Rotor & Betz Limit
A rotor takes energy out of the wind by slowing it down. Slow it too little and most of the wind passes unused; stop it and no new air can flow through. Somewhere in between lies the maximum: the Betz limit. See how much the rotor slows the wind, the power and thrust it gets, the wind behind it, and how fast the blades turn.
Why 59.3% Is the Maximum
Think of the rotor as a disc that brakes the air. The air that goes through it slows down: to (1 − a)·v at the rotor and to (1 − 2a)·v far behind it. The power taken out is the drop in the air’s kinetic energy, ½·ṁ·(v² − vwake²), where ṁ is the mass of air flowing through every second.
Brake harder and each kilogram gives more energy, but fewer kilograms get through, because the slow air piles up in front of the rotor and the wind flows around it. Balancing the two gives the most power at a = 1/3: the wind behind the rotor is a third of the free wind and the rotor catches 16/27 = 59.3% of the power in the wind. This is the Betz limit (Albert Betz, 1920).
From Braking the Wind to the Betz Limit, Step by Step
CpAssumptions of This Model
| Assumption | In reality… |
|---|---|
| The rotor is an ideal disc (actuator disc): the same braking over its whole area, no friction, no swirl in the wake, air that does not compress | real blades lose a few percent to friction, tip vortices and the wake’s rotation: modern turbines reach Cp ≈ 0.45–0.50, not 0.593 |
| The wind behind the rotor follows momentum theory: vwake = (1 − 2a)·v | above a ≈ 0.4 the wake becomes turbulent and mixes with the outside air; the formulas then underestimate the thrust (Glauert’s correction), and at a = 0.5 they break down |
| Rotor speed and torque from the tip-speed ratio you enter, at the power of the ideal disc | the best λ depends on the blade design; above rated wind speed the blades pitch to keep the power and the rotor speed constant |
| Steady, uniform wind straight at the rotor | gusts, wind shear over the rotor height and yaw misalignment lower the power and add fatigue loads |
Key Relations
The power is the kinetic energy the air loses per second on its way through the rotor.
The force needed to slow the air down; the tower and the foundation have to carry it.
Setting the slope of Cp to zero gives the best braking.
Big rotors turn slowly with an enormous torque, which is why turbines need a gearbox or a very large generator.
Sources & Verification
- A. Betz, “Das Maximum der theoretisch möglichen Ausnützung des Windes durch Windmotoren”, Zeitschrift für das gesamte Turbinenwesen 26, 307–309 (1920).
- J. F. Manwell, J. G. McGowan & A. L. Rogers, Wind Energy Explained, 2nd ed., Wiley, 2009 — ch. 3, one-dimensional momentum theory, the Betz limit and the turbulent wake state.
- T. Burton, N. Jenkins, D. Sharpe & E. Bossanyi, Wind Energy Handbook, 2nd ed., Wiley, 2011 — actuator disc, tip-speed ratio.
Frequently Asked Questions
What is the Betz limit and where does it come from?
The most power an ideal rotor can take from the wind: 16/27, about 59.3%. A rotor works by slowing the air; slowing it more takes more energy from each kilogram but lets fewer kilograms through. The best balance is when the rotor slows the wind by a third (a = 1/3), leaving a third of the speed behind it.
How fast does the wind blow behind a wind turbine?
In the ideal case at the Betz optimum, a third of the free wind: 10 m/s becomes 6.7 m/s at the rotor and 3.3 m/s behind it. That slow wake is why turbines in a wind farm are placed several rotor diameters apart.
How large is the thrust force on a wind turbine?
Large: slowing the air pushes back on the rotor. A 112 m rotor at the Betz optimum in a 10 m/s wind feels about 536 kN, the weight of about 55 tonnes; a 236 m offshore rotor at 11 m/s about 2.7 MN, the weight of about 280 tonnes. The tower and foundation have to carry it.
Why do big wind turbines turn so slowly?
The blade tips move about 6 to 9 times faster than the wind (the tip-speed ratio). For a 112 m rotor at 10 m/s with a ratio of 7 the tips move at 70 m/s (252 km/h), yet that is only about 12 turns a minute, because the circle is so large. The slow rotation means a huge torque: about 2.9 MN·m on the shaft.