Weir Flow Calculator
Calculate volumetric flow rate over rectangular, triangular, or trapezoidal weirs from the head of water above the weir crest. Used in irrigation canals, wastewater treatment, and stream gauging.
Last updated: September 2026
Formula below · 2 sources (asme.org, Wikipedia) · Updated Sep 2026
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About this calculator
A weir is an overflow structure that relates the head H over its crest to the discharge Q. With C_d the dimensionless discharge coefficient and g = 9.81 m/s²: rectangular sharp-crested weir with end contractions (Francis), Q = (2/3) × C_d × (L − 0.1nH) × √(2g) × H^1.5, where L is crest length and n the number of end contractions, C_d ≈ 0.61–0.62 (equivalent to Q ≈ 1.83 (L − 0.1nH) H^1.5); 90° V-notch, Q = (8/15) × C_d × √(2g) × tan 45° × H^2.5, with C_d ≈ 0.58–0.59 (Q ≈ 1.38 H^2.5); trapezoidal (Cipolletti, 1:4 side slopes), Q = (2/3) × C_d × L × √(2g) × H^1.5, with C_d ≈ 0.63 (Q ≈ 1.86 L H^1.5); broad-crested, Q = C_d × L × √g × (2H/3)^1.5 = 1.705 × C_d × L × H^1.5, with C_d ≈ 0.85–1.0. Enter the coefficient that matches the weir type you choose. Head H is measured upstream of the weir at a distance of at least 4×H_max to avoid drawdown effects.
How to use
Example — rectangular weir: L = 2.0 m, H = 0.3 m, Cd = 0.62, end contractions n = 2, g = 9.81 m/s². Effective width = L − 0.1 × n × H = 2.0 − 0.06 = 1.94 m. √(2g) = 4.429. Q = (2/3) × 0.62 × 1.94 × 0.3^1.5 × 4.429 = 0.4133 × 1.94 × 0.1643 × 4.429 ≈ 0.584 m³/s. For a 90° V-notch with the same head and Cd = 0.585: Q = (8/15) × 0.585 × 0.3^2.5 × 4.429 = 0.312 × 0.04930 × 4.429 ≈ 0.068 m³/s — much lower, as expected for a narrow V-notch. A broad-crested weir 2 m long with Cd = 0.85: Q = 1.705 × 0.85 × 2 × 0.1643 ≈ 0.476 m³/s.
Frequently asked questions
What is the difference between a rectangular weir and a triangular V-notch weir for flow measurement?
A rectangular weir has a horizontal crest and passes larger flows efficiently, making it suitable for rivers, irrigation canals, and stormwater channels where high discharge must be measured. A triangular V-notch weir has greater sensitivity at low flows because even a small head change produces a measurable discharge change, making it ideal for stream gauging and laboratory measurements. The rectangular formula uses H^1.5 while the triangular formula uses H^2.5, so the V-notch discharge rises more steeply with head, giving better resolution at small flows. For a wide range of flows, a Cipolletti (trapezoidal) weir combines both benefits.
How does the discharge coefficient affect weir flow calculations?
The discharge coefficient Cd accounts for the difference between ideal theoretical discharge and actual flow, caused by streamline curvature, viscosity, surface tension, and approach velocity. For sharp-crested rectangular weirs, Cd is typically 0.61–0.65; broad-crested weirs have Cd around 0.848. An incorrect Cd introduces a direct proportional error into Q, so a 5% error in Cd means a 5% error in computed flow rate. Calibrated weir structures in flow measurement applications should have Cd determined by in-situ rating against a reference meter rather than using tabulated values, especially when approach velocity and channel geometry vary from standard conditions.
Why must the head be measured far enough upstream from the weir crest?
Close to the weir, the water surface curves downward (drawdown) as it accelerates toward the crest, making the local water depth lower than the true hydraulic head driving the flow. Measuring head within this zone underestimates H and therefore significantly underestimates Q. Standard practice (ISO 1438) requires the gauge point to be located at least 4 to 5 times the maximum expected head upstream of the crest, where the water surface is still approximately horizontal and hydrostatic pressure conditions apply. A stilling well connected by a small tap to the channel wall at that location eliminates wave interference and gives the most repeatable readings.