1. Introduction
In the last 30 years, strict government policies have been developed to restrict the emission levels of pollutants that mainly affect the emission of nitrogen oxides (NOxs), carbon monoxide (CO), hydrocarbons (HCs), and particulate matter (PM). Implementing these environmental policies is also intended to reduce the consumption of fossil fuels and encourage the use of alternative low carbon fuels, such as biodiesel, gas-to-liquid, hydrotreated vegetable oil, and farnesane, which also contribute to reducing pollutant emissions.
Manufacturers of fuel injection systems develop these systems focusing on the improvement of the combustion process for reducing pollutant emissions according to the requirements of environmental regulations, increasing their efficiency and, therefore, the brake engine efficiency. Numerous researchers have shown that fuel injection parameters significantly impact pollutant emissions [
1,
2,
3,
4]. Research on the efficiency of injection systems focuses on improving fuel atomization and optimizing the injection process to achieve the best possible combustion in the cylinder. The rate of fuel injection (RoI) is closely related to the heat release rate of the fuel; therefore, analyzing the parameters that directly or indirectly affect the injection rate makes it possible to know how the fuel heat will be released during the combustion process.
Alternative fuels are one of the most promising ways to reduce pollutant emissions, along with modern catalytic aftertreatment systems [
5,
6,
7,
8,
9,
10]. Traditional biodiesels have been a promising substitute for diesel because of their good thermal efficiency and low CO and SOx emissions [
8]. However, other aspects related to crop and food availability have led to the search for alternative fuels, including third-generation synthetic fuels such as farnesane or hydrotreated vegetable oil (HVO). These synthetic fuels have high thermal efficiency and lower emissions than traditional biodiesel [
11,
12,
13,
14]. The influence of biodiesel properties on injection parameters has been extensively studied in traditional biodiesels [
2,
3,
15,
16,
17], although it has been studied to a lesser extent on these synthetic fuels [
18,
19,
20].
Varying injection parameters such as injection timing, injection pressure, and fuel spray duration is one of the strategies to improve engine performance [
2,
21,
22,
23,
24] that can be combined with the use of alternative fuels to reduce emissions. In injection systems, the fuel injection rate depends on different control variables, such as injector nozzle geometry and injector type, engine load, properties of the fuel used, and injection characteristics (multiple injections, injection pressure, injection duration, etc.) [
25,
26,
27,
28]. This large variability of study parameters complicates experimentation. Most researchers use robust test benches and fuel injection rate indicator equipment [
29] to reproduce the operating situations of injection systems in vehicles. However, these facilities are costly and complex to maintain, making it difficult for universities or companies with fewer resources to access this type of facility.
Injection rate simulations are a valuable tool for the reduction of experimental time and error reduction, allowing the simulation of operating conditions that would be dangerous for the experimental equipment. Simulations of injection rates can be performed with zero-dimensional (0D) models [
30,
31], or multidimensional models in combination with other mathematical and software tools [
32,
33,
34]. 0D models are simple to implement and require limited experimental trials [
29].
In 0D models, the injection time variation is the most commonly used parameter for studies focused on combustion behavior, emissions, and fuel consumption [
29]. Although other injection parameters, such as injection pressure, injector energizing time [
29,
35,
36,
37], or injector nozzle geometry [
38,
39] have also been varied.
Regarding the injector geometry, some studies have reported the influence of the number of holes in the cavitation processes with different fuels [
40,
41], and recommended a study with single-hole nozzles to know the behavior of the injection rate more deeply. The variation of nozzle diameters provides valuable information on how geometrical factors can influence the injection rate and ultimately affect the performance and efficiency of the fuel injection system.
