Numerical research of biodiesel spray coupled cavitation by different environmental simulations under high injection pressure

Abstract

The effects of injection pressure and ambient pressure on biodiesel spray characteristics were studied by coupling the in-nozzle flow, and a constant-volume combustion bomb experimental platform was built to verify the accuracy of the coupled numerical model. Spray tip penetration (STP), spray velocity, and droplet size/number distribution of biodiesel were calculated at different injection pressures (200, −250, −300, and −350 MPa) and environmental pressures (0.1, −0.5, −1.0, and −2.0). The sensitivity to injection pressures and environmental pressures was analyzed with particle concentration as the characteristic parameter. With the increase of injection pressure, the STP of biodiesel increased by 5%, which resulted in a faster decrease in the increasing rate of STP (V). Under the same conditions, with the increase of injection pressure, the axial and radial velocities of fuel droplets gradually rose, and the high-speed area was enlarged, while the number of droplets and Sauter mean diameter (SMD) space generally decreased. Compared with the injection pressure, the environmental pressure significantly inhibited the STP and velocity of biodiesel, and the increasing rate of STP (V) decreased gradually and the number of droplet particles and SMD rose with the increase of the environmental pressure.

INTRODUCTION

In direct injection diesel engines, the fuel spray and atomization characteristics play important roles in air–fuel mixture formation, which is decisive for combustion efficiency and emissions.^1–4 The fuel injection pressure to meet the future national emission requirements V/VI is ultrahigh (& GT 200 MPa). It also leads to an intensification of fuel cavitation flow in the jet hole, an intense fuel–air interaction after the jet, and extremely complicated atomization characteristics.^5–9 Therefore, coupling the in-nozzle flow to study the spray characteristics of biodiesel under ultrahigh injection is of great significance.

However, the actual nozzle aperture is very small, which makes it very difficult to capture the internal transient flow and external spray characteristics at the same time. Wang et al. studied the spray characteristics of the curvature inlet radius on the fuel using a two-step method, and found the spray penetration length increased, and the spray cone angle and atomization quality decreased with the rise of the inlet radius.¹⁰ Limin Geng studied the spray characteristics with different aspect ratios by using the moving grid, and found the in-nozzle cavitation flow and Sauter mean diameter (SMD) decreased, and the spray penetration moment increased when the nozzle l/d increased from 4 to 8.¹¹ Yu used large eddy simulation to study the microcharacteristics of fuel under different injection pressures, and found entrainment vortices appeared at the edge of the larger velocity gradient between jet and air, which helped to improve the spray quality.¹²

In fact, when the internal combustion engine is working, not only nozzle factors (e.g., inlet fillet radius, nozzle hole length diameter ratio, and nozzle cone angle) impact the biodiesel spray, but also the environmental parameters inside the combustion chamber somehow affect the spray characteristics. Hawi studied the effect of environmental density on diesel and biodiesel spray in a fast compressor.¹³ When the injection pressure increased from 15 to 25 kg/m³, the spray cone angle of diesel and biodiesel rose by 9.6% and 13.8%, respectively, and the spray penetration moment decreased by 9.2% and 13.1%, respectively.¹³ Li et al. used Mie scattering technology to study the fuel spray characteristics under nonevaporating conditions.² However, with the rise of ambient temperature, the spray penetration moment of the fuel increased, which can produce high-quality fuel mixture.

In summary, experts and scholars pay more attention to the nozzle factors in the simulation research, but ignore the internal flow of the nozzle hole when studying the influence of environmental parameters. Therefore, we coupled the internal nozzle flow, studied the spray characteristics under ultrahigh injection pressure and ambient pressure, and built a constant-volume bomb test system to verify the reliability of the numerical simulation model. To understand the spray characteristics of biodiesel in detail, we introduced the formula into the spray tip penetration (STP) research, and counted the particle velocity, number, and diameter distribution within the 5 mm diameter range along the axis. Finally, sensitivity analysis of ambient pressure and injection pressure was carried out with the reference of particle concentration.

