Mathematical Modeling of Sprinkler Irrigation Process Using Stationary Machines

General Background: Sprinkler irrigation is widely used in agriculture to distribute water across cultivated land. Specific Background: Uniform irrigation depth is essential to avoid areas of excessive or insufficient watering that may reduce crop productivity and waste water resources. Knowledge Gap: Analytical approaches for evaluating spatial water distribution generated by stationary sprinkler systems remain limited. Aims: This study develops a mathematical model to represent the irrigation process produced by stationary sprinkler devices and to analyze droplet distribution over an irrigated field. Results: The model describes droplet dispersion using probability density functions where angular flow is assumed uniform and radial distance follows a normal distribution, incorporating nozzle characteristics and device geometry. Simulations generate spatial irrigation patterns and calculate indicators such as coefficient of variation and uniformity coefficient. Novelty: The study proposes a probability-based framework linking droplet trajectories with machine parameters for evaluating irrigation distribution. Implications: The model supports analysis of device spacing and operational settings to improve irrigation uniformity and agricultural water management. Highlights:• Probability Density Functions Represent Droplet Trajectories Produced by Rotating Heads.• Simulation Outputs Quantify Irrigation Depth Variability Across Cultivated Land.• Device Spacing and Operating Parameters Determine Field-Scale Uniformity Indicators. Keywords: Sprinkler Irrigation, Mathematical Modeling, Probability Density Function, Irrigation Uniformity, Agricultural Engineering.