Investigation of Factors Influencing Solenoid Valve Speed Response Characteristics of the Common Rail Injector

Abstract

The dynamic injection characteristics of high-pressure common rail fuel injection systems are determined by the speed response performance of the solenoid valve. A simulation model has been established for investigating the influence mechanism and change law of characteristic parameters on speed response characteristics of the solenoid valve. The speed response characteristics of the solenoid valve, including the average opening speed, the average closing speed, the maximum opening speed, and the maximum closing speed, caused by the changes of characteristic parameters such as pre-tightening force of the solenoid valve spring, mass of the solenoid valve moving parts, diameter of the outflow orifice, diameter of the inflow orifice, diameter of the control piston, and pressure in the common rail, have been studied. The correlation analysis of the influence factors is carried out by using the experimental design method based on the response surface model, and the correlation coefficients between each factor and the speed response characteristics of the solenoid valve are obtained. The results show that both single factors and interaction factors of the parameters are correlated with the speed response characteristics of the solenoid valve. The results of this paper can provide a theoretical reference for the design and optimization of the high-pressure common rail injector.

1. Introduction

As a promising diesel injection system, the high-pressure common rail system can achieve high-pressure and even ultra-high-pressure injection. It can be used to flexibly adjust the injection timing. In addition, the fuel can be injected multiple times. As a critical component of this system, the injection performance of the electronic fuel injector is significantly impacted by the dynamic response characteristics of the solenoid valve [1,2,3]. Yin et al. [4] developed a simulation model of the solenoid valve which was used for an injection system in Matlab/Simulink environment. The opening and closing characteristics of the solenoid valve were investigated based on the mathematical model. The influence of coil voltage and air gap on the opening and closing characteristics were analyzed. Vrublevskyi et al. [5] presented a new technique in order to manufacture magnetic cores, for the purpose of improving the injection characteristics of an electrically controlled diesel injector. The results indicated that the dynamic response of the diesel injector could be increased by using the proposed method. In this scheme, fuel injection was divided, which results in precise fuel injection control and improved engine performance. Hung et al. [6] investigated the electromagnets controlling the opening and closing of natural gas injectors. A mathematical model was developed to represent the working process of the electromagnets. Simulation methods were applied to analyze the main factors influencing performance. Ultimately, they identified the critical parameters affecting the performance of the electromagnets. Liu et al. [7] introduced a new permanent magnet high-speed actuator (NPMHEA) to improve the performance of electromagnetic actuators in diesel injection systems. A finite element coupling model was created to study the critical factors influencing actuator performance. Additionally, a multi-objective optimization approach was used to identify optimal parameters, enhancing the overall efficiency of the actuator. Wang et al. [8] studied the electromagnetic and dynamic behavior of solenoid valves in high-pressure fuel systems. The research focused on how the design factors, such as core shape and electrical inputs, influence electromagnetic force and valve function. The results showed that higher ampere-turns increased the force until magnetic saturation occurred. Dynamic analysis examined the impact of currents, spring pre-load, and stiffness on valve performance. The findings provide guidance for improving solenoid valve designs to enhance efficiency and reliability in diesel engines. Huber et al. [9] modeled and analyzed the dynamics of the solenoid valve in diesel injectors. The study emphasized transient effects, experimental validation, and design improvements to optimize injector performance. Yang et al. [10] studied the spill control behavior of JP-8 and conventional diesel fuels in a common rail injection system. The research compared fuel flow characteristics, pressure dynamics, and their effects on system efficiency. The goal was to optimize injection system performance by considering the distinct properties of JP-8 and diesel fuels. The results would improve fuel adaptability and control, enhancing engine performance and emissions in future applications. Wang et al. [11] explored how the geometric features of high-pressure injectors, such as the injection nozzle, affect injection behavior. The study emphasized the link between injector design and engine performance. The aim was to enhance fuel atomization, improve combustion efficiency, and reduce emissions for better diesel engine performance. Hung et al. [12] explored how design factors affect the performance of compressed natural gas injection systems. It examined elements like the pressure regulator and various component configurations, such as the volume of common rail and spring characteristics. The findings suggested that tuning these parameters can enhance the stability and accuracy of the gas injection process, leading to improved system performance and efficiency. Dhanji et al. [13] examined how split injection strategies affect spray characteristics with a high-pressure solenoid injector. It looked at the influence of injection timing, fuel pressure, and atomization on spray penetration and droplet size. The findings helped optimize injection parameters to enhance combustion efficiency and reduce emissions in engines. Gao et al. [14] conducted a detailed study on the novel voice coil motor’s behavior, specifically designed for direct injection fuel injectors in aircraft engine gasoline cylinders. By analyzing the structural parameters of the voice coil motor, the researchers identified their significant impact on motor performance. Additionally, the equivalent magnetic circuit method was used to thoroughly analyze and determine critical influencing parameters. Through the optimization of these key parameters, the system performance of the voice coil motor was enhanced. Işikli et al. [15] studied the impact of the armature, its pin, and the solenoid valve on fuel delivery and return volumes. The analysis focused on several common rail injectors operating under different pressure levels. The study offered data-driven insights to enhance the precision of fuel injection, ultimately reducing fuel consumption and improving energy efficiency. Teoh et al. [16] explored the influence of the direct injection system on injection efficiency through experiments on rail pressure and engine speed. The findings indicated that modifying injection strategies effectively reduces the peak of low-heat release rates. Furthermore, optimizing injector parameters, included open time, low time, and duty cycle of injector signals, enhanced the indicated mean effective pressure while lowering injector power consumption. Ma et al. [17] conducted experiments to study low-temperature injection and spray behavior in high-pressure common rail systems. The study analyzed the influence of environmental factors, pressure in common rail, and injection pulse width on fuel injection quantity and spray characteristics.

