## 4 Results and Discussion
### Leak Source Localization  
A cruciform array of twelve sensors was used to locate the LS. Centered on one sensor, five sensors were mounted at 10 cm intervals along the axial (z) axis, and six sensors were mounted at 30\({}^{\circ}\) intervals along the tangential (\(\theta\)) axis (\(\theta\) ranges from 0 to 180 degrees). Two regions of the cylinder surface were selected for the location of the sensor array, namely the lower and upper regions shown in the first column of Figure 6 (hereafter referred to as Zonel and Zone2, respectively). To locate the angular position of the LS, we first estimated the minimum and its position and then shifted \(\Delta\theta\) to match the angular dependence of the \(A_{\rm obs}\) value with the \(A_{\rm sim}(\theta)\) values calculated by Equation (53) (Figure 6a,c). To locate the axial position, we estimated \(A_{\rm 0,axl}\) by applying the exponential decay to the observed values of \(A_{\rm obs}\) and then shifted \(\Delta z\) to match the axial dependence of the \(A_{\rm obs}\) value with the \(A_{\rm sim}(\eta)\) values calculated by Equation (57) (Figure 6b,d). The matching results for finding the position of the LS location are listed in the second and third columns of Table 2.  
As listed in Table 2, the error of the LS location is less than 2% for the angular location and less than 1.0% for the axial location. The theoretical model proposed in this study provides a very accurate solution for the LS location. Furthermore, the \(A_{\rm 0,ang}\) and \(A_{\rm 0,axi}\) obtained from the localization processes are listed in Table 2. For a given set of experimental conditions, the two values (\(A_{\rm 0,ang}\), \(A_{\rm 0,axi}\)) agree well within an error of 10%. The  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Array** & **P\({}_{\rm 0}\)** & **\(\phi\)** & **\(\eta\)/mm** & **A\({}_{\rm 0,ang}\)/\(A_{\rm 0,axi}\)** & **Reynolds NO/k** \\
**Position** & **(bar)** & **(error\%)** & **(error\%)** & **(error\%)** & **A\({}_{\rm 0,ang}\)/\(A_{\rm 0,axi}\)** & **Theory/Estimation** \\ \hline  & 2.07 & \(-\)6.1\({}^{\circ}\) (1.7) & 5.0 (0.5) & 0.05/0.05 & 5.3/5.5–5.9 \\ Zone1 1  & 3.06 & \(-\)6.1\({}^{\circ}\) (1.7) & 5.0 (0.5) & 0.09/0.08 & 7.2/7.0–8.4 \\  & 3.97 & \(-\)6.1\({}^{\circ}\) (1.7) & 5.0 (0.5) & 0.13/0.12 & 8.5/8.5–11.7 \\ \hline  & 2.01 & 3.6\({}^{\circ}\) (1.0) & 7.0 (0.7) & 0.12/0.11 & 13.3/8.0–11.0 \\ Zone2 2  & 3.01 & 3.6\({}^{\circ}\) (1.0) & 7.0 (0.7) & 0.26/0.25 & 18.1/18.1–23.0 \\  & 4.01 & 3.6\({}^{\circ}\) (1.0) & 7.0 (0.7) & 0.40/0.38 & 21.5/28.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results for the localization of the leak source determined by fitting the observed signal amplitudes to the simulated values. The leak source is located at \(\theta=0^{\circ}\) and \(\eta=0\).  
Figure 6: Fitting the angular (**a,c**) and the axial (**b,d**) dependences of the observed mean maximum amplitudes (circles) and \(\sigma\)s (bars) obtained from the sensor array in Zonel (**a,b**) and Zone2 (**c,d**) to the simulated values (solid lines) (three different gas pressures used).  
Reynolds number was estimated by comparing the \(A_{0,\text{ang}}\) and \(A_{0,\text{axi}}\) values obtained from the localization processes with the experimental values shown in Figures 3f and 5f, respectively. As listed in Table 2, the estimated Reynolds number is very close to the theoretical value calculated from the leak hole size and the initial pressure in the + cylinder given by Equation (54). The results suggest that the amplitudes observed from the localization process can provide quantitative information about the hole diameter of the LS if the initial pressure is known.