SIMULATION MODEL ESTABLISHMENT Coupled simulation method

In Fluent research, the Eulerian method is used to simulate the in-nozzle transient flow, while the Lagrangian method is used for spray simulation (Figure 1). Thus, it is difficult to achieve real-time coupling simulation due to the different requirements of grid accuracy and computational steps for the two approaches.¹⁴ The coupling simulation is divided into two steps. First, the flow characteristic parameters at the nozzle outlet, such as mass flow, turbulent kinetic energy, velocity, gas–liquid volume fraction, and pressure, are calculated and stored in the Noz file. Second, when the spray simulation is carried out, the Noz file is imported as the initial boundary condition, and the simulation results are corrected at each time step. The specific spray morphology calculated is shown in Figure 2.

Figure 1. Schematic diagram of the coupling model

Figure 2. Spray development simulated with spray model coupled with nozzle cavitating flow

Mathematical–physical model

A typical single-hole nozzle model was used for simulation, and the specific settings were shown in Table 1. The nozzle diameter D was 0.2 mm, the length L of the transition fillet was 1 mm, and the pressure boundary conditions were set as inlet and outlet boundaries with a back pressure of 0.1 MPa. To speed up the calculation, 1/2 of the nozzle model was used to generate calculation grids without affecting the calculation accuracy (Figure 3A). To test the grid dependence and find the most suitable grid model for the numerical calculation, we used five dimensions of grids (0.5, 0.25, 0.1, 0.05, and 0.025 mm) in the simulation, corresponding to the grid numbers 9317, 17,749, 34,524, 112,389, and 165,695, respectively. The mean mass flows of the five different grids were determined from the calculation under the condition of the in-nozzle flow (Figure 4). The mean mass flows at grid numbers of 112,389 and 165,695 were very close. Hence, considering the calculation accuracy, cost, and other factors, we adapted the grid number of 112,389 to validate the spray model and analyze the related parameters.

Table 1 Parameters of independence analysis spray model

Nozzle parameters	Value
Injection pressure (MPa)	30
Ambient pressure (MPa)	0.1
Fuel	Biodiesel
Ambient temperature (K)	300
Nozzle number	Single
Orifice diameter (mm)	0.2

Figure 3. Computational grid of nozzle flow and atomization

Figure 4. Variation of average mass flow at orifice outlet with grid number

Since the high-pressure spray experiment was carried out in a constant volume, the spray calculation domain was set as a cylinder in a radius of 100 mm and a height of 250 mm, and the structured grids of the cylinder were generated on Gambit (Figure 1B). The nozzle outlet was located in the center of the top surface of the cylinder. With the spray development characteristics considered, the calculation grids were densified near the nozzle outlet and the cylinder axis. Similarly, STP was considered as the characteristic parameter in the fuel spray simulation (Figure 5). The number node of the spray domain at 323,000 can meet the requirement of mesh convergence.

Figure 5. Variation of spray tip penetration moment of fuel with grid number

Mathematical model of nozzle flow

The mixing model in Fluent was adopted, which assumed that the incompressible liquid fuel A was the main phase and the gaseous fuel steam was the secondary phase. The cavitation flow of the nozzle was simulated by solving the continuity equation, momentum equation, second volume fraction equation, and cavitation model.¹⁵

• (1)

  The continuity equation is [Image Omitted. See PDF] where $${v}_{{\rm{m}}}$$ is the average mass velocity and $${\rho }_{{\rm{m}}}$$ is the mixing density.

• (2)

  The momentum equation is [Image Omitted. See PDF]where $${u}_{{\rm{m}}}$$ is the mixed viscosity, $${v}_{{\rm{d}}k}$$ is the slip velocity of the second phase, and $${v}_{{\rm{d}}k}={v}_{k}-{v}_{{\rm{m}}}$$.

• (3)

  The volume fraction equation is

[Image Omitted. See PDF]

The cavitation model is the Singhal complete cavitation model, which takes into account the effects of phase transition on cavitation flow, cavitation dynamics, and the presence of noncondensable gases in the fluid. The mixing density function and vapor transport equation are introduced when using this model: [Image Omitted. See PDF]where f is the mass fraction, and the subscripts v, g, and l represent the vapor, gas, and liquid states, respectively.

The steam transport equation is [Image Omitted. See PDF]where v_v is the velocity vector representing the second phase of the vapor bubble and is the effective diffusion coefficient, and R_e and R_c are the steam generation rate and the condensation rate, respectively.

