Starting and Regulating Characteristics of Electric Pump Feed System for LRE under Different Schemes

Featured Application

The presented results can be applied to the electric pump propulsion system of a rocket engine.

Abstract

Liquid rocket engines (LREs) are essential power sources for access to space. Electric pump feed systems have received noteworthy attention because of their high efficiency, convenient regulation, and simple structure. In this study, an oxidant feed system with two pumps in parallel was established. The centralized parameter method and the distributed parameter method were used for modeling. The dynamic characteristics of different starting schemes and regulating schemes were obtained. The results show that the asynchronous opening of two pumps led to a pressure peak from the second stage to the third stage. Under the low operating conditions, the pump speed of the asynchronous scheme was about 13,300 r/min, the pump speed of the synchronous scheme was about 12,100 r/min, and the pump speed of the joint adjustment scheme was about 24,800 r/min. The joint adjustment of pump speed and valve opening could increase the pump speed by a factor of one-third, while maintaining the efficiency at a high level. The optimal scheme could be selected according to a genetic algorithm-based calculation process, together with the curves of the flow rate and pressure with pump speed and valve opening. This study can contribute to the application of electric pumps for liquid rocket propulsion.

1. Introduction

Powerful propulsion sources are necessary for access to space, such as liquid rocket engines, ramjet engines, and their combinations [1,2,3,4]. The traditional propellant feed system of liquid rocket engines mainly includes a turbine pump cycle and pressure-fed system. The pressure-fed system has the advantages of a simple structure, reliable operation, and convenience for multiple starts. However, the thrust chamber pressure is limited by the structural mass of the tank. The performance of a pressure-fed engine is poor, which is generally suitable for engines with low thrust and multiple starts [5,6]. The turbine pump feed system uses the pump to pressurize the propellant, which greatly reduces the mass of the tank and improves the thrust chamber pressure. These engines have high specific impulse [7,8,9]. However, the structure of these engines is complex, generally including components such as gas generators, turbines, and centrifugal pumps. Their thrust adjustment range is small, which struggles to meet the requirements of multiple starts.

A high-speed and high-performance motor can replace the traditional turbine as the driving machine of a pump in an electric pump feed system [10,11,12]. Previously limited by miniaturized high-energy power supply, accurate propellant control, light high-speed motor, and other technologies, it was once less used in the aerospace field. As technology and manufacturing progressed, especially with regard to the development of high-energy-density batteries, brushless DC motors, and 3D additive manufacturing, its use has gradually emerged in the aerospace field [13,14,15]. It has been demonstrated that the propellant can be pressurized by electric pumps, and nitrogen pumps have been manufactured [16]. Compared with the turbine pump, they have the advantages of high efficiency, convenient adjustment, and simple structure.

Some researchers have studied the characteristics of electric pumps by building mathematical and analytical models. Solda and others [17,18] analyzed the concept and characteristics of an electric pump feed system. They proposed that, in the case of a long operating time, high chamber pressure, or large thrust, the electric pump feed system has more obvious advantages than the pressure-fed system. However, they used an empirical formula to calculate the mass, which is not applicable to all propulsion systems. Wang et al. [19] studied the influence of thrust, mixing ratio, and thrust chamber pressure on the weight of the electric pump feed system. The results showed that there is a critical operating time under different chamber pressures. When the operating time is greater than the critical time, the weight of the electric pump system is light. However, under the conditions of 5.0 kN thrust and a combustion chamber pressure not exceeding 10.0 MPa, the critical time of the propulsion system would be inapplicable for different thrust magnitudes. With the development of key technologies such as batteries, rockets with electric pumps have great potential for development in terms of system weight and system stability [20].

For the electric pump feed system, the weight and the driving method have been studied separately. Liu [21] compared the mass of the electric pump feed system and pressure-fed system. The results showed that the motor mass is mainly affected by thrust and thrust chamber pressure, while the battery mass is also affected by operating time. Nevertheless, this theoretical analysis based on a high-performance battery and high-speed brushless motor needs to be verified by engineering practice. Lu et al. [22] used the maximum torque-to-current ratio as the main strategy for energy efficiency optimization of the motor control system. On the basis of this strategy, a permanent magnet synchronous motor energy efficiency optimization control algorithm was proposed. At present, vector control methods are mostly used to regulate the rotation speed of the electric pump in rockets.

