## 2 Development of the Numerical Model  
### Governing Equations  
In the previous work, the real refrigerant leakage diffusion model proposed by [PERSON] et al. [19] regarded the leakage of high-pressure gas refrigerant in the pipeline as three stages, as shown in Figure 1: (1) Point 0 to point 1 is the gas transport process in the pipeline; (2) point 1 to point 2 is the process of gas leakage from the inside of the pipeline to the outside space through the leakage hole; (3) point 2 to point 3 is the process of the leaking high-pressure gas expanding from the inner wall of the pipeline to the outside world. The diffusion process after refrigerant leakage can also be divided into two stages: In the first stage, due to the high initial pressure in the pipeline of the refrigeration system, the pressure of the refrigerant gas at the leakage hole on the outer wall of the pipeline when leakage occurs is higher than the atmospheric pressure, and the leaking refrigerant gas will continue to expand and compress, which can be described by the mass, momentum, and energy conservation equations, respectively. In the second stage, the expansion and compression processes disappear, and the leaking refrigerant gas has returned to room temperature and normal pressure states. Regardless of the heat transfer between the leaking gas and the external environment, it can be considered as a single turbulent motion process, which can be described by the mass and momentum conservation equations. At the same time, the leaking refrigerant gas will be continuously mixed with the air, and the diffusion flux of the refrigerant gas is calculated using the component transport equation. In fact, after the high-pressure refrigerant gas in the pipeline leaks, there will be a continuous expansion and compression process in the near-field area below the leak hole, and the temperature, pressure, and speed of the refrigerant will undergo very drastic changes. As the refrigerant spreads to the far field, the refrigerant gradually returns to the room temperature and atmospheric pressure state. The improved model has good applicability in describing the expansion process and multi-field coupling such as pressure field and velocity field after the leakage of high-pressure gaseous refrigerant. The accuracy and reliability of this model have been verified in experiments [19]. After the high-pressure refrigerant gas in the pipeline leaks, a series of continuous expansion and compression processes occur in the near-field area below the leakage point. This results in significant fluctuations in the refrigerant's temperature, pressure, and velocity. As the refrigerant diffuses into the far-field region, it gradually equilibrates with the ambient temperature and pressure. During the high-speed jet flow in the near-field, shock waves can form within the flow domain, necessitating the use of a highly refined grid and extremely small-time steps for accurate numerical capture. This study primarily focuses on the diffusion process and deposition characteristics of the refrigerant in the far-field region. To simplify the computational analysis, the numerical model adopts an initial condition where the refrigerant has already equilibrated to room temperature and ambient pressure. The diffusion of the refrigerant in the far-field region is then simulated by solving the governing equations for turbulence and component transport. In addition, we define conditions such as refrigerant inlet flow through UDF. These governing equations are as follows. 1. Continuity equation \[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho u _{j}\big{)}=0\] (1) where \(\rho\) is the density of the mixed gas (kg/m\({}^{3}\)); \(u_{j}\) is the time-averaged velocity of the gas flowing in three directions (\(u_{x},u_{y}\),\(u_{z}\), m/s). 2.