Library / Aerospace / AER-24
Aerospace
Speed for minimum power required
Level-flight speed at which aerodynamic power DV is least, for the parabolic polar.
- Ref
- AER-24
- Rev
- v1
- Field
- Aerospace
- Result
- Vmp [m/s]
Power required is drag times speed, DV. In level flight with the parabolic polar it is least at CL = √(3CD0/k), where the induced drag is three times the parasite drag. The corresponding speed is the minimum-drag speed multiplied by 3^(−1/4) = 0.76, so best endurance is flown slower than best range. Check the stall speed, since the speed can fall below the stall speed for heavy, high-drag aircraft.
Level-flight speed at which aerodynamic power DV is least, for the parabolic polar. It is lower than the minimum-drag speed by a factor 3^(−1/4) ≈ 0.76. A constant propulsive efficiency is needed to turn this into minimum shaft power. Check the stall margin.
Calculator
- Minimum-drag speed, (2W/ρS)^½ (k/CD0)^¼43.3 m/s
- Ratio Vmp/Vmd0.7598
- Lift coefficient at minimum power, √(3CD0/k)1.155
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 aircraft flies at minimum power at 32.902 m/s, 118.4 km/h, at a lift coefficient of 1.155.
- It is 24 % slower than the minimum-drag speed of 43.301 m/s.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| W Weight | 2,000 to 100,000 N | 13.43 | 94.98 | 81.55 | +10% | +4.88 % |
| S Wing area | 5 to 100 m² | 13.16 | 58.86 | 45.7 | +10% | −4.65 % |
| ρ Air density | 0.3 to 1.3 kg/m³ | 31.61 | 65.8 | 34.19 | +10% | −4.65 % |
| CD0 Zero-lift drag coefficient | 0.01 to 0.08 | 23.27 | 39.13 | 15.86 | +10% | −2.35 % |
| k Induced drag factor | 0.02 to 0.1 | 26.86 | 40.17 | 13.31 | +10% | +2.41 % |
Sweep of Weight
| Weight (N) | Minimum-power speed (m/s) |
|---|---|
| 2,000 | 13.43 |
| 14,250 | 35.85 |
| 26,500 | 48.89 |
| 38,750 | 59.12 |
| 51,000 | 67.83 |
| 63,250 | 75.54 |
| 75,500 | 82.53 |
| 87,750 | 88.97 |
| 100,000 | 94.98 |
Sweep one variable
Variables and how each one contributes
Minimum-power speed
What the equation gives you. Everything else feeds it.
Weight
The aircraft weight, in N. Raises the speed with the square root. Raising it by 10 % (to 13,200) moves the result by +4.88 %, from 32.9 to 34.51 m/s.
Air density
The air density, in kg/m³. Lowers the speed with the inverse square root, so the best endurance speed rises with height. Raising it by 10 % (to 1.32) moves the result by −4.65 %, from 32.9 to 31.37 m/s.
Wing area
The wing area, in m². Lowers the speed with the inverse square root. Raising it by 10 % (to 17.6) moves the result by −4.65 %, from 32.9 to 31.37 m/s.
Induced drag factor
The induced drag factor. Raises the speed with its fourth root. Raising it by 10 % (to 0.0495) moves the result by +2.41 %, from 32.9 to 33.7 m/s.
Zero-lift drag coefficient
The zero-lift drag coefficient. Lowers the speed with the inverse fourth root. Raising it by 10 % (to 0.022) moves the result by −2.35 %, from 32.9 to 32.13 m/s.
Worked example with the values above
- Reference speed: √(2W/(ρS)) = √(2 × 12,000 / (1.2 × 16)) = 35.3553 m/s.
- Polar factor: (k/(3CD0))^¼ = (0.045 / 0.06)^0.25 = 0.930605.
- Vmp = 32.9019 m/s.
Assumptions and limits
- A parabolic polar CD = CD0 + kCL², constant over the speed range and with no compressibility rise.
- Steady level, unaccelerated flight with lift equal to weight, at constant density.
Common mistakes
- Using a polar from a different configuration, such as flaps or gear down.
- Forgetting stall and Mach limits on the mathematical solution.
- Mixing up the minimum-drag speed and the minimum-power speed.
Statistics and uncertainty
The parabolic drag polar fits real aircraft well only over the middle of the lift range. Speeds from it are usually accurate to 5 % when CD0 and k come from flight tests; with handbook estimates they can be off by 10 % to 20 %, and the result moves with the one-quarter power of the drag ratio.
Where it is used
- Plotting thrust and power curves and finding best speeds for range and endurance.
- Estimating how stall speed changes in turns.
Maths you need
- Drag polar: CD = CD0 + kCL²
- Level flight: L = W and T = D
- Load factor: n = L/W