The high-pressure common rail injector is a multi-domain system with complex interactions between mechanical, hydraulic, magnetic, and electrical components. Injection dynamics are affected by the opening and closing behaviors of the solenoid valve. The valve’s opening and closing speeds determine the response time during fuel injection. These speeds directly impact the precise control of fuel quantity and injection mode. They serve as the main parameters for indicating the dynamic response characteristics of the high-pressure common rail injector [18,19,20]. The dynamic response characteristics of the solenoid valve for the high-pressure common rail injector are affected by the changes of the characteristic parameters. However, current research mainly focuses on the displacement, response time, and delay time of the solenoid valve, in which the solenoid valve is separate from the injector. The objective of this study is to present a validated simulation model of a high-pressure common rail fuel injection system. By analyzing the speed response characteristics, including the average opening speed, the average closing speed, the maximum opening speed, and the maximum closing speed, the influence law and internal mechanism of the injector characteristic parameters on speed response characteristics of the solenoid valve are studied, which provides a theoretical reference for the design and optimization of a high-pressure common rail injector.

2. Operating Principle of Common Rail Injector

The common rail injector primarily consists of the injector body, solenoid valve assembly, control piston assembly, and needle valve assembly, as shown in Figure 1. After fuel enters the injector, it is divided into two paths: one flows into the fuel chamber via an internal fuel passage, acting on the conical surface of the needle valve; the other enters the control chamber through the fuel inflow orifice, acting on the upper side of the control piston.

When the solenoid valve of the injector is de-energized, the control valve stem presses the ball valve tightly against its seat under the pre-tightening force of the solenoid valve spring, closing the fuel outflow orifice, causing the control chamber to be filled by high-pressure fuel. Since the pressure surface area above the control piston is larger than that of the cone surface of the needle valve in the fuel chamber, the needle valve is pressed against the needle valve seat under the combined effect of the needle valve pre-tightening force of the spring and high-pressure fuel, sealing the nozzle hole, and preventing the injector from spraying fuel.

When the solenoid valve is energized, the armature pulls the control valve stem upward under the influence of the electromagnetic force. This movement causes the ball valve to open, the fuel outflow orifice to open, and the fuel in the control chamber to flow out through the outflow orifice into the fuel tank. Since the diameter of the outflow orifice is larger than that of the fuel inflow orifice, the outflow rate of fuel from the control chamber is greater than the inflow rate. As a result, the pressure in the control chamber rapidly decreases, and the pressure acting on the control piston above it drops quickly. Meanwhile, the fuel in the fuel chamber remains at a high pressure, causing the needle valve and control piston to overcome the pre-tightening of the needle valve spring and move upward. This movement opens the nozzle, and fuel injection begins.

When the solenoid valve is de-energized again, the electromagnetic force disappears, and the control valve stem moves downward under the action of the solenoid valve spring. The ball valve seats, closing the fuel outflow orifice and stopping the fuel flow. The high-pressure fuel entering the control chamber through the fuel inflow orifice causes the pressure in the chamber to rise rapidly. Once the combined force of the fuel pressure acting on the control piston and the force of needle valve spring exceeds the hydraulic pressure on the needle valve seat, the control piston moves the needle valve downward, causing the needle valve to seat and close the nozzle, thereby ending the fuel injection [21,22].