Although 0D models for injection rate are simple, they are less commonly used compared to one-dimensional (1D) models [
29]. Payri R. et al. [
42] developed a 0D model capable of estimating the shape of the injection rate and the amount of fuel mass injected with the lowest possible computational cost. In this work, the rate curve is divided into three parts, and mathematical expressions of straight slopes are used to open and close the fuel injection rate curve, which combines second-order Bézier curves to smooth the corners. Xu et al. [
43] divided the injection process into five stages and used different mathematical strategies to model each stage, even combining the data from other experimental works, such as those of Agarwal et al. [
44], Seykens et al. [
45], and Li et al. [
46]. In Soriano et al. [
31], a model is presented that divides the injection rate curve into three parts: the rising phase (from the beginning of injection until it reaches the maximum value of the injector opening that coincides with the one obtained in Equation (10)), the middle phase (in which the maximum amount of fuel is injected), and the falling phase (since the amount of fuel injected begins to decrease). Gao et al. [
47] developed a model using a similar methodology to Soriano et al. [
31]. They used experimental correlations with parameters related to injector operation, fuel properties, and operating conditions to determine the start of injection time, the end of injection time, and the slopes of the RoI curve. Perini et al. [
48] divided the rate-of-injection curve into four sections (1st needle lift, 2nd full-lift hydraulic transient, 3rd steady state max rate, and 4th descent) to develop the 0D model. The model combines the theoretical injection equations with the information obtained from the experimental tests for each section.
In this research, a 0D model is proposed to obtain the injection rate from certain operating conditions, such as injection pressure, energizing time, and fuel temperature. In addition, the geometry of the injector nozzle and some of the fuel properties are used. Injectors with single-hole nozzles were used to avoid the uncertainties caused by using multiple-hole nozzles on the rate of fuel injection. Data from two synthetic paraffinic fuels have been included so that the model can simulate the rates of these fuels and extend the usefulness of the model obtained. The model proposed in this work is based on other 0D models already published [
42,
43,
47,
48], but particularly, it is an evolution of the model presented by Soriano et al. [
31]. The novelty of this article compared to previous ones lies in the way in which the phases into which the fuel injection rate is divided are established, using mathematical correlations that do not require the calculation of the phase shifts between the energizing signal and the RoI signal.
2. Materials and Methods
2.1. Experimental Installation
Figure 1 shows the experimental setup used in the present investigation, which is described in detail in Soriano et al. [
31].
A pump injection test bench is responsible for supplying the appropriate fuel flow to the high-pressure pump of the injection system so that it raises the pressure and sends the fuel to the rail. The pressure-regulating valve delivers the fuel from the common rail to the injector at the pressure and temperature conditions set for the test. The dyno control system controls the pump pressure, injector pulse, injection frequency and pressure, and injector injection timing.
A fuel injection indicator model EVI-2 K-050-49 (Hantek
®, Quingdao, China) was used to determine the injection rate measurement experiments using the Bosch method [
49]. The EVI control unit displays the static system’s fuel temperature and back pressure.
The injector activation pulse was detected using a Hantek
® CC 65 AC/DC current clamp with a bandwidth of 20 kHz to measure the working intensity of the pulse. The total mass of fuel injected (mf) was measured by a Kern PFB 3000-2 gravimetric balance (Balingen, Germany). All generated signals were viewed with an oscilloscope and captured by a data acquisition system (Yokogawa DL708E, Tokyo, Japan). More details about this experimental setup and the measurement methods used can be found in [
31,
50].
A Bosch 089909/0445110239 solenoid-type injector (Gerlingen, Germany) was used in this work. The three nozzles used have a K-factor of 3.5 and a single orifice (mono-centric). The hole diameters of the nozzles are 115, 130, and 150 µm in diameter, respectively. A Nikon CT-Scan-XT-H-160 X-ray scanner (Tokyo, Japan) was used to obtain the geometrical characteristics of the injector nozzles. This scanner generated three-dimensional (3D) images from two-dimensional (2D) images of the nozzles. In addition, image reconstruction software (VGStudio Max 2.2) was implemented to perform data evaluation to calculate variables such as areas, volumes, porosity, thickness, density, etc.
Figure 2,
Figure 3 and
Figure 4 show the images obtained and the dimensions of each of the nozzles used in the tests.
The measurement errors associated with each component of the experimental installation are detailed in
Table 1.
2.2. Test Fuel
2.2.1. Fuel Properties
One hydrotreated vegetable oil (HVO) fuel, supplied by REPSOL (Madrid, Spain), and one gas-to-liquid (GtL) fuel, supplied by SASOL, were used in this study, the results of which were compared with a commonly used diesel fuel (supplied by REPSOL). The fuels’ characteristics are shown in
Table 2.
