## 8 Fuel System for High Pressure Hydrogen Injection
### Injector Design Considerations
#### 8.1.2 Axial Jet Penetration
Also, the effective pressure ratio decreases with both increasing gas's molar mass and specific heat ratio, as seen from the comparison of different gases in Table 2.  
\[1+\left(\frac{u_{o}^{*}}{u_{a}^{*}}\right)\left(\frac{\kappa_{a}-1}{\kappa_{o}- 1}\right)-\left[\left(\frac{u_{o}^{*}}{u_{a}^{*}}\right)\left(\frac{\kappa_{a }-1}{\kappa_{o}-1}\right)\right]\left(\frac{P_{a}}{P_{o}}\right)^{\frac{\kappa _{a}-1}{2\kappa_{o}}}\left(\frac{P_{eff}}{P_{a}}\right)^{\frac{\kappa_{a}-1}{ 2\kappa_{o}}}-\left(\frac{P_{eff}}{P_{a}}\right)^{\frac{\kappa_{a}-1}{2\kappa _{a}}}=0 \tag{7}\]  
\[Z_{t}=\Gamma\left(d_{e}\right)^{1/2}\times\left(\frac{\kappa_{o}\pi}{4}\frac{ P_{eff}}{P_{a}}R_{a}T_{a}\left(\frac{2}{\kappa_{o}+1}\right)^{\frac{\kappa_{a}}{ \kappa_{o}-1}}\right)^{1/4}t^{1/2} \tag{8}\]  
Accounting for the difference in effective pressure ratio, Table 2 suggests a faster penetration of hydrogen jet relative to nitrogen or methane. This is demonstrated in Figure 10, which compares the theoretical jet penetration of hydrogen to methane as conventional gaseous fuel in HPDI applications. Two comparisons are demonstrated--Figure 10a shows the jet penetration for a variation of injection pressures at a fixed nozzle diameter of 1 mm. Ambient condition was assumed the same as that in Table 2. A comparison with increased hydrogen injection pressure to match the energy flow of methane jet is presented as well. Figure 10b compares the jet penetration for a variation of nozzle diameters at a fixed injection pressure of 150 bar. Similarly, a comparison with increased hydrogen jet orifice diameter to match that of the methane jet's energy flow rate is presented. It is noted that hydrogen with 158 bar and 369 bar injection pressure provide the same theoretical energy flow rate as methane at 150 bar and 350 bar, respectively; nozzle diameters of hydrogen injection need to be increased to 1.03 mm and 1.54 mm for the same theoretical energy flow rate as methane injection with 1 mm and 1.5 mm nozzle diameters, respectively. Equation (8) and Figure 10 show a higher sensitivity of jet penetration to nozzle diameter than to injection pressure, as confirmed by schlieren imaging experiments [98]. In addition, hydrogen jet indeed penetrates faster than methane under the same conditions, which is mainly attributed to a difference in effective pressure ratio, as shown in Table 2. Therefore, the injection pressure of methane jet has to be nearly doubled to provide the same penetration as hydrogen jet. Previous experiments by [PERSON] [102] and large eddy simulation by [PERSON] and [PERSON] [114] have validated the faster jet tip penetration of hydrogen than methane at the same pressure ratio. This needs to be considered when transferring current knowledge from CN \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Reservoir Pressure Ratio (\(P_{o}\)/\(P_{a}\))**} & \multicolumn{4}{c}{**Effective Pressure Ratio (\(P_{eff}\)/\(P_{a}\))**} \\ \cline{2-5}  & **Hydrogen** & **Helium** & **Methane** & **Nitrogen** \\ \hline
2 & 1.58 & 1.46 & 1.34 & 1.27 \\
4 & 2.46 & 2.09 & 1.77 & 1.59 \\
6 & 3.17 & 2.55 & 2.07 & 1.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated effective pressure ratio for different reservoir pressure ratios for hydrogen, helium, nitrogen and methane. to hydrogen. However, hydrogen combustion under engine conditions may affect the jet penetration behaviour, which needs to be further verified.