## 4 Fuel Flow Analysis in the Fuel Atomizer
The flow parameters of the tested fuel injectors during the analysis are presented in the following table (Table 4). Based on the calculations, simulations of fuel flow through a standard and modified atomizer in a Solid Works Flow Simulation environment were undertaken (Figures 3-7). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**P [MPa]** & **Q [m\({}^{3}\)/s]** & **Re\({}_{1}\)** & **Re\({}_{2}\)** & **V\({}_{1}\) [m/s]** & **V\({}_{2}\) [m/s]** \\ \hline
30 & 0.0000186 & 218 & 4,298,326 & 0.0253 & 258 \\
60 & 0.0000408 & 480 & 6,229,029 & 0.0554 & 434 \\
120 & 0.0000602 & 708 & 8,913,541 & 0.0818 & 53 Figure 4: Simulation of fuel flow through a modified atomizer at 500 \(\upmu\)s injection time and fuel temperature of 313 K. (**A**) System pressure of 120 MPa, injected dosage 20.9 mm\({}^{3}\)/H; (**B**) system pressure of 60 MPa, injected dosage 10.2 mm\({}^{3}\)/H; (**C**) system pressure of 30 MPa, injected dosage 2.8 mm\({}^{3}\)/H. Figure 3: Simulation of fuel flow through a standard atomizer with an injection time of 500 \(\upmu\)s and fuel temperature of 313 K. (**A**) System pressure of 120 MPa, injected dosage 20.9 mm\({}^{3}\)/H; (**B**) system pressure of 60 MPa, injected dosage 10.2 mm\({}^{3}\)/H; (**C**) system pressure of 30 MPa, injected dosage 2.8 mm\({}^{3}\)/H. The aim of spiral-elliptical ducts is to trigger eddies along the non-working part of the injector needle. If the fuel moves in the nozzle, it will increase the temperature, due to additional turbulence. A higher temperature will lower fluid kinematic viscosity. Spiral-elliptical ducts increase the flow dimension. The Reynolds number (\(\mathrm{R_{e}}\)) increases according to relationship (8), which has an influence on the fuel stream turbulence. The density of compressible liquids depends on fluid temperature and pressure, and can be considered within approximation (9), where \(\rho_{0}\) is the liquid density under the  
Figure 2:0 Simulation of fuel flow as a function of speed in the zone of valve closing and opening of the atomizer at 120 MPa system pressure. (**A**) Classical atomizer; (**B**) atomizer with spiral-elliptical ducts. Figure 2:1 Simulation of fuel flow as a function of speed in the zone of valve closing and opening of the atomizer at 60 MPa system pressure. (**A**) Classical atomizer; (**B**) atomizer with spiral-elliptical ducts. Figure 2:2 Simulation of fuel flow as a function of speed in the zone of valve closing and opening of the atomizer at 30 MPa system pressure. (**A**) Classical atomizer; (**B**) atomizer with spiral-elliptical ducts. reference pressure P\({}_{0}\), C and B are coefficients, \(\rho_{0}\), B and C depend on the temperature, and P is the calculated pressure [27]. Figure 2:3  
Liquid dynamic viscosity depends on fluid pressure, and can be considered within approximation (10), where \(\mu_{0}\) is the liquid density under the reference pressure P\({}_{0}\), a is a coefficient that depends on the temperature, and P\({}^{\rm i}=10^{5}\) Pa is a constant value. Figure 2:4  
Flow simulation allows us to simulate heat transfer between solids and fluids. Heat transfer conversion has been described by energy conservation Equations (11) and (12). Heat flux is defined by Equation (13). Anisotropic solid media heat conductivity is described by Equation (14), where e is the specific internal energy, Q\({}_{\rm H}\) is the specific heat release per unit volume, \(\lambda_{\rm i}\) is the eigenvalues of the thermal conductivity tensor, Pr is the Prandtl number, h is the thermal enthalpy, \(\sigma_{\rm c}=0.9\) constant, u is the fluid velocity, \(\Omega\) is the angular velocity, and S\({}_{\rm i}\) is the mass distributed external force per unit mass, due to porous media resistance [27]. Figure 2:5  
Figure 2:6  
Figure 2:7  
Figure 2:8  
Figure 8 presents fuel temperature during flow through classical and spiral-elliptical duct atomizers. There is a noticeable difference in heat distribution between nozzles.