Flow equations for parabolic and elliptical weirs

Abstract Steady‐state flow across parabolic and elliptical weirs is addressed. This situation may arise in the cases of actual flow through such cross‐sections, as in an overflow pipe in the side of a vessel, through saturator troughs as employed in the textile finishing industry, or across weirs of these shapes. An exact analytical solution is found in the parabolic case. Specifically, the volumetric flowrate of fluid across a parabolic weir is shown to be directly proportional to the square of the liquid crest height; this result may be useful in certain process or environmental control applications. The solution for the elliptical configuration is a generalization of that presented earlier by Stevens (1957) in his classical paper on circular weirs. Whether the cross‐section is elliptical or circular, the resulting solutions contain the same types of elliptic integrals.