON] et al.'s (2017) results, both for the elliptical and cylindrical configurations. Moreover, the results for the cylindrical case differ less than 3.5% with respect to the results for the same case (2017). Even for the thermal power \(Q\), similar conclusion can be taken. Our simulated cases deviate no more than 4% from what reported in (2017) for all the investigated scenarios. The minor differences between the results may be attributed to a different numerical model generation: in fact, as explained above, differently from [PERSON] et al. (2017), the present authors studied half of the geometrical domain, taking advantage of the symmetry of the system. Therefore, despite the small deviations, the present numerical model can be considered a valid alternative requiring almost half of the computational effort which can have a positive effect on computational efficiency allowing for the possibility of investigating many additional cases.  
Figure 4: Mesh sensitivity analysis: (**a**) Friction factor and (**b**) thermal power.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & 0\({}^{\circ}\) & 15\({}^{\circ}\) & 30\({}^{\circ}\) & 45\({}^{\circ}\) \\ \hline C\({}_{0}\) & 6.4 & - & - & - \\ E1-1 & 5.5 & 5.5 & 5.7 & 5.7 \\ E1-2 & 10.4 & 10.6 & 10.3 & 10.1 \\ E2-1 & 6.1 & 6.2 & 6.2 & 6.2 \\ E2-2 & 10.5 & 11.5 & 11.0 & 13.9 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mesh number elements (\(\bullet 10^{6}\)) for all the simulated cases.