Weir Flow Calculator

Discharge over a thin-plate or broad-crested weir from the measured head, or the head needed to pass a given flow. Rectangular weirs use the Kindsvater-Carter method of ISO 1438, V-notches the Kindsvater-Shen method; Francis, Cipolletti and broad-crested forms are included. Every result is checked against the standard's validity limits.

Inputs

Calculate

Weir type

Kindsvater-Carter: suppressed (b = B) or side-contracted, approach velocity included through h/P.

Length unit

Head over crest (h)

Measured upstream of the plate, at least 4 × h_max back, from the still-water surface to the crest or notch vertex.

Crest width (b)

Approach channel width (B)

Sets the contraction ratio b/B. Equal to the crest width for a suppressed weir (no side contraction).

Crest height above channel bed (P)

Vertical distance from the approach channel bed to the crest (or the notch vertex).

Flow display unit

Results

Flow rate

168.0 L/s

Kindsvater-Carter (ISO 1438)

Flow rate0.1680 m³/s

Discharge coefficient0.6320

Effective head h_e0.2010 m

Head correction k_h1.00 mm

Effective width b_e0.9991 m

Width correction k_b-0.9000 mm

h / P0.4000

b / B1.000

Approach velocity0.2400 m/s

Inputs and outputs explained

Head h. The height of the undisturbed upstream water surface above the crest (rectangular, Cipolletti, broad-crested) or above the vertex of the notch (V-notch). Measure it at least four times the maximum head upstream of the plate, where the surface has not yet started to draw down, and never on the plate itself.
Crest width b. The horizontal length of the crest across the flow. For a Cipolletti weir it is the bottom width of the trapezoid.
Approach channel width B. The width of the channel just upstream of the weir. For the rectangular weir it fixes the contraction ratio b/B that selects the coefficient line; for all types it gives the approach velocity Q / (B·(P + h)).
Crest height P. The height of the crest (or notch vertex) above the bed of the approach channel. h/P is the main approach-velocity parameter: the lower the crest, the faster the water arrives and the higher the effective coefficient.
Notch angle θ. The full included angle of the V. 90° is standard; 20°–100° is the calibrated range. The coefficient and the head correction both depend on it.
Crest length L. Broad-crested weirs only: the length of the horizontal crest in the flow direction, and the radius of its rounded upstream edge. L ≥ 2 H (Bos: H/L ≤ 0.5) lets the flow become parallel over the crest (the condition the critical-depth formula assumes); L − r is the crest length the boundary layer grows over in the round-nosed coefficient.
Discharge coefficient (Ce or Cd). The ratio of the real discharge to the ideal one the integration of Bernoulli gives. For the thin-plate forms Ce is the effective coefficient of ISO 1438 (around 0.58–0.63); the Francis and Cipolletti constants (1.838 and 1.859 m½/s) have the ⅔·√2g folded in; the broad-crested Cd (0.85–1.0) multiplies the ideal 1.705.
Effective head and width. The Kindsvater methods add small fixed corrections — k_h = 1 mm on the head, k_b of a few millimetres on the width — that account for viscosity and surface tension so the coefficient itself stays a clean function of geometry. For the broad-crested weir the equation runs on the total head H = h + V²/2g, found by iteration.
Approach velocity. The mean velocity in the approach channel, Q / (B·(P + h)). The ISO 1438 rectangular and broad-crested forms account for it; the Francis and Cipolletti formulae do not, so the calculator warns when it exceeds about 0.15 m/s.
Head-discharge table. The rating curve of the weir as configured: the discharge for a fixed ladder of heads, with a validity flag per row — the quickest way to see what head range a given weir can measure reliably.

Worked example — rectangular weir in a 1.0 m channel, with a V-notch alternative

The calculator's defaults. A 1.0 m wide concrete channel has a suppressed sharp-crested weir (b = B = 1.0 m) with its crest 0.5 m above the bed, and the gauge 1 m upstream reads a head of 200 mm.

