# Study on the Common Rail Type Injector Nozzle Design Based on the Nozzle Flow Model
## 2 Methodology
### Design of Injector Nozzle Hole  
The nozzle exit flow affects the spray and vaporization of fuel, directly related to engine combustion. The nozzle flow model in Figure 1b is a zero-dimensional model that calculates pressure and velocity at each points of the nozzle flow path. Using the nozzle flow model, the effective mean velocity (\(U_{mean}\)), effective diameter (\(D_{eff}\)), and discharge coefficient (\(C_{d}\)) can be obtained from the geometric shape of the nozzle and the injected fuel quantity from the experiment [10; 11]. In this study, it was modified to predict the discharge coefficient by using inlet injection pressure, instead of the injection quantity obtained through experiments, and nozzle geometry for manufacturing DME injector nozzles. Figure 2 shows the simplified injection pressure characteristics at the nozzle inlet. In the nozzle flow model, the inlet injection pressure is simply assumed as a sine function considering needle movement. \[P_{inj}=P_{1}=P_{2}+C\times\sin(2\pi t/T) \tag{1}\]  
where, \(P_{inj}\) is the injection pressure (Pa), \(P_{2}\) is the pressure of \(5\times 10^{6}\) (Pa) in the combustion chamber, C is the desired maximum pressure (Pa) to the combustion chamber pressure, and T/2 is the injection duration of 1 (ms). Thus, at time of \(t=0.5\) (mm), it becomes peak injection pressure of \(P_{inj}=P_{2}+C\) \begin{table}
\begin{tabular}{c c c c} \hline
**Item** & **Unit** & **Diesel** & **DME** \\ \hline Density (\(\rho_{l}\)) & kg/m\({}^{3}\) & 831 & 667 \\ Kinematic viscosity (\(\
u\)) & cSt & 3 & 0.1 \\ Lower heating value (\(Q_{LHV}\)) & MJ/kg & 42.5 & 27.6 \\ Vapour pressure (\(p_{cap}\)) & Pa & 10,000 & 530,000 \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties of the diesel and DME at \(T=20^{{}^{\circ}}\)[5]. Figure 1: Schematic diagram and SEM image of the injector nozzle. (**a**) Nozzle with needle; (**b**) Nozzle flow model [11; 12]; (**c**) Scanning Electron Microscope (SEM) image of the nozzle hole. If the injection pressure is determined at the nozzle inlet from Equation (1), the initial discharge coefficient is first assumed. Then, the discharge coefficient can be obtained by calculating the equation related to the average flow rate, repeatedly. \[U_{mean}=C_{d}\sqrt{\frac{2\left(P_{inj}-P_{2}\right)}{\rho_{l}}} \tag{2}\]  
\[C_{d}=\frac{1}{\sqrt{1+f+\frac{L_{in}}{D}+K_{i}}} \tag{3}\]  
where, \(U_{mean}\) is the effective mean velocity (m/s), \(C_{d}\) is the discharge coefficient (-) and \(\rho_{l}\) is the density (kg/m\({}^{3}\)) of the fuel as diesel or DME. In addition, \(f\) is friction loss coefficient (-), \(L_{in}\) is the length of the nozzle (mm), \(D\) is the nozzle diameter (mm), and \(K_{i}\) is the entrance loss coefficient (-). The entrance loss coefficient (\(K_{i}\)) is the function of \(R/D\), the ratio of the inlet radius (\(R\)) to the nozzle hole diameter (\(D\)). The friction loss coefficient (\(f\)) in Equation (3) can be determined by [11]:  
\[f=\max(\frac{64}{Re},\ 0.316 Re^{-0.25}) \tag{4}\]  
where, \(Re=vU_{mean}D\) is Reynolds number in the nozzle hole. In order to evaluate whether the flow is cavitating and or not in the nozzle, the vena contracta pressure (\(P_{vena}\)) at point c inside the nozzle in Figure 1b can be obtained from Equation (5) to Equation (7):  
\[C_{c}=\frac{1}{\sqrt{2.6787-11.4\ R/D}} \tag{5}\]  
\[U_{vena}=\frac{U_{vena}}{C_{c}} \tag{6}\]  
\[P_{vena}=P_{inj}-\frac{\rho_{l}}{2}U_{vena}^{2} \tag{7}\]  
where, \(C_{c}\) is the vena contracta coefficient (-), \(P_{vena}\) is the pressure (Pa), and \(U_{vena}\) is the vena contracta velocity (m/s) at point c, as shown in Figure 1b. As the vena contra velocity increases with higher injection pressure, the vena contracta pressure decreases. If pressure (\(P_{vena}\)) at this point is less than the vapour pressure (\(P_{vap}\)) of the fuel, cavitation occurs inside the nozzle.