2.2.2. Fuel Density Estimation
When pressure and temperature conditions are modified, the fuel density also undergoes changes, which impacts the accuracy and stability of the injection rate. To address this aspect, Equation (1) is used for diesel fuel obtained according to the methodology presented by Payri et al. [
51].
In all equations in this section, P0 and T0 refer to the set pressure and temperature conditions.
Equations (2)–(4) for GtL fuel were obtained according to the methodology presented by Outcalt [
52]. Equation
is calculated from the equation proposed by Rackett [
53] and the experimental data in
Table 1, and where
B(
T) is the Tait equation parameter defined by Dymond et al. [
54], valid for any fuel with the characteristics of GtL and HVO fuel. Equations (1)–(4) were used by Soriano et al. [
31] for the development of their model.
For the calculation of the density variation equation with pressure and temperature of the HVO fuel, the same methodology is used as for the GtL fuel, using Equations (2), (4), and (5).
2.2.3. Fuel Dynamic Viscosity Estimation
To determine the equations that determine the behavior of viscosity with temperature, we use the equations presented by Soriano et al. [
29] for diesel fuel (Equation (6)) and GtL (Equation (7)).
In the case of HVO, an exponential expression (Equation (8)) obtained from the experimental data shown in
Table 1 is used.
In these three expressions, the kinematic viscosity is given in cPo, and the fuel temperature is given in °C. To know the variation of viscosity with pressure, Equation (9), proposed by Kousel, is used [
55].
In this work, kinematic viscosity () has been used instead of dynamic viscosity because the former indirectly includes the density value, which allows using one less parameter for the development of correlations.
2.3. Test Plan
In the present work, a solenoid injector with three different single-bore nozzles with diameters of 115 µm, 130 µm, and 150 µm, respectively, was used. To develop the tests, 3 energizing times (ET), 4 injection pressures (Pinj), and 2 fuel temperatures (Tf) were established. The back pressure in the rate of injection indicator was set at 5 MPa for all tests. Fuel temperature was measured at the inlet of the high-pressure pump. Tests were conducted with two fuel temperatures at the inlet of the high-pressure pump to study the effect of fuel temperature at the inlet.
The experimental test plan developed with each fuel is shown in
Table 3. Each of the tests was repeated 5 times, presenting as results in all tests, the average of these 5 experiments.
After performing the experimental tests, the 54 tests from which the model will be obtained are selected. These tests correspond to all those carried out at pressures of 90 and 110 MPa with the three fuels and the three nozzles.
The range of injection pressures tested covers the possible test conditions allowed by the fuel injection system used. However, the values of 50 and 70 MPa are those values where the measured fuel injection rate has more irregularities compared to other higher injection pressure values (90 and 110 MPa). Part of these experimental data were used for model implementation. Different data were used for model validation.
The data required for the development include all the variables that define each test (fuel properties and temperature, orifice diameter, injection pressure, energizing time) as input data of the model. Some variables obtained from the tests, such as the time from the beginning of energizing to the end of the injection rate (hereafter, d0) and the mass injected during the injection event, were used as output of the model.
2.4. Model Proposed Methodology
Theoretically the calculation of the mass injected during fuel injection is obtained by Equation (10). However, assuming this value involves assuming that the fuel injection rate curve is a square pulse, when the rate curve usually has more of a trapezoidal or triangular shape,
where
Cd is the discharge coefficient,
A0 is the geometric outlet section of the nozzle orifice,
Pinj is the injection pressure,
Pback is the back pressure, and
is the fuel density.
For this study, the ranges used for the development of the injection pressure and energizing time tests result in trapezoidal rate curves in all cases, because this model is developed only for rate simulation of this type.
Although it uses concepts developed by other authors [
42,
43,
47,
48], the model proposed is an evolution of the model presented by Soriano et al. [
31]. On this occasion, the model presented divides the injection rate curve into three parts as shown in
Figure 5, but now all these phases are modeled by curved lines using correlations. The three new correlations already include the lags between the energizing signal and the RoI signal, both at the beginning and at the end of the RoI.
In this work, 3 phases are also used to generate the fuel injection rate curve (as in Soriano et al. [
31]) but using a different methodology.
