Library / Hydraulics, open channels and fluid power / HYD-18
Hydraulics, open channels and fluid power
Rectangular sharp-crested weir discharge
Free discharge over a suppressed rectangular thin-plate weir of crest width b under upstream head H above the crest, with negligible approach velocity and an aerated nappe, using a calibrated coefficient.
- Ref
- HYD-18
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
Integrating the flow over the depth of a rectangular opening gives the sharp-crested weir equation Q = (2/3) Cd b √(2g) H^(3/2). H is the head above the crest measured upstream, beyond the drawdown. The nappe must be free and aerated, the approach velocity negligible and the weir suppressed (the crest spans the full width). The coefficient Cd depends on H and the weir height.
Free discharge over a suppressed rectangular thin-plate weir of crest width b under upstream head H above the crest, with negligible approach velocity and an aerated nappe, using a calibrated coefficient.
Calculator
- Discharge per unit width, Q/b0.1638 m²/s
- Discharge if the head doubles0.2316 m³/s
- Discharge in litres per second81.88 L/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A head of 20 cm over a 0.5 m crest passes 0.08188 m³/s (81.88 L/s).
- Doubling the head raises the flow by 2^1.5 = 2.83 times, so a weir is sensitive to head: a 1 % error in head gives a 1.5 % error in flow.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Crest width | 0.05 to 20 m | 0.008188 | 3.275 | 3.267 | +10% | +10 % |
| H Upstream head above the crest | 0.01 to 2 m | 9.154e-4 | 2.589 | 2.588 | +10% | +15.4 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.02614 | 0.1432 | 0.117 | +10% | +4.88 % |
| Cd Discharge coefficient | 0.4 to 0.9 | 0.05282 | 0.1189 | 0.06603 | +10% | +10 % |
Sweep of Upstream head above the crest
| Upstream head above the crest (m) | Discharge (m³/s) |
|---|---|
| 0.01 | 9.154e-4 |
| 0.2587 | 0.1205 |
| 0.5075 | 0.331 |
| 0.7562 | 0.602 |
| 1.005 | 0.9223 |
| 1.254 | 1.285 |
| 1.502 | 1.686 |
| 1.751 | 2.121 |
| 2 | 2.589 |
Sweep one variable
Variables and how each one contributes
Discharge
What the equation gives you. Everything else feeds it.
Discharge coefficient
The discharge coefficient, usually about 0.6. Raises the flow in direct proportion. Raising it by 10 % (to 0.682) moves the result by +10 %, from 0.08188 to 0.09007 m³/s.
Crest width
The crest width, in m. Raises the flow in direct proportion. Raising it by 10 % (to 0.55) moves the result by +10 %, from 0.08188 to 0.09007 m³/s.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the flow with the square root of g. Raising it by 10 % (to 10.79) moves the result by +4.88 %, from 0.08188 to 0.08587 m³/s.
Upstream head above the crest
The upstream head above the crest, in m. Raises the flow with the power 3/2. Raising it by 10 % (to 0.22) moves the result by +15.4 %, from 0.08188 to 0.09446 m³/s.
Worked example with the values above
- Head term: H^(3/2) = 0.2^1.5 = 0.089443.
- Coefficients: (2/3) × 0.62 × 0.5 × √(2 × 9.81) = 0.91542.
- Q = 0.91542 × 0.089443 = 0.081878 m³/s.
Assumptions and limits
- Steady flow with the discharge coefficient suited to the installation and the Reynolds number.
- For weirs: free discharge with an aerated nappe, a thin or broad crest as stated and negligible approach velocity.
Common mistakes
- Measuring the head at the crest of the weir instead of upstream, beyond the drawdown.
- Using a coefficient from a different weir type or scale.
- Using the total head where the static head over the crest is needed, or the reverse.
Statistics and uncertainty
Discharge coefficients are calibrated values, normally good to 1 % to 5 %, and the head enters with a power of 1.5 to 2.5, so a 1 % error in head gives an error in flow of 1.5 % to 2.5 %.
Where it is used
- Measuring flow in pipes, canals, rivers and flumes.
- Setting weir crest levels and sizing overflows.
Maths you need
- Discharge coefficient: the ratio of actual to ideal flow
- Bernoulli equation: energy conservation along a streamline
- Critical flow: flow at its minimum specific energy, with Froude number 1