Computational Evaluation of the Effect of Plunger Spring Stiffness on Opening and Closing Times of the Low-Pressure Gas-Phase Injector

up to 35.21% from the initial value. The obtained results can be useful in modeling the operation of an internal combustion engine or in the operational assessment of the gas injector condition.


Introduction
Due to the continuous tightening of clean exhaust standards [1] and the need to meet global emission standards [2], the use of alternative fuels in transportation is being sought. The largest contributor in this area is liquefied petroleum gas (LPG) [3]. For the most part, the use of LPG in internal combustion engine power systems is done as vapor [4,5], although modern automotive power units require the use of LPG in the liquid phase and with direct injection [6]. Outside of transportation, alternative fuels are used in work machinery [7].
The popularity of alternative LPG vapor power systems is causing an increase in interest in computational methods for evaluating functional parameters. The calculations support the design and manufacturing process as well as are able to evaluate the influence of selected input features of a virtual product on its output parameters. In the case of gas injectors, the output (functional) parameters are mainly opening and closing times and maximum mass flow rate [5,8,9].
Modeling the operation of the low-pressure gasphase injector involving on the combination of mechanical, electrical and hydraulic issues. For the mechanical part, an analytical approach [10,11] or finite element method (FEM) can be used, which is also successfully used in modeling electromagnetic circuits [12]. Analytical models of electromagnetic circuits [13][14][15] give high agreement with FEM calculations and experimental studies. For hydraulic calculations to estimate fuel flow rates, analytical models [16,17], or computational fluid dynamics method (CFD) [18,19] can be used.
Analyzing the results of calculations included in the literature reports, it was observed that the mathematical models presented there show the existence of time delays of plunger displacement in relation to the forcing resulting mainly from the inertia forces and resistance to motion. Therefore, an attempt has been made to computationally demonstrate the effect of the plunger pressure stiffness on the opening and closing times of the low-pressure gas-phase injector as a way to fill the research gap in this subject. Section 2 of the paper presents the object of analysis, Section 3 the mathematical model adopted in the course of the analysis. In Section 4 the conditions necessary to initiate the calculations are described. The results described in Section 5 were aimed to determine the times of opening and closing of the gas injector with variable stiffness of the plunger pressure spring and the accompanying changes in the time-cross section. Section 6 is a summary of the work in which conclusions are presented. The obtained results can be useful in modeling the operation of an internal combustion engine or in the operational assessment of the gas injector condition.

The analysis object
The object of analysis was the Valtek Rail STD Type 30 injector. This injector belongs to the low-pressure gas-phase group. In the unpowered state it is normally closed. When an electrical impulse is applied to the coil terminals, an electromagnetic field is created that lifts the plunger, which is initially pressed against the seat by a spring. After the electric impulse disappears, the injector is closed by the force of the compression spring. A detailed description of the operation will be presented later in this paper. The basic technical data of the analyzed injector have been presented in Table 1. Table 1 The

Modelling of injector operations
The model description is based on the scheme presented in Fig. 1 with the following main simplifying assumptions: − the movement of the plunger is a result of instantaneous values of the acting forces; − the thrust force from the air pressure depends on the position of the plunger; − friction is divided into static, kinetic and viscous, and its value depends on the plunger movement; − vibrations, mechanical disturbances, eddy currents, magnetic saturation, coil temperature change, plunger aerodynamic drag are omitted. Based on the descriptions presented in [5,14,15,21,22] the equilibrium equation of the forces acting on the plunger can be taken as Eq. (1): The waveform of the electromagnetic force Fe resulted from the electromagnetic circuit's characteristic L(x) and the value of the electric current I according to Eq. (2): The variation of current I feeding the circuit was calculated based on Eq. (3): The values of inductance L(x) of the circuit assuming plunger displacement x are written in the form Eq.
The friction force was dependent on the position of the plunger (Eq. (5)): The force from the plunger compression spring was calculated from Eq. (6): The force from gas pressure depended on the position of the plunger (Eq. (7)): The inertia force depended on the mass and acceleration of the plunger (Eq. (8)): Finally, the system of equations necessary to determine the current waveform I and the plunger displacement x will take the form Eq. (9): where the designations are as shown in Fig. 1 and Table 2. Table 2 shows the parameter values and boundary conditions necessary to initialize the calculations. Table 2 Parameter values and initial conditions necessary to initiate the calculation