The characteristics of the centrifugal pump also have a large impact on the performance of the engine [23,24]. Zhou et al. [25] investigated the effects of different flow rates, impeller eccentricities, and whirl ratios on the fluid-induced force of the impeller. The results showed that the extremum value of the fluid-induced force of impeller has a significant positive correlation with the impeller eccentricity. The drawback of this study is that the centrifugal pump speed was much lower than that in the aerospace propulsion. Ramirez and others [26] characterized the cavitation phenomenon in dredging centrifugal pumps. Since part of the energy in the impeller is dissipated by the phase change, the presence of slurry reduces the pressure head to the same extent as the cavitation. Currently, centrifugal pumps are basically used in rocket engines, because the pressure is smoother than that of volume pumps due to the high rotation speed. In addition, the centrifugal pump has a small structure and takes up little space.

At present, although there have been many studies on the electric pump feed system, there is still a lack of analysis on its starting scheme and regulating scheme. The main contribution of the paper is the modeling of an electric pump feed system for liquid propulsion, in which a combination of the centralized parameter method and the distributed parameter method is used. A cooperative adjustment method between the electric pump speed and regulating valve opening based on genetic algorithm is proposed. The effects of different working sequences on the dynamic characteristics of the propulsion system when the dual electric pumps are connected in parallel are analyzed. The study provides some basis for the working sequence setting and highlights the potential application of electric pumps in liquid propulsion.

The paper is organized as follows: Section 2 introduces the composition of the electric pump propulsion system and three typical starting and regulating schemes. Section 3 builds the mathematical models and methods used for the electric pump propulsion system. Section 4 analyzes the differences in the dynamic characteristics of the three typical starting and regulating schemes, as well as the coregulation method of the electric pump speed and the regulating valve opening based on the genetic algorithm. Section 5 presents the conclusion.

2. System Composition and Working Sequence

2.1. Composition of Electric Pump Feed System

The propellant of the engine is hydrogen peroxide (90 wt.%) and kerosene. This study focused on a hydrogen peroxide feed system including a gas tank, an oxidant tank, two electric pumps, motors, motor controllers, oxidant valves, pipelines, and other components, as shown in Figure 1.

The pressurized gas system contains a gas tank, a pneumatic valve, a pressure regulator, etc. Since the system uses an electric pump to pressurize the propellant, the pressure of the regulator outlet does not need to be too high. This pressure is only needed to ensure that no cavitation occurs at the pump inlet. The gas tank contains high-pressure nitrogen. The nitrogen passes through the pressure regulator and reaches the required pressure.

The oxidant tank is mainly used to store high-concentration hydrogen peroxide. Downstream of the tank is a pneumatic valve, which is mainly used to isolate the propellant in the tank and the downstream pipe. The high reliability of the pneumatic valve can ensure the effectiveness of the isolation. After the propellant passes through the pneumatic valve, it is divided into two paths and enters the centrifugal pump separately. The centrifugal pump is a device that pressurizes the propellant from a low pressure to a high pressure. There are solenoid valves at the inlet and outlet of the centrifugal pump to control the propellant. The power source to drive the centrifugal pump is a high-speed permanent magnet synchronous motor (PMSM) [27]. The propellant passing through the two electric pumps converges into the flow regulating valve. The flow-regulating valve follows the principle of cavitation to control the flow into the thrust chamber. The flow of propellant in the line is controlled by changing the throat area of the flow-regulating valve.

The thrust chamber mainly consists of the injector, the catalytic bed, the combustion chamber, and the nozzle. Injectors enable the impact and atomization of the propellant, which plays a very important role in combustion. The catalytic bed is the device that catalyzes the decomposition of high-concentration hydrogen peroxide into high-temperature oxygen and water vapor. High-concentration hydrogen peroxide is storable at room temperature, green, and nontoxic [28,29], which has led to its widespread use. The combustion chamber is the main place where the hydrogen peroxide decomposition products are burned with kerosene [30,31]. The main purpose of the nozzle is to enable the expansion of the high-pressure gas expand, thus increasing the gas velocity, which in turn generates thrust. The temperature of the high-concentration hydrogen peroxide decomposition products is much higher than the ignition point of kerosene; thus, no ignition device is needed [32]. The decomposition products of high-concentration hydrogen peroxide and kerosene are capable of self-ignition and continuous combustion.

