Library / Hydraulics, open channels and fluid power / HYD-16
Hydraulics, open channels and fluid power
Differential-pressure Venturi meter discharge
Volume flow through a Venturi from the upstream-to-throat pressure difference Δp, throat area At and diameter ratio β, for steady incompressible full-pipe flow with equal-elevation taps and a calibrated discharge coefficient.
- Ref
- HYD-16
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
A Venturi meter speeds the flow through a throat, lowering the pressure there. Bernoulli and continuity give Q = Cd At √[2Δp/(ρ(1 − β⁴))], where β is the throat-to-pipe diameter ratio and Cd corrects for friction. The flow varies with the square root of the pressure difference. Use a calibrated Cd; typical values are close to 0.98 for a well-made meter at high Reynolds number.
Volume flow through a Venturi from the upstream-to-throat pressure difference Δp, throat area At and diameter ratio β, for steady incompressible full-pipe flow with equal-elevation taps and a calibrated discharge coefficient.
Calculator
- Throat speed, Q/At3.92 m/s
- Pipe speed, Q β²/At0.98 m/s
- Velocity-of-approach factor, 1/√(1 − β⁴)1.033
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 7,500 Pa difference gives 0.0392 m³/s (39.2 litres per second), with a throat speed of 3.92 m/s.
- Because Q follows √Δp, doubling the flow needs four times the pressure difference, which limits the range of one meter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| At Throat area | 1e-5 to 1 m² | 3.92e-5 | 3.92 | 3.92 | +10% | +10 % |
| Δp Upstream-to-throat pressure difference | 10 to 1e7 Pa | 0.001431 | 1.431 | 1.43 | +10% | +4.88 % |
| ρ Liquid density | 500 to 14,000 kg/m³ | 0.01048 | 0.05544 | 0.04496 | +10% | −4.65 % |
| β Throat-to-pipe diameter ratio | 0.1 to 0.9 | 0.03796 | 0.06472 | 0.02677 | +10% | +1.58 % |
| Cd Discharge coefficient | 0.5 to 1.1 | 0.02 | 0.044 | 0.024 | +10% | +10 % |
Sweep of Upstream-to-throat pressure difference
| Upstream-to-throat pressure difference (Pa) | Discharge (m³/s) |
|---|---|
| 10 | 0.001431 |
| 56.23 | 0.003394 |
| 316.2 | 0.008049 |
| 1,778 | 0.01909 |
| 10,000 | 0.04526 |
| 56,230 | 0.1073 |
| 316,200 | 0.2545 |
| 1,778,000 | 0.6036 |
| 1e7 | 1.431 |
Sweep one variable
Variables and how each one contributes
Discharge
What the equation gives you. Everything else feeds it.
Discharge coefficient
The discharge coefficient, close to 1 for a good meter. Raises the flow in direct proportion. Raising it by 10 % (to 1.078) moves the result by +10 %, from 0.0392 to 0.04312 m³/s.
Throat area
The throat area, in m². Raises the flow in direct proportion. Raising it by 10 % (to 0.011) moves the result by +10 %, from 0.0392 to 0.04312 m³/s.
Upstream-to-throat pressure difference
The upstream-to-throat pressure difference, in Pa. Raises the flow with the square root of the pressure difference. Raising it by 10 % (to 8,250) moves the result by +4.88 %, from 0.0392 to 0.04111 m³/s.
Liquid density
The liquid density, in kg/m³. A denser liquid gives less flow for the same pressure difference, with a square-root effect. Raising it by 10 % (to 1,100) moves the result by −4.65 %, from 0.0392 to 0.03738 m³/s.
Throat-to-pipe diameter ratio
The throat-to-pipe diameter ratio, between 0 and 1. A larger ratio reduces the speed-up from pipe to throat, so a given pressure difference means more flow. Raising it by 10 % (to 0.55) moves the result by +1.58 %, from 0.0392 to 0.03982 m³/s.
Worked example with the values above
- Approach factor: 1 − β⁴ = 1 − 0.0625 = 0.9375.
- Speed term: √[2 × 7,500 / (1,000 × 0.9375)] = 4 m/s.
- Q = 0.98 × 0.01 × 4 = 0.0392 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