Figure 6 shows the variation of the injection pulse and the injected mass as a function of time for a fuel injection rate event. This figure also shows the three phases into which the fuel injection rate curve is to be divided:
The first phase describes the time lag between the beginning of the injector energizing until the beginning of the fuel injection ramp. This time is referred to the ‘needle lift’ (tnl), corresponding to a situation where the needle is moving to let the fuel through.
The second phase into which the rate curve is divided describes the steady state of the rate curve. In this phase, the needle is entirely open. The time this phase occurs has been termed ‘holding injection’ (thi) since it corresponds to the time in which the injection rate curve is stable.
The third phase describes the decrease in fuel flow during injector closure. This time is called ‘needle closure’ (tnc) since it corresponds to a situation in which the needle is moving to close the fuel outlet. It must be considered that the closing process does not consider the residual injections that occur at the end of the injection process.
In addition to the time that each phase lasts, another time parameter has been defined to describe the sum of time that these three phases last, which has been called ‘test time’ (d0) and which corresponds to the time from the beginning of the energizing to the end of the injection rate.
An analysis of the representativeness variables in the numerical models allows us to know these variables’ dependence on the injection rate. Therefore, the first step in model development is determining the relevance of the variables used for model prediction. The variables involved in the injection process in this work are energizing time (ET) measured in ms, injection pressure (P
inj) measured in MPa, fuel properties (specifically, the kinematic viscosity (
, measured in mm
2/s), hole diameter (z) measured in µm, test time (d
0) measured in ms, and fuel temperature (T
f) measured in °C. Contrary to Soriano et al. [
31], in this case, the back pressure is constant (5 MPa) as well as the number of holes (1 single hole); therefore, these are two variables that will not participate in the modeling of the rate curve. The representative variables were verified separately in each phase to improve the model’s ability to reproduce the fuel injection rate curve.
Figure 6 shows the relevance of the participating variables in this study, indicating which should be included in the prediction models for each phase.
In the case of the ‘needle lift’ phase,
Figure 6a shows that d
0 is the most relevant factor due to the stability or low variability of the data. This fact indirectly implies that the most significant variability or dependence is due to P
inj and the kinematic viscosity of the fuel.
Figure 6d shows that the most representative variables are P
inj and d
0, followed by hole diameter, fuel viscosity, and energizing time. In all cases, this is because the slope of the rate curve is similar in almost all tests.
In the case of the ‘holding injection’ phase (
Figure 6b), the variables of major dependence to keep the injection stable during this phase are the hole diameter and the injection pressure.
Figure 6e indicates that the most representative variable is the energizing time since it generates a more noticeable change in the time the injector is open and, as such, in the duration of the injection time.
Finally, for the ‘needle closure’ phase (
Figure 6c), the most relevant variable is again d
0, along with energizing time.
Figure 6f shows that the specific control variable in the equation is energizing time, followed directly by P
inj and d
0.
In summary, in all phases, the injection rate depends on test time, injection pressure, injector energizing time, nozzle bore diameter, and kinematic viscosity. However, in none of the phases is fuel temperature identified as an essential dependent variable.
Once the variables that provide the most remarkable dependence on the behavior of the rate in the injection process have been found, a numerical statistical model is generated that simulates the injection rate concerning the dependent variables. This model establishes a mathematical dependence correlation according to the variables involved in each phase.
The linear predictive model obtained for the ‘needle lift’ (phase 1) is shown in Equation (11), and the values of the coefficients are shown in
Table 4.
Equation (12) provides the mathematical correlation describing the holding injection phase, and the values of the coefficients are shown in
Table 5.
The analysis based on independent values for the needle closure phase provided the mathematical expresion shown in Equation (13), and the values of the coefficients are shown in
Table 6.
2.5. Models Accuracy
An average absolute error of 1.84%, related to the values presented in
Table 1, was obtained in the evaluation process. The coefficient of determination (R
2) values were 90.02%, 97.24%, and 90.45% for the phases ‘needle lift’, ‘holding injection’, and ‘needle closure’, respectively. The high R
2 values indicate an excellent agreement between the experimental data and those obtained from the presented prediction models. A confidence value of 92% was applied in each case to validate the accuracy of the proposed predictive models.