EXACT THEORETICAL RATING LAW FOR SHARP-CRESTED ELLIPTIC, SEMI-ELLIPTIC, AND CIRCULAR WEIRS

Abstract

The study derives an exact theoretical (loss-free) stage-discharge law for sharp-crested elliptic and semi-elliptic weirs, with the circular weir obtained as a rigorous specialization. The formulation cleanly separates universal hydraulic scaling from a dimensionless geometry-flow depth kernel, which is make analytically explicit by converting the defining integral into an exact Euler–Beta series valid on the full admissible range.

The kernel is shown to satisfy the two endpoint constraints by construction, square-root onset at small flow depth and the full-height anchor equal to 4/15, providing a transparent, first-principles reference for analysis and calibration. Building on this foundation, the study constructs a compact Padé-type surrogate with four-significant-figure coefficients that preserves the governing physics and achieves uniform, sub-0.05% deviation from the exact series; the worst case, = 0.04%, is near the upper range; even smaller at full height.

The result is a unified, practice-ready evaluator: differentiable for sensitivity, numerically stable across the entire range, and straightforward to implement in design tools and real-time control.

Because the theory isolates geometry from losses, discharge coefficients can be appended multiplicatively without contaminating the kernel, preserving the correct ordering between theoretical and actual discharge.

A table and error curve benchmark the Padé law against the exact series, confirming near-reference accuracy with computational economy suitable for design charts, calibration workflows, and embedded applications.

This work thus replaces empirical or fitted surrogates, with a shape-exact analytic kernel and a compact Padé evaluator of the theoretical discharge. Multiply the theoretical flow-rate relationship by the site-calibrated discharge coefficient C_d to obtain the operational stage-discharge relationship.

The in-depth sensitivity analysis section rigorously examines how variations in key physical and geometric parameters affect the theoretical discharge over elliptic and circular weirs.

In addition, the authors advance a suite of semi-empirical C_d -models, eight models, rigorously derived from canonical closed-form mathematical kernels, such as Hill/Michaelis–Menten saturations and stretched-exponential Weibull forms, and carefully tailored to the hydraulics of elliptic, semi-elliptic, and circular weirs. The in-depth analysis of the elasticity of C_d on each of the model parameters is also presented.