## 1 Introduction  
Diesel engines are usually used in many applications for transport, industry, power generation, agricultural and military sectors. Clean combustion achieved by adaptive control of engine operation using the injection rate-shaping has received nowadays major significance to ensure increased efficiency and low levels of emissions. By adopting a modern combustion system design accompanied with an advanced fuel injection system, high power and torque densities together with low fuel consumption and emissions can be achieved [1]. Renewable and sustainable alternative fuels such as biofuels can be a solution to the increased demand for fuels and the depletion of oil resources. Also, the association of advanced fuel formulations with highly efficient engine combustion architectures can be a suitable solution for the challenges related to efficiency and emission targets in the short-medium term [2]. The transport sector is the primary consumer of petroleum-derived fuels. Diesel engines are constructed to run on diesel fuel and the alternative fuel should have similar characteristics. Although many studies have been carried out on alternative fuels for diesel engines, the main obstacle which must be overpassed is the elevated cost of these fuels and the difficulties associated with engine changes enabling the operation with these fuels. Some studies have shown that by fuelling the engine with blends of crude palm oil or biodiesel with diesel, a simultaneous reduction of HC, NOx and Soot emissionsis possible [3; 4]. In this sense, biodiesel is actually considered the only alternative fuel that can be used without or with small modifications to the diesel engine having a significant effect on emissions mitigation [5]. In the last decades, several experimental and numerical studies have been developed aimed to examine the effect of various shapes of the rate of injection on diesel engine operation. Most of them showed that the adoption of single injection strategies with modified shapes of the rate of injection can be a very efficient method for decreasing NO\({}_{\mathrm{x}}\) emissions at some loading conditions [6]. Also, the combustion-generated noise can be reduced if there is a lower initial injected fuel mass, thus slowing down the early rapid rise of the heat release and peak pressure rise rate [7]. In other studies, the adoption of different shapes of the rate of injection has been achieved considering split injection strategies [8; 9; 10]. Changing the main injection shape has an important effect on the air-fuel mixing process, ignition delay and pollutant emissions. According to [8], it is likely to attain low NO\({}_{\mathrm{x}}\) and Soot emissions at the same time by using the proper rate shape. The experimental study conducted by [PERSON] et al. [11] shows that boot and ramp rate of injection profiles have an important influence over the premixed combustion phase and the ignition location without changing the injection pressure or the final fuel mass flow rate. According to [12], NO\({}_{\mathrm{x}}\) forms in zones that are close to the stoichiometric and Soot forms in zones with equivalence ratios between two and four. Therefore, the variation of fuel delivery through the change of injection rate shape can be a valuable method of emission control. [PERSON] et al. [13; 14] observed important reductions in NO\({}_{\mathrm{x}}\) emissions with increasing boot length or decreasing boot pressure. Less efficient mixing conditions are provided when the boot shape length is increased, which amplifies Soot emissions drastically. An improved mixing can be obtained if the boot pressure is increased, which provides rapid burning rates, with higher NO\({}_{\mathrm{x}}\) and lower Soot emissions. Using computational fluid dynamics (CFD) software, [PERSON] et al. [15] observed notable differences in the ignition delays among the considered rate shapes with the same start and duration of injection.