## 4 Effect of the Injection Rate Shape
### Evaporation Process  
We next studied the evaporation process influenced by the fuel injection rate by considering the distribution of droplet sizes. This section presents the simulation results of the evaporation ratio under different injection rate shape conditions, as shown in Figure 16. The simulation results show that different injection rate shapes result in different evaporation ratios. The results of the evaporation ratios are shown in the top graph, and the injection mass flow rates are shown in the bottom graph, where the black, red, blue, and yellow lines represent the simulation results of the RECT, QIGD, GIGD, and GIGD injection rates, respectively. Figure 16 shows that the RECT and the QIGD injection rate shapes undergo more rapid evaporation than the other injection rate shapes. As demonstrated by the RECT and the QIGD injection rate shapes, the evaporation ratio increases to nearly one before approximately 0.01 ms. For the GIGD and GIGD injection rates, the evaporation ratio increases to nearly one at approximately 0.1 ms. These results are due to both injection rate shapes having a quickly increasing initial injection rate, thereby resulting in high rail pressure, which can improve the evaporation rate because higher rail pressure results in a high shear of the fuel particles, which can change the fuel state from liquid to gas very quickly with higher mass flow rates, as well as accelerate the fuel evaporation process. To better understand the spray breakup characteristics, the SMD is an important parameter that should be considered to reflect the spray performance. The size of the SMD is related to droplet breakup, in which a smaller SMD result in better droplet breakup. Due to the lack of experimental data to calibrate the simulated SMD results, only the relationship between the SMD and injection rate have been considered. Figure 17 shows the relationships of the SMD values for different injection rate shapes, where the black, red, blue, and yellow lines represent the simulation results of the RECT, QIGD, GIGD, and GIGD injection rates, respectively. The results show that the SMD decreases rapidly when the injection rate increases sharply, at the same time, the SMD gradually decreases as the injection rate gradually increases. This occurs because the droplet tends to breakup under conditions with higher injection rates. A high initial injection rate can clearly reduce the SMD. As can be considered from the high initial injection rate conditions (RECT, QIGD), the SMD decreases to nearly zero at approximately 0.02 ms; with gradual increases in the injection rate conditions (GIGD, G  
Figure 16: Comparing the evaporation ratios from the simulation results. **(a)** Evaporation ratio and (b) injection mass flow rate. nearly zero at approximately 0.1 ms. These values are worth noting for the RECT and QIGD injection rates. Although the initial injection rates are not the same, the injection rate is sufficiently higher to result in a rapid decrease in the SMD. These behaviors support the spray breakup phenomenon. The evaporation rate is higher for the droplet under a higher initial injection rate, as shown in Figure 16. From Figure 17, it can be concluded that the initial injection rate is the main factor affecting the size of the droplets, as an increase in the injection rate results in a decrease in the SMD. Therefore, under higher initial injection rates, the droplets will become smaller and lead to faster evaporation. The fuel evaporation phenomena can be understood by considering the temperature distribution phenomena. Figures 18 and 19 show the simulation results comparing the temperature distribution with different injection rate shapes during injection and after the EOI. The spray images were obtained from the cut-plane in the direction of the spray. The gradient color region represents the temperature data and shows the temperature value as a gradient color bar in the bottom right corner.