Contraction ratio b/B = 1.0 (suppressed) and h/P = 0.20 / 0.50 = 0.40, so the ISO 1438 coefficient line is Ce = 0.602 + 0.075 · 0.40 = 0.6320.
Corrections: k_b = -0.9 mm for b/B = 1 gives an effective width b_e = 0.9991 m; k_h = 1 mm gives an effective head h_e = 0.201 m.
Discharge Q = 0.6320 · ⅔ · √(2·9.80665) · 0.9991 · 0.201^1.5 = 0.1680 m³/s = 168.0 L/s. The approach velocity Q / (B·(P + h)) = 0.240 m/s is already accounted for through the h/P term.
Francis cross-check: for a suppressed weir Q = 1.838 · b · h^1.5 = 0.1644 m³/s, -2.1 % against the ISO value — the classic formula has no approach-velocity term, which is exactly the size of the gap at this h/P.
At the low end of the range: the same rectangular weir at its minimum reliable head of 30 mm passes only 9.8 L/s. A 90° V-notch with its vertex 0.6 m above the bed in a 1.2 m channel (Ce = 0.5779, k_h = 0.88 mm) reads that flow at a head of 138 mm — 4.6 × more head to measure (h/P = 0.23, fully contracted), which is why a V-notch resolves small flows so much better than a wide rectangular crest.

These figures are computed by the same functions the calculator runs; the defaults reproduce step 3, and switching the type to "Francis" reproduces step 4. The standards quote ±1–2 % for a well-built, well-gauged thin-plate weir inside its limits.

Assumptions and limits

Free (modular) flow: the downstream water level stays below the crest — ISO 1438 asks for at least 50 mm of clearance — and the underside of the nappe is fully aerated. A clinging (unvented) nappe passes more flow than the equations say; a submerged weir passes less than the upstream head indicates, so the free-flow equation over-reads.
A thin-plate weir has a square-edged crest 1–2 mm thick, bevelled on the downstream side; a thick or rounded crest behaves as a broad-crested weir with a different coefficient.
The approach channel is straight, uniform and long enough (≥ 10 × B) for a smooth, symmetrical velocity distribution; the head is read ≥ 4 × h_max upstream in a stilling well, not at the plate.
Kindsvater-Carter is valid for h ≥ 0.03 m, b ≥ 0.15 m, P ≥ 0.10 m, h/P ≤ 2.5 and, for contracted weirs, (B − b)/2 ≥ 0.10 m; coefficient lines are tabulated at b/B = 0, 0.2, 0.4 … 1.0 and interpolated between them. Kindsvater-Shen is for fully contracted notches: 20° ≤ θ ≤ 100°, 0.05 ≤ h ≤ 0.38 m, h/P ≤ 0.4, h/B ≤ 0.2, P ≥ 0.45 m, B ≥ 0.9 m.
Francis and Cipolletti are fixed-coefficient formulae (USBR: ±5 % for Cipolletti) that neglect approach velocity; use them for h ≥ 0.06 m, h ≤ b/3, P ≥ 2h (3h for a suppressed Francis weir, which has no end contractions to hide an approach-velocity error) and side contractions ≥ 2h.
The broad-crested form assumes critical flow with parallel streamlines over the crest, which needs H/L ≤ 0.5, i.e. L ≥ 2 H (Bos §4.1.4). The round-nosed Cd is eq. 4-2 of the 1976 edition of Bos, Discharge Measurement Structures (quoting BS 3680-4B; the form behind ISO 4374), with the boundary-layer factor x = 0.005 of a well-finished field structure; its limits are h ≥ max(0.06 m, 0.05 L), 0.05 ≤ H/L ≤ 0.5, H/P ≤ 1.5, P ≥ 0.15 m, r ≥ 0.11 H and a crest at least 0.30 m, H and L/5 wide, and its uncertainty is ±2(21 − 20 Cd) % (eq. 4-3). The 1989 edition replaces the coefficient by Cd = 0.93 + 0.10 H/L with ±(3|H/L − 0.55| + 4) %; the two agree closely on short crests at moderate H/L (0.977 against 0.964 on the worked example) but differ by up to about 10 % at low H/L on long crests. Square-edged or very short crests need a measured or manually entered Cd.
Water at ordinary temperatures. Viscosity and surface-tension effects are absorbed by the k_h and k_b corrections and the minimum-head rule; the methods are not calibrated for viscous or foaming liquids.

