Establishment and Solution of the First Model
The Setting of Initial Condition for Simple High-Pressure Tubing
As is shown in Figure 1, we establish the simplest model and assume that each component will not deform and there is no leakage problem during the operation of the high-pressure fuel pump. There are no bubbles in the high-pressure fuel pipe, injection nozzle, and high-pressure fuel pump.
In this model, the fuel enters the High Pressure Injection Pump for sale through A and the fuel supply time is controlled by the one-way valve switch. The fuel injector works 10 times per second. After the one-way valve’s opening, the fuel injector will be closed for 10 ms. During each operation, the fuel injection time is 2.4 ms, and the fuel injection rate from nozzle B is shown in Figure 2. The pressure provided by the high-pressure fuel pump at inlet A is 160 MPa, and the initial pressure in the high-pressure fuel pipe is 100 MPa. We will study how to set the opening time of the one-way valve to keep the pressure of the high-pressure tubing at 100 MPa and adjust the opening time of the one-way valve to make the pressure of the high-pressure tubing go from 100 MPa to 150 MPa through different time periods.
In the high-pressure tubing system, the entry and ejection of fuel are carried out according to the corresponding law. Under the control of the one-way valve switch, there is a time interval between the last entry of fuel and the next entry of fuel, and the fuel intake within the time interval is 0. The fuel injection pattern is similar. Therefore, it is not suitable to use common linear expressions to effectively describe the law of fuel inlet and fuel outlet in high-pressure tubing. In this case, for the convenience of research, we discretize the time, so as to better analyze the influence of fuel inlet and fuel outlet on the pressure in high-pressure tubing.
The pressure in the high-pressure tubing varies with time, and the change of some parameters of the tubing at one time will affect the pressure calculation of the high-pressure tubing at the next time. We calculate the pressure change of the high-pressure tubing by establishing the difference equation due to the discretization of the time.
Under the condition given by the difference equation of pressure and time relationship of high-pressure tubing, we can use the enumeration method to start the one-way valve opening time from 0, traverse it with 0.01 ms as step size, and substitute it into the corresponding difference equation to calculate the pressure change of high-pressure tubing. The enumeration method is to use the computer operation speed and the high accuracy of the characteristics, to solve the problem of all possible circumstances, not to miss the test, from which to find the answer to meet the requirements. Enumeration method sacrifices time for comprehensiveness as discussed elsewhere .
2.1.2. Establishment of the First Model
(1) Derivation of Elastic Modulus Formula. The elastic modulus of liquid is the main factor that affects the analysis of the hydraulic system as discussed in . The elastic modulus data with the pressure changing from 0 to approximately 200 MPa whose changing length is 0.5 MPa are given in the research data. Therefore, regression analysis can be conducted to obtain the linear relationship between the elastic modulus and fuel pressure, providing more comprehensive data for the following research. Since the fuel pressure is between 100 and 160 MPa, in order to obtain a better linear relation of fitting degree, we conduct regression analysis in SPSS according to the elastic modulus of 90∼170 MPa in the research data. The relation between elastic modulus and fuel pressure is as follows:
In the formula, is the elastic modulus, and is the fuel pressure.
(2) Derivation of Fuel Density Formula. The pressure change of fuel is proportional to the change of density as discussed in , so we can get
In the formula, is fuel density and is fuel pressure.
(3) Calculation of Fuel Injection Amount in High-Pressure Tubing. According to the given data, the working cycle of the fuel injector is 100 ms, and there is only 2.4 ms for fuel injector to work in the state of fuel injection. When the fuel injector works, the fuel injection rate varies with the opening time of the valve.
According to Figure 2, the piecewise function of fuel injection rate and time when the injector works is as follows:
After integration, the relationship between fuel injection amount and time is as follows:
In equations (5) and (6), is the fuel injection rate of the injector, is the fuel injection amount of the fuel injector, and is the length of the opening time of the injector.
- Establishment of the Difference Equation. In order to make the result as accurate as possible and improve the efficiency, we set up the difference equation whose step size () is 0.01 ms. Before establishing the difference equation, the values of some relevant parameters need to be calculated. The results are shown in Table 1.
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