Weir Discharge Coefficients — Reference
All values for free-flow (non-submerged), well-ventilated nappe, with the head H measured upstream of the drawdown (typically 4·H upstream of the crest). Submergence corrections required when Hdownstream/Hupstream > 0.6.
Sharp-Crested (Suppressed) Rectangular Weir
Q = Cd · L · H3/2 (US units, Q in cfs, L & H in ft)
| Source | Cd (US) | Notes |
|---|---|---|
| Rehbock formula (suppressed) | 3.27 + 0.40·H/P | P = weir height; valid 0.03 ≤ H/P ≤ 1.0 |
| Kindsvater-Carter (typical sharp-crested) | 3.33 | Common textbook design value |
| USBR (broad sharp-crest) | 3.32–3.36 |
Side-contracted (Francis): subtract 0.1H per contraction from L → effective length Le = L − 0.1nH (n = 1 or 2 contractions).
Broad-Crested Weir / Spillway
Q = Cd · L · H3/2 (US units)
| Crest geometry | Cd (US) | Notes |
|---|---|---|
| Square-edged broad-crest | 2.6–3.1 | varies with H/Lcrest |
| Rounded upstream edge (r = 0.1L) | 3.0–3.3 | |
| Ogee spillway (design head Hd) | 3.95 | maximum at design head; lower at partial heads |
| Ogee spillway (H = 0.5·Hd) | 3.6 | |
| Roadway / parking lot overtopping | 2.7–3.0 | FHWA HEC-22, paved |
V-Notch (Triangular) Weir — Free Discharge
Q = (8/15)·Cd·tan(θ/2)·√(2g)·H5/2, often simplified to Q = K·H5/2
| Notch angle (θ) | Cd | K (US, Q in cfs, H in ft) |
|---|---|---|
| 22.5° | 0.611 | 0.497 |
| 30° | 0.585 | 0.685 |
| 45° | 0.581 | 1.035 |
| 60° | 0.577 | 1.443 |
| 90° (most common) | 0.578 | 2.49 |
| 120° | 0.580 | 4.34 |
Valid for H > 0.2 ft and H/P < 0.4 (P = weir height above channel floor). Below H = 0.2 ft, surface tension errors dominate — use a reduced coefficient or a different measurement device.
Cipolletti (Trapezoidal) Weir
Side slopes 1H:4V, designed so contraction loss compensates for end contractions. Q = 3.367 · L · H3/2, with Cd ≈ 0.63 (SI form).
SI form (metric) of the same equations
| Weir | Equation (SI: Q in m³/s, L & H in m) |
|---|---|
| Sharp-crested rect. | Q = (2/3) · Cd · L · √(2g) · H3/2; Cd ≈ 0.62 |
| Broad-crested | Q = Cd · L · √g · (2H/3)3/2; Cd ≈ 0.85–1.0 |
| V-notch (90°) | Q = (8/15) · 0.578 · tan(45°) · √(2g) · H5/2 = 1.36 · H5/2 |
Sources: USBR Water Measurement Manual 3rd ed. (2001), Chapter 7. Bos, M.G. (1989), Discharge Measurement Structures, ILRI Publication 20. Brater & King, Handbook of Hydraulics, 7th ed. USGS WSP 200 series.
When to use a weir vs. orifice vs. flume
Weirs are the right choice when you have free fall downstream (no tailwater backing into the structure) and a wide range of flows to measure. V-notch wins for low flows where a wide rectangular notch would have near-zero head; rectangular sharp-crest wins for higher flows where the notch would be impractically tall. Broad-crested structures (including spillways) handle very high flows where the structural mass and submergence tolerance both matter. Switch to orifice flow when the device is fully submerged and you need a known driving head — common in detention pond outlet structures.
For full pond and outlet-structure design with stage-storage and stage-discharge routing, see HydroComplete — the SaaS sister product to PE-Calc.