Weirs are open-channel devices and are not part of the SimuPipe pressure-network solver; for closed-pipe flow measurement see the ISO 5167 orifice plate calculator. The solver's own accuracy record is on the validation page.

References

The equations, coefficient tables and limits on this page come from:

ISO 1438:2017. Hydrometry — Open channel flow measurement using thin-plate weirs. iso.org — the Kindsvater-Carter rectangular and Kindsvater-Shen triangular methods, coefficient tables, k_h / k_b corrections and the installation limits used for the warnings.
US Bureau of Reclamation. Water Measurement Manual, 3rd ed. rev. (2001), Chapter 7 — Weirs. usbr.gov — the same two methods in US units (Table 7-1 and Figure 7-4 reproduce the Ce and k_b values used here), the Francis and Cipolletti equations and their limits.
Kindsvater, C. E. and Carter, R. W. (1957). "Discharge characteristics of rectangular thin-plate weirs". Journal of the Hydraulics Division, ASCE 83(HY6). — the original rectangular thin-plate study behind the effective-coefficient method.
Shen, J. (1981). Discharge Characteristics of Triangular-Notch Thin-Plate Weirs. USGS Water-Supply Paper 1617-B. pubs.usgs.gov — the triangular-notch coefficient and head-correction curves the V-notch fits reproduce.
Bos, M. G. (ed.) (1976). Discharge Measurement Structures, 1st ed., ILRI Publication 20, Wageningen. edepot.wur.nl — §4.1 of the 1976 first edition (the linked PDF): the round-nose broad-crested weir discharge equation (4-1), the boundary-layer discharge coefficient (4-2) used here, its uncertainty (4-3) and the limits of application (§4.1.4). The 1989 third edition gives a different coefficient, Cd = 0.93 + 0.10 H/L, with its own uncertainty expression.
ISO 4374:1990. Liquid flow measurement in open channels — Round-nose horizontal broad-crested weirs. — the standard for round-nosed horizontal broad-crested weirs.
Chow, V. T. (1959). Open-Channel Hydraulics, McGraw-Hill. — the critical-depth derivation of the ideal broad-crested coefficient 1.705 and the general theory of sharp-crested weirs.

How a weir measures flow

A weir is a plate or wall across an open channel that the water must flow over. Upstream of it the water backs up to a level set only by the discharge, so measuring that level — the head above the crest — measures the flow. The relation between head and discharge depends on the shape of the opening and on how cleanly the water separates from the crest, which is what the discharge coefficient and the standards' installation limits are about.

Rectangular sharp-crested weir (Kindsvater-Carter, ISO 1438)

Integrating the velocity √(2g·y) over a rectangular opening gives the ideal discharge ⅔·√(2g)·b·h^1.5; the real nappe contracts, so a coefficient of about 0.6 multiplies it. Kindsvater and Carter made that coefficient a clean function of geometry by moving the viscous and surface-tension effects into two small fixed corrections:

Q = C_e \cdot \tfrac{2}{3} \sqrt{2g} \; (b + k_b) \, (h + k_h)^{3/2}

$Q$ — discharge (m³/s)
$C_e$ — effective discharge coefficient, a straight line in h/P whose slope and intercept depend on b/B (0.602 + 0.075·h/P for a suppressed weir)
$g$ — gravitational acceleration, 9.80665 m/s²
$b$ — crest width (m)
$k_b$ — width correction (m): +2.4 mm for a narrow contracted weir, rising to +4.2 mm at b/B = 0.8, −0.9 mm for a suppressed weir
$h$ — measured head above the crest (m)
$k_h$ — head correction, 0.001 m

