Library / Dimensional analysis and similarity / DIM-23
Dimensional analysis and similarity
Propeller shaft torque from the torque coefficient
Shaft torque of a propeller from its non-dimensional torque coefficient, the fluid density, the rotational speed in revolutions per second and the diameter.
- Ref
- DIM-23
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Qs [N·m]
Qs = CQ ρ n² D⁵ gives the shaft torque of a propeller from its torque coefficient CQ, taken from propeller data at the required advance ratio, blade geometry and pitch. n is the speed in revolutions per second and D the diameter. At matched coefficients, with s, d and z the speed, diameter and density ratios, the torque scales as z s² d⁵.
Shaft torque of a propeller from its non-dimensional torque coefficient, the fluid density, the rotational speed in revolutions per second and the diameter.
Calculator
- Shaft power, 2πnQs18.85 W
- Torque with the speed doubled, 4Qs0.6 N·m
- Torque with the diameter doubled, 32Qs4.8 N·m
- Rotational speed1,200 rpm
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.
- With CQ = 0.01 at 1,200 rpm, a 0.5 m propeller in a fluid of 1.2 kg/m³ needs 0.15 N·m at the shaft, which is 18.85 W.
- Torque rises with the fifth power of the diameter, so a larger propeller needs a much stronger shaft and gearbox even at a lower speed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D Propeller diameter | 0.05 to 10 m | 1.5e-6 | 480,000 | 480,000 | +10% | +61.1 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 0.0125 | 250 | 250 | +10% | +10 % |
| n Rotational speed | 1 to 200 rev/s | 3.75e-4 | 15 | 15 | +10% | +21 % |
| CQ Torque coefficient (from propeller data at the advance ratio) | 1e-4 to 0.2 | 0.0015 | 3 | 2.998 | +10% | +10 % |
Sweep of Propeller diameter
| Propeller diameter (m) | Shaft torque (N·m) |
|---|---|
| 0.05 | 1.5e-6 |
| 1.294 | 17.4 |
| 2.537 | 505 |
| 3.781 | 3,710 |
| 5.025 | 15,380 |
| 6.269 | 46,470 |
| 7.512 | 114,900 |
| 8.756 | 247,100 |
| 10 | 480,000 |
Sweep one variable
Variables and how each one contributes
Shaft torque
What the equation gives you. Everything else feeds it.
Torque coefficient (from propeller data at the advance ratio)
The torque coefficient at the operating advance ratio, a plain number from propeller data. Raises the torque in direct proportion. Raising it by 10 % (to 0.011) moves the result by +10 %, from 0.15 to 0.165 N·m.
Fluid density
The fluid density, in kg/m³. Raises the torque in direct proportion. Raising it by 10 % (to 1.32) moves the result by +10 %, from 0.15 to 0.165 N·m.
Rotational speed
The rotational speed, in revolutions per second (rpm divided by 60). Raises the torque with the square of the speed. Raising it by 10 % (to 22) moves the result by +21 %, from 0.15 to 0.1815 N·m.
Propeller diameter
The propeller diameter, in m. Raises the torque with the fifth power of the diameter, so diameter is the strongest lever. Raising it by 10 % (to 0.55) moves the result by +61.1 %, from 0.15 to 0.2416 N·m.
Worked example with the values above
- Speed and size terms: n²D⁵ = 20² × 0.5⁵ = 12.5 m³/s².
- Multiply by the density and coefficient: 0.01 × 1.2 × 12.5.
- Qs = 0.15 N·m.
Assumptions and limits
- CQ is taken at the operating advance ratio for the actual blade geometry and pitch.
- Axial inflow, with n in revolutions per second and the density at the operating state.
Common mistakes
- Putting the speed in rpm instead of revolutions per second.
- Mixing the torque coefficient with a power coefficient: CP = 2πCQ.
- Using the coefficient at the wrong advance ratio.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (CQ × 1, ρ × 1, n × 2, D × 5). The diameter error counts five times and the coefficient depends on the advance ratio, so these two dominate.
Where it is used
- Sizing the shaft, gearbox and motor for a propeller or fan.
- Converting a measured torque into a coefficient for scaling to another speed or size.
Maths you need
- Torque: turning moment, force times distance
- Advance ratio: axial speed relative to the blade-tip speed