Broad-Crested Weir Calculator
Calculate the volumetric flow rate over a broad-crested weir, or solve for any unknown: head, weir length, or discharge coefficient. Enter your dimensions, choose metric or imperial units, and get discharge, critical depth, flow velocity at the crest, Froude number, and a full worked solution. Switch the solve mode to find whichever variable you need.
Formula
Worked example
Weir length L = 2.0 m, head H = 0.5 m, Cd = 1.705 (theoretical). Q = 1.705 * 2.0 * sqrt(9.807) * 0.5^1.5 = 1.705 * 2.0 * 3.132 * 0.3536 = 3.794 m^3/s (approx). Critical depth Hc = (2/3) * 0.5 = 0.333 m. Velocity at crest V = 3.794 / (2.0 * 0.333) = 5.69 m/s. Fr = 5.69 / sqrt(9.807 * 0.333) = 5.69 / 1.807 = 3.15 - note: a Froude number this high signals that the short crest is not fully establishing critical flow; increase crest width or use a longer weir.
What is a broad-crested weir?
A broad-crested weir is a flat-topped hydraulic control structure built across an open channel. Unlike a sharp-crested (thin-plate) weir where the nappe springs free of the crest, a broad-crested weir has a crest length long enough - typically at least twice the upstream head (b >= 2H) - for the flow to pass through a region of critical depth before it drops away downstream. This critical-flow condition is the key feature: because critical depth depends only on the discharge and channel width, the structure acts as a self-calibrating flowmeter. Broad-crested weirs are used worldwide for irrigation offtakes, flood-control spillways, stream gauging stations, and fish-passage structures because they are robust, easy to construct, and tolerant of sediment and debris compared with thin-plate alternatives.
The discharge formula and how it works
The standard discharge equation for a broad-crested weir under free-flow (unsubmerged) conditions is Q = Cd * L * sqrt(g) * H^(3/2), where Q is the volumetric flow rate, Cd is the discharge coefficient, L is the weir crest length perpendicular to flow, g is gravitational acceleration (9.807 m/s^2 or 32.174 ft/s^2), and H is the upstream head above the crest. The theoretical value of Cd is derived from critical flow theory as (2/3)^(3/2) * sqrt(g) = 1.705 (in SI units). In practice Cd ranges from about 1.40 for a rough-edged crest up to 1.80 for a streamlined ogee profile. The H^(3/2) power law means that a 10% increase in head produces about a 15% increase in discharge, so accurate head measurement is more important than precisely knowing Cd. Critical depth at the crest is Hc = (2/3) H, and mean flow velocity there is V = Q / (L * Hc). The Froude number Fr = V / sqrt(g * Hc) should equal 1.0 under ideal critical-flow conditions. This calculator verifies Fr and flags values outside the 0.85-1.25 range as a sign of submergence or an insufficiently long crest. The formula is the same in metric and imperial units; Cd has different numerical values in each system (1.705 m^0.5/s in SI, approximately 3.087 ft^0.5/s in US customary when combined with g = 32.174 ft/s^2).
Inputs, measurement, and design checks
Head H must be measured at a cross-section 3-4 crest widths upstream of the weir face where the velocity distribution is uniform. The minimum weir height P above the channel bed should satisfy P >= 2H so that the approach velocity head is less than 5% of the total head. The crest width b (the dimension parallel to flow) must be large enough to establish critical flow: the rule of thumb is b >= 2H and b <= 10H. At b > 10H, boundary-layer growth along the crest can depress Cd. Submergence check: the submergence ratio S = downstream head / upstream head must stay below 0.67 (often written H2/H1 < 0.67) for free-flow conditions. When S exceeds 0.67 the downstream water begins to back up, critical depth is drowned out, and the standard equation over-predicts discharge. Field installations should include a downstream gauge to monitor S.
Applications and accuracy
Broad-crested weirs are the preferred measurement structure for irrigation canals and small rivers because they handle sediment and floating debris better than thin-plate weirs, and they tolerate small errors in head measurement more gracefully than flumes. A field installation with a staff gauge accurate to 5 mm carries a discharge uncertainty of about 3-5% (because the H^1.5 exponent amplifies head error). Laboratory-grade point gauges (accuracy 1 mm) reduce total error to around 1.5%. Back-calculating Cd from a concurrent discharge measurement (using dilution gauging or an electromagnetic meter) is the most reliable way to calibrate a site-specific weir coefficient. Labyrinth weirs - in which the plan shape is folded to multiply the effective crest length - use the same formula with an effective L equal to the total unfolded length, giving 3.5-4 times the discharge of a straight weir of the same channel width.
