## 4 Effect of the Injection Rate Shape  
The injection rate shape is an important factor that affects the spray formation. After leaving the injector, the fuel spray atomizes into droplets, vaporizes, and mixes with the air. In this work, models with different injection rate shapes are analyzed for their spray and fuel mixing behavior using the basic conditions from the previous work on ECN case No. 2, as shown in Table 2. The four injection rate shapes shown in Figure 12 were selected for this study. The rectangular injection rate (RECT) shape is the constant injection rate and is a commonly used injection rate shape. The other three shapes consist of the Quick Increase Gradual Decrease injection rate (QIGD), the Gradual Increase Gradual Decrease injection rate (GIQD), and the Gradual Increase Quick Decrease injection rate (GIQD). These shapes were designed to examine the effects of an increase and a decrease of the injection rate over a short injection duration to study the influence of spray and combustion behavioral control factors based on previously reviewed research. The above factors include the peak injection rate, the initial injection rate, and the injection velocity of each injection period, which are useful for understanding the spray and fuel mixing behavior. The four injection rate shapes must have the same injection duration and fuel quantity to be used as a benchmark. The three shapes (QIDG, GIGD, and GIQD) had peak injection rates higher than the experimental conditions in case No. 2, which means that the rail pressure would be higher than 150 MPa. As shown in the simulation model, these injection rate shapes can create a peak rail pressure of approximately 600 MPa. A maximum rail pressure of 600 MPa is too high for current engine technology. In this study, the maximum rail pressure generated by the injection rate was determined from the study conditions according to the constant value of the injection duration with the same amount of fuel as used in the experimental data to ensure that the model calibrated from the experiment data would be accurate. The injection pressure can be increased by changing the injection rate of the fuel pump and adjusting the injector area. If the injection pressure is too high, the ignition delay period will be shorter. This may cause a homogeneous decrease in mixing and reduce the combustion efficiency. In practice, changing the injection rate and injector area is necessary to help reduce these effects. Figure 11: Comparison of the spray distribution angle for different rail pressures. (**a**) Spray angle and (**b**) spray cone angle. As a result that the newly created injection rate shape for this study is determined by a constant injection duration and fuel mass (including under ambient conditions, which are similar to those of the experimental data), under some conditions, this novel injection rate provides a peak injection rate that is higher than the injection rate of the experimental data used to calibrate the model. This means that higher injection rates will also result in higher rail pressure. Although this study explored the maximum rail pressure conditions up to 600 MPa, the results calibrated with the experimental data for rail pressure of 50, 100, and 150 MPa showed that the model can provide reliable simulation results, even under conditions where the pressure is different. In particular, the simulation results show that the study conditions at a maximum rail pressure of 150 MPa can provide better simulation results than under conditions of a lower maximum rail pressure. This means that the newly created model is effective for high rail pressure conditions. In addition, in this paper, all case studies (including the cases of maximum rail pressure conditions of 600 MPa) use ambient conditions. The injection timing and the fuel mass are the same as those of the experimental conditions at a rail pressure of 150 MPa because the simulation results for the calibration under rail pressure conditions of 150 MPa give good results for both quantity and volume.