3. Establishment and Verification of Simulation Model

Sole reliance on experimental study is inadequate for an in-depth analysis of the high-pressure common rail system response and injection characteristics. Simulation methods enable the discovery of the intrinsic mechanisms underlying the speed response characteristics of the injector solenoid valve [23,24,25]. The simulation methods for the common rail fuel injection system mainly include mathematical modeling [26] and modeling with commercial software such as AMESim [27], GT-Power [28], GT-SUITE [29], and Matlab/Simulink [30]. AMESim is a one-dimensional simulation code in which each physical component is represented by simplified icons and is composed of one or more lumped parameter models. It is capable of modeling a system with hydraulic, mechanical, and electrical components. As the high-pressure common rail system is a complex nonlinear system involving the coupling of electric fields, magnetic fields, mechanical motion, and flow fields, this software is suitable for modeling of this system. It is widely used in modeling the common rail system [31,32,33]. Figure 2a shows the developed AMESim (ver-2021.2) model for a high-pressure common rail fuel injection system and the description of each component of the model. The components include basic hydraulic parts, mechanical parts, and electrical signals, as shown in the figure. To validate the model’s accuracy, experiments are carried out using the injection system test bench, as illustrated in Figure 2b. What needs to be explained is that the components of the same name in the simulation model correspond to the components of the same name in test bench, and their functions are the same.

In the experiment, the EFS 8200 series control system has been used to control and regulate the rotation speed of the high-pressure fuel pump. Using the EFS 8422 to control the volume control valve of the pump, so as to adjust the quantity of supplied fuel of the pump, further realizes the regulation of rail pressure and ensures the precise control of the pump. The IPOD control module has been used to drive the solenoid valve of the injector to control the injection timing, injection pulse width, and the number of injections. The Kistler 4067 high-pressure sensor has been used to measure the inlet pressure of the injector. The DL 750 oscilloscope has been used to record, output, and print the measurement pressure signal of sensors in the experiment. The main components of the test bench setup and measuring instruments are listed in

Table 1. Main components of the test bench setup and measuring instruments.

Setup Name	Application and Accuracy
EFS 8200 series	Performance test of CRS
EFS 8244	Rail pressure control
IPOD	Solenoid valve drive
Flow sensor LWGY-10	0–50 L/min, 0.1 L/min
Low-pressure sensor AOB-131	0–10 MPa, accuracy 0.02 MPa
High-pressure sensor Kistler 4067	0–200 MPa, accuracy 10 Pa
EMI2	0–600 mm3, accuracy 0.6 mm3
Oscilloscope DL 750	0–100 MHz, accuracy 10 MS/s

.

The simulation results are compared with experimental measurements, as depicted in Figure 3. The comparison between the measured and simulated rail pressure is illustrated in Figure 3a, while the comparison between the measured and simulated injection rate of the system is depicted in Figure 3b. Meanwhile, the quantities of fuel injection are calculated as the average of the results from 100 cycles during the experiment to ensure accuracy. These quantities, measured under different rail pressures and injection pulse widths, are listed in

Table 2. Comparisons between measured and simulated fuel injection quantities.

Rail Pressure (MPa)	Injection Pulse Width (μs)	Measured Value (mm3)	Simulated Value (mm3)
60	800	32.78	33.12
	2000	72.95	71.03
	3200	108.45	109.29
100	800	43.05	43.55
	2000	89.05	92.99
	3200	143.95	142.64
140	800	51.92	51.96
	2000	111.61	110.90
	3200	168.46	170.03

. The accuracy of the simulation model is expressed using three quantitative indices: maximum relative error (MRE), R², and Nash–Sutcliffe Efficiency (NSE) [34]. Among these, the accuracies of the fuel injection rate and fuel injection quantity are quantified using R² and MRE, respectively. The accuracy of the common rail pressure is evaluated using both NSE and MRE. After analyzing the data presented in Figure 3 and Figure 4, along with Table 2, it is determined that the MRE of the fuel injection quantity is 4.4%, the R² of the injection rate is 98.05%, the NSE of the rail pressure is 0.753, and the MRE of the rail pressure is 2.66%. The comparison results demonstrate that the rail pressure fluctuation derived from the simulation model of the high-pressure common rail injection system corresponds well with the experimental measurements. Additionally, the system-simulated injection rate shows a high degree of agreement with the experimental data, validating the high accuracy of the developed simulation model.