Broad-crested weir discharge coefficient (Cd) guide
| Crest profile / condition | Cd (SI, m^0.5/s) | Notes |
|---|---|---|
| Sharp-cornered rectangular crest | 1.44 - 1.50 | Baseline geometry; nappe not fully attached |
| Rounded upstream edge (r >= 0.1H) | 1.60 - 1.70 | Standard design for irrigation and gauging |
| Theoretical free-flow (critical depth) | 1.705 | Derived: (2/3)^1.5 * sqrt(9.807) |
| Streamlined ogee or shaped crest | 1.70 - 1.80 | Optimized profile, smooth approach |
| Rough masonry or concrete (ks > 3mm) | 1.40 - 1.55 | Surface friction reduces efficiency |
| Submerged conditions (S > 0.67) | < 1.40 | Standard equation no longer valid; use modular correction |
Typical Cd values used in the formula Q = Cd * L * sqrt(g) * H^(3/2). Field calibration can refine these by 5-10%.
Frequently asked questions
What is the standard discharge coefficient for a broad-crested weir?
The theoretical free-flow coefficient derived from critical-depth theory is Cd = 1.705 (SI units, m^0.5/s). In practice, Cd ranges from about 1.40 for a sharp-cornered, rough-surfaced crest to 1.80 for a streamlined ogee crest with a rounded upstream face. The most common design value is 1.60-1.70 for a rectangular crest with a rounded upstream edge. When building a measurement structure, calibrate Cd against a separate discharge measurement if accuracy better than 5% is required.
Why does the Froude number matter for a broad-crested weir?
The flow over a broad-crested weir reaches critical depth (Froude number = 1.0) at the crest. This critical-flow condition is what makes the structure a reliable flowmeter: once critical depth is established, the discharge depends only on the upstream head and is independent of downstream conditions. If Fr deviates significantly below 1.0, the downstream water level may be submerging the crest, breaking the critical-flow condition and causing the standard formula to over-estimate discharge. A Froude number well above 1.0 suggests the crest is too short to fully establish critical flow.
What is the difference between a broad-crested and a sharp-crested weir?
A sharp-crested (thin-plate) weir has a crest thin enough that the water springs free in a jet (nappe) and does not touch the downstream face. Its discharge coefficient is around 0.61. A broad-crested weir has a crest long enough (b >= 2H) for the flow to pass through critical depth. Its formula Q = Cd * L * sqrt(g) * H^1.5 uses a Cd of about 1.705 (SI). Broad-crested weirs are more robust and sediment-tolerant; sharp-crested weirs are more precise at low flows because the nappe shape is well-defined.
How does submergence affect accuracy?
When the downstream (tailwater) depth rises above 67% of the upstream head (submergence ratio S > 0.67), the downstream water drowns out the critical-flow zone on the crest. The standard free-flow equation then over-predicts discharge. Between S = 0.67 and S = 0.95 a modular-limit correction factor (typically 0.85-0.95 times the free-flow Q) should be applied. Above S = 0.95 the weir is acting as a submerged orifice and the standard weir equation is not valid at all.
What is critical depth and how is it calculated?
Critical depth Hc is the flow depth at which the Froude number equals 1.0, meaning kinetic and potential energy are in balance. For a rectangular channel (which is the assumed cross-section of a broad-crested weir), critical depth is Hc = (Q^2 / (g * L^2))^(1/3). The broad-crested weir formula assumes this critical depth occurs at the crest and equals (2/3) of the upstream head H, so Hc = (2/3) H. You can verify this directly: Hc = (Q / (L * sqrt(g)))^(2/3) should match (2/3) H within a few percent.
How do I convert the discharge coefficient between metric and imperial units?
The formula Q = Cd * L * sqrt(g) * H^1.5 is dimensionally consistent: Cd has units of m^0.5/s in SI and ft^0.5/s in US customary. The theoretical value 1.705 m^0.5/s is equivalent to 1.705 * sqrt(3.28084) = 1.705 * 1.8117 = 3.087 ft^0.5/s. If you see a Cd of about 3.09 cited in US engineering references and 1.705 in metric references, they are the same weir coefficient expressed in different unit systems. This calculator handles the conversion automatically when you switch unit systems.
What weir height P is needed to reduce the approach velocity effect?
The standard recommendation is P >= 2H, where P is the height of the weir crest above the channel bed and H is the upstream head. When P >= 2H, the ratio H / (H + P) is at most 0.33, which means the approach velocity head is less than about 3% of the total head. Larger P values further reduce approach velocity but add to construction cost. If the channel is very shallow relative to H and P cannot meet this criterion, apply an approach velocity correction by computing the total head E = H + V1^2 / (2g) and substituting E for H in the formula.
Sources
- Chanson, H. (2004). Hydraulics of Open Channel Flow, 2nd edition. Butterworth-Heinemann. Broad-crested weir theory, Chapter 17.
- U.S. Army Corps of Engineers, HEC-RAS Hydraulic Reference Manual - Weir Flow Coefficients
- Ponce, V.M., Online Channel Hydraulics - Broad Crested Weir Calculator, San Diego State University