# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 4 Conclusions  
The need for research on nozzles used in agricultural applications arises from the pursuit of increased efficiency in the process of applying plant protection agents (understood as saving the used substance and achieving better surface coverage) while reducing risks to the environment and human health. The numerical approach proposed in this study enables the modeling of the flow process and the initial formation of a liquid jet in this type of nozzle. This method allows for the verification of key parameters for various nozzle geometries without incurring the costs associated with preparing and conducting a series of experimental tests on a test stand. The conducted research focused on a hollow cone nozzle designed for the application of plant protection agents in orchards and marked the beginning of analytical work on developing a comprehensive multiphase model combining water and air phases. The authors proposed a model that utilizes the VOF method. The distribution of flow-related parameters in selected sections of the computational domain was obtained. The analysis focused on the variation of velocity and pressure within the flow channels of the nozzle. The obtained results indicate that the critical area is the chamber at the nozzle exit, formed between the vortex insert and the ceramic stem. Along the longitudinal axis of the nozzle, the TKE value in this region reached 35 m\({}^{2}\)/s\({}^{2}\). The cone angle of the spray differed by approximately 10% from the nominal value, while the mass flow rate differed by approximately 16.7%. These differences may be attributed to minor inaccuracies in the solid model representing the real nozzle geometry. A conical stream develops in the flow zone (several millimeters behind the sprayer orifice), where the outer region exhibits the highest velocities, attaining speeds between 10 and 15 m/s. Further away, the velocity decreases due to intensive mixing with the air. A significant impact on flow parameters is also exerted by various model settings related to turbulence, phase interaction, and mixing, as well as the physics of liquid jet breakup into droplets. This issue will be the subject of further analysis by the authors of this paper. The obtained results and observed phenomena inside and just beyond the nozzle provide a solid foundation for continuing research aimed at improving the numerical model and expanding the analysis to additional aspects of the spraying process. In the next stages, models based on liquid jet breakup and individual droplet generation are planned--specifically, the use of the DPM combined with atomization submodels such as Kelvin-Helmholtz (KH) or Taylor analogy breakup (TAB). This will allow tracking the trajectories and velocities of individual droplets after leaving the nozzle and determining dispersion parameters in space. The next step will involve introducing variable environmental conditions, such as wind speed, humidity, and temperature, enabling the simulation of spray drift and a better assessment of plant coverage efficiency under field conditions. Modeling droplet evaporation in flight, which significantly impacts crop protection effectiveness and environmental safety, may be particularly interesting. From a design perspective, simulations of alternative nozzle geometry variants are also planned--for example, by changing the number and inclination angle of the holes in the swirl plate or modifying the geometry of the swirl insert. This will make it possible to identify configurations that promote greater stability of the liquid cone and a lower turbulence level. Conceptualization, P.K. ; methodology, P.K. and Z.C. ; software, P.K. ; validation, P.K., Z.C., and S.P. ; formal analysis, P.K. ; investigation, P.K. and Z.C. ; data curation, Z.C. ; writing--original draft preparation, P.K. ; writing--review and editing, Z.C. and S.P. ; visualization, P.K. and Z.C. ; supervision, S.P. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding.