## 2 Materials and Methods
### Calculation Model
#### 2.2.2 Establishing the Elastic Constant of the Spring
The flow model adopted in the simulation with ANYSYS Fluent, as well as the values of the adopted parameters are presented in Table 4. It was considered that the pressure at the valve inlet increases proportionally with the distance between the seat and the valve. Regarding the selection of turbulence models, the impact of different turbulence models on the accuracy of calculations is very important. The performed analysisinvolving the simulation of a pressure relief valve handling water and air as fluids, the choice of the k-epsilon turbulence model aligns well with common practices for such scenarios in CFD, being used in many scientific papers, such as [1; 8; 23; 24; 25; 26]. Also, this model makes a balance between accuracy and computational efficiency, making it well-suited for engineering analyses [8].The choice to use non-equilibrium wall functions in the simulation, particularly in the context of analyzing air/water flow through a discharge valve, is typically motivated by the flow characteristics near the solid boundaries (such as the valve surface) and the computational efficiency considerations [23]. The concept of wall functions is commonly employed in simulations to model the near-wall behavior of the flow. Near the walls, the turbulence intensity is typically high, and resolving all the turbulent scales in this region would be computationally expensive. Wall functions allow the simulation to model the effects of turbulence close to the wall without explicitly resolving the near-wall region [27; 28].The non-equilibrium wall function approach, in the context of the k-epsilon model, refers to a modification of the standard wall functions to account for non-equilibrium effects. In the standard equilibrium wall functions, it is assumed that the flow near the wall reaches a state of equilibrium, meaning that the production of turbulence (turbulent kinetic energy) is balanced by its dissipation. However, in some flows, especially those with strong adverse pressure gradients \begin{table}
\begin{tabular}{c c} \hline \hline The flow model & k-epsilon [1; 8; 23; 24; 25; 26], Realizable, non-equilibrium wall function [23] \\ \hline \multirow{4}{*}{Environment: air/water} & Air density: 1.204 kg/m\({}^{3}\) \\  & Water density: 998.2 kg/m\({}^{3}\) \\ \cline{1-1}  & Air dynamic viscosity: 1.813\(\cdot\)10\({}^{-5}\) kg/(m\(\cdot\)s) \\ \cline{1-1}  & Water dynamic viscosity: 1.003\(\cdot\)10\({}^{-3}\) kg/(m\(\cdot\)s) \\ \hline \multirow{2}{*}{Connection conditions} & Inlet pressure: (4.1-4.4) MPa \\ \cline{1-1}  & Outlet pressure: 0.1 MPa \\ \hline \hline Type of analysis & Steady, 300 iteration \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The model adopted in the flow simulation. Figure 8: Position of the adjusting nozzle: position S0 (**a**), position S1 (**b**), position S-1 (**c**). or complex geometries, the flow near the wall may deviate from equilibrium [28; 29].The non-equilibrium wall function approach considers the departure from equilibrium in the near-wall region. It involves additional terms or corrections in the wall functions to account for the non-equilibrium effects on the turbulence quantities (k-epsilon) near the wall. These modifications aim to improve the accuracy of the simulation in capturing the turbulence behavior in challenging flow conditions [29; 30; 31]. The use of non-equilibrium wall functions in simulations, especially in the context of the k-epsilon turbulence model, can offer several advantages in capturing more realistic turbulent flows, particularly near walls: improved accuracy in complex flows; better representation of near-wall effects; applicability to challenging geometries; enhanced predictions of separated flows; reduced sensitivity to grid resolution; consistency with experimental data; enhanced physical realism. The fluid flowing through the safety relief valve is assumed to be incompressible in the simulation, as in [7; 23].