For the suppressed weir (crest spanning the full channel, b = B) the coefficient line is

C_e = 0.602 + 0.075 \, \frac{h}{P} \qquad (b = B)

$P$ — crest height above the approach-channel bed (m); h/P carries the approach-velocity effect
$B$ — approach channel width (m); b/B is the contraction ratio that selects the coefficient line

The older Francis formula, still common in US practice, folds the coefficient and the ⅔·√2g into one constant and handles end contractions by shortening the crest by 0.1h per side:

Q = 1.838 \, (b - 0.2h) \, h^{3/2}

$1.838$ — 3.33 ft½/s converted to SI (m½/s)
$b - 0.2h$ — effective crest width for two end contractions (m); use b alone for a suppressed weir

V-notch (triangular) weir (Kindsvater-Shen)

A triangular notch has no crest length, so the discharge grows with the 2.5 power of head: at low flows the head is still large enough to read accurately, which is why V-notches are the standard choice below a few tens of litres per second.

Q = C_e \cdot \tfrac{8}{15} \sqrt{2g} \; \tan\!\left(\tfrac{\theta}{2}\right) (h + k_h)^{5/2}

$\theta$ — full included angle of the notch (degrees); 90° is standard, 20°–100° is calibrated
$C_e$ — effective coefficient, 0.578 at 90°, from the Kindsvater-Shen curve (a function of θ alone for a fully contracted notch)
$k_h$ — head correction (m), about 0.9 mm at 90° and larger for narrow notches

For a 90° notch the equation collapses to the familiar Q ≈ 1.38·h^2.5 in SI units. Full contraction — the condition the coefficient curve was measured under — needs h/P ≤ 0.4, h/B ≤ 0.2, P ≥ 0.45 m and B ≥ 0.9 m.

Cipolletti (trapezoidal) weir

Cipolletti sloped the sides at 1 horizontal to 4 vertical so that the extra area in the triangles makes up for the end contractions, and the weir behaves like a suppressed rectangular one of width b:

Q = 1.859 \, b \, h^{3/2}

$1.859$ — the USBR constant 3.367 ft½/s in SI units (1.859 m½/s), velocity of approach neglected, accuracy about ±5 %
$b$ — bottom (crest) width of the trapezoid (m)

Broad-crested weir

A crest long enough in the flow direction for the streamlines to become parallel forces the flow through critical depth, two thirds of the total head. Writing the discharge at that section gives

Q = C_d \left(\tfrac{2}{3}\right)^{3/2} \sqrt{g} \; b \, H^{3/2} = 1.705 \, C_d \, b \, H^{3/2}

$C_d$ — discharge coefficient accounting for the boundary layer on the crest and abutments (below), about 0.85 for a square-edged crest at low H/L
$H = h + V^2/2g$ — total head on the crest (m): measured head plus the velocity head of the approach flow, solved by iteration (Bos's approach-velocity coefficient Cv applied exactly)
$V$ — approach velocity Q / (B·(P + h)) (m/s)

The 1.705 is (⅔)^1.5·√g — the ideal coefficient for critical flow — and is where the "C = 1.7" of textbook broad-crested formulae comes from. For a round-nosed horizontal crest Bos (eq. 4-2, from BS 3680-4B) gives the coefficient from the boundary-layer displacement on the crest and the abutments:

C_d = \left[1 - \frac{2x(L - r)}{b}\right]\left[1 - \frac{x(L - r)}{h}\right]^{3/2}

$x$ — boundary-layer factor: 0.005 for a well-finished concrete field structure, 0.003 for a laboratory weir
$L$ — crest length in the flow direction (m), between 2 and 20 × H (Bos: 0.05 ≤ H/L ≤ 0.5)
$r$ — radius of the upstream nose (m), at least 0.11 × H

Which weir?

