# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 3 Results and Discussion  
As a result of the 3D calculations performed in transient conditions, a range of parameters related to the liquid flow in the specified computational domain was obtained. The results are presented for an operating pressure of 500 kPa and a time of \(t=15.64\) ms, which elapsed since the liquid began entering through the defined inlet surface. The mass flow rate of the nozzle as a function of operating pressure, as declared by the manufacturer, is presented in Figure 9. This parameter for the adopted working pressure was verified on an experimental test stand designed for visualizing the sprayed water jet (Figure 9). Measurements were conducted by the authors for the pressure value of 5 bar. The obtained experimental data were then compared with the manufacturer's specifications to ensure consistency and validate the experimental setup. Figure 9: (**a**) A diagram of the mass flow rate as a function of the working pressure of the liquid at the sprayer inlet and (**b**) a photo of the test stand for observing the liquid jet: 1—head with the tested  
nozzle; 2--water measuring cylinder; 3--manometer; 4--stroboscopic lamp; 5--fluid jet observation area (manufacturer data: [47]). The study involved measuring the outflow time of a specified liquid volume that flowed into a glass measuring cylinder. The resulting mass flow rate (0.012 kg/s) coincided with the manufacturer's declared value. From numerical calculations for the tested geometry at an overpressure of 500 kPa, a mass flow rate of 0.010 kg/s was obtained. The observed difference from the nominal value of 16.7% can most likely be attributed to imperfections in the mapping of the nozzle geometry. A profilometer was used to identify complex shapes, while the remaining ones were measured with an ENGINDOT TACKLIFE DC02 digital caliper, offering a measurement accuracy of \(\pm\)0.03 mm. In the solid model imported into the numerical model, sharp edges were defined in many places, which, for technological reasons, have rounded edges in reality. Another factor that may affect the observed differences could be inconsistencies in the initial conditions defined in the analysis, in particular, in terms of phase interactions compared to their actual progression. Detailed verification and validation procedures are crucial for ensuring the reliability of CFD simulations, especially those utilizing commercial software [48]. Therefore, the shape of the sprayed jet was also observed and compared with the jet obtained from numerical calculations. The numerical results obtained, although generally consistent with experimental data and the nozzle's catalog specifications, show certain discrepancies. A potential source of these deviations lies in the geometric representation of the nozzle. In the CAD model, locally idealized sharp edges were assumed, whereas the real component features smooth rounding due to manufacturing processes. The use of transient simulation with a time step of \(10^{-5}\) s allowed the capture of dynamic changes in the outlet region. However, the number of iterations per time step (20) and the applied discretization schemes may have limited the accuracy in resolving turbulent phenomena. The applied \(k\)-\(\omega\) SST turbulence model with the production limiter is known for its good stability in near-wall flows, but its ability to capture complex vortex structures may be limited compared to more computationally intensive models such as large eddy simulation (LES) or detached eddy simulation (DES). Figure 10 presents a comparison of the fluid jet obtained from CFD calculations and from experiments. The spray cone angle of the liquid on the test stand was consistent with the value specified by the manufacturer (\(80^{\circ}\)). In the case of the liquid jet obtained from the numerical calculations, this angle was approximately 10% larger. This results from the complexity of the model factors related to the physics of the spraying process and the applied submodels that influence the equations characterizing multiphase flow.