# Shock Initiation of a Satellite Tank under Debris Hypervelocity Impact
## 3 Initiation Criterion for the Liquid Hydrazine Tank
### Verification of the Initiation Criterion  
WSTF (1991) performed the HYP-2 test where an aluminum projectile with the diameter of 0.318 cm was shot into a steel vessel filled with liquid hydrazine at the velocity of 6100 m/s [6]. The vessel is a cylinder with 7.6 cm in diameter, 8.9 cm in height and 0.89 mm in wall thickness. The initiation criterion for the liquid hydrazine tank is used to calculate the power density and to judge whether an explosion occurs in the HYP-2 test. A flow chart of how the calculation process proceeds is shown in Figure 3.  
The material parameters of the projectile, vessel and liquid hydrazine used in the calculations are shown in Table 2[19; 20]. The power density calculated by the initiation criterion for the liquid hydrazine tank is \(P_{d}\) = 2658 kbar\({}^{2}\)\(\upmu\)s. This value was obtained using the process in Figure 4 as follows:  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]  
\[P_{2}=7830(4550+1.616u_{2})u_{2}. \tag{28}\]  
Based on the continuous conditions on the interface, the following can be derived:  
\[P_{1}=P_{2}, \tag{29}\]  
\[u_{1}+u_{2}=6100, \tag{30}\]  
\[2700[5240+1.338(6100-u_{2})](6100-u_{2})=7830(4550+1.616u_{2})u_{2}, \tag{31}\]  
\[u_{2}=1975.87\;\text{m}/\text{s},P_{2}=1.197\times 10^{11}\;\text{Pa}. \tag{32}\]  
Figure 3: Flow chart of the calculation process of initiation criterion for liquid hydrazine tank. The pressure decreases to \(P_{3}\) when the shock wave propagates to the inside of the tank wall:  
\[P_{3}=P_{2}e^{-a\delta}=1.197\times 10^{11}\times e^{-0.45\times 0.89}=8.0\times 10^{10 }\ \mathrm{Pa}. \tag{33}\]  
Given the shock wave pressure \(P_{3}\), the particle velocity \(u_{3}\) on the inside of tank wall can be calculated:  
\[7830(4550+1.616u_{3})u_{3}=8.0\times 10^{10}, \tag{34}\]  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]0  
When the debris reaches the inside of tank wall, the shock wave pressure, shock wave velocity and particle velocity in debris are \(P_{4}\), \(D_{4}\), and \(u_{4}\). \[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]1  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]2  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]3  
The debris velocity \(u_{p}\) can be solved by continuous boundary conditions:  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]4  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]5  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]6  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]7  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]8  
\[P_{1}=2700(5240+1.338u_{1})u_{1}, \tag{27}\]9  
TDL simulation was also carried out to determine the power density in the HYP-2 test [8], which was 2400 kbar\({}^{2}\)\(\upmu\)s. The calculated power density is larger than the TDL simulation, the reasons are as follows: the vessel is filled with 1/3 nitrogen and 2/3 liquid hydrazine actually, the release wave is generated at the gas-liquid interface in the simulation, when the release wave meets the shock wave, the shock wave pressure decreases and the pressure duration shortens accordingly. However, the influence of the release wave is not taken into account in the initiation criterion. The relative error between the initiation criterion calculation and TDL simulation is 9.71%, verifying the effectiveness of the initiation criterion for the liquid hydrazine tank. The two power densities are both smaller than the minimum power density \(P_{dmin}=3.98\times 10^{5}\ \mathrm{kbar^{2}}\ \upmu\)s, indicating that the shock wave produced by the projectile impact was insufficient to initiate an explosion, which is consistent with the HYP-2 test results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Material** & \(\rho\) **(kg/m\({}^{3}\))** & \(a\) **(km/s)** & \(b\) & \(u_{p0}\) **(m/s)** & \(r_{p}\) **(cm)** & \(\alpha\) **(mm\({}^{-1}\))** & \(\delta\) **(mm)** \\ \hline Projectile & 2700 & 5.24 & 1.338 & 6100 & 0.159 & — & — \\ Vessel & 7830 & 4.55 & 1.616 & — & — & 0.45 & 0.89 \\ Hydrazine & 1008 & 1.50 & 1.500 & — & — & — & — \\ \hline \end{tabular}
\end{table}