# Analysis on the Exhaust Air Recirculation of the Ventilation System in Multi-Story Building
## 2 Materials and Methods
### Velocity Profile and Grid Independence Test for CFD Analysis  
Vertical velocity profile shows the distribution of wind speed on the ground based on elevation; wind speed depends on elevation. To establish the boundary conditions for the external wind, the study considered a vertical velocity profile and created the user defined file (UDF). The velocity profile used the mathematical distribution model expressed in the law of exponent, as shown in Equation (3). In case of an urban downtown area where ground roughness is huge and uneven, the law of exponent shown in Equation (3) is usually used [25]. \[V\left(Z\right)=V_{R}\times\left(Z/Z_{R}\right)^{\alpha} \tag{3}\]  
where \(V\left(Z\right)\) refers to the average wind speed at elevation \(Z\), \(V_{R}\) refers to the average wind speed at reference height \(Z_{R}\), and exponent \(\alpha\) refers to vertical velocity profile exponent. The velocity profile by elevation, and based on the shape of vertical velocity profile, a different \(\alpha\) value (vertical velocity profile exponent) can be produced. Once \(\alpha\) is determined, the wind speed at an arbitrary elevation can be determined. KBC 2016, the structural design criteria in South Korea, uses the law of exponent as the velocity profile, and by considering \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Variable** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \begin{tabular}{c} Ventilation rate \\ (m\({}^{3}\)/h) \\ \end{tabular} & 150 & 250 & 400 & 150 & 250 & 400 & 150 & 250 & 400 & 150 \\ \hline \begin{tabular}{c} Wind direction \\ \end{tabular} & S, \(\theta\) = 180\({}^{\circ}\) & SW, \(\theta\) = 225\({}^{\circ}\) & N, \(\theta\) = 360\({}^{\circ}\) & SW, \(\theta\) = 225\({}^{\circ}\) \\ \hline \begin{tabular}{c} Installation status of \\ the guide panel \\ \end{tabular} & \multicolumn{6}{c}{Installed} & Not installed \\ \hline \begin{tabular}{c} Turbulent model \\ \end{tabular} & \multicolumn{6}{c}{RNG k-\(\epsilon\)turbulence model} \\ \hline \begin{tabular}{c} Number of elements \\ \end{tabular} & \multicolumn{6}{c}{About 4,500,000} \\ \hline \begin{tabular}{c} Exhuast and intake \\ ports condition \\ (Ventilation system) \\ \end{tabular} & \begin{tabular}{c} Velocity inlet and \\ outlet \\ \end{tabular} & \multicolumn{6}{c}{Case 1, 4, 7 and 10, \(U_{max}\) = 4.35 m/s} \\ \cline{3-6}  & \multicolumn{6}{c}{Case 2, 5 and 8, \(U_{max}\) = 7.25 m/s} \\ \cline{3-6}  & \multicolumn{6}{c}{Case 3, 6 and 9, \(U_{max}\) = 11.59 m/s} \\ \hline \begin{tabular}{c} Wind direction: South, \(\theta\) = 180\({}^{\circ}\) \\ \end{tabular} & \multicolumn{6}{c}{Wuid direction: South, 0 = 225\({}^{\circ}\)} \\ \cline{3-6}  & \multicolumn{6}{c}{Wuid direction: North, 0 = 36\({}^{\circ}\)} \\ \hline \
\begin{tabular}{c} (\(V_{R}\) = 2.3 m/s, \(R\) = 10 m) \\ \end{tabular} & \multicolumn{6}{c}{Wuid direction: North, 0 = 36\({}^{\circ}\)} \\ \cline{3-6}  & \multicolumn{6}{c}{Wuid direction: North, 0 = 36\({}^{\circ}\)} \\ \hline
\begin{tabular}{c} (\(V_{R}\) = 2.3 m/s, \(R\) = 10 m) \\ \end{tabular} & \multicolumn{6}{c}{(\(V_{R}\) = 2.3 m/s, \(R\) = 10 m)} \\ \hline \end{tabular}
\end{table} 
Table 1: Cases and boundary condition for CFD analysis. this, the index \(\alpha\) was set to 0.22 based on the ground surface conditions and the intensity of illumination in the surrounding area [26; 27]. Figure 4 shows the CFD analysis result after applying the UDF file to realize a vertical wind profile based on the surrounding environment. In this study, since the recirculation rate of exhaust air was evaluated by CFD analysis, a grid independence test was performed to increase the reliability of the analysis results. The grid independence test was performed with three different mesh element quantities, including 0.4, 2.5, and 4.5 million elements. Figure 5 establishes that the 2.5 and 4.5 million element simulations have approximately the same velocity profile at Line 1, 2, and 3.