4. Analysis of the Influence Factors for the Speed Response Characteristics of the Solenoid Valve

According to the structural principle and working process of the high-pressure common rail injector, the processes of fuel pressure build-up and relief in the control chamber are critical to fuel injection. The control chamber is formed between the control piston and the injector body. The control piston and the solenoid valve work together, and its size affects the volume of the control chamber, which in turn affects the pressure fluctuation in the control chamber. The control chamber is connected to the common rail through a high-pressure pipeline. The pressure in the common rail fluctuates during fuel injection, thus the speed response characteristics of the solenoid valve are different under different pressures in the common rail. The inflow orifice affects the flow area of the fuel in the injector flowing into the control chamber, and then influences the flow rate of high-pressure fuel in the control chamber, while the outflow orifice affects the discharge rate of high-pressure fuel in the control chamber. The pre-tightening force of the solenoid valve spring provides the downward force of the initial control valve, and the mass of the moving parts mainly affects the inertia of the solenoid valve, both of which affect the speed response characteristics of the solenoid valve. This paper focuses on these influence parameters, which have an effect on the speed response characteristics of the solenoid valve.

4.1. Pre-Tightening Force of the Solenoid Valve Spring

The effects of the pre-tightening force of the solenoid valve spring on its average and maximum speeds are demonstrated in Figure 4. As illustrated in Figure 4a, the average opening speed of the solenoid valve is relatively higher when the pre-tightening force of the solenoid valve spring is smaller, and the average opening speed of the solenoid valve decreases with the increase of the pre-tightening force of the solenoid valve spring. The average closing speed of the solenoid valve increases with the increase of the pre-tightening force of the solenoid valve spring. Based on Hooke’s Law, with the spring stiffness of the solenoid valve unchanged, the spring pre-compression increases linearly as the pre-tightening force of the solenoid valve spring increases. For the same solenoid valve stroke, the spring resistance acting on the control valve stem increases with the pre-tightening force of the solenoid valve spring, leading to an increase in the opening time of the solenoid valve. Consequently, the average opening speed of the solenoid valve decreases as the pre-tightening force of the solenoid valve spring increases. After the coil is de-energized, the solenoid valve is influenced only by the hydraulic force and the downward restoring force of the spring. During the closing process of the solenoid valve, the closing time decreases as the pre-tightening force of the solenoid valve spring increases, thus the average closing speed of the solenoid valve increases. As the pre-tightening force of the solenoid valve spring increases, the maximum opening speed of the solenoid valve decreases linearly, and the maximum closing speed increases linearly with the increase in the pre-tightening force, as shown in Figure 4b.

As the pre-tightening force of the solenoid valve spring increases, the opening resistance of the solenoid valve increases, reducing the resultant force acting on the valve and decreasing its opening acceleration. Consequently, the maximum opening speed of the solenoid valve decreases with the increase in the pre-tightening force. Once the control valve stem reaches its maximum stroke, the pressure differential between the control chamber and the low-pressure chamber remains constant, but the elastic potential energy of the spring increases with the pre-tightening force of the solenoid valve spring. Therefore, as the pre-tightening force of solenoid valve spring increases, the restoring force of the spring grows, resulting in a higher resultant force during the closing process. This causes more elastic potential energy to be converted into the kinetic energy of the moving parts of the solenoid valve, increasing the descending acceleration of the solenoid valve, thus increasing the maximum closing speed of the solenoid valve.

4.2. Mass of Solenoid Valve Moving Parts

The variations in the average speed and maximum speed of the solenoid valve with respect to the mass of its moving parts are shown in Figure 5. It can be seen from Figure 5a that the average opening speed and the average closing speed of the solenoid valve both decrease when the mass of solenoid valve moving parts increases, and the reduction amplitude of the average speed of the solenoid valve changes in a power function relationship with the increase of the mass of the solenoid valve moving parts. The start of opening and closing moments of the solenoid valve are influenced by the moments when the solenoid valve coil is energized and de-energized, and have nothing to do with the mass of the solenoid valve moving parts. The mass of the moving parts mainly affects the inertia of the solenoid valve. When the mass of the solenoid valve moving parts increases, it causses the inertia of the solenoid valve to increase.

