## 3 Research Results  
### Simulation of the Fuel Mass Stream through the Spray Holes  
The fuel flow through the multi-hole injector nozzle can be compared to the flow rate when measured with a measuring orifice, e.g., a small orifice (Figure 1). The fuel mass flow \(q_{mi}\) through the spraying hole, similarly to the measuring orifice or the nozzle, is determined by the relationship [9; 22; 23]:  
\[q_{mi}=\mu_{r}\epsilon A_{r}\sqrt{2\Delta p\rho} \tag{1}\]  
where \(A_{r}\)--spray hole cross-section, \(\mu_{r}\)--flow number, \(\Delta p\)--pressure difference in the injector nozzle space (sack) \(p_{i}\) and in the combustion chamber \(p_{c}\), \(\epsilon\)--expansion number and \(\rho\)--density of the flowing fuel, determined for the conditions directly in front of the orifice. The area of the spray hole can be calculated from the formula:  
\[A_{r}=\frac{\pi d_{rt}^{2}}{4} \tag{2}\]  
where \(d_{rt}\)--spray hole diameter at the temperature \(t\). Both cylindricity and circularity errors occur during production [35] (Figure 2). In addition, there are material des decrements operation, as well as the deposit formation. Moreover, this diameter changes with temperature; therefore, the hole diameter \(d_{rt}\) at temperature \(t\) is:  
\[d_{rt}^{2}=k_{t}d_{r}^{2} \tag{3}\]  
The spray hole dimensions change during the measurement with the flow of the medium at a higher temperature. The thermal expansion of the hole considers the correction factor for thermal expansion \(k_{t}\):  
\[k_{t}=1+2\alpha_{t}(t_{1}-20^{\circ}) \tag{4}\]  
Figure 1: Cross-section of a measuring nozzle compliant with ISO 5167-1:1991 (**a**) and a cross-section of an injector nozzle (**b**): \(d\)—diameter of the throat flow opening, \(D\)—diameter of the pipeline before the measuring nozzle, \(r_{k}\)—radius of the nozzle edge rounding on the inlet side, \(d_{r}\)—diameter of the nozzle hole, \(l_{r}\)—nozzle hole length, \(1\)—spherical surface of the injector nozzle. Figure 2: The solid model of the injector nozzle (**a**) and image of cross-section of the nozzle hole with rounding (**b**). where \(\alpha_{t}10\mid \begin{array}{c}t_{1}\\ 20\end{array} \) is the average coefficient of thermal expansion per 1 \({}^{\circ}\)C from the temperature of 20 \({}^{\circ}\)C to \(t_{1}\). There is also an increase in fuel temperature by 1 K at \(x\) for a pressure increase of 10 MPa [23], whereby the formula can by:  
\[t_{px}=t_{d}+(p_{px}-p_{o})/10 \tag{5}\]  
where \(t_{px}\)--fuel temperature in place of \(x\), \(t_{d}\)--fuel inlet temperature, \(p_{px}\)--fuel pressure at \(x\) and \(p_{o}\)--ambient pressure. In addition, the injector nozzle and the fuel heat up from the working medium in the combustion chamber by the Clapeyron equation [22]. The flow number \(\mu_{r}\) can be an experimentally determined coefficient for similar flows. In general, it is calculated from the formula [9]:  
\[\mu_{r}=k_{1}\,k_{2}\,k_{3}\mu_{o} \tag{6}\]  
where: \(k_{1}\)--correction factor of viscosity taking into account the influence of Reynolds number, \(k_{2}\)--correction factor of roughness taking into account the influence of roughness of the internal surface of the pipeline in which the differential device is installed, \(k_{3}\)--correction factor of the inlet edge bluntness, \(\mu_{o}\)--the computational number of the flow corresponding to the actual value \(\mu\) determined in the smooth bore for the highest Reynolds number (Re\({}_{\text{max}}\)). The correction factor of viscosity \(k_{1}\) depends on the Reynolds number and the quotient of the flow diameter of the spray hole and the inflow channel \(D_{t}\), which can be written as:  
\[k_{1}=f(\frac{d_{rt}^{2}}{D_{t}^{2}},R_{e}) \tag{7}\]  
The correction factor for viscosity can be read from the diagram for the calculated Reynolds number and the quotient \(l_{rt}/d_{rt}\) ([23], p. 449) or a relationship can approximate it.