## 3 Results  
### Experimental Results  
As stated in the previous section, to develop the model, it is necessary to know the injected mass of fuel. In Supplementary Materials, Table S1 shows the fuel mass values of each experimental test injected in one injection. These values have been obtained from the average of the five tests carried out for each condition established in the test plan.  
As an example, Figure 7a,b represent the mean value of each of the tests performed with its confidence interval for the tests performed at a fuel temperature of 20 \({}^{\circ}\)C (Figure 7a) and at a temperature of 40 \({}^{\circ}\)C (Figure 7b) with an injection pressure of 90 MPa. It is observed that the fuel temperature in the ranges used minimally affects the fuel injection rate. At the same time, the nozzle bore size and ET are the variables that cause the most variability in the results. Logically, the larger bore-size nozzle produces a higher fuel injection rate. Regarding the fuels, in all cases, more diesel fuel is injected, followed by HVO, and finally, GtL fuel, although the difference between them is slight. This result may be because cavitation is higher with GtL [56] and HVO fuels [41].  
In Supplementary Materials, Tables S2-S4, show the injection timing values concerning the injector energizing of the experimental tests with every nozzle. In each cell, it is indicated at the beginning and the end of each phase in which the fuel injection rate curve has been divided, concerning the beginning of the energizing.  
As an example, Figure 8a,b represent the mean value of each of the tests performed at a fuel temperature of 20 \({}^{\circ}\)C, with ET of 1 ms (Figure 8a) and ET of 2 ms (Figure 8b) for all fuels and all nozzles, but only for the first phase, since it is the phase that most depends on the variation of the parameters of this study. In both figures, it is observed that  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(c_{1}\) & 42.452 & \(c_{10}\) & 0.003340 \\ \(c_{2}\) & 74.621 & \(c_{11}\) & 1.4541 \\ \(c_{3}\) & 0.003723 & \(c_{12}\) & 0.01556 \\ \(c_{4}\) & 1.975 & \(c_{13}\) & 62.628 \\ \(c_{5}\) & 0.02427 & \(c_{14}\) & 0.003203 \\ \(c_{6}\) & 73.444 & \(c_{15}\) & 0.002795 \\ \(c_{7}\) & 31.159 & \(c_{16}\) & 1.4098 \\ \(c_{8}\) & 0.000062 & \(c_{17}\) & 0.01392 \\ \(c_{9}\) & 31.228 & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Value of the coefficients of Equation (13).  
Figure 7: Behavior of the injection rate with its control parameters: (a) fuel temperature to 20 \({}^{\circ}\)C, (b) fuel temperature to 40 \({}^{\circ}\)C.  
the increase in pressure reduces the duration of the first phase and makes it more stable, even independent of the properties of the fuels. Figure 8a,b are similar, so the duration of energizing does not affect this first phase. Regarding the size of the hole, especially at low pressures, the tests with the largest nozzle (150 \(\upmu\)m) have a longer duration of the first phase. This is because with a larger hole, at equal pressures, the fuel output speed is lower, which causes the needle to take longer to fully open the injector.  
From the results shown in Tables S2-S4, it can be concluded that:  
* In the first phase (needle lift), both the time at which the injection event starts and the time at which it ends decrease with increasing pressure, temperature, and ET, and are more pronounced in HVO.
* In the second phase (holding injection), the start time of this phase evolves, as indicated for the end of the previous phase, and the end time of this phase depends almost entirely on the injection pressure (and, of course, on the ET).
* In the third phase (needle closure), the duration of this phase depends mainly on the injection pressure and ET.  
The effect of temperature is more pronounced in the first phase, but in the following phases, the effect is practically null. Something similar occurs with the use of different fuels, the most prominent effect on the duration time of the phases occurs in the first phase.