## 2 The Improved Gas Leakage Model of Pressure Pipeline
### The Verification of the Improved Gas Leakage Model
#### 2.4.1 Comparison between Numerical and Experimental Results of the Mach Disk Position and Diameter  
When the initial leakage pressures were 1 and 3 Mpa, the state parameters of the leaked hydrogen and the leakage mass flow rates calculated by the hole model and the improved leakage model are shown in Table 3. It can be seen from Table 4 and the above description from Section 2.1 that in the hole model, the gas in the leakage section was considered to be in the sonic flow state, and the leakage mass flow rate also reached the maximum value under the above pressure conditions; thus, the Mach disk could not be obtained in the flow area. However, the leakage mass flow rate and the gas pressure at the outlet of the leakage section calculated by the improved leakage model were lower than the calculation results of the hole model, and the leakage velocity at the outlet of the leakage section was higher than the local velocity of sound. The reason for this difference was that when the diameter of the leakage section was 1 mm, the pressure of the gas in the leakage section was higher, and the gas was still in a subsonic flow state, and the leakage mass flow rate had not yet reached the maximum value. When the gas was close to the outlet of the leakage section, the gas underwent a secondary expansion process due to the pressure difference between the inside of the pipeline and the external space, causing its pressure to reduce and velocity to increase. At this time, a choked flow was formed in the leakage section, and the leakage mass flow rate would not increase with the decrease in the gas pressure. The variation of the Mach number of the leaked hydrogen in the local flow area based on the improved gas leakage model is shown in Figure 5. It can be seen that under the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **Parameter** & **Pressure (MPa)** & **Temperature (K)** & **Velocity (m/s)** \\ \hline \multirow{4}{*}{Hole model} & Initial leakage value & 1 & 300 & 0 \\  & Critical leakage value & 0.53 & 249.32 & 1206.31 \\  & Maximum leakage mass flow (g/s) & & 0.48 & \\  & Leakage section outlet & 0.40 & 234.32 & 1371.85 \\  & Leakage mass flow (g/s) & & 0.41 & \\  & Initial leakage value & 3 & 300 & 0 \\  & Critical leakage value & 1.58 & 249.04 & 1213.71 \\ Hole model & Maximum leakage mass flow (g/s) & & 1.44 & \\  & Leakage section outlet & 1.18 & 233.53 & 1381.14 \\  & Leakage mass flow (g/s) & & 1.24 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Leakage gas state parameters obtained from the hole model and the improved leakage model. condition of initial leakage pressures of 1 and 3 Mpa, the Mach numbers of the gas at the outlet of the leakage section were 1.17 and 1.18, respectively. The leaked hydrogen was injected into the external space at supersonic velocity and continued to expand, and a series of expansion waves were formed in the flow area. After the leaked gas through the expansion wave, its temperature and pressure decreased rapidly, the Mach number increased, and the maximum Mach number values reached 3.03 and 4.72 at different pressure conditions, respectively. When the pressure of the leaked gas was lower than the atmospheric pressure, the gas began to be compressed by the surrounding environment and moved toward the direction of the jet axis. At this time, the expansion waves were converted into compression waves, and when countless compression waves met, a normal shock wave was formed in the direction of the vertical gas jet, also known as a Mach disk. After the leaked gas through the Mach disk, the temperature and pressure increased rapidly, the Mach number decreased, and the gas changed from the supersonic flow state to the subsonic flow state. The Mach disk formed by the leaked gas in the flow area was closely related to the gas state parameters at the outlet of the leakage section, so the position and diameter of the Mach disk were selected to verify the accuracy of the gas state parameters calculated by the improved leakage model.