Library / Fluid Mechanics / FLU-09
Fluid Mechanics
Speed from a pressure difference
Bernoulli’s equation rearranged: pressure difference converts into flow speed.
- Ref
- FLU-09
- Rev
- v1
- Field
- Fluid Mechanics
- Result
- v [m/s]
ΔP is the difference between the stagnation (total) pressure and the static pressure, in pascals, and ρ the density of the fluid. This is Bernoulli’s equation solved for the speed. It is what a pitot-static tube measures, and it also gives the throat speed in a Venturi meter.
Bernoulli’s equation rearranged: pressure difference converts into flow speed. Used in Venturi meters and pitot tubes.
Calculator
- Speed in sea-level air (1.225 kg/m³) for the same ΔP90.35 m/s
- Pressure difference needed for twice the speed20,000 Pa
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 pitot tube reading 600 Pa in sea-level air indicates about 31 m/s (113 km/h). The reading is a pressure and the speed follows from the density, so the density must be right.
- The same 5000 Pa pushes air to 90 m/s but water to only 3.2 m/s, a factor √(1000/1.225) ≈ 29 lower.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρ Fluid density | 0.1 to 20,000 kg/m³ | 0.7071 | 316.2 | 315.5 | +10% | −4.65 % |
| ΔP Pressure difference | 1 to 1e7 Pa | 0.04472 | 141.4 | 141.4 | +10% | +4.88 % |
Sweep of Pressure difference
| Pressure difference (Pa) | Flow speed (m/s) |
|---|---|
| 1 | 0.04472 |
| 7.499 | 0.1225 |
| 56.23 | 0.3354 |
| 421.7 | 0.9184 |
| 3,162 | 2.515 |
| 23,710 | 6.887 |
| 177,800 | 18.86 |
| 1,334,000 | 51.64 |
| 1e7 | 141.4 |
Sweep one variable
Variables and how each one contributes
Flow speed
What the equation gives you. Everything else feeds it.
Pressure difference
The pressure difference between the stagnation point and the static pressure, in Pa. Square root: four times the pressure difference is needed to double the speed. A 10 % rise in ΔP raises v by about 5 %.
Fluid density
The density of the fluid, in kg/m³ (1.225 for sea-level air, 1000 for water). Inverse square root: a fluid ten times denser gives a speed three times lower for the same ΔP, so water flows slowly where air flows fast.
Worked example with the values above
- Double the pressure difference: 2ΔP = 2 × 5,000 = 10,000 Pa.
- Divide by the density: 10,000 / 1,000 = 10 m²/s².
- v = √10 = 3.162 m/s.
Assumptions and limits
- Steady, incompressible flow along a streamline (for air, below about Mach 0.3).
- Negligible viscous loss between the two measuring points, and a fluid that comes fully to rest at the stagnation point.
Common mistakes
- Leaving out the factor 2 in 2ΔP.
- Entering ΔP in kPa when the formula needs pascals.
Statistics and uncertainty
v ∝ √(ΔP/ρ), so a 2 % error in ΔP gives a 1 % error in speed. Low speeds mean tiny pressures, where a small zero offset in the sensor can be a large share of the signal: at 10 m/s in air ΔP is only 61 Pa, so a 5 Pa offset gives a 4 % speed error. Air density changes with altitude and temperature, so true airspeed needs a density correction.
Where it is used
- Measuring airspeed with a pitot tube.
- Finding flow speed from the pressure drop in a Venturi meter.
Maths you need
- Bernoulli’s equation: p + ½ρv² = constant
- Square roots: √(2ΔP/ρ)