## 4 Fuel Flow Analysis in the Fuel Atomizer  
In order to illustrate the phenomena occurring before the injection process, a fuel flow model was made for a standard and modified atomizer according to the received patent Osipowicz [19]. The analysis was performed using Solidworks Flow Simulation. This program enables the study of a wide range of fluid flow and heat transfer phenomena. The model has a demonstrative function, and its task is to examine whether the changes made have affected the fuel turbulence in the atomizer. The flow was tested when the atomizer was opened at the moment when the needle was raised. This experiment is the first stage of the design and implementation of spiral-elliptical ducts in modern fuel injectors. Based on the tests carried out, it can be concluded that the tested fuel injectors are in working order. The simulation of the fuel flow through the atomizer was carried out for three measurements with the same actuation times, with 500 us fuel injectors at 30, 60, and 120 MPa pressure (Tables 2 and 3). During the tests the phenomena prevailing in the atomizer were described. The following figure shows the tip of the fuel injector divided into zones (Figure 2). \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Pressure [MPa]** & **Injection Time [μs]** & **Range [mm3/H]** & **Dosage [mm3/H]** \\ \hline
1 & 135 & 780 & 34.71–49.69 & 35.3 \\
2 & 30 & 420 & 0.31–3.89 & 0.7 \\
3 & 80 & 260 & 0.31–4.09 & 1.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Measurements of the injection doses according to the standard test of the standard fuel injector. Figure 2: Fuel atomizer divided into zones: (**I**) The non-working part of the needle, (**II**) the needle well, (**III**) the outlet from the atomizer. For the analysis of phenomena occurring in the atomizer of a modern fuel injector, general equations of the motion of viscous fluids, resulting from the momentum conservation principle, can be used. These phenomena are described by the Navier-Stokes equations in scalar form, where \(\mathrm{V_{x,\,y,\,z}}\) are fluid velocities in a Cartesian system, \(\overset{\rightarrow}{V}\) is the fluid flow velocity, \(\mathrm{p}\) is the pressure, \(\mathrm{X,\,Y,\,Z}\) are the individual mass forces in the Cartesian system components, \(\overset{\rightarrow}{F}\) is the individual mass force, and \(\rho\) is the fluid density:  
\[\frac{dV_{x}}{dt}=X-\frac{1}{\rho}\cdot\frac{\partial p}{\partial x}+v\cdot \Delta V_{x}+\frac{1}{3}\cdot v\cdot\frac{\partial}{\partial x}(div\overset{ \rightarrow}{V}), \tag{3}\]  
\[\frac{dV_{y}}{dt}=Y-\frac{1}{\rho}\cdot\frac{\partial p}{\partial y}+v\cdot \Delta V_{y}+\frac{1}{3}\cdot v\cdot\frac{\partial}{\partial y}(div\overset{ \rightarrow}{V}), \tag{4}\]  
\[\frac{dV_{z}}{dt}=Z-\frac{1}{\rho}\cdot\frac{\partial p}{\partial z}+v\cdot \Delta V_{z}+\frac{1}{3}\cdot v\cdot\frac{\partial}{\partial z}(div\overset{ \rightarrow}{V}), \tag{5}\]  
and in vector form:  
\[\frac{d\overset{\rightarrow}{V}}{dt}=\overset{\rightarrow}{F}-\frac{1}{ \rho}\cdot grad(p)+v\cdot\Delta\overset{\rightarrow}{V}+\frac{1}{3}\cdot v \cdot grad(div\overset{\rightarrow}{V}). \tag{6}\]  
The following data was used to make the model: Channel (atomizer) parameters, flow rate (fuel injection dose), and fuel parameters (density and viscosity). The Bernoulli equation was used to calculate the speed of fuel flow through the atomizer and stream, where \(\mathrm{V,\,V_{1},V_{2}}\) are fluid flows, \(\mathrm{p_{1}}\) and \(\mathrm{p_{2}}\) are the pressure, \(\rho\) is the fluid density, \(\mathrm{\mu}\) is the fluid dynamic viscosity coefficient, and \(\mathrm{d_{0}}\) is the flow dimension. \[\frac{\rho V_{1}^{2}}{2}+p_{1}=\frac{\rho V_{2}^{2}}{2}+p_{2}, \tag{7}\]  
\[R_{e}=\frac{\rho\cdot V\cdot d_{0}}{\mu}. \tag{8}\]  
Analyzing the figure, the pressure in the atomizer in zones I and II is at the same level. Based on relationships (7) and (8), the Reynolds number and fuel flow rate will also be the same.