# Launch Dynamic Simulation of a Compressed-Air Launcher for Fire Suppression
## 3 Calculation Model  
The flow of the high-pressure compressed air gas behind the projectile in the barrel is a complex and transient process with variable boundaries [9]. To simplify the calculation model, the following assumptions are introduced:  
1. The gas in the barrel behind the projectile is an ideal gas, and the gas state parameters are uniformly distributed in the barrel behind the projectile, and it is in a state of thermal equilibrium at any time. 2. Due to the transient launch time, the heat exchange between the system and the outside world is ignored. The gas expansion process is frictionless and adiabatic. Hence, it can be considered an isentropic process. 3. The gas flow along the barrel is a one-dimensional quasi-steady flow, and the influence of the gas jet on projectile motion is not considered. Figure 1: Schematic diagram of compressed-air launcher. The calculation model is shown in Figure 2. The inner diameter of the barrel is \(D\). The projectile mass is \(m_{p}\). The initial loading position is \(l\), and the projectile travel is \(L\). The initial pressure, volume, and mass of the gas in the gas cylinder are \(p_{0}\), \(V_{0}\), and \(m_{0}\), respectively, and they satisfy the ideal gas state equation:  
\[p_{0}V_{0}=m_{0}RT_{0}, \tag{1}\]  
where \(R\) is the gas constant, and \(T_{0}\) is the initial temperature of the high-pressure gas. The high-pressure gas is injected into the launch barrel via a diameter \(d\) nozzle. The thickness of the nozzle is ignored in the calculations. The gas pressure, volume, and mass in the barrel behind the projectile are \(p_{1}\), \(V_{1}\), and \(m_{1}\), respectively. They also follow the ideal gas state equation:  
\[p_{1}V_{1}=m_{1}RT_{1}, \tag{2}\]  
where \(T_{1}\) is the temperature of the high-pressure gas in the barrel behind the projectile. By investigating and discussing the high-pressure gas in the barrel behind the projectile, the parameters of the gas state in the barrel at the initial time and at a certain time have the following relationship:  
\[\frac{p_{10}V_{10}}{p_{1}V_{1}}=\frac{m_{10}T_{10}}{m_{1}T_{1}}, \tag{3}\]  
where the subscript 0 indicates the initial state. For an adiabatic process in a closed system, the following relation exists for the system at any time:  
\[\left(\frac{T_{1}}{T_{10}}\right)^{\gamma}=\left(\frac{p_{1}}{p_{10}}\right)^{ \gamma-1}, \tag{4}\]  
where \(\gamma\) is the adiabatic exponent of the gas, and the value is set as 1.4 in the calculations. By substituting Equation (4) into Equation (3), the following can be obtained:  
\[\left(\frac{p_{10}}{p_{1}}\right)^{\frac{1}{\gamma}}=\frac{m_{10}}{m_{1}}\cdot \frac{V_{1}}{V_{10}}, \tag{5}\]  
Hence, the gas pressure in the barrel behind the projectile at a certain moment can be expressed as:  
\[p_{1}=p_{10}\bigg{(}\frac{V_{10}}{m_{10}}\cdot\frac{m_{1}}{V_{1}}\bigg{)}^{ \gamma}. \tag{6}\]  
The initial gas volume and mass in the barrel behind the projectile are  
\[V_{10}=\frac{\pi}{4}D^{2}l, \tag{7}\]  
Figure 2: Calculation model of compressed-air launcher. Figure 2.0  
where \(\rho_{a}\) is the air density, set as 1.293 kg/m\({}^{3}\). At moment \(t\), the projectile is moving to position \(x\). The gas volume and mass in the barrel behind the projectile can be written as  
Figure 2.1  
and  
Figure 2.2  
where \(q\) is the gas flow rate entering the barrel through the nozzle. It has different forms under supersonic and subsonic conditions, which are written as follows [10]:  
\[q=\left\{ \begin{array}{cc}S\sqrt{\frac{2\gamma}{\gamma-1}p_{0}\rho_{0}\left[ \left(\frac{p_{1}}{p_{0}}\right)^{\frac{2}{\gamma}}-\left(\frac{p_{1}}{p_{0}} \right)^{\frac{\gamma+1}{\gamma}}\right]}&\text{when }\frac{p_{1}}{p_{0}}<\left( \frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}\\ \qquad\qquad S^{*}\sqrt{\gamma\left(\frac{2}{\gamma+1}\right)^{\frac{2}{ \gamma-1}}p_{0}\rho_{0}}&\text{when }\frac{p_{1}}{p_{0}}\geq\left( \frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}},\end{array} \right. \tag{11}\]  
where \(\rho_{0}\) is the gas density in the gas cylinder.