Library / Dimensional analysis and similarity / DIM-24
Dimensional analysis and similarity
Propeller shaft power from the torque coefficient
Shaft power of a propeller, the torque times the angular speed, written with the power coefficient C_P = 2πC_Q so that it scales with the cube of the speed and the fifth power of the diameter.
- Ref
- DIM-24
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- P [W]
The shaft power of a propeller is the torque times the angular speed, P = 2πnQs. Writing Qs = CQ ρ n² D⁵ gives P = CP ρ n³ D⁵ with the power coefficient CP = 2πCQ. At matched coefficients, with s, d and z the speed, diameter and density ratios, power scales as z s³ d⁵. n is the speed in revolutions per second.
Shaft power of a propeller, the torque times the angular speed, written with the power coefficient C_P = 2πC_Q so that it scales with the cube of the speed and the fifth power of the diameter.
Calculator
- Power coefficient, CP = 2πCQ0.06283
- Shaft torque, P/(2πn)0.15 N·m
- Power with the speed raised by 10 %, 1.331P25.09 W
- Power in kilowatts0.01885 kW
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 absorbs 18.85 W at the shaft.
- Power rises with the cube of the speed and the fifth power of the diameter: a 10 % increase in speed needs 33 % more power, and a 10 % larger propeller needs 61 % more.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D Propeller diameter | 0.05 to 10 m | 1.885e-4 | 6.032e7 | 6.032e7 | +10% | +61.1 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 1.571 | 31,420 | 31,410 | +10% | +10 % |
| n Rotational speed | 1 to 200 rev/s | 0.002356 | 18,850 | 18,850 | +10% | +33.1 % |
| CQ Torque coefficient (from propeller data at the advance ratio) | 1e-4 to 0.2 | 0.1885 | 377 | 376.8 | +10% | +10 % |
Sweep of Rotational speed
| Rotational speed (rev/s) | Shaft power (W) |
|---|---|
| 1 | 0.002356 |
| 25.88 | 40.82 |
| 50.75 | 308 |
| 75.63 | 1,019 |
| 100.5 | 2,392 |
| 125.4 | 4,643 |
| 150.3 | 7,992 |
| 175.1 | 12,650 |
| 200 | 18,850 |
Sweep one variable
Variables and how each one contributes
Shaft power
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 power in direct proportion. Raising it by 10 % (to 0.011) moves the result by +10 %, from 18.85 to 20.73 W.
Fluid density
The fluid density, in kg/m³. Raises the power in direct proportion. Raising it by 10 % (to 1.32) moves the result by +10 %, from 18.85 to 20.73 W.
Rotational speed
The rotational speed, in revolutions per second (rpm divided by 60). Raises the power with the cube of the speed, so speed is a strong lever. Raising it by 10 % (to 22) moves the result by +33.1 %, from 18.85 to 25.09 W.
Propeller diameter
The propeller diameter, in m. Raises the power 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 18.85 to 30.36 W.
Worked example with the values above
- Power coefficient: CP = 2π × 0.01 = 0.062832.
- Speed and size terms: n³D⁵ = 20³ × 0.5⁵ = 250 m⁵/s³.
- P = 0.062832 × 1.2 × 250 = 18.85 W.
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.
- Using a torque coefficient with a power formula that already contains the factor 2π, or the reverse.
- Treating this as the electrical input power: motor and controller losses add to it.
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 × 3, D × 5). The diameter (fifth power) and speed (cube) errors dominate, and the coefficient depends on the advance ratio.
Where it is used
- Estimating the power a propeller absorbs and checking that a motor or engine can supply it.
- Scaling power with speed, size and density using matched coefficients.
Maths you need
- Shaft power: torque times angular speed
- Power coefficient: shaft power in dimensionless form