## 4 Effect of the Injection Rate Shape
### Mixture Properties  
For a deeper understanding of the diesel spray mixing process, the mixture properties are studied in this section by comparing their equivalence ratios and turbulence kinetic energy (TKE) values at different injection rates. The mixture properties are analyzed by considering the region of the equivalence ratio predicted from the 3D model, which plays an important role in the analysis of the diesel spray mixing process. Therefore, the equivalence ratio is calculated from the basic data of the mass fraction. In this study, the conditions of the case study are the conditions under which the ambient component does not contain oxygen (r\({}_{\rm O_{2}}=0\)). We applied the following equation of chemical combustion:  
\[{\rm Fuel}+{\rm O_{2}}\rightarrow{\rm CO_{2}}+{\rm H_{2}O}. \tag{9}\]  
Figure 19: Comparison of temperature distribution after the EOI for different injection rate shapes. Here, we assume that by replacing the oxygen ratio with the total gas ratio (including the oxygen ratio), the general air contains 21% oxygen and 79% other gases. Therefore, the mass fraction is given as Fuel/(Fuel + O\({}_{2}\) + other gases), where the equivalence ratio is equal to 1. From the simulation results, the different injection rate shapes show different spray behaviors and equivalence ratio histories, although the injection duration and injection mass quantity are the same. Figures 20 and 21 provide a comparison of the spray behavior and equivalence ratio histories of the simulation results during injection (Figure 20) and after the EOI (Figure 21). The spray images were obtained from the cut-plane in the direction of spray. The gradient color region represents the equivalence ratio data and shows the equivalence ratio values as a gradient color bar in the bottom right corner, while the black particles are the liquid fuel data from the simulation. Figure 20 shows the spray behavior and equivalence ratio histories for the four injection rates. It can be observed that although the injection mass and time for all simulated cases are the same, the spray behavior and equivalence ratio histories for each injection rate shape are very different. There is a high fuel distribution near the nozzle exit of the injection rate shapes with a high peak rail pressure (QIGD, GIGD, and GIQD), which is different from the case with the lower peak rail pressure (RECT). In general, fuel distribution is influenced by changes in the fuel injection rates. These phenomena are clearly reflected in Figure 21. Figure 21 clearly shows that the GIQD injection rate with the highest injection rate at the EOI provides the leanest equivalence ratio near the nozzle exit. This is due to the influence of the injection rate, which affects the fuel breakup and the surrounding air crossflow. The momentum arising from increasing the injection rates yields a complementary  
Figure 21: Comparison of the spray behavior and equivalence ratio histories after the EOI for different injection rate shapes. momentum. The initial injection rate is related to the droplet breakup. For example, for high initial injection rate conditions (RECT and QIGD), the black particles represent the liquid fuel intensely near the nozzle exit and descend quickly when the injection time has passed. This is an example of quick SMD reductions, as shown in Figure 17.  
For a better understanding of the spray behavior in diesel spray mixing, TKE is an important influencing factor that can explain the phenomenon of the equivalence ratio. TKE can reflect the intensity of the turbulent movement in the cylinder. Figure 22 shows a comparison of TKE with different injection rates. The TKE is displayed as the average value of the TKE in the control volume model, where the black, red, blue, and yellow lines represent the simulation results of the RECT, QIGD, GIGD, and GIQD injection rates, respectively. The top graph shows the TKE and the bottom graph shows the injection mass flow rate. The injection rate has a significant relationship with the TKE.