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Aerospace
Ideal actuator-disk power in forward flight
Ideal power put into the flow by a propeller of thrust T in a freestream V, as thrust times the speed through the disk, V + vi.
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- AER-54
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- v1
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- Aerospace
- Result
- Pideal [W]
The ideal power put into the air by an actuator disk is thrust times the speed through the disk: Pideal = T(V + vi). Of this, TV is useful thrust power and Tvi is the induced loss, the kinetic energy left in the wake. The ideal propulsive efficiency is V/(V + vi). Profile drag of the blades, swirl and mechanical losses make the shaft power larger than the ideal figure.
Ideal power put into the flow by a propeller of thrust T in a freestream V, as thrust times the speed through the disk, V + vi. The useful power is TV, and the rest, Tvi, is the induced loss. Profile drag, swirl and mechanical losses increase the shaft power actually needed.
Calculator
- Useful thrust power, TV12,000 W
- Induced power loss, Tvi3,000 W
- Ideal propulsive efficiency, V/(V + vi)0.8
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.
- The disk needs 15 kW, of which 12 kW does useful work and 3 kW is lost in the wake.
- The ideal efficiency is 80 %; a real propeller would be lower, as it also has blade drag and swirl.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Thrust | 10 to 100,000 N | 250 | 2,500,000 | 2,500,000 | +10% | +10 % |
| V Freestream speed | 0 to 150 m/s | 3,000 | 93,000 | 90,000 | +10% | +8 % |
| vi Induced velocity at the disk | 0 to 50 m/s | 12,000 | 42,000 | 30,000 | +10% | +2 % |
Sweep of Induced velocity at the disk
| Induced velocity at the disk (m/s) | Ideal power (W) |
|---|---|
| 0 | 12,000 |
| 6.25 | 15,750 |
| 12.5 | 19,500 |
| 18.75 | 23,250 |
| 25 | 27,000 |
| 31.25 | 30,750 |
| 37.5 | 34,500 |
| 43.75 | 38,250 |
| 50 | 42,000 |
Sweep one variable
Variables and how each one contributes
Ideal power
What the equation gives you. Everything else feeds it.
Thrust
The thrust, in N. Scales the power in direct proportion. Raising it by 10 % (to 660) moves the result by +10 %, from 15,000 to 16,500 W.
Freestream speed
The freestream speed, in m/s. Raises the useful power TV and the total. Raising it by 10 % (to 22) moves the result by +8 %, from 15,000 to 16,200 W.
Induced velocity at the disk
The induced velocity at the disk, in m/s. Raises the induced loss and the total power. Raising it by 10 % (to 5.5) moves the result by +2 %, from 15,000 to 15,300 W.
Worked example with the values above
- Speed through the disk: V + vi = 20 + 5 = 25 m/s.
- Thrust: T = 600 N.
- Pideal = 600 × 25 = 15,000 W.
Assumptions and limits
- An ideal actuator disk with uniform loading and steady incompressible flow, with no swirl.
- Rotor in hover or propeller in axial flight, with consistent revolutions-per-second definitions.
Common mistakes
- Using revolutions per minute where the formula needs revolutions per second.
- Using the disk area of one rotor for a multi-rotor system.
- Treating the ideal power as the shaft power, which ignores the losses.
Statistics and uncertainty
Momentum theory sets a lower bound on power: real propellers need 10 % to 30 % more than the ideal figure for blade profile drag, swirl and tip losses. Figure of merit of good rotors is 0.7 to 0.8, and propeller efficiency peaks around 0.85 to 0.88.
Where it is used
- Estimating the power needed for propellers, helicopters and drones.
- Reducing test data to coefficients and efficiency.
Maths you need
- Momentum theory: thrust equals mass flow times velocity change
- Advance ratio: J = V/(nD)
- Disk loading: thrust per unit disk area