The moment that the solenoid valve reaches the maximum lift and the moment that it is completely closed are both delayed, resulting in the opening and closing time of the solenoid valve being prolonged, so as the mass of solenoid valve moving parts increases, the average opening and closing speeds of the solenoid valve decrease. It can be seen from Figure 5b that the maximum opening speed and the maximum closing speed of the solenoid valve decrease with the increase of the mass of the solenoid valve moving parts, and the maximum speed of the solenoid valve decreases with the increase of the mass of the moving parts in a power function relationship. The opening and closing motions of the solenoid valve involve variable acceleration processes. Therefore, the maximum opening speed is attained when the valve reaches its maximum stroke, while the closing speed reaches its peak at the moment of full closure. Given that the speed is zero at both the opening and closing moments of the solenoid valve, the increase in the mass of the solenoid valve moving parts leads to an extension of the opening time. As a result, both the maximum opening speed and the maximum closing speed of the solenoid valve decrease with the increase in the mass of the moving parts.

4.3. Diameter of Outflow Orifice

Figure 6 illustrates the impact of changes in the diameter of outflow orifice on both the average speed and the maximum speed of the solenoid valve. The average opening speed of the solenoid valve increases with the increase in the diameter of the outflow orifice, while the variation in the diameter of the outflow orifice has a minimal effect on the average closing speed, as demonstrated in Figure 6a. The diameter of the outflow orifice influences the discharge rate of the high-pressure fuel in the control chamber, thereby serving as a throttling function in fuel pressure relief. With the increase in the diameter of the outflow orifice, the fuel flow area expands, leading to a higher flow rate per unit time. This accelerates the discharge rate of the high-pressure fuel in the control chamber through the outflow orifice, thereby shortening the opening time of the solenoid valve and increasing the average opening speed. After the solenoid valve reaches the maximum stroke, the discharge rate of the high-pressure fuel in the control chamber through the outflow orifice stabilizes. The change in the diameter of the outflow orifice has a negligible effect on both the closing moment and the moment of complete closure. Thus, when the coil is de-energized, the closing time of the solenoid valve is scarcely influenced by the diameter of the outflow orifice, resulting in a minimal impact on the average closing speed of the solenoid valve.

It can be seen from Figure 6b that the maximum opening speed of the solenoid valve increases linearly with the increase in the diameter of the outflow orifice, whereas the maximum closing speed is only slightly influenced by the diameter of the outflow orifice. During the opening process of the solenoid valve, it is influenced by hydraulic pressure, spring resistance, and electromagnetic force. As the diameter of the outflow orifice increases, the discharge rate of high-pressure fuel accelerates, resulting in a shorter opening time. The hydraulic pressure acting on the solenoid valve increases during the opening process, which in turn raises the upward opening force and the opening acceleration. Consequently, with the increase in the diameter of the outflow orifice, the maximum opening speed of the solenoid valve also increases. After the coil is de-energized, the solenoid valve is influenced only by the fuel hydraulic pressure and the spring restoring force. When the solenoid valve reaches the same stroke, the spring restoring force remains constant. The closing speed of the solenoid valve differs slightly due to the impact of high-pressure fuel flowing through the outflow orifice with varying diameters. Consequently, the maximum closing speed of the solenoid valve exhibits minimal variation with changes in the diameter of the outflow orifice.

4.4. Diameter of Inflow Orifice

The variations of the average speed and maximum speed of the solenoid valve with the diameter of the inflow orifice are shown in Figure 7. As shown in Figure 7a, the average opening speed of the solenoid valve increases with the increase in the diameter of the inflow orifice, whereas the average closing speed decreases as the diameter of the inflow orifice increases. The diameter of the inflow orifice affects the flow area through which fuel enters the control chamber of the injector, thereby impacting the flow rate of high-pressure fuel supplied to the control chamber. The flow area of the inflow orifice increases with the increase in diameter. During the opening of the solenoid valve, with the diameter of the inflow orifice remaining unchanged, the flow rate of high-pressure fuel entering the control chamber accelerates. Consequently, the flow of high-pressure fuel entering the control chamber from the inflow orifice increases per unit time, which slows the reduction in fuel pressure within the control chamber. Meanwhile, the fuel entering from the outflow orifice into the low-pressure chamber pressure rises, thereby increasing the hydraulic pressure acting on the solenoid valve. This causes the solenoid valve to reach its maximum stroke earlier. Since the diameter of the inflow orifice does not influence the opening moment of the solenoid valve, the opening time decreases and the average opening pressure increases as the diameter of the inflow orifice increases. When the diameter of the outflow orifice remains constant and the diameter of the inflow orifice increases, the pressure drop in the control chamber rises over the same period.

