Wind Turbine Power Formulas: Betz Limit
Wind turbine power formulas: swept area, power in the wind, power with the power coefficient, the Betz limit and rotor speed from the tip speed ratio.
The formulas
| To find | Formula |
|---|---|
| Swept area | A = π × D² / 4 |
| Power in the wind | Pwind = ½ × ρ × A × v³ |
| Turbine power | P = ½ × ρ × A × v³ × Cp |
| Betz limit | Cp max = 16 / 27 = 0.593 |
| Rotor speed from tip speed ratio | ω = λ × v / R, and N = ω × 60 / (2π) |
ρ is the air density, 1.225 kg/m³ at sea level and 15 °C, and it falls with altitude and temperature. D is the rotor diameter in metres, R the radius, v the wind speed in m/s, Cp the power coefficient (usually 0.35 to 0.45 for a good turbine) and λ the tip speed ratio. Power rises with the cube of the wind speed, so a small change in speed matters a lot. A real turbine reaches its rated power at a rated wind speed and then limits it, so the cubic law does not apply above that speed. The electrical output is also reduced by the gearbox and generator efficiency.
Worked example: a 10 m rotor in an 8 m/s wind
- A = π × 10² / 4 = 78.54 m²
- Pwind = 0.5 × 1.225 × 78.54 × 8³ = 24.63 kW
- With Cp = 0.4, P = 9.852 kW
- The Betz limit gives at most 14.6 kW
Worked example: rotor speed
A tip speed ratio of 7 at 8 m/s.
- ω = 7 × 8 / 5 = 11.2 rad/s, which is 107 rpm
Power by wind speed (Cp = 0.4, no rated limit)
| Wind speed | Power |
|---|---|
| 4 m/s | 1.232 kW |
| 6 m/s | 4.156 kW |
| 8 m/s | 9.852 kW |
| 10 m/s | 19.24 kW |
| 12 m/s | 33.25 kW |
Frequently asked questions
What is the Betz limit?
No turbine can take more than 16/27 (59.3%) of the kinetic energy of the wind, because the air must keep moving after it passes the rotor.
Why does the wind speed matter so much?
The power in the wind is proportional to the cube of its speed. Doubling the speed gives eight times the power.
Does the air density matter?
Yes, in direct proportion. At a high altitude or in hot weather the density, and so the power, is lower.
What is the tip speed ratio?
The ratio of the speed of the blade tip to the wind speed. A turbine has its best Cp at one ratio, which sets the rotor speed for each wind speed.
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