Weir Flow Calculator

Estimate discharge over V-notch, rectangular contracted, and rectangular suppressed (Francis) weirs from the measured head above the crest. Imperial forms use field constants; metric forms expose the discharge coefficient directly.

V-notch Imperial

2.49coef

Rectangular (Francis)

3.33coef

Typical C_d

0.58–0.62

g (metric)

9.81m/s²

Units Imperial uses field constants; metric uses C_d·√(2g)

Weir type Picks the formula applied

Notch angle θ (degrees) Full included angle of the V-notch

Head H (ft) Vertical depth of water above the crest, measured upstream

Discharge coefficient C_d Used in the metric form; Imperial uses field constants

Flow rate Q

0.44cfs

V-Notch, Imperial field form

Secondary units

198GPM

Weir type

V-Notch

Formula applied

Q = 2.49 tan(θ/2) H^2.5

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Tip Measure H upstream of the nappe drawdown, typically at least 3–4 H upstream of the crest. Measuring too close to the plate systematically under-reads head and under-reports flow.

Tip Why these coefficients? They fall out of integrating the velocity profile across the weir opening:

Rectangular weir (width L, head H) v(y) = √(2g(H − y)) ← ideal velocity at depth y Q = ∫₀ᴴ L · v(y) dy = L · √(2g) · ∫₀ᴴ √(H − y) dy = L · √(2g) · (2/3) · H^(3/2) The (2/3) is the integration of the velocity profile over the head. The Francis coefficient C_d ≈ 0.62 corrects for vena contracta and end contractions; the metric form Q = (2/3) · C_d · √(2g) · L · H^(3/2) lumps everything into the (2/3) · √(2g) · C_d ≈ 1.84 SI constant. V-notch (full angle θ) For a triangular cross-section the area grows linearly with depth, so the integration becomes ∫ y · √(H − y) dy and yields: Q = (8/15) · C_d · √(2g) · tan(θ/2) · H^(5/2) The (8/15) — vs the (2/3) for a rectangular weir — is exactly that triangle integral.

Worked example

90° V-notch, H = 0.5 ft, Imperial form.

1. Evaluate terms θ/2 = 45° → tan(45°) = 1 H^2.5 = 0.5^2.5 = 0.1768 2. Apply formula Q = 2.49 × tan(θ/2) × H^2.5 Q = 2.49 × 1 × 0.1768 Q ≈ 0.44 cfs ≈ 198 GPM

Formulas

Weir type	Imperial	Metric	Notes
V-Notch	Q = 2.49 tan(θ/2) H^2.5	Q = (8/15) C_d √(2g) tan(θ/2) H^2.5	Low-flow sensitivity.
Rectangular Contracted	Q = 3.33 (L − 0.2H) H^1.5	Q = (2/3) C_d √(2g) L H^1.5	Reduces L for end contractions.
Rectangular Suppressed (Francis)	Q = 3.33 L H^1.5	Q = (2/3) C_d √(2g) L H^1.5	Crest spans full width.
Cipolletti	Q = 3.367 L H^1.5	—	Trapezoidal notch; contraction correction is built in.

Common mistakes

Measuring H at the crest. Drawdown near the plate gives a false low head. Move the staff gauge at least 3–4 H upstream.
Submerged nappe. Once downstream water touches the underside of the nappe, sharp-crested equations stop applying.
Unventilated nappe. A rectangular weir spanning the full channel without side air access behaves differently from the Francis form.
Wrong notch angle. θ in the formula is the full included angle of the V, not half-angle. Using half doubles the error.

Warn Weir equations assume sharp crests, free discharge, and calm approach. For high-accuracy measurement follow ISO 1438, ASTM D5640, or a published standard for the installation.

FAQ

What is a weir used for?

A weir measures or regulates open-channel flow. Liquid passes over a known crest shape and the upstream head above the crest is used to estimate discharge without a mechanical flowmeter.

When should I use a V-notch vs rectangular weir?

Use a V-notch for lower flows or where sensitivity to small head changes matters. Use a rectangular weir for higher flows or where a wider crest suits the channel.

What is the discharge coefficient?

C_d corrects the ideal equation for viscosity, crest geometry, aeration, and contraction. It comes from calibration or published standards and is most visible in the metric form.

How accurate is weir flow measurement?

Very accurate when installation and operation are controlled. Poor head measurement, submergence, debris, or a worn crest can introduce large errors even if the formula is right.

What are the installation requirements for a weir?

Install the plate level, plumb, and sharp-crested, with a calm upstream reach, the head gauge well upstream of drawdown, and free aerated discharge downstream.

Sources

USBR Water Measurement Manual ISO ISO 1438 — thin-plate weirs ASTM D5640 weir practice USGS Water-Supply Paper 2175 — Measurement & Computation of Streamflow (Vol. 2, weirs & flumes)

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