## 8 Fuel System for High Pressure Hydrogen Injection
### Injector Design Considerations
#### 8.1.1 Injection Rate  
Under choked flow conditions, the maximum flow velocity and thus the maximum mass flow rate (\(\dot{m}_{\text{max}}\)) across the nozzle are limited by the speed of sound. By assuming that the enthalpy is only temperature dependent and incorporating the compressible flow theory under choked condition, the maximum mass flow rate can be approximated using Equations (1)-(3), with the subscripts \(o\) and superscript \({}^{*}\) representing the infinite reservoir and speed of sound condition, respectively [98].  
Figure 8: GDI injector modification design with a brass coaxial single-nozzle cap and its detailed section view.  
These equations should be applicable for different nozzle geometries, using the nozzle area at sonic condition. The validity of these equations for high pressure hydrogen injection was tested by measuring the mass flow using a straight single-nozzle electrohydraulic-actuated injector by [PERSON] et al. [98]. A range of pressure ratios and nozzle diameters was tested and the agreement between the measured and calculated average mass flow rates was better than 90% in most cases. The agreement becomes worse, if the injection duration is shorter than 5 ms, attributed to the needle opening transient.  
\[\dot{m}_{\text{max}} =\rho^{*}u^{*}A^{*} \tag{1}\] \[=\left[\rho_{o}\cdot\left(\frac{2}{\kappa+1}\right)^{1/(\kappa-1) }\right]\left[\sqrt{\frac{2\kappa}{\kappa+1}\frac{P_{o}}{\rho_{o}}}\right]A^{*}\] (2) \[=P_{o}A^{*}\sqrt{\frac{\kappa}{R_{o}T_{o}}}\cdot\left(\frac{2}{ \kappa+1}\right)^{(\kappa+1)/2(\kappa-1)} \tag{3}\]  
where \(A=\) area  
\[P =\text{pressure}\] \[\rho =\text{density}\] \[u =\text{velocity}\] \[T =\text{temperature}\] \[R =\text{specific gas constant (i.e., universal gas constant/molar mass)}\] \[\kappa =\text{specific heat ratio}\]  
Equation (3) was used to calculate the sensitivity of maximum injection rate to nozzle diameter and injection pressure (Figure 9a) and to estimate the theoretical minimum injection duration for engine operation (Figure 9b), assuming a single-hole injector. The fuel temperature was fixed at 298 K and a minimum injection pressure of 150 bar was postulated to ensure that a critical pressure ratio condition of 2 is reached for choked flow and underexpanded jet, assuming a back pressure representative of conditions around the top dead center in CI engines. The injection duration was derived from the nozzle mass flow using an injection quantity of \(\sim\)7 mg of hydrogen per injection, assuming the load and efficiency from a previous hydrogen DI SI study [73] in a 2.2 L four-cylinder engine (i.e., IMEP of 6 bar and ITE of \(\sim\)45% at 2000 rpm). The injection rate of hydrogen jet, as shown in Figure 9a, increases proportionally to injection pressure and with the square of nozzle diameter. The indicative minimum injection duration, as shown in Figure 9b, therefore decreases inversely proportional to higher injection pressure and quadratically with larger nozzle diameter. Despite the low density of hydrogen, with a nozzle diameter of 1 mm, the theoretical minimum injection duration as short as 5\({}^{\circ}\) CA can be achieved at injection pressure of 350 bar. At moderate injection pressure of 200 bar, the flow rate is sufficient to deliver the fuel mass within 10\({}^{\circ}\) CA. However, it should be noted that the calculation is just a theoretical indication using the maximum flow rate and the actual injection duration is expected to be longer due to the injector opening and closing transient [98]. Apart from delivering the required fuel quantity for a certain load, nozzle size and arrangement as well as the injection pressure need to be optimized for the engine geometry in order to achieve the best performance.