Weir Flow Calculator

A sharp-crested weir converts a single depth measurement into a discharge. Because the velocity of a freely falling jet varies with the square root of head above the crest, integrating over the flow depth gives a discharge scaling with head to the 3/2 power for a rectangular notch. For a triangular V-notch the flow width also grows with depth, adding one further power of head and producing a 5/2 exponent. The discharge coefficient, near 0.62 for a sharp-crested rectangular weir and about 0.58 to 0.60 for a 90 degree V-notch, absorbs contraction of the nappe, viscous and surface tension effects, and the velocity of approach in the upstream pool. The steeper exponent is what makes the V-notch attractive at low flow: a small change in discharge produces a comparatively large change in head, so measurement resolution improves where it is most needed. Practical head ranges are roughly 0.06 to 0.38 m for a 90 degree V-notch and 0.05 to 0.60 m for a rectangular weir. Accuracy depends far more on installation than on arithmetic. The crest must be level and genuinely sharp, the nappe must be ventilated so it springs clear rather than clinging to the plate, head should be read about 4 to 5 times the maximum head upstream of the crest to stay outside the drawdown zone, and tailwater must remain below the crest. Submergence, a rounded or fouled crest, or sediment accumulating in the approach pool can each shift the indicated flow by well over 10 percent.

The 3/2 power relationship for flow over a weir was derived by the Italian mathematician and physicist Giovanni Poleni, whose De Motu Aquae Mixto of 1717 integrated the Torricelli velocity distribution across the depth of the overfall and obtained the two-thirds coefficient that still appears in the rectangular weir equation. Poleni was working on the hydraulics of Italian rivers and irrigation canals, where measuring canal deliveries was a legal question as much as an engineering one. Reliable coefficients required experiment rather than derivation. James Bicheno Francis, chief engineer of the Proprietors of the Locks and Canals at Lowell, Massachusetts, ran an extensive programme of full-scale weir tests in order to apportion Merrimack River water among the textile mills, publishing the Lowell Hydraulic Experiments in 1855. His results yielded the widely used correction that reduces effective crest length by 0.1 times the head for each end contraction. The V-notch weir was introduced by James Thomson, professor of engineering at Belfast and later Glasgow and elder brother of Lord Kelvin, who reported experiments on triangular notches in the late 1850s and demonstrated their advantage at small discharges. Twentieth century refinement came from Theodor Rehbock in Germany during the 1920s and from Carl Kindsvater and Rolland Carter, whose 1957 work on effective head and effective length underlies the modern ISO 1438 and ASTM weir standards.