The above-described rocket engine was used in a combined power vehicle in operating conditions at high altitude with a high Mach number. It had a maximum thrust of 33.6 kN and an operating condition adjustment ratio of 4:1. The engine consisted of three different operating stages. To ensure the performance of combustion, the mixing ratio was set to 7.5, which is basically near the optimal mixing ratio. With such a large thrust, it is difficult for one electric pump to achieve such a large power. Therefore, two electric pumps were used in parallel in this study, concentrating on the starting and regulating scheme between the two electric pumps. The technical parameters of the system are shown in

Table 1. Basic technical parameters of the oxidant system.

Parameters	Value	Unit
Thrust of the engine	8.4–33.6	kN
Operating time	80	s
Total flow of the engine	3–12	kg/s
Oxidant flow of the engine	2.6–10.6	kg/s
Stage of operation	3	/
Adjustment ratio	4:1	/
Mixture ratio	7.5	/
Maximum supply pressure	≮6.9	MPa
Total power of electric pumps	100	kW

.

The system underwent the operating process shown in

Table 2. Working process of the oxidant system.

Period	Time (s)	Oxidant Flow (kg/s)
1st	0–40	2.6
2nd	40–45	5.3
3rd	45–80	10.6

. The first and third stages of the operation were relatively long. The first stage denotes the minimum operating condition, and the third stage denotes the maximum operating condition. The second stage with a time of 5 s denotes an intermediate operating condition. In the above operating process of the system, the flow regulation error is not to exceed 5%. The over regulation time from one operating point to another is not to exceed 1.5 s.

2.2. Working Sequence of Oxidant System

In order to analyze the influence of different starting schemes and different regulating schemes on the dynamic characteristics of the engine, three schemes were set up, namely, Scheme ASYNC, Scheme SYNC, and Scheme JOINT. Scheme ASYNC and Scheme SYNC adjust only the pump speed during the starting and regulating of the propulsion system, while the regulating valve opening is constant. The difference between these two schemes is whether the two pumps are started synchronously or asynchronously. Scheme JOINT adjusts the pump speed and the regulating valve opening at the same time, and the working sequence is the same as Scheme SYNC. The flow in the first stage and the second stage is small; thus, a single pump can be started during these stages. In the third stage, the two pumps work at the same time, i.e., Scheme ASYNC. Since the states of the two pumps are consistent, the two pumps can be kept in a working state at all three stages considering the symmetry, i.e., Scheme SYNC. Considering that the regulating valve can adjust the throat area, the pump speed and the opening of the regulating valve can be adjusted at the same time, i.e., Scheme JOINT. The working sequence diagrams in the asynchronous (Scheme ASYNC) and synchronous (Scheme SYNC) cases are shown in Figure 2 and Figure 3, respectively. The working sequence diagram under the condition of joint adjustment for pump speed and valve opening (Scheme JOINT) is shown in Figure 4.

3. Models and Methods

The electric pump feed system includes a gas tank, a pressure regulator, an oxidant tank, two electric pumps, a regulating valve, and other components. The control equations and parameters of each component are introduced below.

3.1. Electric Pump

According to the law of energy conservation, the power generated by the impeller of a centrifugal pump is consumed in the energy growth of liquid flow [33,34]. The pressure head of the pump is mainly determined by four factors: the speed of the pump, the change rate of the pump speed, the mass flow through the pump, and the change rate of the mass flow through the pump. The effect of speed and mass flow on the actual working process is usually considered. The pressure head function of the centrifugal pump is obtained by the curve fitting method, as shown in Equation (1). The form of the dynamic equation of centrifugal pump torque is the same as that of pressure head.

$$\{\begin{array}{l}\mathsf{\Delta }p=f(Q,N)\\ T=f(Q,N)\end{array},$$

(1)

where, $Q$ is the volume flow, and $N$ is the pump speed. According to the definition of $\theta$ and specific speed ($N_S$) in Equation (2), the head function ($WH$) and torque function ($WT$) are defined in Equation (3) [8].