Spray calculation model

The fuel spraying process is very complicated and involves basic physical processes, such as primary crushing, secondary crushing, heat transfer evaporation, turbulent diffusion, and collision polymerization. Therefore, when the fuel is atomized, fuel droplets, fuel vapor, and a surrounding multiphase spray domain composed of gas will appear.¹⁶ In the numerical calculation, thus, basic fluid governing equations were established first, including continuity equation, momentum conservation equation, energy conservation equation, component transport equation, and realizable k − E turbulence equation. Then the DPM was used to establish a spray simulation model (Table 2).

Table 2 Spray calculation model

Model type	Model name
Turbulence model	Realizable k − E
Jet breakup model	KH-RT broken model
Collision model	O-Rourke model
Turbulence diffusion model	DRW model
Drag model	Dynamic drag model

Abbreviations: DRW, discrete random walk; KH-RT, Kelvin–Helmholtz and Rayleigh–Taylor.

The Kelvin–Helmholtz and Rayleigh–Taylor (KH-RT) model takes into account the effects of turbulence and cavitation on droplet breakage in spray flow, and has high accuracy in spray simulation calculation under high pressure.¹⁷ Therefore, the KH-RT model is adopted.

The KH model states that the breakup of oil droplets is controlled by the maximum surface wave growth rate $${{\rm{\Lambda }}}_{\text{KH}}$$ and the corresponding wavelength $${{\rm{\Omega }}}_{\text{KH}}$$. When the amplitude of the droplet surface and small stable wave reaches the fastest growing wavelength $${{\rm{\Lambda }}}_{\text{KH}}$$, split occurs. During the breakup of oil droplets, the radius R₀ of the split subdroplets is considered to be proportional to the wavelength $${{\rm{\Lambda }}}_{\text{KH}}$$ of the rapidly growing superficial wave: [Image Omitted. See PDF]where B₀ is the model constant.

The oil droplet radius r varies continuously as follows: [Image Omitted. See PDF]where crushing time $${\tau }_{\mathrm{KH}}=3.72{B}_{{1}^{r}}/{\Lambda }_{\mathrm{KH}}{\Omega }_{\mathrm{KH}}$$, C₂ is an adjustable parameter of the model.

Moreover, $${{\rm{\Omega }}}_{\text{KH}}$$ and $${{\rm{\Lambda }}}_{\text{KH}}$$ are calculated as follows: [Image Omitted. See PDF]

In addition, the rapid deceleration caused by drag in droplet motion generates unstable RT waves, which also play a role in droplet breakup. The fragmentation length L of the liquid core region in the RT model is [Image Omitted. See PDF]

VALIDATION OF COUPLING SPRAY MODEL

To ensure the correctness of the spray model coupled with nozzle flow, we used the direct imaging technology to verify it on the constant-volume projectile experimental platform. The specific experimental layout is shown in Figure 6. This device mainly involves a fuel injection system, a constant-volume bomb, a data acquisition and control system, and a high-speed camera. The fuel injection system consists of a CB08 high-pressure fuel pump, CH028 fuel injectors (both from Bosch), a variable frequency motor, and a high-pressure common rail. The fuel pump is driven by a 3-kW YX3 pusi three-phase asynchronous motor, and the speed is adjusted within 0–1000 revolutions through a frequency converter. The high-pressure common rail can reach the maximum rail pressure of 180 MPa. The signal control system is a signal transmission program based on LabView. The electronic control unit outputs fuel control signals to drive the fuel pump, motor, and fuel injectors to work together. The AL-130U210C high-speed camera works at a resolution of 1280 × 720 and 2000 shooting frames per second.

Figure 6. Overall layout of the constant-volume incendiary bomb test bench. ECU, electronic control unit; LED, light-emitting diode; PC, personal computer.

The material used for verification is No. 0 diesel, and the materials for simulation calculation are biodiesel. The specific physical and chemical properties are shown in Table 3.

