# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 3 Results and Discussion
The numerical calculations provided a distribution of key parameters related to the flow inside the computational domain, namely pressure and fluid velocity. In addition, thanks to the calculations performed with the VOF model, it was possible to obtain the water volume fraction distribution inside the air-filled domain. The last analyzed parameter was turbulence kinetic energy (TKE). This parameter was important because the sprayer contains a vortex insert, which significantly alters the flow parameters. In the first part, selected parameter values for the defined horizontal cross-sections of the sprayer, as shown in Figure 8, were analyzed. The first of the A-A planes (Figure 11) corresponds to the cylindrical space directly behind the inlet surface. Behind the main inlet, the fluid accelerates toward three openings located on the disk with the stem, and in the A-A cross-section, the velocity reaches a value of 0.13 m/s. Inside this section, the pressure is almost exactly equal to the preset initial value of 500 kPa. A slight gradient reaching single pascals results from the gradual acceleration of the liquid toward the orifice. Another B-B section under consideration (Figure 12) corresponds to three kidney-shaped channels. The fluid flows through them in the direction of the annular chamber. A pressure drop of up to 232 Pa due to flow resistance was observed. At the walls of the holes, the velocity was close to zero due to the formation of the wall layer. In the middle of the hole, the velocity was still low at 0.70 m/s. After flowing through the openings, the liquid flows into the annular chamber, which corresponds to the C-C plane (Figure 13). Its pressure was homogeneous at approximately 500 kPa, which corresponded to the inlet pressure. The velocity inside the chamber dropped slightly to a maximum value of approximately 0.60 m/s. Further down the nozzle, the liquid moves towards the vortex insert. In the middle of the cylindrical channel (plane D-D in Figure 14), the velocity increases at two locations corresponding to the grooves in the vortex insert. The maximum value reached nearly 2 m/s. The static pressure did not change significantly and was approximately 500 kPa. Figure 11: (**a**) Distribution of velocity and (**b**) pressure in the horizontal plane A-A. Figure 10: Comparison of the jet (**a**) from CFD calculations—opening angle 88\({}^{\circ}\) and (**b**) from the experiment—opening angle 80\({}^{\circ}\). Figure 14: (a) Distribution of velocity and (b) pressure in the horizontal plane D-D.  
Figure 12: (a) Distribution of velocity and (b) pressure in the horizontal plane B-B. Figure 13: (a) Distribution of velocity and (b) pressure in the horizontal plane C-C.  
One particularly interesting area inside the nozzle is the vortex insert. The E-E plane runs through the center of the height of the outflow channels from this element (Figure 15). The shaping of the channels results in a significant acceleration of the fluid inside the chamber formed between the ceramic insert and the stem. Near the center of the chamber, the velocity is close to 16 m/s. It should be emphasized that in the very center, the velocity is half as low, measuring approximately 8 m/s. Due to the acceleration of the fluid, the static pressure decreases significantly, reaching negative values (below the accepted atmospheric pressure). The pressure dropped to approximately \(-13\) kPa. Next, the liquid flows into the orifice at the end of the sprayer, with a diameter of 1.18 mm. The pressure and velocity in the F-F plane corresponding to this location are shown in Figure 16. The water stream, pre-shaped in the vortex chamber, accelerates at the beginning of the orifice to a value exceeding 21 m/s. This value is achieved near the walls of the orifice. In the core of the flow, the velocity gradually decreases to a value close to zero. The acceleration of the fluid results in a pressure drop in accordance with Bernoulli's principle. However, the negative pressure at the orifice inlet decreases from 13 kPa in the vortex chamber to 10 kPa.