## 1 Introduction
Regarding the injector geometry, some studies have reported the influence of the number of holes in the cavitation processes with different fuels [40; 41], and recommended a study with single-hole nozzles to know the behavior of the injection rate more deeply. The variation of nozzle diameters provides valuable information on how geometrical factors can influence the injection rate and ultimately affect the performance and efficiency of the fuel injection system. Although 0D models for injection rate are simple, they are less commonly used compared to one-dimensional (1D) models [29]. [PERSON] et al. [42] developed a 0D model capable of estimating the shape of the injection rate and the amount of fuel mass injected with the lowest possible computational cost. In this work, the rate curve is divided into three parts, and mathematical expressions of straight slopes are used to open and close the fuel injection rate curve, which combines second-order Bezier curves to smooth the corners. [PERSON] et al. [43] divided the injection process into five stages and used different mathematical strategies to model each stage, even combining the data from other experimental works, such as those of [PERSON] et al. [44], [PERSON] et al. [45], and [PERSON] et al. [46]. In [PERSON] et al. [31], a model is presented that divides the injection rate curve into three parts: the rising phase (from the beginning of injection until it reaches the maximum value of the injector opening that coincides with the one obtained in Equation (10)), the middle phase (in which the maximum amount of fuel is injected), and the falling phase (since the amount of fuel injected begins to decrease). [PERSON] et al. [47] developed a model using a similar methodology to [PERSON] et al. [31]. They used experimental correlations with parameters related to injector operation, fuel properties, and operating conditions to determine the start of injection time, the end of injection time, and the slopes of the RoI curve. [PERSON] et al. [48] divided the rate-of-injection curve into four sections (1 st needle lift, 2 nd full-lift hydraulic transient, 3 rd steady state max rate, and 4 th descent) to develop the 0D model. The model combines the theoretical injection equations with the information obtained from the experimental tests for each section. In this research, a 0D model is proposed to obtain the injection rate from certain operating conditions, such as injection pressure, energizing time, and fuel temperature. In addition, the geometry of the injector nozzle and some of the fuel properties are used. Injectors with single-hole nozzles were used to avoid the uncertainties caused by using multiple-hole nozzles on the rate of fuel injection. Data from two synthetic paraffinic fuels have been included so that the model can simulate the rates of these fuels and extend the usefulness of the model obtained. The model proposed in this work is based on other 0D models already published [42; 43; 47; 48], but particularly, it is an evolution of the model presented by [PERSON] et al. [31]. The novelty of this article compared to previous ones lies in the way in which the phases into which the fuel injection rate is divided are established, using mathematical correlations that do not require the calculation of the phase shifts between the energizing signal and the RoI signal.