## 3 Research Results
### Simulation of the Fuel Mass Stream through the Spray Holes
The correction factor \(k_{2}\) depends on the quotient of the flow diameter of the nozzle hole and the inlet channel, the quotient of the surface roughness \(\Delta\) and the channel diameter \(D_{t}\), and the Reynolds number. Correction factor for nozzle holes \(k_{2}\) was calculated from the dependence [23]:  
\[k_{2}=\frac{1}{\sqrt{1.5+\frac{l_{rt}}{d_{rt}\left(1.14+2\lg\frac{d_{rt}}{ \Delta}\right)^{2}}}} \tag{8}\]  
where \(l_{rt}\)--nozzle hole length at temperature \(t\) and \(\Delta\)--hole surface roughness. The roughness of the surface of the injector nozzle sack was determined according to industry requirements and measurements using a roughness gauge for the sections of the spray channels. The correction factor for the inlet edge defocus \(k_{3}\) depends on the quotient of the flow diameter of the nozzle hole and the inlet channel, the channel diameter in front of the hole \(D_{t}\) and the edge radius \(r_{k}\) on the upstream inlet side of the spray hole. The correction multiplier \(k_{3}\) was determined based on the quotient of the edge rounding radius and the diameter of the spray hole \(r_{k}/d_{rt}\) and the hole module \(m\):  
\[m=\frac{d\,\frac{2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\The design number of flow \(\mu_{o}\) corresponds to the actual value determined in a smooth pipeline for the highest Reynolds number (\(Re_{max}\)) and the quotient of the flow diameter of the spray hole and the inflow channel.  
The Reynolds number can be calculated for the diameter ahead of the nozzle hole at temperature \(t\) or for the diameter \(d_{rt}\) (radius \(r_{rt}\)):  
\[Re_{d}=\frac{c_{g}4r_{ht}}{v_{p,t}}=\frac{c_{g}4r_{ht}\rho_{p,t}}{\eta_{p,t}} \tag{11}\]  
where \(c_{g}\)--average fuel flow velocity in the minimum cross-section, \(r_{ht}\)--hydraulic radius, \(v_{p,t}\)--fuel kinematic viscosity for temperature \(t\) and pressure \(p\) and \(\eta_{p,t}\)--absolute viscosity for temperature \(t\) and pressure \(p\). The kinematic viscosity can be calculated from the relationship:  
\[v_{p,t}=\frac{\eta_{p,t}}{\rho_{p,t}} \tag{12}\]  
The expansions number \(\varepsilon\) takes into account the change in the specific volume of the compressible fluid related to the pressure reduction as it flows through the spray hole. Fuel is a compressible liquid subordinated to [PERSON]'s law with the modulus of elasticity [23]. The relationship between temperature and fuel density was also determined by the following relationship [3; 23]:  
\[\rho_{p,t}=\rho_{15}-\Delta\rho(t-15) \tag{13}\]  
where \(\Delta\rho\) is the reduction of fuel density after heating by 1 \({}^{\circ}\)C, \(\Delta\rho=6.5\) kg/m\({}^{3}\). The viscosity of the fuel is important due to the use of different liquid fuels of different viscosities for the supply of reciprocating internal combustion engines. The pressure losses on the way from the injection pump drive to the injector nozzle were calculated for two engine types in [19; 20]. The absolute pressure loss \(\Delta\omega\) in the spray hole depends on the type of nozzle hole, its constriction, the flow rate and the differential pressure. For the shape compared to the orifice, it is calculated based on the formula [22]:  
\[\Delta\omega=\frac{\sqrt{1-\beta^{4}}-k_{2}2}{\sqrt{1-\beta}+k_{2}\beta^{2}}\Delta p \tag{14}\]  
where \(\beta\)--spray hole narrowing is equal. The narrowing \(\beta\) of the spray opening is defined by:  
\[\beta=\frac{d_{rt}}{D_{t}} \tag{15}\]  
Figure 3a shows the influence of surface roughness on pressure losses when flowing through the spray hole. This relationship has a minimum of 0.05 \(\upmu\)m. Figure 3b shows the effect of the spray hole length on pressure losses. As the surface roughness increases and the spray hole length increases, the irreversible pressure loss increases non-linearly with related variables.