Physics calculator

Weir Discharge Calculator

Compute thin-plate weir discharge. Rectangular Q=(2/3) C_d L √(2g) H^{3/2}. V-notch Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}. Runs locally. Not Manning, not Q=Av, not Darcy, and not an orifice.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 15/15 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: Thin-plate weir discharge or head. Rectangular and 90° V-notch (θ adjustable).
Scope: Thin-plate weir; free nappe
Verification: Engine tested · 15/15 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 1438 — Hydrometry — Open channel flow measurement using thin-plate weirs — Rectangular and triangular thin-plate weirs
ISO 80000-4 — Mechanics — Volume flow rate
Chow, V.T. Open-Channel Hydraulics — Sharp-crested weirs
Sources
Evidence: 1 golden · 9 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 163 CURRENT) @ 2026-09-17T03:09:03.307Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

RectangularQ = (2/3) C_d L √(2g) H^{3/2}

Triangular / V-notchQ = (8/15) C_d √(2g) tan(θ/2) H^{5/2}

iSI g = 9.80665 m/s² unless given. Default C_d = 0.62 (rectangular) or 0.58 (90° V-notch). Francis 3.33 L H^{3/2}, Kindsvater–Carter contractions, Cipolletti, Manning, Q=Av, Darcy, and orifice plates are out of scope.

How to use

Choose the weir shape

Rectangular (suppressed thin-plate) or triangular V-notch. This is not a Manning or orifice page.

Enter head H or discharge Q

Give one, not both. H is the upstream head above the crest.

Enter L or θ

Rectangular needs crest length L. V-notch uses θ (default 90°).

Optional C_d

Leave blank for 0.62 / 0.58. C_d must be in (0, 1].

Example calculations

Common configurations with formula and result.

Rectangular 0.2 m head

L=1 m, H=0.2 m, default C_d=0.62

Q = (2/3)(0.62)(1)√(2g) (0.2)^{3/2}

Q in m³/s

90° V-notch

θ=90°, H=0.2 m, default C_d=0.58

Q = (8/15)(0.58)√(2g) tan(45°) (0.2)^{5/2}

Q in m³/s

Weir Discharge calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 15/15 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: Thin-plate weir. Rectangular: Q=(2/3) C_d L √(2g) H^{3/2}. Triangular: Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}.
What it calculates: Thin-plate weir discharge or head. Rectangular and 90° V-notch (θ adjustable).
Inputs: mode?
H | Q
L?
theta_deg?
Cd?
g?
Outputs: Q
H
Cd
g
L?
theta_deg?
Formula: Q = (2/3) C_d L √(2g) H^{3/2} | Q = (8/15) C_d √(2g) tan(θ/2) H^{5/2}
Assumptions: Thin-plate weir; free nappe
SI g=9.80665 unless given
Rectangular: Q=(2/3) C_d L √(2g) H^{3/2}; not Francis contractions
Triangular: Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}
Not Manning, not Q=Av, not Darcy, not an orifice
Units: SI; L and H in m; Q in m³/s; θ in degrees; g in m/s²
Boundary conditions: missing H or Q → MISSING_REQUIRED_INPUT
rectangular missing L → MISSING_REQUIRED_INPUT
H, Q, L, Cd, θ, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
H and Q together, or θ_deg and θ_rad → MIXED_INPUT_ENCODING
C_d > 1 → VALUE_OUT_OF_RANGE
Francis, Cipolletti, Manning, or orifice as mode → INVALID_MODE
Example: L=1 H=0.2 → Q=(2/3)(0.62)(1)√(2g) H^{3/2}
Validation cases: 3 published on this page · 15/15 tests · Production surface contract 3/3 · View evidence
L=1 H=0.2 → Q = (2/3) 0.62 L √(2g) H^{3/2}
mode=triangular H=0.2 → Q = (8/15) 0.58 √(2g) tan(θ/2) H^{5/2}
L=1 H=0.2 Cd=3.33 → VALUE_OUT_OF_RANGE
Sources: ISO 1438 — Hydrometry — Open channel flow measurement using thin-plate weirs — Rectangular and triangular thin-plate weirs
Supports: Thin-plate weir discharge from head over a sharp crest
ISO 80000-4 — Mechanics — Volume flow rate
Supports: Q is volume flow; H is a length (head)
Chow, V.T. Open-Channel Hydraulics — Sharp-crested weirs
Supports: Rectangular (2/3) C_d L √(2g) H^{3/2}; triangular (8/15) C_d √(2g) tan(θ/2) H^{5/2}
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute thin-plate weir discharge from upstream head.

Supported and not supported

Supported — rectangular and triangular (V-notch) thin-plate weirs · invert H from Q · API physics.fluid.weir

Not supported — Manning, Q=Av identities, Darcy–Weisbach, orifice plates, Francis 3.33 L H^{3/2}, Kindsvater–Carter contractions, Cipolletti, submerged nappe

Agent / API notes

Capability id: physics.fluid.weir · tool id: weir-discharge · pin 1.0.0.

{ "mode": "rectangular", "L": 1, "H": 0.2 }

V-notch: { "mode": "triangular", "H": 0.2 } (θ defaults to 90°). Invert: { "L": 1, "Q": 0.1638 }. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/weir-discharge. There is no /calc/fluid.

Other calculators in this family: Manning formula, Flow rate, Darcy–Weisbach, Orifice discharge, Pump power .

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Frequently asked questions

Key distinctions behind the calculation.

Is this the Manning formula?

No. Manning is uniform open-channel flow from n, A, R, and S at /calc/physics/manning-formula. A weir is a thin-plate overflow.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate.

Can I use the US Francis formula 3.33 L H^{3/2}?

No. That customary coefficient is out of scope. C_d must be in (0, 1]. Contractions (Kindsvater–Carter) are also out of scope.

Is this an orifice?

No. A hole under head or a pipe orifice plate is /calc/physics/orifice-discharge. A weir is a thin-plate overflow.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine weir wrapper.