$$\{\begin{array}{l}\theta =\pi +\mathrm{atan}2\left(\frac{\nu }{\alpha }\right)\\ N_S=\frac{N_{\mathrm{R}}\cdot \sqrt{Q_{\mathrm{R}}}}{H_{\mathrm{R}}^{0.75}}\end{array},$$

(2)

$$\{\begin{array}{l}WH(N_S,\theta )=\frac{h}{{\alpha }^2}/(1+\frac{{\nu }^2}{{\alpha }^2})\\ WT(N_S,\theta )=\frac{\beta }{{\alpha }^2}/(1+\frac{{\nu }^2}{{\alpha }^2})\end{array},$$

(3)

where the subscript $\mathrm{R}$ represents the rated operating condition, $H$ refers to the pressure head, and $h$,$\nu$,$\alpha$, and $\beta$ are the dimensionless pressure head, volume flow, pump speed, and torque, respectively.

When two pumps are operating simultaneously under maximum operating conditions, they need to meet the requirements of the maximum pressure and flow rate of the system. The maximum flow rate of the system was 10.6 kg/s. Therefore, the flow rate of a single pump at the design point was 5.3 kg/s and the pressure head was 597.7 m. According to the current operating conditions of most centrifugal pumps, the pump speed under maximum operating conditions is 36,000 r/min. The structural parameters of the electric pump are shown in

Table 3. Main parameters of electric pump structure.

Parameters	Value	Unit
Pressure of pump outlet	8.5	MPa
Rated pump flow	5.3	kg/s
Number of poles	2	/
Number of slots	18	/
Rated voltage	540	VDC
Rated speed	36,000	r/min
Rated power	50	kW
Stator outer diameter	135	mm
Stator inner diameter	45	mm

.

The electric pump structure is shown in Figure 5, and it is driven by a canned motor. The impeller and the rotor are all in the propellant. This avoids the dynamic seal structure; thus, no propellant leakage occurs. A reliable dynamic seal in a high-speed rotating part is very difficult [35]. Furthermore, a small portion of propellant in the pump backflows through the shaft into the pump inlet. This takes heat away through the propellant from the stator coil.

In order to achieve stable operation under different operating conditions, the pump characteristic curves at different speeds were calculated in this study. The pump characteristics are described by the curve of the pump head with the flow rate and the curve of the pump efficiency with the flow rate. The pump characteristic curves are shown in Figure 6 and Figure 7. The speed ranged from 15,000 r/min to 36,000 r/min, thus covering the regulating range in this study. Several characteristic curves obtained from 3D numerical simulation and their interpolations were used to describe the pump characteristics, instead of using 3D numerical simulations of real-time interactive data for the whole system. This is because the real-time interaction data would lead to a large amount of computation for the whole system. Using the characteristic curves and their interpolations reduces the accuracy to some extent; however, this method is acceptable in the whole-system simulation.

3.2. 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).

$$\dot{m}=\frac{1}{\sqrt{k}}\cdot A\cdot \mathsf{\Psi }\cdot \sqrt{\frac{2\cdot p_{up}\cdot {\rho }_{up}}{k_{dp}}},$$

(4)

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 $\mathsf{\Psi }$ 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).

$$\dot{m}=\mu A_C\sqrt{2\rho (p_{up}-p_S)}$$

(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. Main parameters of the regulating valve and the pipelines.

Parameters	Value	Unit
Throat diameter of regulating valve	10.47	mm
Maximum regulating valve flow	10.6	kg/s
Flow coefficient of regulating valve	0.8	/
Nominal diameter of gas valve	10	mm
Nominal diameter of valve before and after pump	25	mm
Nominal diameter of gas pipeline	10	mm
Nominal diameter of main liquid pipeline	40	mm

.

