## 2 The Improved Gas Leakage Model of Pressure Pipeline
### The Improved Gas Leakage Model and Its Corresponding Derivation  
The improved gas leakage model was proposed based on the above hole model, and its detailed derivation was as follows. The leakage of gas in the pipeline was considered in three stages. The schematic diagram is shown in Figure 2. Zero to one is the gas transportation process in the pipeline. The flow direction of the gas was considered to  
Figure 1: Schematic diagram of the gas leakage process in the hole model. be parallel to the axis of the pipeline, and the heat transfer between the leakage gas and the external space was ignored, but the effect of the frictional resistance between the gas and the pipeline wall cannot be ignored. This process was treated as an adiabatic process. One to two is the movement process of the gas inside the pipeline to the leakage hole. The gas streamline changes due to the pressure differences inside and outside the leakage section. At the same time, the change in the flow area makes the temperature and pressure of the leaked gas decrease and the velocity of the leaked gas increase, which can be called a throttling process. In this process, the viscous resistance, orifice loss and heat transfer between the gas and the pipeline wall can be approximately ignored. Thus, it was solved as an isentropic process. Two to three is the secondary expansion process of the refrigerant gas in the leakage section. Because the initial leakage pressure of the refrigerant in the refrigeration system is high, the pressure of the gas in the leakage section is still higher than the atmospheric pressure after the throttling process, and the refrigerant gas is in an under-expanded state (The gas with a certain initial pressure leaks or sprays from the pipeline or container, and the pressure of the gas at the nozzle outlet or the outlet of the leakage section is still higher than the ambient pressure, which is called insufficient expansion, also known as the under-expansion state). When the under-expanded gas is close to the outlet of the leakage section, it will expand further. In the improved model, the gas transportation, throttling and expansion process were regarded as a one-dimensional motion along the gas flow direction, the leakage gas was regarded as the compressible fluid and the ideal gas state equation was adopted. The compression factor \(Z\) was introduced to reduce the deviation caused by the calculation of the actual gas leakage. The gas transport process zero to one can be described by the energy and momentum equations, respectively, and the gas motion equation can be obtained by integrating the established equations:  
\[\frac{k+1}{k}\ln(\frac{p_{0}T_{1}}{p_{1}T_{0}})+\frac{M}{RG^{2}}\left(\frac{ p_{1}{}^{2}}{T_{1}}-\frac{p_{0}{}^{2}}{T_{0}}\right)+\frac{4 fL_{e}}{D}=0 \tag{2}\]  
where \(G\) is the mass flow rate per unit cross-sectional area of the pipe, kg/(m\({}^{2}\cdot\)s); \(f\) is the friction coefficient of the pipe wall; \(L_{e}\) is the effective length of the pipe, m; \(D\) is the pipe diameter, m; \(k\) is the isentropic index of the leakage gas; \(M\) is the kilomolar mass of the leakage gas, kg/kmol; \(R\) is the gas constant, kJ/(kmol\(\cdot\)K). Introducing the sound velocity \(c=\sqrt{kRT}\) and Mach number \(Ma=\frac{v}{c}\) into the energy conservation equation, assuming that \(f\) is a constant value during the flow of the gas in the  
Figure 2: Schematic diagram of the gas leakage process in the improved model. pipeline, the relationship between the gas state parameters at points zero and one can be obtained by solving the equations. \[\left\{ \begin{array}{ll}\frac{T_{1}}{T_{0}}=\frac{Y_{0}}{Y_{1}}\\ \frac{\rho_{1}}{\rho_{0}}=\frac{Ma_{0}}{Ma_{1}}\sqrt{\frac{Y_{1}}{Y_{0}}}\\ \frac{\rho_{1}}{P_{0}}=\frac{Ma_{0}}{Ma_{1}}\sqrt{\frac{Y_{0}}{Y_{1}}}\end{array} \right. \tag{3}\]  
where \(Y_{i}=1+\frac{k-1}{2}{Ma_{i}}^{2}\).