## 2 Materials and Methods
### Model Proposed Methodology
[31], in this case, the back pressure is constant (5 MPa) as well as the number of holes (1 single hole); therefore, these are two variables that will not participate in the modeling of the rate curve. The representative variables were verified separately in each phase to improve the model's ability to reproduce the fuel injection rate curve. Figure 6 shows the relevance of the participating variables in this study, indicating which should be included in the prediction models for each phase. * In the case of the 'needle lift' phase, Figure 6a shows that d\({}_{0}\) is the most relevant factor due to the stability or low variability of the data. This fact indirectly implies that the most significant variability or dependence is due to P\({}_{\text{inj}}\) and the kinematic viscosity of the fuel. Figure 6d shows that the most representative variables are P\({}_{\text{inj}}\) and d\({}_{0}\), followed by hole diameter, fuel viscosity, and energizing time. In all cases, this is because the slope of the rate curve is similar in almost all tests. * In the case of the 'holding injection' phase (Figure 6b), the variables of major dependence to keep the injection stable during this phase are the hole diameter and the injection pressure. Figure 6e indicates that the most representative variable is the energizing time since it generates a more noticeable change in the time the injector is open and, as such, in the duration of the injection time. * Finally, for the 'needle closure' phase (Figure 6c), the most relevant variable is again d\({}_{0}\), along with energizing time. Figure 6f shows that the specific control variable in the equation is energizing time, followed directly by P\({}_{\text{inj}}\) and d\({}_{0}\). In summary, in all phases, the injection rate depends on test time, injection pressure, injector energizing time, nozzle bore diameter, and kinematic viscosity. However, in none of the phases is fuel temperature identified as an essential dependent variable. Figure 6: Description of relevance of variables of linear prediction models: (**a,d**) needle lift phase, (**b,e**) holding injection phase, and (**c,f**) needle closure phase. Once the variables that provide the most remarkable dependence on the behavior of the rate in the injection process have been found, a numerical statistical model is generated that simulates the injection rate concerning the dependent variables. This model establishes a mathematical dependence correlation according to the variables involved in each phase. The linear predictive model obtained for the 'needle lift' (phase 1) is shown in Equation (11), and the values of the coefficients are shown in Table 4.  
\[ \begin{array}{c}t_{nl}\approx-a_{1}-a_{2}*ET+a_{3}*P_{inj}+a_{4}*v_{f}-a_{5}* z-a_{6}*d_{0}-a_{7}*P_{inj}{}^{2}-a_{8}\\ *v_{f}{}^{2}+a_{9}*2^{2}+a_{10}*d_{0}{}^{2}+a_{11}*ET*d_{0}+a_{12}*P_{inj}*d_{0 }-a_{13}\\ *v_{f}*d_{0}+a_{14}*z*d_{0}\end{array}  \tag{11}\]  
Equation (12) provides the mathematical correlation describing the holding injection phase, and the values of the coefficients are shown in Table 5.  
\[ \begin{array}{c}t_{hi}\approx b_{1}-b_{2}*ET-b_{3}*P_{inj}-b_{4}*v_{f}-b_{5}* z+b_{6}*d_{0}+b_{7}*ET^{2}-b_{8}*P_{inj}{}^{2}\\ +b_{9}*v_{f}{}^{2}+b_{10}*2^{2}-b_{11}*d_{0}{}^{2}+b_{12}*ET*P_{inj}+b_{13}*ET* v_{f}\\ -b_{14}*ET*z+b_{15}*ET*d_{0}+b_{16}*P_{inj}*v_{f}+b_{17}*P_{inj}*z-b_{18}\\ *P_{inj}*d_{0}+b_{19}*v_{f}*z+b_{20}*v_{f}*d_{0}+b_{21}*z*d_{0}\end{array}  \tag{12}\]  
The analysis based on independent values for the needle closure phase provided the mathematical expresion shown in Equation (13), and the values of the coefficients are shown in Table 6.  
\[ \begin{array}{c}t_{nc}\approx c_{1}+c_{2}*ET-c_{3}*P_{inj}+c_{4}*v_{f}-c_{5}* z-c_{6}*d_{0}+c_{7}*ET^{2}+c_{8}*2^{2}+c_{9}*d_{0}{}^{2}\\ -c_{10}*ET*P_{inj}+c_{11}*ET*v_{f}-c_{12}*ET*z-c_{13}*ET*d_{0}+c_{14}\\ *P_{inj}*d_{0}-c_{15}*v_{f}*z-c_{16}*v_{f}*d_{0}+c_{17}*z*d_{0}\end{array}  \tag{13}\]