The general calculation equations of flow and enthalpy of pipeline are as follows [36]:

$$\{\begin{array}{l}\mathrm{d}m=\rho \cdot \frac{A}{\sqrt{k_f}}\cdot \sqrt{\frac{2\cdot \mathrm{d}p\cdot k_{dp}}{\rho }}\\ \mathrm{d}{h}_S=\mathrm{d}m\cdot h_S\end{array},$$

(6)

where $\mathrm{d}m$ is the mass differential, $\mathrm{d}{h}_S$ 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

$$\mathsf{\Phi }=\frac{dU}{d\tau }+{\displaystyle \sum _j^{}{(h_S+\frac{c_f^2}{2}+gz)}_{out}{\dot{m}}_{out}}-{\displaystyle \sum _i^{}{(h_S+\frac{c_f^2}{2}+gz)}_{in}{\dot{m}}_{in}}+p_i$$

(7)

where $\mathsf{\Phi }$ 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 the gravitational potential energy, ${\dot{m}}_{out}$ represents the mass flow out of the control body, ${\dot{m}}_{in}$ represents the mass flow into the control body, and $p_i$ represents the internal work done per unit time.

The mass equation in the gas tank is

$$\frac{\mathrm{d}m}{\mathrm{d}\tau }=V[{(\frac{\partial \rho }{\partial p})}_t\cdot \frac{\mathrm{d}p}{\mathrm{d}\tau }+{(\frac{\partial \rho }{\partial t})}_p\cdot \frac{\mathrm{d}t}{\mathrm{d}\tau }],$$

(8)

where $m$ represents the gas mass in the tank, $\rho$ represents the gas density, $V$ represents the gas volume, $p$ represents the gas pressure, and $t$ represents the gas temperature.

The energy equation in the gas tank is

$$\mathrm{d}U={\displaystyle \sum {\dot{m}}_i}{h}_S{}_i+\delta q-p\frac{\mathrm{d}V}{\mathrm{d}\tau },$$

(9)

where $U$ refers to the internal energy of the gas in the tank, $h_S$ is the specific enthalpy of the gas, $p$ represents the pressure of the gas, $V$ is the volume of the gas, and $q$ refers to the heat exchange between the gas and the wall.

The propellant tank is ellipsoidal, and it is mainly used to store hydrogen peroxide. The volume of the propellant tank is 500 L and that of the pneumatic cushion is 75 L. The pneumatic cushion pressurizes the propellant to prevent the cavitation phenomenon of the pump. The gas tank is spherical, and it is mainly used to store the high-pressure nitrogen. The volume of the gas tank is 30 L, and the initial pressure is 15 MPa. The main parameters of propellant tank and gas tank are shown in

Table 5. Main parameters of the propellant tank and the gas tank.

Parameters	Value	Unit
Volume of propellant tank	500	L
Pneumatic cushion of propellant tank	75	L
Allowable pressure of propellant tank	1	MPa
Wall thickness of propellant tank	2	mm
Volume of gas tank	30	L
Pressure of gas tank	15	MPa
Heat transfer coefficient of gas tank	40	W/(m2·K)

.

The valves and tanks were modeled using the centralized parameter method, and the pipelines were modeled using the distributed parameter method. The centralized parameter method treats the component as a point and ignores the internal dynamic process of the component. The components of the system were modeled in AMESim software, and the calculations were performed through this platform. In particular, the characteristic curves of the pump were obtained via a numerical simulation and imported into AMESim. After the parameters of each component were determined, the system was completed. The high-pressure gas supplied by the gas tank enters the oxidant tank, and the oxidant enters the two pumps from the tank. The two pumps are connected in parallel, and the oxidant finally enters the thrust chamber.

The integration of the propulsion system components was performed on the AMESim platform. The liquid medium was modeled using a two-phase flow model because the pump and regulating valve may experience a cavitation state during operation. The pumps were modeled using pressure head curves and efficiency curves and imported into the platform. A permanent magnet synchronous motor model and control signals were used to regulate the pump speed. The AMESim platform has an intelligent and robust integral solver that adjusts the algorithm to the characteristics of the model itself in real time. The accuracy of the computational solution in the propulsion system was set to 1 × 10⁻⁶. This platform has been successfully applied in a variety of different fields, covering multidomain modeling and dynamic simulations of complex systems including mechanical, fluid, thermal analysis, electromagnetic, and control systems.

