Normalised numerical analysis of uniformity coefficients for a newly referenced sprinkler area under square and equilateral triangle spacing arrangements

Water distribution uniformity is essential in sprinkler irrigation systems, as it underpins water-use efficiency and is strongly influenced by sprinkler spacing. To support the optimal design of solid-set sprinkler systems, this study presents a theoretical-numerical approach, employing a highly refined integration step (IS), for analysing water uniformity using Christiansen’s uniformity coefficient (CU). A normalised approach is introduced, which considers a dimensionless description of the sprinkler radial pattern (RP) of water application rate (WAR) and a newly defined reference area assigned to each sprinkler, resulting in a square, for the square spacing arrangement, and in a hexagon, for the equilateral triangle spacing. The RP-WAR is described by a parabolic function parameterised by a shape parameter a, spanning convex and concave patterns. For each radial pattern, the dimensionless overlap fractions (OF) and sprinkler spacings required to achieve target CU values of 90 and 95 % are presented. Results showed that CU values of about 95 % can be obtained with OF in the range [0.419-0.631] for square spacing, and in the range [0.294-0.586] for triangle spacing, while for CU ≈ 90 %, the required OF are in the range [0.363-0.562] and [0.242-0.508], respectively. Error analysis revealed that IS, which can be interpreted as the sampling density in experimental runs, influences CU estimation. Employing IS corresponding to 13 sampling points along the radius offers an effective compromise between effort and accuracy, with relative errors in CU estimation around ±5%. Numerous applications were performed and the suggested procedure was verified according to literature data.