# Effects of Valve, Armature, and Armature Pin Guidance on Diesel Injector Performance
## 2 Materials and Methods
### Geometry and Case Setup  
Backflow amounts between the valve, armature, and armature pin were compared through simulations of the inlet-outlet boundary conditions. The design was according to the seven different experimental groups outlined in Table 1, and flow rates, pressures, and velocities were computed for each group. This study employed the finite-volume method to solve the conservation of mass and momentum equations using Ansys Fluent. The mass conservation and momentum equations are provided in Equations (3) and (4), respectively. Additionally, the pressure drop can be determined using Equation (5), while the Reynolds number (\(Re\)) can be calculated using Equation (6). \[\
abla.\left(\rho\overset{\rightarrow}{V}\right)=0, \tag{3}\]  
\[\rho\left(\overset{\rightarrow}{V}.\
abla\right)\overset{\rightarrow}{V}=- \
abla_{p}+\
abla.\left(\mu\
abla\overset{\rightarrow}{V}\right)+\rho \overset{\rightarrow}{S}, \tag{4}\]  
\[\Delta\text{P}=P_{2}-P_{1}=\frac{8\mu LV_{\text{avg}}}{R^{2}}=\frac{32\mu LV_{ \text{avg}}}{D^{2}}, \tag{5}\]  
\[Re=\frac{\rho V_{\text{avg}}D}{\mu}, \tag{6}\]  
where \(\rho\) is the fluid density, \(\overset{\rightarrow}{V}\) is the fluid velocity vector, \(P\) is fluid pressure, and \(\overset{\rightarrow}{S}\) is the gravity acceleration vector. Simulation studies corresponding to the groups listed in Table 1 were conducted using Ansys Fluent. Given that \(Re\leq 2300\), the flow was determined to be laminar. The analyses were performed under steady-state conditions. The inner and intermediate pipe inlet pressure were determined as 500 kPa. The outlet pressure of both was 100 kPa. The SIMPLE (semi-implicit method for pressure-linked equations) algorithm was utilized as the solution method. Because of its ease of converge, first-order upwind spatial discretization  
Figure 4: Level of diameters in group 3 considering total tolerance (A: valve inner diameter, B: armature outer diameter, C: armature inner diameter, D: armature pin outer diameter). was preferred for momentum and pressure. After convergence was achieved, the analysis solution was continued with second-order upwind spatial discretization. The test fluid properties in Section 2.1 were used as the fluid. Figure 5 illustrates the fuel flow path between the armature, armature pin, and valve (a) and the model geometry used for the simulation study (b). Backflow occurs through three pathways: between the valve and armature, between the armature and armature pin, and through the valve holes. A continuous flow between the armature and armature pin is independent of the energizing time. However, flow through the valve holes only begins after the armature is lifted, also continuing through the valve holes. Because liquid typically follows the path of least resistance, the primary backflow of fuel occurs through the valve holes when the injector remains energized for an extended duration. However, if the energization time is brief, the backflow from  
Figure 5: (**a**) Fuel flow path between valve, armature, and armature pin; (**b**) definition of inner pipe and intermediate pipe. the valve holes may be less than the flow between the valve and armature, as well as between the armature and armature pin. In the absence of energization, which represents a significant portion of the injector's lifespan, backflow occurs solely between the armature and armature pin. The importance of continuous leakage (backflow) over the injection rate has also been mentioned in the study by [PERSON] et al. They noted that static condition leakages (backflow) are higher than those in dynamic conditions in solenoid injectors [27]. To maintain consistency across all experimental groups, the energization time was set to a minimum, and as the valve holes were uniformly produced in the same size across all groups, backflow from these holes was disregarded. This approach ensures that the influence of the valve holes remains neutral. Consequently, only the fuel flow paths between the valve, armature, and armature pin were considered in the simulation geometry.