# Numerical Evaluation of Hot Air Recirculation in Server Rack
## 3 Numerical Analysis
### Mesh Sensitivity Study  
The internal volume of the server was extracted using the surface wrapper feature of Star CCM+. The volume of interest is the volume of the surface wrapper extracted from the self-contained volume of the server. In other words, all outer surfaces of the internal parts of the server were surface wrapped to form a volume within the server. The method was used to create seed points placed at two defined locations within the server and extract the volume. A surface mesh wrapper was used concurrently with trimmer volume meshes (hexahedral mesh with a minimal cell skewness angle close to zero to target orthogonal meshes around concave cells). The prism layer with a total thickness of \(1\times 10^{-4}\) m from the surface and seven inflation layers, produces near-wall cells of \(y^{+}=5\) the values used. The subsequent objective was to determine the optimal mesh grid size that would prevent a significant decrease in the model's accuracy resulting from the discretization of the computational domain [29]. Therefore, a grid convergence index (GCI) was used to evaluate the discretization error of the computational meshes to ensure the accuracy and convergence of the numerical solutions [30]. The Richardson extrapolation method was used in the analysis, and cell size was used to obtain the minimal refinement ratio. A refinement ratio \(r=G\_course/G\_fine\) of 1.34 is proposed by [PERSON] et al. [28] for lowering grid sizes [31; 32; 33]. \[GCI_{\text{fine}}\ =\ \frac{1.25|\varepsilon|}{r^{p}-1} \tag{10}\]  
\[\varepsilon=(f_{2}-f_{1})/f_{1} \tag{11}\]  
where is \(r\) the refinement ratio between two successive grids, and N cells is the number of grid elements. The GCI values were calculated using different mesh sizes to test the mesh convergence. The outlet pressure from the far field was used. The initial boundary condition at the outflow was set to a zero-gauge pressure. A three-grid investigation of grid convergence was used in this study, and the base sizes selected from which the mesh cells were generated were f_1, f_3, and f_5, as listed in Table 4. The outlet pressure is represented by phi and its extrapolated value is extrapolated by phi. Where \(r\) is the refinement ratio between two successive grids, and N cells is the number of grid elements. The grid convergence index (GCI) is a percentage, and the asymptotic GCI indicates its asymptotic value. When its value was approximately unity, it denoted a grid-independent solution. \(p\) indicates the order attained during the simulation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(f\)} & **Base Size** & **Cell Mesh Size** & **Outlet Pressure** & **Fan Thrust** & **Error** & **Error** \\  & **[mm]** & **[-]** & **[Pa]** & **[N]** & **(Outlet Pressure)** & **(Fan Thrust)** \\ \hline
1 & 5.5 & \(1.220578\times 10^{6}\) & 2.254 & \(1.251\times 10^{-3}\) & 0.000\% & 0.000\% \\
2 & 5 & \(1.246704\times 10^{6}\) & 2.124 & \(1.231\times 10^{-3}\) & 5.768\% & 1.599\% \\
3 & 4.5 & \(1.287264\times 10^{6}\) & 2.114 & \(1.223\times 10^{-3}\) & 0.471\% & 0.650\% \\
4 & 4 & \(1.799871\times 10^{6}\) & 2.107 & \(1.245\times 10^{-3}\) & 0.331\% & \(-1.799\%\) \\
5 & 3.5 & \(1.962441\times 10^{6}\) & 2.130 & \(1.235\times 10^{-3}\) & \(-1.092\%\) & 0.803\% \\
6 & 3 & \(1.989244\times 10^{6}\) & 2.095 & \(1.240\times 10^{-3}\) & 1.643\% & \(-0.405\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational grid sensitivity results. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**J** & **Kt** & **Kq** & **Eta** \\ \hline
0.01 & 0.0614 & 0.1496 & 0.00019 \\
0.3 & 0.0473 & 0.1314 & 0.013 \\
0.625 & 0.0318 & 0.116 & 0.0048 \\
1.55 & \(-\)0.0127 & \(-\)0.0521 & \(-\)0.017 \\
2 & \(-\)0.0343 & \(-\)0.1503 & \(-\)0.03 \\
2.6 & \(-\)0.063 & \(-\)0.2813 & \(-\)0.045 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The power flow curve of the fan model.