V-notch for small flows and wide turndown (a 90° notch reads 1 L/s at 55 mm and 100 L/s at 350 mm); rectangular for larger flows where the head would otherwise become impractical; Cipolletti where a simple, self-correcting contracted weir is wanted; broad-crested where debris, a long service life or an existing structure rules out a thin plate. In every case the head gauge and the approach conditions decide the accuracy more than the choice of formula.

The derivations, installation rules and a worked comparison of the methods are in the article how weirs measure flow. For flow measurement inside a pipe use the orifice plate calculator; for the units, the engineering unit converter. To size the pipework that carries the measured flow onwards, try SimuPipe.

Frequently Asked Questions

How do I calculate the flow rate over a weir?

Measure the head h — the height of the still upstream water surface above the crest — at least four times the maximum head upstream of the plate, then apply the equation for the weir shape. Rectangular: Q = Ce·⅔·√(2g)·(b + k_b)·(h + k_h)^1.5 with Ce ≈ 0.602 + 0.075·h/P for a suppressed weir; 90° V-notch: Q ≈ 1.38·h^2.5; Cipolletti: Q = 1.859·b·h^1.5; broad-crested: Q = 1.705·Cd·b·H^1.5. This calculator applies the ISO 1438 / USBR versions of those equations with their corrections and limits.

Rectangular or V-notch weir — which should I use?

A V-notch for small flows: because its discharge goes with h^2.5, the head stays large enough to read accurately when the flow is small, and one 90° notch covers roughly 1 to 100 L/s. A rectangular weir for larger flows, where a V-notch would need an impractical head; a 1 m suppressed crest passes about 170 L/s at 200 mm of head. Cipolletti weirs sit between the two for contracted installations.

What is the difference between a suppressed and a contracted weir?

A suppressed weir's crest spans the full channel width (b = B), so the nappe is not contracted at the sides; it needs an aeration vent under the nappe because the sidewalls trap air. A contracted weir is narrower than the channel, and the nappe pulls in from each end, reducing the effective width. ISO 1438 handles both through the contraction ratio b/B, which selects the coefficient line and the width correction k_b; the Francis formula instead shortens the crest by 0.1h per end.

Where should the head be measured?

Upstream of the plate, at a distance of at least four times the maximum head, where the surface has not yet begun to draw down towards the crest — a stilling well connected to the channel side is standard. Measuring at the plate reads the already-dropped surface and under-estimates the flow. The zero of the gauge must be set to the crest level (or the notch vertex) carefully: a 2 mm zero error is a 1.5 % flow error at 200 mm of head on a rectangular weir and 2.5 % on a V-notch.

How accurate is a weir flow measurement?

ISO 1438 puts the uncertainty of the discharge coefficient at about 1 % for a rectangular weir and 1–2 % for a V-notch, when the weir is built, installed and gauged to the standard's rules; the Francis and Cipolletti formulae are around 3–5 %. Head-measurement error usually dominates in practice, because the discharge goes with h^1.5 or h^2.5 — a 1 % head error is a 1.5 % or 2.5 % flow error.

What happens if the weir is submerged or the nappe is not aerated?

The equations assume free (modular) flow: the downstream level stays below the crest and air circulates freely under the falling sheet of water. If the tailwater rises above the crest, the weir is submerged: the downstream water backs the flow up, and at a given upstream head the weir passes less flow than the free-flow equation gives, so the equation over-reads (submergence corrections such as Villemonte's exist but are approximate). The opposite happens if the nappe clings to the plate because the space beneath it is not vented: the flow is then higher than computed, typically by several percent. Keep at least 50 mm of clearance below the crest and vent suppressed weirs.

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