# Numerical Modelling of the Multiphase Flow in an Agricultural Hollow Cone Nozzle
## 1 Introduction  
Agricultural sprayers are commonly used for the application of crop protection products. Depending on the application, they differ significantly in geometry, which affects the size of the resulting droplets [1, 2]. The nozzle geometry also influences droplet velocities [3]. These sprayers can be categorized into two main types: field sprayers, used for crops in open fields, and orchard sprayers, employed in orchards and vineyards. Regardless of the sprayer design or application environment, the quality of spraying directly affects the efficiency of chemical application, helping to minimize product usage and reduce environmental impact. Numerous studies focus on evaluating spray distribution quality on crop surfaces. Improper spraying can lead to over- or under-application of crop protection products, affecting treatment efficiency and cost. Moreover, external factors such as wind can cause significant spray drift. A computational fluid dynamics (CFD) model for predicting pesticide flow in orchards and spray drift from an air-assisted orchard sprayer is presented in [4]. In recent years, extensive research has been conducted on optimizing the spraying process of chemical agents during agricultural treatments. Regardless of sprayer configuration or working fluid parameters, the applied liquid jet often faces challenges related to wind drift and limited plant penetration. This is primarily because the process of liquid jet formation, followed by its breakup and entrainment with air, is directly linked to overall spray quality. Dimensionless similarity numbers, such as the Ohnesorge number and the Weber number, are commonly used to evaluate liquid jet breakup into droplets in spray process modeling. These numbers enable the scaling of physical phenomena and facilitate comparisons between different systems regardless of their specific dimensions or fluid properties. The Ohnesorge number characterizes the ratio of viscous forces to inertial forces during the spray process. High values of this number indicate that viscous forces dominate, leading to the formation of smaller droplets and a more uniform spray distribution. In contrast, the Weber number represents the ratio of inertial forces to surface tension forces. High Weber number values suggest that inertial forces dominate, promoting liquid jet breakup and resulting in smaller droplet formation. Key parameters influencing droplet size include nozzle geometry (such as orifice diameter and shape and cone angle at the nozzle outlet), liquid operating pressure, and ambient atmospheric conditions (temperature and humidity) during spraying. An analysis of the design and operational parameters of plant protection equipment using computational fluid dynamics (CFD), with particular attention to factors such as operating pressure, application height, nozzle type, fluid density, and viscosity, is presented in [5]. In agricultural spraying, plant protection products are dissolved in water along with adjuvants, which are substances added to the spray solution to enhance the effectiveness of the plant protection products. These substances increase surface wettability and reduce surface tension, facilitating the spreading of the spray solution over the plant's surface and improving coverage of hard-to-reach areas. The effect of adjuvant quantity on spray drift for different nozzles was analyzed in the paper [6]. A mixture of different chemicals affects the resulting viscosity of a liquid and its surface tension. High-viscosity liquids are less prone to fragmentation. Larger internal forces counteract the stretching of the liquid, leading to the formation of larger droplets. In contrast, high surface tension means that the liquid forms a more stable surface, making it more difficult to break up. Droplets formed from liquids with high surface tension tend to be larger and more spherical. However, water is often used in experimental spray tests due to the fact that plant protection products represent a health exposure for researchers.