Table 3 Some physical and chemical properties of experimental materials

Parameters	B100	D100
Density (g/L)	880	730
Dynamic viscosity (MPa·s)	518	323
Surface tension (mN/m)	38.6	26.3

Figure 7 compares the spray shape of biodiesel taken by a direct imaging method and calculated on Fluent. Clearly, the spray shape calculated on Fluent is similar to the spray shape shot. In 0–0.2 ms during the test and simulation, the cone angle did not change significantly, and the atomization area was small. At 0.2–0.8 ms, the horizontal expansion ability of the simulated spray pattern was enhanced, the radial STP was shortened, and the spray end was in an inverted cone shape, which are similar to the spray development in the experiment.

Figure 7. Comparison of spray morphology experiment (A) and simulation (B)

Figure 8 compares the STP calculated by Fluent and the experimental results processed by MATLAB. Clearly, the STP curves from the experiment and simulation basically overlap, despite certain errors within 5%. The main reasons are that first the consideration of the first fuel break is imperfect, and the software calculation model has certain errors. In general, the calculated values well agree with the experimental values. Therefore, the model as-established better reflects the spray development process, which further verifies the reliability of the spray calculation model.

Figure 8. Comparison of spray tip penetration experiment and simulation

In view of the shortage of the experimental equipment, we conducted SMD and velocity verification between the calculation results of the spray model and the experimental research results in Geng et al.¹⁸ By setting the same boundary conditions (injection pressure and ambient pressure), we calculated the particle velocity of spray and the SMD of droplets in the ±5 cm circular section at 40 cm in the axial direction of the nozzle. As shown in Figure 9, the particle velocity in the experiment in Geng et al.¹⁸ is only 7 m/s away from the speed in our study. The measured SMD is slightly larger than the simulated SMD, with an error of 10% (Figure 9). The simulation results of the two parameters are basically consistent with the experimental results, which verifies the effectiveness of the calculation model.

Figure 9. Comparison of Sauter mean diameter experiment and simulation

RESULT ANALYSIS AND COMPARISON

We first analyzed the in-nozzle flow, and then further studied the spray characteristics of biodiesel under different injection pressures and ambient pressures, including STP, spray velocity, spray particle number, and diameter distribution. The specific working conditions of simulation calculation are shown in Table 4. The simulation calculation of each working condition was repeated three times to ensure that the error of the simulation result was within the allowable range.

Table 4 Simulation calculation condition

Nozzle parameters	Value
Fuel	Biodiesel
Injection time (ms)	1
Injection pressure (MPa)	250, 300, 350
Ambient pressure (MPa)	0.1, 0.5, 1

Three-dimensional simulations of two-phase flow in orifice

To more accurately analyze the spray characteristics under high-pressure conditions, we simulated the flow at the nozzle holes under high spray pressures (200, 250, 300, and 350 MPa) based on 350 MPa. Figure 10 shows the variation of cavitation flow area of the nozzle hole. Clearly, high injection pressure induces cavitation, which generally occurs near the nozzle hole inlet and extends along the nozzle wall toward the outlet. Moreover, cavitation extends to the nozzle outlet in a larger area at higher injection pressure. Figure 11 shows the changing rules of mass flow rate and average velocity at the nozzle outlet. The injection pressure has a certain influence on the mass flow rate and average velocity at the nozzle outlet, which basically show a linear uptrend. Therefore, the injection pressure greatly influences the in-nozzle flow, and it is necessary to carry out a cavitation-coupling analysis.

Figure 10. Cavitation distribution under different jet pressures: (A) 200 MPa, (B) 250 MPa, (C) 300 MPa, and (D) 350 MPa

Figure 11. Variation of average velocity and average mass flow at nozzle outlet with injection pressure

Spray tip penetration

Figure 12 shows the change curve of biodiesel STP with time. Clearly, the injection pressure has an obvious promoting effect on STP. The greater injection pressure leads to the more STP increment. The increase of STP is significantly inhibited by the environmental pressure. The greater environmental pressure results in more obvious inhibition. At t = 0.2 ms, STPs under three pressures are 71.9, 77.8, and 83.6 mm, respectively; at the ambient pressure of 0.1 MPa, the STP is 180.1 mm at t = 1.0 ms (Figure 12A). When the ambient pressure increases to 1 MPa, the STP decreases by 48% to 93 mm. At the same time, to further study the impacts of injection pressure and ambient pressure on STP, we introduced $$v=\unicode{x02206}s/\unicode{x02206}t$$ to reflect the STP growth rate in different time periods (Figure 13). With the increase of injection pressure, V decreases faster, and an inflection point appears at 0.3 ms (Figure 13A). In addition, at the greater environmental pressure, V gradually decreases and fluctuates to a certain extent after 0.8 ms (Figure 13B). This phenomenon can be explained from the perspective of droplet momentum.¹ Under higher fuel injection pressure, the initial momentum and initial velocity of droplets both increase, STP gradually increases, and V remains large.¹⁹ However, due to the entrainment of air, biodiesel droplets constantly exchange energy, so V gradually decreases. In addition, the increase of ambient pressure raises the density of ambient gas and the resistance of spray axial movement, and leads to the rapid decline of V.⁴

Figure 12. Influence of influencing factors on spray characteristics spray penetration moment. (A) Injection pressure and (B) ambient pressure.

Figure 13. Spray tip penetration change rate under different conditions. (A) Injection pressure and (B) ambient pressure. 

Spray velocity variation

When the biodiesel spray moves forward axially, it will be entrained by air, generating vortices on both sides. The value of spray speed can better explain the entrainment intensity of fuel droplets by air.²⁰ Figure 10 shows the distributions of x − z surface velocity field under four injection pressures, and further intercepts the velocity distributions of particles within the ranges of x = +5, y = +5, and Z of 20, 40, 60, and 80 mm. The high-speed area is concentrated in the center near the nozzle end, and the speed gradually decreases from inside to outside with the extension of axial distance. With the increase of injection pressure, the maximum velocity in the core area of spray increases from 328 to 399 m/s. This is because with the increase of injection pressure, the internal and external pressure difference is enlarged, and the initial kinetic energy and velocity of the oil jet leaving the nozzle also increase, so the maximum velocity is enlarged.²¹ In addition, when the environmental pressure increases, the spray particle diffusion area and velocity at the four axial positions gradually decrease (Figure 12). This is mainly because with the increase of environmental back pressure, the axial and radial resistance of fuel droplets rise, the energy exchange with air, and the fuel atomization are both accelerated, so the speed and diffusion area gradually decrease.

Droplets size/number distribution

Figure 14A–D shows the droplet size/quantity distributions at four positions (20, 40, 60, and 80 mm away from the nozzle) under four injection pressures. Clearly, the particle quantity distribution is the most when it is 20 mm away from the nozzle under four injection pressures, and decreases step-by-step along the radial direction. At the same time, the number of particles in the range of 8–12 μm is the largest (Figure 14A). The number of particles in the range of 6–10 μm accounts for the majority (Figure 15B,C). In 12 days, the diameter of 2–3 μm is mainly distributed in 2–3 μm, indicating increasing the injection pressure can reduce the particle diameter and improve the spray quality.³ This is because at the initial stage of spray, the particle velocity and momentum of biodiesel are large and the interaction time with air is short, so the particle diameter and quantity are large. With the axial development of spray, the velocity and momentum are greatly reduced and enhanced by the air entrainment, respectively. Therefore, the fuel droplet accelerates the liquefaction, which is reflected in the different reductions of particle diameter and quantity in space.

Figure 14. Velocity distributions at different injection pressures. (A) 0.1 MPa, (B) 0.5 MPa, (C) 1.0 MPa, and (D) 1.5 MPa.

Figure 15. Particle distribution under different injection pressures. (A) 200 MPa, (B) 250 MPa, (C) 350 MPa, and (D) 300 MPa.

Figures 15 and 16 show the change curve of overall SMD at four measurement positions under different pressures. Clearly, the droplet particles of biodiesel gradually decrease along the axial direction in space, and the SMD of droplet particles decreases with the increase of injection pressure, which is the same as a reported trend, indicating that increasing injection pressure can improve the spray quality.

Figure 16. SMD changes at different positions. SMD, Sauter mean diameter.