4. Results and Discussion

4.1. Comparison of Pump Speed Adjustment Schemes

While adjusting the pump speed, the regulating valve in front of the thrust chamber was kept under the maximum opening state. The flow into the thrust chamber could be adjusted only by the pump speed. The flow in the first stage and the second stage was small; hence, a single pump could be started in the first stage and the second stage. In the third stage, the two pumps worked at the same time. This was determined as Scheme ASYNC. Since the states of the two pumps were consistent, the two pumps could be kept in an operating state in all three stages considering the symmetry. This was determined as Scheme SYNC. These two schemes are studied below.

According to the mass flow requirements of the system, the speed of the pump in all three stages was determined by continuously adjusting the speed, as shown in

Table 6. Speed comparison between Scheme ASYNC and Scheme SYNC.

Period	Scheme ASYNC		Scheme SYNC
	Quantity of Working Pumps	Speed (r/min)	Quantity of Working Pumps	Speed (r/min)
1st	1	13,300	2	12,100
2nd	1	22,300	2	19,600
3rd	2	36,000	2	36,000

. The speed regulation curve is shown in Figure 8, and the system started from 1 s.

In Scheme ASYNC, the pump speed in the first stage was 13,300 r/min, which was about 37% of the rated speed, and the mass flow was 50% of the rated mass flow. In Scheme SYNC, the speed of the two pumps in the first stage was 12,100 r/min, about one-third of the rated speed. Under low operating conditions, the efficiency of the pump would be low. In the second stage, the pump speed was about 62% of the rated speed in Scheme ASYNC, and the pump speed was about 54% of the rated speed in Scheme SYNC. In the third stage, both pumps operated under the rated operating conditions.

The comparison between the pre-regulating pressure and pre-injection pressure of the two schemes is shown in Figure 9. In the start-up stage, there was an obvious pressure peak of the pre-injection pressure in Scheme SYNC. The processes from the first stage to the second stage and from the second stage to the third stage were relatively stable in the two schemes. In terms of response time, the pressure building time of Scheme SYNC was shorter than that of Scheme ASYNC in terms of pre-regulating pressure and pre-injection pressure. This was mainly due to the simultaneous operation of two pumps, and the regulation range of each pump was lower than that of Scheme ASYNC.

The mass flow curve of the system is shown in Figure 10, and the pressure of the pump outlet is shown in Figure 11. The steady-state flow of the two schemes conformed to the state value of the design point. In Scheme ASYNC, there was a large peak in the flow of the continuously operating pump from the second stage to the third stage. Meanwhile, there was also a pressure peak from the second stage to the third stage for the other pump, which was mainly caused by starting up at this time. That is, the pump going from non-operating to maximum operating conditions would result in a larger water hammer affecting the other pump. This is mainly due to the instantaneous flow increase in the regulating valve inlet causing a pressure increase in the pump outlet. Furthermore, the start-up time of the pump had an obvious impact on this peak.

The pressure of the pump inlet in the two schemes is shown in Figure 12. The inlet pressure of the continuously operating pump in Scheme ASYNC dropped when the other pump was opened. It was reduced to about 0.25 MPa, which is also higher than the cavitation pressure of the pump. This pressure drop was mainly due to the opening of the other pump. At this time, there was a pressure peak when the other pump was started. The pressure before the pumps in Scheme SYNC was relatively stable. There was only one start-up pressure peak, while there were no pressure peaks when changing operating conditions later. This result applies to the propulsion system consisting of two electric pumps in parallel. In addition, the regulating valve needed to operate in a cavitation state to ensure that pressure fluctuations in the thrust chamber did not affect the pump outlet pressure.

4.2. Multivariable Design of Joint Adjustment Scheme

Under the condition of pump speed adjustment only, the regulating valve in front of the thrust chamber was kept under the maximum opening state. The flow could be adjusted only by the pump speed. Under the condition of joint adjustment, the pump speed and the regulating valve opening were adjusted at the same time to meet the requirements. Since there are two independent variables, many combinations of pump speed and valve opening could meet the same requirements.

