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Aerospace
Rotor hover figure of merit
Ratio of the ideal minimum hover power T^{3/2}/√(2ρA) to the actual shaft power P of an isolated rotor out of ground effect, with negligible freestream.
- Ref
- AER-55
- Rev
- v1
- Field
- Aerospace
- Result
- FM
Figure of merit compares an actual rotor with an ideal one in hover. Momentum theory gives the least power for thrust T as Pideal = T^(3/2)/√(2ρA), and FM = Pideal/P = T^(3/2)/[P√(2ρA)]. It is dimensionless and below one. Well-designed rotors reach 0.7 to 0.8. Use the total disk area of the rotor and conditions out of ground effect with negligible forward speed.
Ratio of the ideal minimum hover power T^{3/2}/√(2ρA) to the actual shaft power P of an isolated rotor out of ground effect, with negligible freestream. It is below one and typically 0.6 to 0.8 for good rotors. Use the total disk area of that rotor.
Calculator
- Ideal hover power, T^(3/2)/√(2ρA)37,950 W
- Disk loading, T/A600 Pa
- Ideal induced velocity in hover, √(T/(2ρA))15.81 m/s
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 rotor has a figure of merit of 0.9487: it needs 40 kW where the ideal rotor would need 37.95 kW.
- The remaining 5.13 % of the power goes to blade profile drag, swirl and non-uniform inflow.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Thrust | 100 to 100,000 N | 0.008069 | 255.2 | 255.1 | +10% | +15.4 % |
| P Shaft power | 1,000 to 1,000,000 W | 0.03795 | 37.95 | 37.91 | +10% | −9.09 % |
| A Disk area | 0.1 to 200 m² | 0.1342 | 6 | 5.866 | +10% | −4.65 % |
| ρ Air density | 0.3 to 1.3 kg/m³ | 0.9115 | 1.897 | 0.9859 | +10% | −4.65 % |
Sweep of Shaft power
| Shaft power (W) | Figure of merit |
|---|---|
| 1,000 | 37.95 |
| 2,371 | 16 |
| 5,623 | 6.748 |
| 13,340 | 2.846 |
| 31,620 | 1.2 |
| 74,990 | 0.506 |
| 177,800 | 0.2134 |
| 421,700 | 0.08999 |
| 1,000,000 | 0.03795 |
Sweep one variable
Variables and how each one contributes
Figure of merit
What the equation gives you. Everything else feeds it.
Thrust
The rotor thrust, in N (equal to the weight in hover). Raises FM with the power of 3/2 for a given shaft power. Raising it by 10 % (to 2,640) moves the result by +15.4 %, from 0.9487 to 1.094.
Shaft power
The measured shaft power, in W. A larger power lowers the figure of merit in inverse proportion. Raising it by 10 % (to 44,000) moves the result by −9.09 %, from 0.9487 to 0.8624.
Air density
The air density, in kg/m³. A lower density lowers the ideal power’s denominator and raises the ideal power, lowering FM for a given P. Raising it by 10 % (to 1.32) moves the result by −4.65 %, from 0.9487 to 0.9045.
Disk area
The total rotor disk area, in m². A larger disk lowers the ideal power and so lowers FM for a given shaft power. Raising it by 10 % (to 4.4) moves the result by −4.65 %, from 0.9487 to 0.9045.
Worked example with the values above
- Ideal power: T^1.5/√(2ρA) = 117,575.5 / √(9.6) = 37,947.33 W.
- Shaft power: P = 40,000 W.
- FM = 37,947.33 / 40,000 = 0.948683.
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 disk area for one blade or for only part of a multi-rotor system.
- Using it in forward flight or near the ground, where the ideal power is different.
- Mixing up thrust and weight when the rotor accelerates.
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