Sensitivity analysis of maximum particle concentration

To better characterize the impacts of the two factors (fuel injection pressure and ambient pressure) on spray quality, we conducted a sensitivity analysis on the maximum particle concentration of the fuel under different conditions. This is because the maximum concentration of spray particles can better reflect the mist injection, the internal gas–liquid distribution status, and the amount of fuel evaporation, and thus provides a reference and basis to improve the oil–gas mixture and combustion performances of diesel engines under specific working conditions. At 1 ms after the start of spraying, the spray basically penetrates a distance over 50 mm and has reached the area around the combustion chamber. Therefore, 1 ms is selected as the time point for the sensitivity analysis of spray characteristics. Sensitivity is mathematically defined as the sensitivity of function F to its variable $$\varnothing =\frac{\partial F/\partial x}{F/x}$$, which reflects the relative change rate of the function value to the independent variable.

Figure 17 shows the influence of injection pressure and ambient pressure on the maximum particle concentration at the injection time of 1 ms. The particle concentration roughly increases linearly with the increase of ambient pressure, and linearly decreases with the rise of fuel injection pressure (Figure 15A–C).

Figure 17. Influence of (A) injection pressure and (B) ambient pressure on particle concentration at 1 ms

Figure 18 compares the sensitivity of maximum particle concentration to fuel injection pressure and to ambient pressure. As the injection pressure rises, the sensitivity of particle concentration gradually increases. When the injection pressure increases in the range of below 300 MPa, the sensitivity to the maximum particle concentration rapidly rises. At above 300 MPa, this uptrend becomes more obvious. Like the influence of injection pressure, the sensitivity of the maximum particle concentration also rises rapidly as the environmental pressure rises. However, when the injection pressure is between 300 and 350 MPa, the sensitivity to the ambient pressure of 0.1 MPa is less than that to the injection pressure of 250 MPa, and the sensitivity to the back pressure is 0.5–1 MPa. These results indicate that ambient pressure plays a more significant role than injection pressure in promoting fuel evaporation and reducing fuel particle concentration.

Figure 18. Sensitivity comparison of maximum particle concentration for (A) injection pressure and (B) ambient pressure

Figure 19 shows the average sensitivity of the two factors (e.g., fuel injection pressure and ambient pressure) to the maximum particle concentration and the variation ranges in all simulated conditions. The average sensitivity maximizes to 0.36 with the environmental pressure, and the range of variation is also the largest, reaching 0.24. The second is the injection pressure, with an average sensitivity of 0.35 and a slightly smaller variation range of 0.23 than the environmental pressure. Under different working conditions, the sensitivity of the maximum particle concentration to ambient pressure is relatively high. In comparison, the average effect of injection pressure on the maximum particle concentration is slightly smaller, and the impact of ambient temperature is much lower than that of injection pressure.

Figure 19. Comparison of the average value and variation range of the sensitivity of different factors to particle concentration

RESULT ANALYSIS

The Euler–Lagrange method was used to study the biodiesel spray characteristics at ultrahigh injection pressure. At the same time, a test platform of constant-volume projectile spray was built to verify the model.

• 1.

  A flow model in the jet hole was established. The use of 110,000 grids clearly shows that the injection pressure greatly influences the cavitation inside the jet hole, the average velocity at the outlet, and the mass flow rate. Meanwhile, a test platform for spray characteristics of constant-volume projectile was built, and spray images under different spray conditions were obtained by using the direct imaging technology. The experimental results agree well with the numerical data (spray penetration, SMD, and macrospray image), which verifies the accuracy of the spray model under ultrahigh spray pressure.

• 2.

  With the increase of injection pressure, the STP of biodiesel increased slightly, the increasing rate showed a turning point after 0.3 ms, and remained stable after 0.8 ms. The increasing environmental pressure significantly inhibited the STP of biodiesel, and the increasing rate V continued to decrease.

• 3.

  The spray pressure can increase the axial and radial spray diffusion areas, and the vortex generated is conducive to fuel fragmentation, while the environmental pressure obviously inhibits spray velocity. In addition, the spatial distribution of spray particle diameter gradually decreases under ultrahigh injection pressure.

• 4.

  Under different working conditions, the ambient pressure has the greatest influence on the sensitivity of the maximum particle concentration, followed by the injection pressure, indicating that increasing the ambient pressure can rapidly improve the spray quality compared with the injection pressure.

ACKNOWLEDGMENTS

This work was carried out with financial support from the National Natural Science Foundation of China (Grant No. 21905031) and the Science Foundation of Changzhou University (Grant No. ZMF18020299).