Weir Flow Calculator: Rectangular, V-Notch and Cipolletti Discharge
Discharge over a sharp-crested rectangular weir (suppressed or contracted), a V-notch weir of any angle (Kindsvater-Shen) or a Cipolletti weir from the head and crest size, in cfs, L/s, gpm and m³/h, with the Kindsvater-Carter check, the head for a target flow, validity checks and a rating table.
How it works
Suppressed rectangular (Francis): Q = 3.33 L H^1.5 · Contracted rectangular: Q = 3.33 (L − 0.2H) H^1.5 · Cipolletti: Q = 3.367 L H^1.5 · V-notch (Kindsvater-Shen): Q = 4.28 Ce tan(θ/2) (H + kh)^2.5, with Ce and kh from the notch angle (90°: Ce ≈ 0.578, kh ≈ 0.0029 ft) · Kindsvater-Carter check (suppressed): Q = (3.22 + 0.40 H/P)(L − 0.003)(H + 0.003)^1.5 · Q in cfs, L, H, P in feet; 1 cfs = 28.317 L/s = 448.83 gpm
A sharp-crested weir turns a flow measurement into a depth measurement. Water spilling over a thin, level crest forms a free-falling sheet (the nappe) whose discharge depends only on the head above the crest and the shape of the opening, so a gauge in a stilling well upstream reads the flow directly. The calculator uses the equations of the US Bureau of Reclamation Water Measurement Manual. Suppressed rectangular weirs (crest across the full channel width) and contracted weirs (crest narrower than the channel, so the flow contracts at both ends) use the Francis equation, where each end contraction removes 0.1 × H from the effective length. The Cipolletti weir has 1:4 sloping sides that offset the end contractions, giving Q = 3.367 L H^1.5. V-notch weirs use the Kindsvater-Shen equation for any angle, with the effective discharge coefficient Ce and head correction kh taken from fits to the USBR calibration curves; for a standard 90° notch the Cone formula Q = 2.49 H^2.48 is shown as a cross-check. Because flow grows with H^2.5, a V-notch still gives a readable head at very small flows, which is why 90° notches are standard for springs, small streams and treatment plants. If you enter the crest height above the channel bed, a suppressed weir is also rated with the Kindsvater-Carter method, whose coefficient rises with H/P and so includes the velocity of approach; the Francis value increasingly under-reads as the head approaches one third of the crest length. The validity checks follow USBR limits: head at least 0.2 ft (61 mm), head no more than L/3 for rectangular and Cipolletti weirs, V-notch head up to 1.25 ft, crest at least 2 × the head above the bed, and free (unsubmerged) flow.
Worked example
With the defaults, a 3 ft suppressed rectangular weir with 0.5 ft of head passes Q = 3.33 × 3 × 0.5^1.5 = 3.33 × 3 × 0.3536 = 3.532 cfs (100.0 L/s, 1,585 gpm). With the crest 1.5 ft above the bed, the Kindsvater-Carter coefficient is 3.22 + 0.40 × 0.5/1.5 = 3.353, giving 3.585 cfs, 1.5% more because it includes the approach velocity. To pass 10 cfs the head would have to rise to 1.00 ft, exactly the L/3 limit for a 3 ft crest, so a longer crest is advisable. A 90° V-notch at 0.5 ft of head carries 0.444 cfs (12.56 L/s) by Kindsvater-Shen and 0.446 cfs by the Cone formula.
Sources: USBR Water Measurement Manual, ch. 7 sec. 7: V-notch weirs of any angle (Kindsvater-Shen limits) · USBR Water Measurement Manual, ch. 7 sec. 9: standard contracted rectangular weirs, Q = 3.33 (L − 0.2h) h^1.5 · USBR Water Measurement Manual, ch. 7 sec. 10: standard suppressed rectangular weirs (limits) · USBR Water Measurement Manual, ch. 7 sec. 11: 90° V-notch Cone equation Q = 2.49 h^2.48 · USBR Water Measurement Manual, ch. 7 sec. 12: Cipolletti weir Q = 3.367 L h^1.5 · LMNO Engineering: V-notch weir calculator (Ce and kh fits by angle) · CED Engineering: Sharp-crested weirs for open channel flow measurement (worked examples)
Frequently asked questions
What is the formula for flow over a rectangular weir?
For a sharp-crested suppressed weir the Francis equation is Q = 3.33 L H^1.5 in cfs and feet (about Q = 1.84 L H^1.5 in m³/s and metres). With end contractions use the effective length L − 0.2H. For better accuracy, especially when the crest is low, the Kindsvater-Carter method adds a coefficient that grows with head over crest height.
How do I calculate flow over a V-notch weir?
A 90° V-notch passes Q = 2.49 H^2.48 cfs with H in feet (the Cone formula). For any angle from 25° to 100° use the Kindsvater-Shen equation Q = 4.28 Ce tan(θ/2)(H + kh)^2.5; for 90°, Ce is about 0.578 and kh about 0.003 ft. A 60° notch carries roughly 58% of the flow of a 90° notch at the same head.
Where should I measure the head?
Upstream of the weir, at a distance of at least four times the maximum head, where the water surface has not yet started to draw down toward the crest. Use a stilling well or a staff gauge zeroed exactly to the crest level, because a few millimetres of error change the flow by several percent.
What if the downstream water rises above the crest?
The weir is then submerged and the free-flow equations over-estimate the flow. Keep the tailwater at least 0.2 ft (6 cm) below the crest, raise the weir, or rate it with a submergence correction or a field calibration.
Rectangular, V-notch or Cipolletti: which should I use?
Use a V-notch for small or widely varying flows (roughly 0.05 to 4 cfs for a 90° notch), a rectangular or Cipolletti weir for larger flows. Suppressed rectangular weirs need ventilation under the nappe because the sidewalls trap air; contracted and Cipolletti weirs ventilate themselves.
Sharp-crested vs broad-crested weirs?
These equations are for a thin plate crest. Broad-crested weirs, spillways and culvert inlets have different coefficients (roughly Q = 2.6 to 3.1 L H^1.5 in US units) and are rated with other methods.
Related calculators
Preliminary engineering estimate using the standard USBR sharp-crested weir equations; final design or a legal flow record must be checked by a licensed engineer, and a weir that does not meet the installation criteria needs a field calibration.