During the closing process of the solenoid valve, the hydraulic resistance acting on the solenoid valve increases due to the high-pressure fuel flowing through the outflow orifice, thereby extending the closing time. Consequently, the average closing time of the solenoid valve decreases as the diameter of the inflow orifice increases. It can be seen from Figure 7b that the maximum opening speed of the solenoid valve increases linearly with the increase in the diameter of the inflow orifice, whereas the maximum closing speed decreases as the diameter of the inflow orifice increases. Although the opening time of the solenoid valve decreases with the increase in the diameter of the inflow orifice, the larger fuel input and smaller output to the control chamber result in a slower pressure drop within the chamber. Additionally, the outflow pressure through the outflow orifice is higher, leading to an increased opening acceleration. Consequently, the maximum opening speed of the solenoid valve increases with the increase in the diameter of the inflow orifice. The hydraulic resistance during the closing process of the solenoid valve increases due to the increase in the diameter of the inflow orifice. Since the closing process is mainly governed by the spring restoring force and hydraulic pressure, and the spring restoring force is independent of the diameter of the inflow orifice, the net force acting on the solenoid valve during the closing process decreases, leading to a reduction in the closing acceleration. As a result, the maximum closing speed decreases with the increase in the diameter of the inflow orifice.

4.5. Diameter of Control Piston

Figure 8 illustrates the impact of the diameter of the control piston on both the average speed and maximum speed of the solenoid valve. As shown in Figure 8a, the average opening speed of the solenoid valve increases when the diameter of the control piston increases, while the average closing speed exhibits minimal variation with the increase in the diameter of the control piston. The diameter of the control piston primarily influences the volume change of the control chamber. The larger the control chamber volume, the more effectively it stabilizes pressure fluctuations. As the diameter of the control piston increases, the control chamber volume of the injector also increases. With the diameter of the outflow orifice unchanged, the flow area for high-pressure fuel to discharge through the outflow orifice remains constant. When the ball valve of the solenoid valve moves away from the valve seat to open the outflow orifice, the fuel discharge flow rate in the control chamber remains unchanged. However, the increase in the diameter of the control piston leads to a larger control chamber volume, which slows the reduction of fuel pressure within the control chamber. Consequently, the hydraulic pressure of the fuel discharging through the outflow orifice increases, resulting in a higher net force acting on the solenoid valve during the opening process, thus reducing the opening time. Therefore, as the diameter of the control piston increases, the average opening speed of the solenoid valve increases. Given the relatively small volume of the control chamber, its ability to stabilize the high-pressure fuel is not very pronounced. As a result, the variation in the diameter of the control piston has little impact on the solenoid valve closing process, and the average closing speed remains largely unchanged with the increase in the diameter of the control piston.

Variations in the diameter of the control piston have a minimal effect on the maximum speed of the solenoid valve, as demonstrated in Figure 8b. An increase in the diameter of the control piston results in a shorter opening time for the solenoid valve, a slower pressure drop in the control chamber, a higher hydraulic pressure acting on the solenoid valve, an increased upward force, and a greater opening acceleration. However, during the upward opening process of the solenoid valve, the effects of force and opening time on the motion speed counterbalance each other. As a result, the variation in the diameter of the control piston has a minimal impact on the maximum opening speed of the solenoid valve. With the increase in the diameter of the control piston, the change in the closing time of the solenoid valve is minimal. After the solenoid valve reaches its maximum stroke, the discharge of high-pressure fuel in the control chamber stabilizes, and the net force acting on the solenoid valve during the closing process remains almost constant. As a result, the maximum closing speed of the solenoid valve exhibits little variation with an increase in the diameter of the control piston.

4.6. Pressure in the Common Rail

The variations of the average speed and maximum speed of the solenoid valve with the pressure in the common rail are shown in Figure 9. The average opening speed of the solenoid valve increases with the increase in pressure in the common rail, while the average closing speed decreases with the increase in pressure in the common rail, as shown in Figure 9a. The injector control chamber is connected to the common rail via a high-pressure fuel pipeline. When the solenoid valve is de-energized and the injector is not in operation, the pressure in the control chamber is equal to the pressure in the common rail. Consequently, as the rail pressure increases, the fuel pressure in the control chamber also rises. Before the ball valve of the solenoid valve is lifted from its seat, the fuel pressure acting on the ball valve through the outflow orifice increases. As the rail pressure increases, the opening moment of the solenoid valve advances. Once the ball valve opens the outflow orifice and fuel starts to depressurize in the control chamber, the hydraulic pressure acting on the solenoid valve increases with the rise in the rail pressure, which in turn increases the upward force on the solenoid valve. As a result, the moment when the solenoid valve reaches its maximum stroke is advanced. Therefore, with an increase in the rail pressure, the solenoid valve opening time decreases, and the average opening speed increases. Since the armature of the solenoid valve reaches and maintains its maximum stroke, the fuel hydraulic pressure acting on the armature is in a balanced state.