Results and discussion
In each analyzed case, the system of differential Eq. (9) was solved numerically in Matlab/Simulink [23]. The implicit trapezoidal method combined with backward differentiation was used. The integration step was varied, with the mini-minimum value set at 1×10 -7 s). Matlab/Simulink software allows easy implementation of empirical models, as confirmed in [5,24,25].
In the initial part of the analysis, assuming the data and boundary conditions based on Fp Fm of the system to a pulsed electric voltage forcing U = 12 V with duration tinj = 5×10 -3 s, with spring stiffness C = =780 N/m was obtained. The impulse in the form of electric voltage U (from-segment A-D with length tinj = 2.21×10 -3 s) is an excitation (Fig. 2). In response, the electric current waveform I has characteristic breakthroughs (points B` and C`) at the points of beginning of travel (point B) and reaching by the plunger the maximum opening xmax (point C). The breakthrough separates the coil inductance values at constant (segments A-B and C-D) and variable plunger position (segment B-C). When an electrical pulse appears at the injector coil terminals, the plunger responds with a delay tro (Fig. 2). This is due to the need to overcome spring preload, inertia resistance and plunger friction. When the coil supply is lost (points D and D`), the plunger is moved by the force from the compression spring. In this case, the closing delay is not visible, hence the response time cannot be determined.
The plunger lift x enabled determination of the time to full opening (ttfo =3.30×10 -3 spoint C in Fig. 1) and closing (ttfc = 2.21×10 -3 spoint D). Comparing the results with the manufacturer's data (Table 1), there was a 2.77% longer time to full opening and 0.50 % shorter time to full closing. Based on this, the proposed injector performance model was found to be representative for comparison purposes.
In the main part of the analysis, the times to full opening and closing of the injector were determined with changes of the stiffness of the compression spring in the range (100...2000) N/m with a step of 100 N/m. The results of the calculations for the times to full opening ttfo and closing ttfc are shown in Table 3. Table 3 Determined times to full opening and closing of the injector depending on the stiffness spring In order to graphically represent the results of the calculations, Fig. 3 was created. The obtained values for the time to full opening ttfo were approximated by a linear function, obtaining high agreement (R 2 = 0.9995). In the case of closing times, a power approximation function was used, where a high agreement was also observed (R 2 = 0.9992). Analyzing the differences in the ttfo times, little change is apparent in the spring stiffness range assumed in the calculations. This means that the electromagnetic field generated by the coil is sufficient even at a stiffness of C = 2000 N/m. Considering the time ttfo as a whole, its 18.85% elongation at 2000 N/m relative to the stiffness of 100 N/m was noted. In the case of time ttfc in the considered spring stiffness range, the time was reduced by 80.17% at 100 N/m relative to 2000 N/m.
The plunger compression spring stiffness loss or gain can occur in the course of operation. On the one hand, wear and tear can cause a decrease in stiffness, while on the other hand, contaminants between the spring coils can cause an increase in stiffness.
Variation in the times ttfo and ttfc can fundamentally affect the functional capability of the injector. In Fig. 4, selected plunger lift waveforms are shown to illustrate the differences in the area fields underneath them. The surface area indirectly illustrates the injector output.

Conclusions
The paper presents calculations regarding the influence of the pressure spring stiffness on the opening and closing times of the low-pressure gas-phase injector. Based on the analyzes, the following conclusions were drawn.
1. The proposed model description of the operation of the gas injector showed the correlation of the results with the manufacturer's data. There was an reduction in the time to fully open by 2.77 % and increase in the time to fully close by 0.50 %. The model was found to be representative for comparison purposes. In the paper, an attempt was made to computationally demonstrate the effect of the plunger pressure stiffness on the opening and closing times of the low-pressure gasphase injector as a filling of the research gap in this subject. Based on the presented mathematical model describing the operation of the injector, firstly the results were related to the manufacturer's technical data showing a shorter time to full opening by 2.77% and a longer time to full closing by 0.50%. On this basis, it was considered that the proposed model can be used for comparison purposes. In the assumed range of compression spring stiffness (100... 2000) N/m, it was shown that as the stiffness increases, the time to fully open decreases by 18.85%, while the time to fully closed decreases by 80.17%. Additionally, it is shown that the time-cross section as the stiffness of the compression spring increases can decrease up to 35.21% from the initial value. The obtained results can be useful in modeling the operation of an internal combustion engine or in the operational assessment of the gas injector condition.