TURBINE OPERATION/ENERGY OUTPUT
Actuator disk/Betz law
The ideal braking of the wind
The more kinetic energy a wind turbine pulls out of the wind, the more the wind will be slowed down as it leaves the left side of the turbine in the picture. If all the energy from the wind would be extracted, the air would move away with the speed zero, i.e. the air could not leave the turbine. In that case no energy would be extracted at all, since all of the air would obviously also be prevented from entering the rotor of the turbine. In the other extreme case, the wind could pass though the tube above without being hindered at all. In this case it would not extracted any energy from the wind.
Therefore it can be assumed that there must be some way of braking the wind which is in between these two extremes and is more efficient in converting the energy in the wind to useful mechanical energy. It turns out that there is a surprisingly simple answer to this: An ideal wind turbine would slow down the wind by 2/3 of its original speed. To understand why, the fundamental physical law for the aerodynamics of wind turbines should be used: the Betz law.
The Betz' law says that you can only convert a maximum of 16/27, or else 59.3%, of the kinetic energy in the wind to mechanical energy using a wind turbine.
The Betz' law was first formulated by the German Physicist Albert Betz in 1919. His book "Wind-Energie" published in 1926 gives a good account of the knowledge of wind energy and wind turbines at that moment. It is quite surprising that one can make such a sweeping, general statement which applies to any wind turbine with a disc-like rotor. To prove the theorem requires a bit of math and physics, but don't be put off by that, as Betz himself writes in his book.
The proof of the Betz theorem can be found in the Reference Manual (Proof of Betz law).
- Principles of energy extraction: general notes
- Actuator disk model: graphs for velocities, pressures
- Flow assumptions: assumptions and first equations
- Fluid dynamic conservation laws: more equations, induction velocity
- Induction: axial induction factor, equations
- Thrust and power coefficient: definitions
- Results (A): maximum energy extraction
- Results (B): Betz maximum
- Results (C): velocity and pressure distribution