## 2 The Improved Gas Leakage Model of Pressure Pipeline
### The Numerical Simulation  
#### 2.3.1 Governing Equations of the Diffusion Process  
The actual diffusion process of the leakage gas in the external space can be considered as a single-phase, turbulent flow and heat transfer process, thus, the mass, momentum and energy conservation equations were adopted. At the same time, the leaked gas was continuously mixed with the air in the external space, and the diffusion flux of the leaked gas was calculated by the component transport equation. The general form of the governing equations is as follows:  
Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho u_{ j}\big{)}=0 \tag{12}\]  
where \(\rho\) is the density of mixed gas composed of leaked gas and air; \(x_{j}\) is the three diffusion directions x, y and z of the leaked gas in external space; \(u_{j}\) is the corresponding velocity component of the three directions.  
Momentum equation:  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\bigg{(}\mu_{t}\frac{\partial u_{i}}{\partial x_{j}}\bigg{)}+ \frac{\partial}{\partial x_{j}}\bigg{(}\mu_{t}\frac{\partial u_{j}}{\partial x _{i}}\bigg{)}+(\rho-\rho_{a})g_{i} \tag{13}\]  
where \(\mu_{t}\) is the turbulent viscosity of the mixed gas; \(g\) is the acceleration of gravity; \(p\) is the pressure of the mixed gas; \(\rho_{a}\) is the density of the air.  
Energy equation:  
\[\frac{\partial(\rho T)}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{j}T\big{)}=-\frac{P}{c_{P}}\frac{\partial u_{j}}{\partial x_{j}}+ \frac{1}{c_{P}}\frac{\partial}{\partial x_{j}}\bigg{(}k_{t}\frac{\partial T}{ \partial x_{j}}\bigg{)}+\frac{c_{P_{t}}-c_{P_{a}}}{c_{P}}\bigg{(}\rho D_{t} \frac{\partial\omega}{\partial x_{j}}\bigg{)}\frac{\partial T}{\partial x_{j} }+\phi \tag{14}\]  
\[\phi=2\mu_{t}\bigg{(}\frac{\partial u_{j}}{\partial x_{j}}\bigg{)}^{2}+\mu_{t} \bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x _{i}}\bigg{)}^{2} \tag{15}\]  
where \(T\) is the temperature of the mixed gas; \(k_{t}\) is the turbulent thermal conductivity; \(c_{P}\) is the specific heat capacity of the mixed gas; \(c_{P_{i}}\) is the specific heat capacity of the leaked gas; \(c_{P_{j}}\) is the specific heat capacity of the air; \(\omega\) is the mass fraction of the leaked gas in the mixed gas.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **The Current Hole Model** & **The Improved Gas Leakage Model** \\ \hline \multirow{4}{*}{The schematic figure} & & & \\  & The transport process & The same model and solution method were used. \\  & The leakage process & The gas pressure at the outlet of the leakage & The relationship between the gas flow \\  & The leakage process & section was directly assumed based on the & \\  & & given critical pressure ratio formula. & at point two can be solved. \\ Difference & & The change in the gas state parameters in the & The expansion process of the gas in the \\  & The expansion process & interval from the leakage port in the pipeline & leakage section was considered, and the \\  & to the outer wall of the pipeline was ignored. & state parameters of the gas at the outlet of \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the current hole model and the improved leakage model.  
Component transport equation:  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{j}\omega\big{)}=\frac{\partial}{\partial x_{j}}\Bigg{(}\rho D_{t}\frac{ \partial\omega}{\partial x_{j}}\Bigg{)} \tag{16}\]  
where \(D_{t}\) is the turbulent mass diffusion coefficient.