Weir Flow Discharge Calculator
Estimate volumetric flow rate (discharge) over a weir structure given its width, head height, and discharge coefficient. Used by hydraulic engineers designing spillways, irrigation channels, and stormwater outlets.
About this calculator
A weir is a barrier across a channel that controls and measures flow. For a rectangular weir, discharge is calculated using the formula Q = Cd × L × √(2g) × H^1.5, where Cd is the discharge coefficient (typically 0.6–0.62), L is the weir crest width in metres, g is gravitational acceleration (9.81 m/s²), and H is the head height above the crest in metres. The H^1.5 exponent reflects how flow accelerates non-linearly as water depth increases. The discharge coefficient accounts for real-world energy losses due to contraction and friction at the weir edge. Triangular and trapezoidal weirs use modified versions of this relationship. Accurate measurement of head height H is critical, as small errors are amplified by the 1.5 power relationship.
How to use
Suppose you have a rectangular weir with width L = 2.0 m, head height H = 0.4 m, and a discharge coefficient Cd = 0.62. Plug into Q = Cd × L × √(2 × 9.81) × H^1.5: √(2 × 9.81) = √19.62 ≈ 4.429. Then H^1.5 = 0.4^1.5 = 0.253. So Q = 0.62 × 2.0 × 4.429 × 0.253 ≈ 1.39 m³/s. This means roughly 1.39 cubic metres of water flows over the weir every second under these conditions.
Frequently asked questions
What is a typical discharge coefficient value for a rectangular weir?
For a sharp-crested rectangular weir, the discharge coefficient Cd typically ranges from 0.60 to 0.62. This value accounts for the contraction of the water jet (vena contracta) as it passes over the crest and energy losses due to friction. Broad-crested weirs have lower coefficients, often around 0.50–0.55. The exact value depends on weir geometry, crest condition, and approach velocity, and is sometimes determined experimentally for critical applications.
How does head height affect weir discharge?
Head height H has a disproportionately large effect on discharge because it appears raised to the power of 1.5 in the weir formula. Doubling the head height increases discharge by a factor of 2^1.5 ≈ 2.83, not simply 2. This nonlinear sensitivity makes accurate measurement of H essential — even a few centimetres of error can lead to significant underestimation or overestimation of flow. Staff gauges or pressure transducers are commonly used to measure head precisely in field conditions.
When should I use a triangular weir instead of a rectangular weir?
A triangular (V-notch) weir is preferred for measuring low flow rates because its geometry concentrates flow at the notch apex, giving a measurable head even at very small discharges. Rectangular weirs perform better at higher flow rates where a wide crest provides stable measurement. Trapezoidal weirs (Cipolletti weirs) offer a compromise, maintaining more accurate measurement across a broader range of flows. The choice depends on the expected flow range, required accuracy, and the hydraulic conditions of the channel.