When the coil is de-energized, the electromagnetic force disappears, and the solenoid valve closes under the influence of hydraulic resistance and the spring restoring force. As the rail pressure increases, the fuel hydraulic resistance also increases, which results in a reduction of the net force acting on the solenoid valve during closure, thereby prolonging the closing time. Therefore, with an increase in the rail pressure, the average closing time of the solenoid valve decreases. When the pressure in the common rail increases, the maximum opening speed of the solenoid valve increases, whereas the maximum closing speed decreases linearly with the rise in rail pressure, as illustrated in Figure 9b. After the solenoid valve opens, as the pressure in the common rail increases, the fuel hydraulic pressure increases, resulting in an increase in the upward force acting on the solenoid valve moving parts, which consequently accelerates the opening. Although the opening time decreases as rail pressure rises, the impact of force on the solenoid valve motion speed is more important. Therefore, with the increase in rail pressure, the maximum opening speed of the solenoid valve increases. During the solenoid valve seating and closing process, the fuel hydraulic resistance increases as rail pressure rises. Because the spring restoring force is unaffected by rail pressure, the downward force acting on the solenoid valve moving parts diminishes, leading to a reduction in closing acceleration. Although the closing time increases with the rise in rail pressure, the impact of force on the solenoid valve closing speed is more dominant. As a result, with the increase in rail pressure, the maximum closing speed of the solenoid valve decreases.

5. Analysis of the Correlation Between the Influencing Factors

The analysis of the influencing factors on speed response characteristics of the solenoid valve has revealed the influence of individual factor variations on the speed response. However, given the potential complex interaction among parameters, single-factor analysis alone is inadequate for fully explaining the influence patterns of each factor on speed response characteristics of the solenoid valve and the magnitude of their respective impacts. As the above six parameters vary across different ranges, it is difficult to analyze their interaction relationships, and conventional experimental methods would require a large number of trials and are insufficient for uncovering the underlying patterns. The response surface modeling (RSM) method based on the central composite face-centered (CCF) design is a statistical approach for modeling and optimizing processes where multiple variables influence a response [35]. In CCF, factorial, center, and axial points are chosen, positioned on the faces of the experimental space to ensure all factors remain within defined ranges. This design is particularly useful for maintaining realistic predictions and avoiding inaccuracies when extrapolation beyond practical limits is undesirable. By fitting a second-order polynomial model, CCF captures linear, interaction, and quadratic effects of variables, enabling detailed analysis of the response surface, and the model is then used to identify optimal conditions for the response through statistical tools like regression analysis and optimization techniques. CCF is straightforward, minimizes experimental effort, and maintains practical feasibility, making it ideal for systems requiring precise optimization within strict variable bounds. Hence, this paper adopts the principles of experimental design and applies the RSM method based on CCF design, taking each influence factor as the independent variable and the speed response characteristics of the solenoid valve as the response variable.

The definition of correlation coefficient can be found in Reference [36]. The correlation coefficients between the 21 factors formed by the six parameters considering the interaction are shown in Figure 10. A positive coefficient indicates that increasing the factor will enhance the solenoid valve speed, while a negative coefficient suggests that increasing the factor will reduce the solenoid valve speed. The absolute value of the coefficient reflects the extent of the factor’s influence on the solenoid valve speed response, with a larger coefficient signifying a greater impact on the speed response. It can be observed that not only do individual factors exhibit a correlation with the solenoid valve speed response characteristics, but the interactions between different parameters also influence the solenoid valve speed characteristics.

