**1. Introduction**

Diesel engines find widespread applications in many industries, including transportation, agriculture, and power generation, among others. Soot and NOx emissions, which are products of fuel combustion in these engines, pose a threat to the environment and the health of living organisms. Over the years, strict global regulations have been set to reduce the negative impact of such emissions. These stringent legislations require manufacturers to design cleaner and more e fficient engines [1,2]. Many technologies, such as diesel particulate filters (DPFs) [1] and selective catalytic reduction (SCR) [3], have been developed to reduce emissions from diesel engines. In addition, controlling the combustion process (e.g., by using low-temperature combustion (LTC) [2], homogeneous charge compression ignition (HCCI) [4], reactivity-controlled compression ignition (RCCI) [5], and premixed charge compression ignition (PCCI)) have also attracted significant research interest [2]. Research on the fuel mixing process for preparing diesel combustion components in cylinders through e ffective injection rate adjustment is an important consideration. In this study, we only consider injection rate adjustment in the fuel injection system without changing the air intake and other contexts of the

system; thus, this study does not modify the existing injection system too greatly. This is due to the significant influence of the air–fuel mixing process on combustion e fficiency and exhaust emissions in diesel engines.

The latest developments in fuel injection strategy e fficiency tend to focus on the injection rate to increase fuel mixing e fficiency and reduce emissions. Notably, the combustion time will be shorter for cases with high initial injection rates. Generally, a high initial injection rate will result in better atomization and air entrainment. The duration of combustion decreases as the mixture of air and fuel improves, meaning that combustion occurs faster. Conversely, under a low initial injection rate, the initial atomization of the fuel and premixing are not good. The slow initial injection fuel droplets will combine with faster fuel droplets, which will result in a larger droplet size that yields a poor spray breakup. The combustion period will then be longer as more time is required to inject the fuel to atomize and evaporate it for combustion. In the past, researchers were interested in investigating the injection rates to improve fuel injection strategies. Juneja et al. [6] noted that increasing the injection rate after a previous injection is su fficient to increase the collision frequency and formation of large droplets, resulting in high momentum and greater penetration. Liu et al. [7] found that a higher peak injection rate yielded a higher spray tip penetration, peak entrainment rate, and entrainment rate after the end of injection (EOI). In addition, Arsie et al. [8] suggested that the start of injection is the main parameter that a ffects the impingement phenomenon, whereas Kun Lin Tay et al. [9] found that the start of combustion for each rate shape is di fferent, although the injection duration and start of injection are same due to the start of pressure rise in each case. The peak in-cylinder pressures are higher when the start of combustion is advanced due to the injection rate shaping. The combustion duration will be shorter with a higher initial injection velocity. Apart from that, the results from numerical study on the effects of boot injection rate shapes by co-workers of Balaji Mohan and Kun Lin Tay [10–12], showed that NOx decreased but large soot particles occur due to low injection velocity and narrow soot distribution when the main injection velocity is higher [11]. Dezhi Zhou et al. [12] found that higher boot injection velocity and shorter boot injection duration resulted in shorter ignition delay and more fuel burning at the premixed combustion stage. This suggests that a higher injection pressure will often lead to a better spraying process, that this is one of the most e ffective ways to meet the e fficiency requirements, and that this process has a potential benefit in diesel engine performance [13–15]. Agarwal et al. [16] found that increasing the injection pressure reduces the number and mass of particles and increases the diesel spray velocity, which improves the atomization and evaporation process. In addition, Shuai et al. [17] applied a numerical simulation to examine the e ffects of injection time and injection rate shape on the performance and exhaust emissions of compression ignition engines. The authors found that CO, UHC, and soot emissions can be reduced by using rectangular-type and boot-type rate shapes instead of other types. Based on these previous studies, the injection rate is clearly an important parameter that requires more attention. Injection rate parameters, such as injection velocity, injection mass quantity, and injection duration, have a significant influence on the fuel mixing and combustion process [18]. Many of these investigations considered how the diesel spray mixing process behavior can increase mixing e fficiency by studying the e ffect of the injection rate shape. Attempts to increase fuel mixing efficiency by determining the injection rate shape remain unsatisfactory. Although there have been extensive studies in the past on the influence of injection rates, most of these studies were interested in investigating the influence of injection rates on combustion e fficiency and engine emissions [9–12,17]. Few studies have investigated the influence of injection rates on spray mixing behavior.

Based on our previous study [19], we examined the spray mixing characteristics under di fferent injection rate shapes using a modified one-dimensional spray model. This one-dimensional spray model can analyze the spray penetration, entrainment rate, and velocity over a cross-sectional area, as well as the equivalence ratio distributed along with the spray's axial distance. This model does not consider breakup and evaporation. Instead, it reveals general information on both the liquid and the vapor. The fuel and air are assumed to be immediately mixed uniformly, so turbulent mixing cannot be analyzed in detail. A 3D-CFD model is better for researching a spray that includes two-phase flow characteristics.

Creating a spray model using CFD has become an e ffective way to study diesel spray to analyze the mechanism of fuel–air mixing and atomization. The results of many previous studies on creating a spray model using CFD show that the various parameters of fuel injection can be e fficiently monitored and predicted [20–22]. Nevertheless, there are few studies on the injection rate shape's e ffects on spray mixing. Unlike a quasi-steady-state spray, it is di fficult to obtain clear spray images that include liquid and vapor information under varying injection rates, so there is a lack of experimental data to validate the 3D-CFD spray model with varying injection rates. Consequently, most numerical studies focus on the e ffects of injection rates on combustion processes and do not o ffer an in-depth understanding of the spray behaviors under di fferent injection rates.

The objective of the current study is to analyze the e ffect of di fferent injection rate shapes on the diesel mixing process using a numerical modeling method. In this work, the "CONVERGE" CFD code was adopted for a constant-volume combustion chamber with a single hole injector, particularly to study the spray breakup and spray mixing behavior. Validation of the model results involved a comparison of the spray shape and spray penetration with the experimental data from previous researchers of the Sandia National Laboratory, taken from the ECN website [23]. We found that the modified CFD spray model can predict spray behavior. Four injection rate shapes were used to analyze the e ffects of injection rate shapes on diesel spray mixing behavior to understand the mixing process in-depth, including the microscopic spray characteristics, evaporation process, and mixture properties. The results of this study are expected to provide useful insights for developing an e ffective fuel injection rate design for future diesel engines.
