Analytical solutions for sprinkler irrigation uniformity under idealised uniform radial water application rate: comparison of square and equilateral triangular layouts

Solid-set sprinkler irrigation systems are widely used in arid and semi-arid regions due to their ability to achieve uniform water application and support automation. However, accurate design is essential to optimise water use and energy efficiency. This study presents a novel analytical approach for assessing sprinkler performance under an idealised uniform radial pattern of water application rate (RP-WAR), considering both square and equilateral triangular layouts. Partial overlap areas were derived to determine weighted statistics, including Christiansen's uniformity coefficient (CU). For both sprinkler layouts, the dimensionless overlap fractions (OF) and sprinkler spacings required to achieve target CU values of 90 and 95 % were determined. A target CU of 95 % is achieved at OF values of 0.555 and 0.640 for square and equilateral triangular layouts, respectively, whereas a CU of 90 % corresponds to OF values of 0.546 and 0.441. Using 20 CU data from the literature, the procedure was verified, resulting in a relative error ranging from - 7.70 % to 3.57 %. Under different RP-WAR scenarios and sprinkler layouts, efficiency indices were introduced to quantify the trade-offs between the required number of sprinklers and energy savings, with a normalisation procedure that facilitates comparative analysis. Weighted aggregation (single-index) and Pareto-ranking methods provided complementary approaches for sprinkler decision-making, alongside crop uniformity requirements and soil hydraulic properties. Results showed that uniform RP-WAR generally maximise energy efficiency, while convex patterns show higher efficiency than concave patterns in both square and equilateral triangular layouts.