In the case of joint adjustment, $A$ represents the area of the regulating valve, while $A/A_{\max }$ represents the opening of the regulating valve, with a value ranging from 0 to 1. Under the maximum operating conditions, the pump speed was 36,000 r/min and the valve opening was 1. In the second stage, i.e., in the medium state, the pump speed ranged from 24,000 to 33,000 r/min, and the valve opening ranged from 0.5 to 0.7. Multiple combinations of pump speed and valve opening were calculated. The results are shown in Figure 13 and Figure 14, where p₀ represents the pressure of the pump outlet, and q_m represents the mass flow of the pump. At a certain valve opening, the outlet pressure and mass flow of the pump increased with the increase in speed. At a certain speed, the pressure of the pump outlet remained basically unchanged with the change in valve opening, whereas the mass flow of the pump increased almost linearly with the increase in valve opening.

The available combinations of pump speed and valve opening can be found in Figure 13 and Figure 14. In Figure 14, the mass flow required was 5.3 kg/s, and the mass curve intersecting with this straight line indicates the available combination. For example, under 33,000 r/min and 0.52, the pressure of the pump outlet reached about 7.8 MPa. This high pressure caused a large pressure loss in the regulating valve. For example, under 25,500 r/min and 0.67, the pressure of the pump outlet was about 4.7 MPa. This low pressure resulted in incomplete cavitation at the regulating valve.

In the first stage, that is, the low operating conditions, the pump speed ranged from 20,000 r/min to 29,000 r/min, and the valve opening ranged from 0.2 to 0.4. Multiple combinations of pump speed and valve opening were calculated. The results are shown in Figure 15 and Figure 16. In the case of a certain opening, the outlet pressure and mass flow of the pump increased with the increase in speed. At a certain speed, the pressure of the pump outlet decreased slightly with the increase in valve opening. The mass flow increased almost linearly with the increase in valve opening. A greater pump speed led to a more obvious increase in mass flow.

Similarly, the combination of pump speed and valve opening available under low operating conditions can be found in Figure 15 and Figure 16, and the curve intersecting with the aim line indicates the available combination. It can be clearly seen that the intersection range of the curve and aim line under low operating conditions was smaller than that under medium operating conditions. The range of available combination points under low operating conditions was smaller than that under medium operating conditions.

In order to obtain the optimal combination of pump speed and valve opening in the electric pump feed system, this study used a genetic algorithm to perform the calculation process, as shown in Figure 17. First, the target flow rate for this engine was determined. For this study, the target flow rate in the first stage was 2.6 kg/s and that in the second stage was 5.3 kg/s. Then, the available combination range of pump speed and valve opening was obtained using the flow rate curve with pump speed and valve opening. That is, the pump speed and valve opening could only be taken within this range. Within this range, a combination of pump speed and valve opening was identified. On the basis of the pressure curve with pump speed and valve opening, the pump outlet pressure at the specific pump speed and valve opening was obtained. At this point, it was determined if the pump outlet pressure could meet the requirements of the engine system. If not, the process was calculated using a genetic algorithm. The main elements of the genetic algorithm include selection, crossover and mutation. A new pump outlet pressure was obtained again, and then the judgment continued until it met the system requirements. In this method, a combination of constraints and existing curves is needed to meet the requirements.

4.3. Comparison between Pump Speed Adjustment and Joint Adjustment Schemes

According to the mass flow and pressure requirements of the system, the operating state in all three stages was determined by continuously adjusting the pump speed and valve opening. The results of Scheme SYNC and Scheme JOINT are compared in

Table 7. Speed comparison between Scheme SYNC and Scheme JOINT.

Period	Scheme SYNC		Scheme JOINT
	Quantity of Working Pumps	Speed (r/min)	Quantity of Working Pumps	Speed (r/min)
1st	2	12,100	2	24,800
2nd	2	19,600	2	29,700
3rd	2	36,000	2	36,000

, and the speed curve is shown in Figure 18. The system started from 1 s.

In Scheme SYNC, the pump speed in the first stage was 12,100 r/min, about one-third of the rated speed. When Scheme JOINT was adopted, the pump speed was 24,800 r/min, about two-thirds of the rated operating condition. Therefore, Scheme JOINT could significantly reduce the operating range of the pump and maintain the efficiency at a high level. In the second stage, the pump speed of Scheme SYNC was 19,600 r/min, about half of the rated operating condition. Under joint adjustment, the pump speed of Scheme JOINT was 29,700 r/min, about 5/6 of the rated operating condition. In the third stage, the pumps of both schemes worked under the rated operating conditions.