It can be seen from Figure 10a that the mass of the solenoid valve moving parts has a correlation coefficient of −0.1004 with the average opening speed, indicating a negative correlation. This variation significantly affects the solenoid valve average opening speed, identifying it as the most influential first-order factor. The pre-tightening force of the solenoid valve spring and pressure in the common rail have correlation coefficients of −0.0348 and 0.0445, respectively, with the average opening speed. This suggests that their variations exert a moderate influence on the solenoid valve average opening speed, categorizing them as secondary first-order influencing factors. The diameter of the outflow orifice, diameter of the inflow orifice, and diameter of the control piston have correlation coefficients of 0.0093, 0.0071, and 0.0012, respectively, indicating a positive correlation with the average opening speed. Their impacts on the solenoid valve average opening speed are minimal. As shown in Figure 10b–d, the mass of the solenoid valve moving parts emerges as the major first-order factor influencing the average closing speed. Meanwhile, the pre-tightening force of the solenoid valve spring and the pressure in the common rail serve as minor factors, and the diameter of the outflow orifice, diameter of the inflow orifice, and diameter of the control piston exhibit less influence on the average closing speed. The pre-tightening force of the solenoid valve spring and the mass of the solenoid valve moving parts are the key first-order factors impacting the maximum opening speed of the solenoid valve. In contrast, the diameter of the outflow orifice, diameter of the inflow orifice, diameter of the control piston, and pressure in the common rail serve as secondary first-order factors influencing the solenoid valve maximum opening speed. The pre-tightening force of the pressure spring, the mass of the solenoid valve moving parts, and the pressure in the common rail are the dominant first-order factors affecting the solenoid valve maximum closing speed. Meanwhile, the diameter of the outflow orifice and the diameter of the inflow orifice are minor factors, and the diameter of the control piston exerts only a minimal influence on the maximum closing speed.

As shown in Figure 10a, the second-order factor 11, derived from the interaction of various influencing parameters, exhibits a correlation coefficient of 0.0234 with the average opening speed of the solenoid valve. This represents the strongest positive correlation among all second-order factors, making it the most significant influencing factor within this category. The correlation coefficients of factors 7, 9, 10, and 16–19 with the average opening speed of the solenoid valve range from 0.0019 to 0.0078, showing a positive correlation. Conversely, factors 13–15 and 21 show negative correlations, with coefficients ranging from −0.0049 to −0.0019. These factors are considered secondary influences among the second-order factors. The absolute values of the correlation coefficients between the other secondary influencing factors formed by the interaction of the parameters and the average opening speed of the solenoid valve are small, and the influences on the average opening speed of the solenoid valve are weak. Similarly, as indicated in Figure 10b–d, factors 7, 8, 10–14, 16–18, 20, and 21 are the primary second-order factors affecting the solenoid valve average closing speed. Factors 9 and 15 serve as secondary factors, whereas second-order factor 19 exerts a minimal influence on the average closing speed. Factors 7, 11, 12, 15, 18, and 19 are the key second-order factors affecting the maximum opening speed of the solenoid valve. Factors 8–10, 13, 14, 16, 17, and 20 serve as minor factors, while second-order factor 21 has an insignificant effect on the maximum opening speed. Factors 7, 8, 11, 13, 15–18, and 20 are the dominant second-order factors influencing the maximum closing speed of the solenoid valve, while the remaining second-order factors have a secondary impact on the maximum closing speed.

6. Conclusions

The opening speed and closing speed of the solenoid valve reflect the speed of the action response during fuel injection, which directly affects the precise control of the injection quantity and injection mode. In this paper, a high-pressure common rail fuel injection system simulation model is developed in AMESim. To validate the model’s accuracy, experiments are carried out using the injection system test bench. The comparison results demonstrate that the simulation model corresponds well with the experimental measurements. The analyzed speed response characteristics including the average opening speed, the average closing speed, the maximum opening speed, and the maximum closing speed. In addition, the influence law and internal mechanism of the injector characteristic parameters on speed response characteristics of the solenoid valve are studied. The results of this paper can provide a theoretical reference for the design and optimization of the high-pressure common rail injector. Specific conclusions are as follows:

(1) The solenoid valve average opening speed decreases as the pre-tightening force of the solenoid valve spring and the mass of the solenoid valve moving parts increase, but it increases with larger diameters of the outflow orifice and inflow orifice, diameter of the control piston, and higher pressure in the common rail. The solenoid valve average closing speed increases as the pre-tightening force of the solenoid valve spring rises, but decreases with the larger mass of the solenoid valve moving parts, diameter of the inflow orifice, and higher pressure in the common rail.

(2) The solenoid valve maximum opening speed decreases as the pre-tightening force of the solenoid valve spring and mass of the solenoid valve moving parts increase, while it increases with larger diameters of the outflow orifice and inflow orifice, and higher pressure in the common rail. The solenoid valve maximum closing speed increases with the pre-tightening force of the solenoid valve spring, while it decreases as the mass of the solenoid valve moving parts, diameter of the inflow orifice, and pressure in the common rail increase.

(3) By applying the experimental design method, the correlation of factors affecting the solenoid valve speed response characteristics is analyzed. The results indicate that not only do individual parameters correlate with the solenoid valve speed response, but interaction factors between different parameters also show a correlation with the solenoid valve speed response. Furthermore, the correlation between the first-order factors and the solenoid valve speed response is stronger than the second-order factors formed by parameter interactions.