Weir Flow Sharp Broad Interactive Calculator

Sharp-Crested Weir Flow

Q = (2/3) × C_d × L × √(2g) × H^3/2

Broad-Crested Weir Flow

Q = C_d × L × H^3/2

Froude Number

Fr = V / √(g × H)

Scenario: Water Rights Compliance Monitoring

Carlos manages a 180-hectare almond orchard in California's Central Valley operating under a senior water right allowing 2.15 m³/s diversion from an irrigation district canal. State regulations require continuous flow measurement with ±5% accuracy to prevent over-appropriation during drought conditions. He installs a 2.5 m wide sharp-crested weir with an ultrasonic head sensor. During a compliance inspection, the sensor reads 0.53 m head height. Using this calculator with C_d = 0.62, Carlos verifies his instantaneous flow is 1.87 m³/s—well within his allocation. The calculation confirms his measurement system is functioning correctly and provides documented evidence of compliance, avoiding potential penalties of $500 per acre-foot of unauthorized diversion.

Scenario: Wastewater Treatment Plant Expansion

Jennifer, a process engineer at a municipal wastewater treatment facility, is designing an upgrade to increase capacity from 45,000 to 68,000 m³/day. The existing effluent measurement system uses a 1.8 m V-notch weir that becomes submerged during peak wet-weather flows, causing measurement errors that trigger regulatory violations. She evaluates replacing it with a broad-crested weir that can handle 0.95 m³/s peak flow (82,080 m³/day) with only 0.35 m head. Using this calculator in broad-crested mode with C_d = 1.72, she determines that a 3.2 m wide weir will maintain free-flow conditions throughout the design range. The calculated Froude number of 0.89 confirms near-critical flow, and the installation reduces head loss by 0.18 m compared to the existing system, improving pump efficiency and reducing energy costs by an estimated $4,200 annually.

Scenario: Stormwater Detention Basin Design

Marcus, a civil engineer with a land development firm, is designing a stormwater detention basin for a 12-hectare commercial development. Local regulations require that post-development peak discharge cannot exceed the pre-development 10-year storm rate of 0.42 m³/s. He designs a compound weir outlet structure with a low-flow 0.3 m wide notch for water quality treatment and a 1.5 m wide rectangular section for flood control. During the design storm, the basin fills to a depth providing 0.28 m head over the rectangular weir. Using this calculator, Marcus confirms the sharp-crested rectangular section discharges 0.385 m³/s at this head (C_d = 0.61 due to end contractions). Combined with 0.047 m³/s through the low-flow notch, total discharge is 0.432 m³/s—meeting the regulatory limit with a 3% margin. This precise sizing prevents costly over-excavation while ensuring the municipality approves the drainage plan.

The fundamental difference lies in the flow regime and pressure distribution over the overflow surface. Sharp-crested weirs use a thin plate (typically 2-6 mm thick) with a beveled or sharp downstream edge, creating a freely falling nappe with atmospheric pressure on both the upper and lower surfaces. This design is ideal for precise flow measurement in the range of 0.015 to 1.5 m³/s and requires only 0.05-0.6 m of head. Broad-crested weirs feature a horizontal crest with length exceeding 0.5 times the head, forcing the flow to reach critical depth (Froude number = 1) over the crest. This geometry makes them better suited for combined flow control and measurement, handling larger flows (0.5-50 m³/s typical) with greater structural stability. Sharp-crested weirs offer 2-3% measurement accuracy but are vulnerable to damage and debris accumulation, while broad-crested weirs provide 3-5% accuracy with superior durability and self-cleaning characteristics. The discharge coefficient also differs fundamentally: sharp-crested C_d values of 0.60-0.65 appear in an equation containing an explicit √(2g) term, while broad-crested C_d values of 1.6-1.8 incorporate this gravitational factor within the coefficient itself.

Accurate head measurement requires careful attention to sensor location and surface wave effects. The head measurement point should be positioned upstream from the weir at a distance of 3-4 times the maximum expected head, but not less than 0.6 m. This location ensures the measurement occurs in the zone of gradually varied flow before the drawdown curve near the weir affects the water surface profile. For sharp-crested weirs, the head is measured from the crest elevation to the undisturbed upstream water surface. Install the measurement staff gauge or pressure transducer in a stilling well connected to the main channel through a small-diameter pipe (25-50 mm) to dampen surface waves and turbulence that can cause reading fluctuations of ±5-10 mm. For automated systems, submersible pressure transducers or ultrasonic level sensors mounted above the water surface provide continuous measurement. Ultrasonic sensors require temperature compensation and should be mounted at least 0.3 m above maximum water level to prevent false readings from spray. The zero reference (weir crest elevation) must be established through optical leveling with ±2 mm accuracy, as a 5 mm error in crest elevation produces a 3-5% flow calculation error. In channels with floating debris or foam, use a float-operated gauge within a stilling well rather than ultrasonic technology, which can lock onto debris or foam rather than the water surface.

