Approximate Closed-Form Solution of the Differential Equation Describing Droplet Flight during Sprinkler Irrigation

Abstract

Sprinkler irrigation is widely used in agriculture because it allows for rational use of water. However, it can induce negative effects of soil erosion and of surface waterproofing. The scholars of these phenomena use the numerical integration of the equation of motion, but if there was an analytical solution, the study would be facilitated, and this solution could be used as software for regulating sprinklers. Therefore, in this study, the solution of the differential equation of the flight of droplets produced by sprinklers in the absence of wind was developed. The impossibility of an exact analytical solution to the ballistic problem due to the variability of the drag coefficient of the droplets is known; therefore, to find the integrals in closed form, the following were adopted: a new formula for the drag coefficient; a projection of the dynamic’s equation onto two local axes, one tangent and one normal to the trajectory and some linearization. To reduce the errors caused by the latter, the linearization coefficients and their calculation formulas were introduced through multiple non-linear regressions with respect to the jet angle and the initial droplet speed. The analytical modeling obtained, valid for jet angles from 10° to 40°, was compared to the exact numerical solution, showing, for the total travel distance, a high accuracy with a mean relative error MRE of 1.8% ± 1.4%. Even the comparison with the experimental data showed high accuracy with an MRE of 2.5% ±1.1%. These results make the analytical modeling capable of reliably calculating the travel distance, the flight time, the maximum trajectory height, the final fall angle and the ground impact speed. Since the proposed analytical modeling uses only elementary functions, it can be implemented in PLC programmable logic controllers, which could be useful for controlling water waste and erosive effects on the soil during sprinkler irrigation.