# Research Progress of Related Technologies of Electric-Pneumatic Pressure Proportional Valves
## 3 Related Technology Research Progresses
### Modeling of Gas Flow Characteristics  
In terms of the analysis of gas flow characteristics of pneumatic pressure proportional valve, many scholars studied the flow characteristics of the gas passing through the valve port and the thermodynamic properties of the gas in the cavity [32]. By simplifying the complex process of flowing through the valve port of the gas, the process is considered as a constant flow of the nozzle in the research of [PERSON]. If it is assumed that the specific heat of gas is constant, the one-dimensional constant drop flow equation of the ideal gas passing through the contraction nozzle can be presented like:  
\[\dot{m}_{t}=\left\{ \begin{array}{ll}\frac{A}{\sqrt{RT_{u}}}\sqrt{\frac{2k}{ k-1}}P_{u}\sqrt{\left(\frac{P_{d}}{P_{u}}\right)^{2/k}-\left(\frac{P_{d}}{P_{u}} \right)^{(k+1)/(k)}}&C_{t}<\frac{P_{d}}{P_{u}}\leq 1\\ \frac{A}{\sqrt{RT_{u}}}\,P_{u}\sqrt{k\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)} }&\frac{P_{d}}{P_{u}}\leq C_{t}\end{array} \right. \tag{2}\]In the formula, \(R\) is the gas constant, \(T_{u}\) is the upper temperature of the gas, \(m\) is the gas mass flow through the valve port, \(P_{u}\) and \(P_{d}\) are the downstream pressure of the contraction pipe, and \(C_{t}\) is the critical pressure ratio. For air, \(C_{t}\) = 0.528. When the pressure difference between upstream and downstream is very small as \(\Delta\mathrm{P}=(P_{u}-P_{d})\to 0\), this excessive gain will enlarge the error in numerical calculation. At the same time when \(\Delta P\to 0\), the gas flow rate approaches 0, and the flow of gas is laminar flow, expressing the gas flow in the quality problems. In order to solve this problem, the following modified formula is needed:  
\[\dot{m}_{t}=\left\{ \begin{array}{ll}ka\cdot A\cdot T_{u}{}^{-0.5}(P_{u}-P_{d })&\beta_{lam}\leq\frac{P_{d}}{R_{u}}<1\\ A\cdot P_{u}\cdot\sqrt{\frac{2k}{k-1}\cdot\frac{1}{K\cdot T_{u}}\left[\left( \left(\frac{P_{d}}{P_{u}}\right)^{2/k}-\left(\frac{P_{d}}{P_{u}}\right)^{(k+1) /(k)}\right)\right]}&b\leq\frac{P_{d}}{P_{u}}<\beta_{lam}\\ A\cdot P_{u}\cdot\sqrt{\frac{k}{k\cdot T_{u}}\cdot b^{\frac{k+1}{K}}}&\frac{P_ {d}}{P_{u}}<b\end{array} \right. \tag{3}\]  
In the formula, \(\beta_{lam}\) is the threshold value which is 0.995-0.999, \(b\) is \(C_{t}\) in the formula and \(ka\) is the compensation coefficient. The actual pressure proportional valve orifice is not smooth; it causes the flow loss of the convective gas [37]. Therefore, the actual flow quality is less than the theoretical value of the formula, in order to further improve the flow quality it needs the following formula:  
\[\dot{m}=C_{d}\cdot\dot{m}_{t} \tag{4}\]  
The experimental results showed that the flow coefficient \(C_{d}\) in different types of orifice is different. [PERSON] [37] further suggested that the flow coefficient \(C_{d}\) in the orifice is related to the upstream and downstream pressure ratio, and established the corresponding relation. [PERSON] [38] and [PERSON] [39] established the relationship between the flow coefficient and the orifice geometry. The empirical formula of flow coefficient \(C_{d}\) and upstream and downstream pressure ratio is given by [PERSON] [40]. In addition, some scholars further standardized the calculation of \(C_{d}\) based on the empirical formula of [PERSON]. For the study of gas in the cavity of thermodynamics, both domestic and foreign scholars all referred to the ideal gas state equation, the mass continuity equation, and the first law of thermodynamics. The differential equations of pressure and temperature are established, and then the differential equation is simplified. [PERSON] [14] calculated the gas in the cavity, and the variation law of pressure in the cavity is given by using the ideal gas state equation. According to the law of adiabatic change, the change of pressure and temperature of the gas in the cavity is obtained according to the change rule in research on [PERSON] [41]. [PERSON] thought that the heat exchange between the gas and the inner wall of the cavity could be deduced from the energy equation.