# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 3 Results and Discussion
However, just a few millimeters away from the nozzle exit, the pressure equalizes with the ambient pressure. The analysis of the velocity distribution in the liquid outflow zone from the nozzle orifice, as shown in Figure 19, revealed more details. According to [PERSON]'s principle, due to the local narrowing of the flow in the orifice, the liquid accelerates rapidly and reaches a velocity exceeding 23 m/s. At the nozzle exit, a conical stream forms, with the highest velocities, ranging from 10 to 15 m/s, occurring in the outer region of the jet. The remainder of this zone has an irregular velocity field, varying in value over quite a wide range from 0 to 10 m/s. This is due to the turbulent nature of the flow and the local breakup of the continuous jet into droplets. The last parameter analyzed was turbulence kinetic energy. The higher the TKE value, the more intense the velocity fluctuations and, therefore, the more turbulent the flow. In the context of a two-phase liquid spraying model in a gaseous atmosphere, this parameter indirectly indicates the intensity of phase mixing and particle dispersion. The maximum value of this parameter occurred at the exit of the discharge orifice and was 37.04 m\({}^{2}\)/s\({}^{2}\). In the zone directly behind the nozzle, a high variability in the TKE values was observed. The location of the maximum TKE values (10-15 m\({}^{2}\)/s\({}^{2}\)) approximately coincides with the regions of the highest velocity in this zone, as shown in Figure 9:1 This is due to the progressive breakup of the jet into droplets and the presence of local turbulence. The maximum TKE values are concentrated near the outlet and along the periphery of the spray core, where strong shear layers develop due to the velocity difference between the swirling liquid jet and the surrounding air. These zones of elevated turbulence intensity are closely linked to the beginning of primary atomization, as high-speed shear promotes  
Figure 9:2 Pressure distribution in the vertical plane of the nozzle. instability and fragmentation of the liquid film. Although breakup modeling was not included in the current simulation, the TKE distribution observed here aligns with the expected initiation sites of droplet formation. Figure 21a-c shows the course of velocity, pressure, and turbulence kinetic energy strictly along the \(z\)-axis of the nozzle. The velocity along the chamber axis before the exit orifice (10-15 mm along the \(z\)-axis) reaches values from 6 to 11 m/s. Then, after exiting through the nozzle, it decreases to a low value of about 2 m/s. In this zone, the pressure becomes negative, i.e., it is below the ambient pressure. The intense fluid motion and local vortex phenomena resulting from the geometry of the vortex insert (presence of two inflow channels) result in achieving a high TKE value reaching up to 35 m\({}^{2}\)/s\({}^{2}\). Figure 21d-f present the obtained values of the analyzed parameters across the entire \(yz\) cross-sectional plane of the nozzle. The maximum velocity in the chamber before the exit nozzle reached nearly 25 m/s. Velocities reaching up to 10 m/s were achieved at a distance of 5 mm from this orifice (\(z\) = 20 mm). As the distance increased, the maximum value of this parameter decreased by approximately half to 5-6 m/s. Regarding pressure, its greatest fluctuation occurs in the region corresponding to the chamber before the exit orifice. At that point, the pressure decreases from a value of 500 kPa, close to the inlet value, to locally negative values. The maximum TKE values (approximately 35-36 m\({}^{2}\)/s\({}^{2}\)) correspond to those obtained directly on the \(z\)-axis. A local maximum at \(z\) = 34 mm, which is 19 mm from  
Figure 9:3 Velocity distribution in the vertical plane of the nozzle. Figure 9:4 Distribution of turbulent kinetic energy in the vertical plane. the exit orifice, deserves attention. At that point, this parameter reached a value close to 20 m\({}^{2}\)/s\({}^{2}\), which is twice the value observed in the rest of the outflow region.