Submergence fundamentally alters weir hydraulics by eliminating the free-flow condition and introducing downstream water level as a controlling variable. For sharp-crested weirs, submergence begins when the downstream head exceeds approximately 0.67 times the upstream head, causing the nappe to cling to the downstream face rather than springing clear. The standard discharge equation becomes invalid, and flow must be calculated using submerged weir formulas that require both upstream and downstream head measurements: Q = C_d × L × √(2g) × [(H₁)^3/2 - (H₂)^3/2], where H₁ is upstream head and H₂ is downstream head. Broad-crested weirs experience submergence when tailwater rises above the critical depth on the crest, typically when H₂/H₁ exceeds 0.75-0.85. Under submerged conditions, the weir essentially functions as an orifice, and discharge becomes proportional to the square root of head difference rather than H^3/2. Flow measurement uncertainty increases dramatically under submergence—from ±2-3% for free flow to ±8-15% for submerged conditions—because small errors in either head measurement are magnified. In applications where submergence may occur, designers should either increase weir crest elevation to maintain free flow, install a second downstream measurement point for submerged flow calculations, or use alternative measurement technologies such as acoustic Doppler current profilers that are insensitive to tailwater conditions.

No, triangular and V-notch weirs require different discharge equations because their geometry creates a variable width as head changes, unlike the constant width of rectangular weirs. The standard V-notch equation is Q = (8/15) × C_d × tan(θ/2) × √(2g) × H^5/2, where θ is the notch angle (typically 90 degrees). The H^5/2 exponent versus H^3/2 for rectangular weirs means V-notch sensitivity changes with flow rate—they are extremely sensitive at low flows but less sensitive at high flows. For a 90-degree V-notch with C_d = 0.58, the discharge equation simplifies to Q = 1.38 × H^5/2 in metric units. This design excels for measuring flows spanning a wide range; a 90-degree V-notch can measure from 0.0001 to 0.1 m³/s with good accuracy, whereas a rectangular weir of the same maximum capacity would show poor sensitivity at the low end of this range. The higher power relationship means a 50% flow increase produces a 23% head increase for V-notch versus only 14% for rectangular configuration, providing greater measurement resolution. However, V-notch weirs are more sensitive to installation errors—a 1-degree error in notch angle causes a 0.9% discharge error, and precise machining or field verification of the notch angle is essential. Compound weirs combining a low-flow V-notch with a high-flow rectangular section provide optimal measurement across very wide flow ranges, common in stormwater and combined sewer applications where dry-weather flows may be 1-2% of wet-weather peaks.

Temperature influences weir hydraulics through three distinct mechanisms: water density effects on gravitational acceleration, viscosity changes affecting Reynolds number and discharge coefficient, and thermal expansion of the weir structure itself. The gravitational term in weir equations is actually g × (ρ_water/ρ_ref), where density decreases from 999.8 kg/m³ at 4°C to 995.6 kg/m³ at 30°C—a 0.42% change that produces 0.21% flow calculation difference for the same head. This effect is generally negligible compared to measurement uncertainties. Water viscosity changes from 1.52 × 10^-3 Pa·s at 5°C to 0.80 × 10^-3 Pa·s at 30°C significantly affect Reynolds number, particularly for small weirs with low heads. The discharge coefficient increases with Reynolds number according to C_d ≈ C_∞ × [1 + K/(Re)ⁿ], where Re = (ρ × V × L_c)/μ and L_c is a characteristic length. For a sharp-crested weir at 5°C versus 30°C with the same head, the Reynolds number nearly doubles, potentially changing C_d by 1-2%. The third effect—thermal expansion of the weir plate or crest—becomes significant for long weirs (L greater than 5 m) in environments with extreme temperature variations. Steel expands 12 × 10^-6 per °C, so a 5 m steel weir experiences 0.6 mm length change across a 10°C temperature swing, representing a 0.012% dimension change. For precision flow measurement installations (±1% target accuracy), use temperature-compensated head sensors, establish discharge coefficients through in-situ calibration at operating temperatures, and construct weirs from low-expansion materials like stainless steel or concrete in thermally stable environments.

Weir accuracy degrades over time without regular inspection and maintenance addressing sediment accumulation, structural deformation, biological growth, and debris blockage. Sediment deposition upstream of the weir raises the effective channel bottom, reducing actual head for a given flow and causing discharge to be underestimated. A 20 mm sediment layer on a weir designed for 0.4 m maximum head reduces measured flow by approximately 5%. Establish a minimum quarterly inspection schedule (monthly during high-flow seasons) to measure and remove accumulated sediment, documenting crest elevation with optical leveling to detect settlement or structural movement. Sharp-crested weir plates require particular attention to edge sharpness—rounding or corrosion of the crest edge increases the discharge coefficient unpredictably, causing measurement drift. Edges should be inspected annually and resurfaced or replaced when radius exceeds 1 mm or when corrosion produces visible irregularity. Algae and biological growth on the weir face and in stilling wells alter surface roughness and can block pressure-sensing ports, introducing measurement errors of 10-30%. Chemical treatment (chlorine or copper sulfate at concentrations safe for downstream uses) or physical cleaning every 2-4 weeks prevents biofilm establishment. Debris screens upstream of the weir protect against large objects but require frequent cleaning to prevent flow restriction—a partially clogged screen creates a pressure drop that artificially raises upstream head. For critical applications such as water rights measurement or regulatory compliance monitoring, perform annual flow calibration by comparing weir-measured discharge against an independent measurement method (current meter traverses, dye dilution, or portable flumes) to verify that accumulated maintenance issues haven't shifted the stage-discharge relationship beyond acceptable tolerances.

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About the Author

Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations

Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.

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