## 8 Fuel System for High Pressure Hydrogen Injection
### Injector Design Considerations
#### 8.1.2 Axial Jet Penetration  
[PERSON] and [PERSON] [110] developed a widely used jet tip penetration \((Z_{t})\) prediction model for a gaseous jet as a function of time assuming the conservation of momentum. The validity of this model has been demonstrated for highly underexpanded jets with pressure ratio of up to 70, unaffected by the barrel-shaped shock. This model, described by Equations (4) and (5) [110, 111], assumes a constant momentum discharge rate and density during the injection, as well as self-similar jet velocity and mixing distribution. Since the underexpanded jet flow is choked, the nozzle exit velocity (subscript \(e\)) is assumed to be the sonic speed [112]. The subscript \(a\) represents the ambient condition and \(\Gamma\) is a constant dependent on jet cone angle, as shown in Equation (6) [110]. The sensitivity of \(\Gamma\) to jet cone angle is small. A 50% increase of \(s\) from 0.2 to 0.3 only decreases \(\Gamma\) from 3.04 to 2.89 (i.e., 5% decrease). [PERSON] and [PERSON] [110], therefore, proposed that a universal estimation of 3 for \(\Gamma\) can be used. A study [102] in a constant-volume chamber has also shown a small sensitivity of fully developed underexpanded jet spread angles to ambient conditions. Regardless of the fuel type and pressure ratios from 3 to 12, the jet spreading angles remained similar in the range of 22\({}^{\circ}\)-28\({}^{\circ}\). \[Z_{t} =\Gamma\left(d_{e}\times u_{e}\right)^{1/2}\times\left(\frac{\pi} {4}\frac{P_{o}}{P_{a}}R_{a}\frac{T_{a}}{T_{o}}\left(\frac{2}{\kappa_{o}+1} \right)^{\frac{1}{\kappa_{o}-1}}\right)^{1/4}t^{1/2} \tag{4}\] \[=\Gamma\left(d_{e}\right)^{1/2}\times\left(\frac{\kappa_{o}\pi} {4}\frac{P_{o}}{P_{a}}R_{a}T_{a}\left(\frac{2}{\kappa_{o}+1}\right)^{\frac{ \kappa_{o}}{\kappa_{o}-1}}\right)^{1/4}t^{1/2} \tag{5}\]  
\[\Gamma^{4}+\frac{1.92(1-s)^{2}}{\sqrt{\pi}(2-s)s^{3}}\Gamma^{2}-\frac{24}{\pi( 2-s)s^{3}}=0 \tag{6}\]  
where \(t=\) time after SOI  
\(d=\) diameter  
\(s=\) ratio of jet width to jet penetration distance \((D_{t}/Z_{t})\)  
In reality, the effective pressure (\(P_{\mathit{eff}}\)) at the nozzle exit is less than the fuel supply reservoir pressure due to the high compressibility of gas flow [113], which is not considered in Equation (4). An estimation of the effective pressure depending on the reservoir pressure, ambient pressure and other gas properties was developed by [PERSON] et al. [111] based on the shock tube diaphragm  
Figure 9: Theoretical (**a**) maximum steady injection rate and (**b**) minimum injection duration required for injecting 7 mg hydrogen at 2000 rpm at different injection pressures and nozzle diameters. rupture process as described by Equation (7). The effective pressure ratio to the ambient pressure (\(P_{eff}\)/\(P_{a}\)) in Equation (7) can be approximated using the Newton-Raphson method. This effective pressure can then be substituted to Equation (4) to predict jet tip penetration accounting for the pressure loss at nozzle exit, as shown in Equation (8). Therefore, the jet penetration mainly changes with the injection pressure and nozzle exit diameter at an unchanged operating condition and depends on the gas properties. Table 2 shows a comparison of the effective pressure ratio for hydrogen, helium, methane and nitrogen, using a range of reservoir/ambient pressure ratios. A typical CI engine ambient condition around top dead center was assumed, with ambient density, pressure and temperature at 20.8 kg/m\({}^{3}\), 60 bar and 1000 K, respectively [104]. The reservoir temperature was set at 298 K. It is noted that the effective pressure still increases with the reservoir pressure but at a lower rate and methane has a higher pressure loss than hydrogen at the same condition.