# Shock Initiation of a Satellite Tank under Debris Hypervelocity Impact
## 3 Initiation Criterion for the Liquid Hydrazine Tank
### Stock Wave Pressure  
When the spherical debris impacts the satellite tank vertically at the velocity \(u_{p0}\), two shock waves will be formed. They propagate toward the debris and the tank wall, respectively. Based on the shock wave theory, the impact of debris on the satellite tank is simplified as one-dimensional impact. As shown in Figure 2a, when the debris impacts the outside of tank wall, the shock wave pressure, shock wave velocity and particle velocity in debris are \(P_{1}\), \(D_{1}\), and \(u_{1}\). The shock wave pressure, shock wave velocity and particle velocity in tank wall are \(P_{2}\), \(D_{2}\), and \(u_{2}\). According to the conservation of mass and momentum, the expressions of shock wave pressure in the debris and tank wall can be obtained [14]:  
\[P_{1}=\rho_{p}D_{1}u_{1}, \tag{5}\]  
\[P_{2}=\rho_{t}D_{2}u_{2}, \tag{6}\]  
where, \(\rho_{p}\) is the density of debris and \(\rho_{t}\) is the density of tank wall. The relationships between the shock wave velocity and particle velocity are proposed based on the linear [PERSON] relation:  
\[D_{1}=a_{p}+b_{p}u_{1}, \tag{7}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Test/Designed Test** & **Projectile/Rank Material** & **Velocity (m\(\bullet\)s\({}^{-1}\))** &
\begin{tabular}{c} **Power Density** \\ **(kbar\({}^{2}\)\(\bullet\)\(\bullet\)\(\bullet\))** \\ \end{tabular} & **Results** \\ \hline WSTF\#HYP-2 & Al/Steel & 6100 & \(2.40\times 10^{3}\) & no reaction \\ SAIC & Al/Al & 5000 & \(1.37\times 10^{5}\) & some reaction \\ TITANK & Steel/Ti & 7500 & \(3.98\times 10^{5}\) & explosion \\ \hline \end{tabular}
\end{table} 
Table 1: Conditions and results of hypervelocity impact test on liquid hydrazine [8]. Figure 1: Flow chart for judging the damage consequences of a satellite tank under debris hypervelocity impact. Figure 2: (a) One-dimensional model of the spherical debris impacting the outside of tank wall; (b) One-dimensional model of the spherical debris impacting the inside of tank wall. \[D_{2}=a_{t}+b_{t}u_{2}, \tag{8}\]  
where, \(a_{p}\) and \(b_{p}\) are the Hugoniot coefficients of debris and \(a_{t}\) and \(b_{t}\) are the Hugoniot coefficients of tank wall. The following expressions can be obtained by substituting Equations (7) and (8) into Equations (5) and (6):  
\[P_{1}=\rho_{p}(a_{p}+b_{p}u_{1})u_{1}, \tag{9}\]  
\[P_{2}=\rho_{t}(a_{t}+b_{t}u_{2})u_{2}. \tag{10}\]  
Based on the continuous boundary conditions, the following can be derived:  
\[P_{1}=P_{2}, \tag{11}\]  
\[u_{p0}=u_{1}+u_{2}. \tag{12}\]  
Shock wave attenuates in the tank wall. The pressure decreases to \(P_{3}\) when the shock wave propagates to the inside of the tank wall [15]:  
\[P_{3}=P_{2}e^{-a\delta}, \tag{13}\]  
where, \(a\) denotes the attenuation coefficient of the wall; \(\delta\) denotes the thickness of the wall. Given the shock wave pressure \(P_{3}\), the particle velocity \(u_{3}\) on the inside of tank wall can be calculated:  
\[P_{3}=\rho_{t}(a_{t}+b_{t}u_{3})u_{3}. \tag{14}\]  
When the debris reaches the inside of tank wall, as shown in Figure 2b, its velocity decreases to \(u_{p}\). The shock wave pressure, shock wave velocity and particle velocity in debris are \(P_{4}\), \(D_{4}\), and \(u_{4}\). Figure 1:0  
The debris velocity \(u_{p}\) can be solved by continuous boundary conditions:  
Figure 1:1  
Figure 1:2  
When the debris moves in the liquid, it is only subjected to drag force and its velocity decreases gradually. The following can be deduced from [PERSON]'s second law [16]:  
Figure 1:3  
where, \(V_{p}\) is the debris volume, \(A_{p}\) is the cross-section area of debris, \(C_{D}\) is the drag coefficient. The drag coefficient \(C_{D}\) is a dimensionless number, which depends not only on the debris shape, but also on the Reynolds number of the liquid [17]. The debris moves at high speed in the liquid, part of its kinetic energy is transferred to the liquid. The liquid begins to flow, turbulence is generated around the debris and the Reynolds number Re \(>10^{3}\).