Construction and validation of the improved Ballistic model for sprinkler water trajectory considering jet fragmentation and droplet shape

Spray droplet trajectory can often be used to calculate the water and energy distribution of the nozzle. The spray droplet movement and distribution patterns can greatly contribute to the design of sprinkler systems under windy, evaporative, and sloping conditions. Experimental tests, CFD simulations, and theoretical calculations have been commonly used to explore the characteristics of spray droplets. The ballistic trajectory model in the theoretical calculation has effectively simulated the spray droplet distribution using droplet dynamics, due to the high operational efficiency and small test volume. However, the previous model suffers from an oversimplification of the droplet break-up process and the shape of the motion. It is necessary to further improve the accuracy of the model using more realistic parameters for the jet break-up process and droplet shape. In this study, an energy-weighted droplet equivalence index was proposed to modify the initial conditions of the ballistic trajectory model, the kinematic droplet shape parameters, and the kinematic droplet drag coefficient. A model of spray droplet distribution characteristics was established using the improved equation for the ballistic trajectory. The velocity, particle size, and angle of spray droplet landing were simulated by inputting parameters, such as the nozzle diameter, working pressure, and nozzle elevation angle. The accuracy of the model was verified using an HY50 turbine drive sprinkler. A comparison was made on the differences between the current model and the simulated values of the conventional ballistic trajectory model. The modified model was compared using the characterized drag coefficient under four common operating conditions. An analysis was implemented on the effects of different operating pressures, nozzle diameters, nozzle elevation angles, and mounting heights on the droplet size, velocity, and angle at the end of the range. The results show that the MAE of the improved model was reduced than before by 43.3% and 75.1% (landing velocity), 51.8% and 27.1% (landing position), and 61.4% and 76.1% (landing angle), respectively, in the nozzle diameter of 20 mm and working pressure of 0.35 MPa. Taking the landing velocities from the improved model, the average RMSEs were 0.53, 0.93, and 2.21 m/s, respectively, while the average NRMSEs were 0.10, 0.17, and 0.40, respectively, in the Fukui’s and Li’s models under the four optimal conditions. The greater the nozzle diameter and working pressure were, the greater the droplet particle size and landing velocity were at the end of the range, and the smaller the landing angle was. The greatest effect was found in the nozzle working pressure variation on the droplet particle size, the nozzle diameter variation on the droplet landing velocity, and the nozzle elevation angle on the droplet landing angle. The least effect was found in the nozzle mounting height on the end droplet characteristic parameters. Therefore, the jet fragmentation and droplet shape can be expected to focus on when building the model. But the sprinkler system can be inevitably affected by multiple factors, such as the topography, wind, evaporation, and deflector pipes when operating in the field. The droplet distribution of sprinklers should be simulated by the multiple factors at the farm scale, in order to improve the generalizability and practicality of the current model. This finding can provide new ideas to simulate the spray droplet, water, and energy using distributions of ballistic trajectories.