## 1 Introduction
Using a right-angled triangle shape with greater pressure at the end of the injection event leads to the lowest NO\({}_{\mathrm{x}}\), but the highest Soot emissions. If the peak pressure is provided at the start of the injection event low values of Soot emissions are reported. [PERSON] et al. [16] examined the opportunity of applying a phenomenological multi-zone model modified accordingly to consider various injection rates. Considering the entire engine behaviour, the boot rate with bigger pressure at the end of the injection process was the best shape, while the constant rate was the least effective. [PERSON] et al. [17] used seven rate shapes, the start of injection for each of them being tuned so that all determined the same trigger point of the main heat release. The rectangular shape and the boot-type shape delivering the peak velocity at the end of the injection can reduce Soot, HC, and CO emissions in comparison to the trapezoidal, triangular and respectively to the boot-type shapes, which deliver the peak injection velocity at the start of the injection event. According to [PERSON] et al. [18; 19; 20], the triangular shapes of injection present bigger incylinder pressure rise rates relative to the ramp injection profiles. The duration of the combustion process decreases when the initial injection rate is higher and increases when a larger amount of fuel is introduced in the last part of the injection process. The ignition delay period lengthens when initial injection velocities are decreasing. If boot shapes with higher pressures at the end of the injection event are used, the combustion phasing occurs differently, and the NO emissions are usually decreasing in comparison with the conventional rectangular injection rate. When the boot injection rate is lower, greater Soot particles are formed. [PERSON] et al. [21] performed a numerical study based on the adoption of four injection rate shapes (saddle, rectangle, triangle, and hump) considering constant injected fuel quantity and injection period. The start of the injection event for hump and saddle rates appears before the start of the injection for the triangle rate. Higher NO\({}_{\mathrm{x}}\) emissions were formed for these shapes as a result of the greater high-temperature area and this is the consequence of bigger initial injection pressure promoting improved atomization and reduced ignition delay. The Soot emissions for hump rate were improved compared to the other shapes because of the higher pressure in the later injection period that improves the mixing of fuel and air. Five shapes of the rate of injection were studied in [22]. The highest peak combustion temperature was reached for the triangle shape (highest NO\({}_{\rm x}\) emissions) as a result of the greater rate at a certain injection moment. The rectangle shape represented the optimum for Soot and NO\({}_{\rm x}\) emissions. The slope shape offered a moderate level of HC and the minimum CO emissions. As a result of delayed heat release, the wedge shape led to a low level of NO\({}_{\rm x}\), but a high level of Soot emissions. The trapezoid shape offered the lowest Soot and a moderate level of NO\({}_{\rm x}\) emissions. [PERSON] et al. [23] investigated different boot injection profiles by varying the pressure and length of the shape for a biodiesel-fuelled engine. Simulation results revealed that NO\({}_{\rm x}\) emissions were reduced when a lower fuel quantity has been introduced in the initial stage of injection and thus a lower fuel injection velocity resulted. The long boot length and high boot pressure shapes can be the solution to achieve the compromise between NO\({}_{\rm x}\) and Soot. There are many research papers dedicated to studying the consequences of changing the injection rate shape on diesel engine combustion process and emissions formation using different CFD software. Although CFD tools are highly efficient, they consume a lot of time and computing resources.