## 3 Models and Methods
### Valve, Pipeline, and Tank  
In this study, a quasi-static relationship was used to describe the characteristics of the mass flow and upstream and downstream pressure of the valve. The physical manifestation is a variable-geometry section with a certain throttling effect. The calculation of valve flow is shown in Equation (4). \[\dot{m}=\frac{1}{\sqrt{k}}\cdot A\cdot\Psi\cdot\sqrt{\frac{2\cdot p_{up}\cdot \rho_{up}}{k_{dp}}}, \tag{4}\]  
Figure 6: Pump head vs. flow curve under different speeds. Figure 7: Pump efficiency vs. flow curve under different speeds. where, \(k\) is the pressure drop coefficient of the valve, \(p_{up}\) is the upstream pressure, \(\rho_{up}\) is the upstream density, \(k_{dp}\) is the gain of pressure drop, and \(\Psi\) is the flow coefficient of the valve. The regulating valve meets the law of throat cavitation. In the cavitation state, the flow through the regulating valve is independent of its outlet pressure and is controlled by the inlet pressure only. This ensures that the pressure oscillations in the thrust chamber do not affect the propellant flow. In general, the outlet pressure of the regulating valve should be less than 80% of the inlet pressure to ensure the cavitation condition. The control equation of the regulating valve is shown in Equation (5). \[\dot{m}=\mu A_{C}\sqrt{2\rho(p_{up}-p_{S})} \tag{5}\]  
where \(\mu\) is the flow coefficient of the regulating valve, \(A_{C}\) is the throat area of the regulating valve, \(p_{up}\) is the upstream pressure of the regulating valve, \(p_{S}\) is the saturated vapor pressure of propellant, and \(\rho\) is the density of the propellant. The maximum pressure of the pump outlet is 8.5 MPa and the maximum regulating valve flow is 10.6 kg/s. The maximum diameter of the regulating valve is 10.47 mm. The flow coefficient under cavitation state is taken as 0.8. When the cavitation number is small, the flow coefficient is a fixed value. When the cavitation number is greater than the critical cavitation number, the flow coefficient decreases with the increase in cavitation number. The parameters of the regulating valve are shown in Table 4.  
The general calculation equations of flow and enthalpy of pipeline are as follows [36]:  
\[\left\{ \begin{array}{l l}\mathrm{d}m=\rho\cdot\frac{A}{\sqrt{k_{f}}}\cdot \sqrt{\frac{2\cdot dp\cdot k_{dp}}{\rho}}\\ \mathrm{d}l_{S}=\mathrm{d}m\cdot h_{S}\end{array} \right., \tag{6}\]  
where \(\mathrm{d}m\) is the mass differential, \(\mathrm{d}l_{S}\) refers to the enthalpy flow differential, \(\rho\) represents the upstream density, \(A\) is the cross-sectional area of pipeline, \(k_{f}\) refers to the friction coefficient, \(k_{dp}\) represents the coefficient of pressure drop, \(\mathrm{d}p\) is the pressure differential in the pipeline, and \(h_{S}\) refers to the specific enthalpy. A tank is used to store propellant, which is generally composed of liquid propellant and gas. During flight, the gas from the pressurization system continuously enters the tank. The propellant flows out of the tank under the pressure of the gas in the tank and the suction of the pump [37]. In this study, the centralized parameter method was used to describe each physical quantity. According to the first law of thermodynamics, the energy equation in an open system is  
\[\Phi=\frac{dU}{d\tau}+\sum_{j}\left(h_{S}+\frac{c_{f}^{2}}{2}+gz\right)_{out} \dot{m}_{out}-\sum_{i}\left(h_{S}+\frac{c_{f}^{2}}{2}+gz\right)_{in}\dot{m}_{ in}+p_{i} \tag{7}\]  
where \(\Phi\) represents the heat flow, \(\frac{dU}{d\tau}\) represents the energy increment in the control body per unit time, \(h_{S}\) represents the enthalpy value, \(c_{f}\) represents the flow velocity, \(gz\) represents