# Effects of the Damping Parameters on the Opening and Closing Characteristics of Vent Valves
## 2 Model
### Membrane Assembly
Figure 3: Result of meshing of membrane assembly: (**a**) Results of mesh division on spring side; (**b**) Diaphragm-side mesh partitioning results. In the working process of the membrane assembly, it could be regarded as the membrane and the spring in series, and the compression area s was 1017.88 mm\({}^{2}\). Through Equation (6), the overall stiffness of the membrane assembly could be obtained as  
Figure 4: Deformation of membrane assembly with control pressure at different ambient pressures: (**a**) Both ambient pressure and control pressure are 0.10133 MPa; (**b**) The ambient pressure is 0.10133 MPa, and the control pressure is 0.107 MPa; (**c**) Both ambient pressure and control pressure are 0.07 MPa; (**d**) The ambient pressure is 0.07 MPa, and the control pressure is 0.08 MPa; (**e**) Both ambient pressure and control pressure are 0.0577 MPa; (**f**) The ambient pressure is 0.0577 MPa, and the control pressure is 0.07 MPa; (**g**) Both ambient pressure and control pressure are 0.0472 MPa; (**h**) The ambient pressure is 0.0472 MPa, and the control pressure is 0.053 MPa. K = 6.33 N/mm, the stiffness of the spring K\({}_{\rm b}\) was 2.5 N/mm, and the equivalent stiffness of the film K\({}_{\rm a}\) was 0.169 N/mm:  
\[\text{y}_{2} =160.8\text{P}_{2}-160.7237\text{P}_{0}, \tag{6}\]  
where, y\({}_{2}\) is the displacement of the terminal, P\({}_{2}\) is the control pressure and P\({}_{0}\) is the ambient pressure. \[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial X_{i}}(\rho \varepsilon u_{i})=\frac{\partial}{\partial X_{j}}\left[(\mu+\frac{\mu_{t}}{ \sigma_{\varepsilon}})\frac{\partial\varepsilon}{\partial X_{j}}\right]+C_{1 \varepsilon}\frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon}. \tag{10}\]  
It is known that the working medium of the vent valve is air, whose density is 1.225 kg/m\({}^{3}\) and kinematic viscosity is 1.7894 \(\times\) 10\({}^{-5}\) kg/m\(\cdot\)s. In order to analyze the influence of different pressure differences on the mass flow, the outlet pressure was 0.1 MPa, the inlet pressure was different, and the velocity distribution in the flow field is shown in Figure 6. As can be seen from the Figure 6, the velocity value was the largest at the entrance, and the velocity was evenly distributed at other locations. The mass flow rate at different pressure ratios was obtained at the outlet pressure of 0.1 MPa. To equate the steady flow plate with the damping hole, the mass flow-to-pressure ratio formula was required, and this is shown in Equation (11):  
\[Q_{m}=\left\{ \begin{array}{cc}C_{0}Sp_{0}\sqrt{\frac{2\gamma}{RT_{0}(\gamma- 1)}\left[\left(\frac{P}{P_{0}}\right)^{\frac{2}{\gamma}}-\left(\frac{P}{P_{0} }\right)^{\frac{\gamma+1}{\gamma}}\right]}&\frac{P}{P_{0}}>b\\ \text{CSp}_{0}\sqrt{\frac{\gamma}{RT_{0}}}\left(\frac{2}{\gamma+1}\right)^{ \frac{\gamma+1}{2(\gamma-1)}}&\frac{P}{P_{0}}\leq b\end{array} \right., \tag{11}\]  
where C\({}_{0}\) is 0.72, T\({}_{0}\) is 293.15 K, b is 0.528. the gas constant R is 286.69 J/(kg\(\cdot\)K), the specific heat capacity is at a temperature of 290 K, and the pressure of 0.1 MPa is 1.402 kJ/(kg\(\cdot\)K). According to Equation (11), the equivalent damping hole area of different inlet pressures with an outlet pressure of 0.1 MPa could be obtained. The calculated results were fitted with the ratio of the outlet pressure and inlet pressure by a cubic polynomial. The fitting curve is shown in Figure 7, and the curve's formula is shown in Equation (12). The coefficient of determination R\({}^{2}\) of the fitting equation was 0.9896 \(>\) 0.9, and the fitting result was considered to be good. \[S=0.22378\left(\frac{P_{1}}{P_{0}}\right)^{3}-0.47271\left(\frac{P_{1}}{P_{0 }}\right)^{2}+0.33294\left(\frac{P_{1}}{P_{0}}\right)+0.01623, \tag{12}\]  
Figure 6: Velocity distribution results of steady flow plate when inlet is 0.11 MPa and outlet is 0.1 MPa.