Library / Hydraulics, open channels and fluid power / HYD-20
Hydraulics, open channels and fluid power
Ideal broad-crested rectangular weir discharge
Discharge over a broad horizontal crest of width b with critical flow on the crest, from the upstream total head He above the crest including the approach velocity head.
- Ref
- HYD-20
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
A broad horizontal crest forces critical flow on the crest, where the depth is two thirds of the upstream total head He. The discharge is then Q = b √g (2He/3)^(3/2), with no losses assumed and free discharge. The total head includes the approach velocity head. A calibrated coefficient is needed for real installations, as friction and the crest shape reduce the flow below this ideal value.
Discharge over a broad horizontal crest of width b with critical flow on the crest, from the upstream total head He above the crest including the approach velocity head. It assumes free flow and no losses.
Calculator
- Critical depth on the crest, 2He/30.4 m
- Critical speed on the crest, √(g yc)1.981 m/s
- Discharge per unit width, Q/b0.7924 m²/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 total head of 0.6 m over a 2 m crest gives 1.585 m³/s, with critical flow of 0.4 m depth on the crest.
- Multiply by a calibrated coefficient, typically 0.85 to 0.95, to account for friction and crest shape in a real structure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Crest width | 0.1 to 100 m | 0.07924 | 79.24 | 79.16 | +10% | +10 % |
| He Upstream total head above the crest | 0.02 to 5 m | 0.009644 | 38.12 | 38.11 | +10% | +15.4 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.506 | 2.771 | 2.265 | +10% | +4.88 % |
Sweep of Upstream total head above the crest
| Upstream total head above the crest (m) | Discharge (m³/s) |
|---|---|
| 0.02 | 0.009644 |
| 0.6425 | 1.756 |
| 1.265 | 4.851 |
| 1.888 | 8.842 |
| 2.51 | 13.56 |
| 3.133 | 18.9 |
| 3.755 | 24.81 |
| 4.377 | 31.23 |
| 5 | 38.12 |
Sweep one variable
Variables and how each one contributes
Discharge
What the equation gives you. Everything else feeds it.
Crest width
The crest width, in m. Raises the flow in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 1.585 to 1.743 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 1.585 to 1.662 m³/s.
Upstream total head above the crest
The upstream total head above the crest, in m. Raises the flow with the power 3/2. Raising it by 10 % (to 0.66) moves the result by +15.4 %, from 1.585 to 1.828 m³/s.
Worked example with the values above
- Critical depth: 2He/3 = 2 × 0.6 / 3 = 0.4 m.
- Power 3/2: 0.25298; √g = 3.1321.
- Q = 2 × 3.1321 × 0.25298 = 1.5847 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