The comparison between the pre-regulating pressure and pre-injection pressure of the two schemes is shown in Figure 19. In the start-up stage, there was an obvious pre-injection pressure peak in both schemes. The processes from the first stage to the second stage and from the second stage to the third stage were relatively stable. The pre-injection pressure of both schemes remained basically the same, because it was mainly determined by the thrust chamber. The pre-regulating pressure of Scheme JOINT was significantly higher than that of Scheme SYNC, especially in the first and second stages; thus, it could ensure that the regulating valve is in an obvious cavitation state.

The mass flow and outlet pressure of the pump are shown in Figure 20. The steady-state flow of the two schemes could meet the design requirements. In terms of flow oscillations, the amplitude was larger in Scheme JOINT than that in Scheme SYNC; however, the time to reach a stable value was smaller in Scheme JOINT than in Scheme SYNC. In other words, the response time of the propulsion system was faster and the overshoot was larger in Scheme JOINT. The pressure of the pump outlet in Scheme JOINT was significantly higher than that in Scheme SYNC, mainly due to the decrease in valve opening and the increase in pump speed. In terms of pressure overshoot, the overshoot was smaller in Scheme JOINT than in Scheme SYNC when the maximum operating conditions were reached.

Overall, Scheme JOINT adopted the mode of joint adjustment of pump speed and valve opening, revealing multiple working points that could meet the requirements. The adjustable range was large, and the optimal scheme could be selected according to the calculation results. Compared with Scheme SYNC, Scheme JOINT could mainly improve the pump speed and the pressure of the pump outlet, enabling the regulating valve to work in a significant cavitation state. In this way, the pressure drop requirements of the catalytic bed, regulating valve, and injector could be met. Under the low operating conditions, the speed of Scheme SYNC was about 12,100 r/min and that of Scheme JOINT was about 24,800 r/min, about one-third and two-thirds of the rated operating condition, respectively. The joint adjustment could reduce the regulation range of the pump and maintain its efficiency at a high level. During starting and regulating, the two pumps operated in the same state under the schemes of joint adjustment and the pump speed adjustment.

In the analysis and comparison of the three schemes, the maximum thrust of the rocket engine was 33.6 kN, and the power of each electric pump was 50 kW. The above results are can be used for an evaluation of the magnitude of thrust and power, whereas their applicability to small thrust engines needs further study.

5. Conclusions

Under the condition of pump speed adjustment, the schemes of asynchronous and synchronous starting and regulating of the two pumps were analyzed. In addition, the pump speed adjustment was compared to the joint adjustment of pump speed and valve opening. Furthermore, a genetic algorithm-based calculation method was implemented.

Under the low operating conditions, the speed of Scheme ASYNC was about 13,300 r/min and that of Scheme SYNC was about 12,100 r/min, close to one-third of the rated operating condition. In Scheme SYNC, since the two pumps were in the same operating state, the overall operating conditions were stable. In scheme ASYNC, there was a pressure peak from the second stage to the third stage due to the asynchronous opening of the two pumps. At the same time, the opening time of the second pump needed to be strictly controlled. Compared with the pump speed adjustment, the joint adjustment of pump speed and valve opening could mainly improve the pump speed and the pressure of the pump outlet, enabling the regulating valve to work in a significant cavitation state. In this way, the pressure drop requirements of catalytic beds, regulating valves, and injectors could be met. The joint adjustment could reduce the regulation range of the pump by a factor of one-third, as well as maintain the pump efficiency at a high level.

In addition, the joint adjustment of pump speed and valve opening revealed multiple combinations of pump speed and valve opening that could meet the requirements. Using the genetic algorithm-based calculation method proposed in this study, together with the curves of flow rate and pressure with pump speed and valve opening, the optimal combination of parameters satisfying the engine flow demand while minimizing energy consumption could be obtained. This method is applicable to propulsion systems with different thrust levels; however, the pressure peak resulting from different schemes may vary with different levels of thrust. Moreover, the dynamic characteristics of the propulsion system when more than two electric pumps are used in parallel are not clear. These issues deserve further investigation.