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 you wonder about the stream tube in the picture, you have not read the page on how the wind turbine deflects the wind ). Stream tube around turbine
If we tried to extract all the energy from the wind, the air would move away with the speed zero, i.e. the air could not leave the turbine. In that case we would not extract any energy 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 our tube above without being hindered at all. In this case we would likewise not have extracted any energy from the wind.
We can therefore assume 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, we have to use the fundamental physical law for the aerodynamics of wind turbines:
Betz' Law
Betz: "Wind Energie" from 1926 Betz' law says that you can only convert less than 16/27 (or 59%) of the kinetic energy in the wind to mechanical energy using a wind turbine.
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. Betz' own proof of the theorem is included in the Reference Manual on this web site.
© Copyright 1997-2003 Danish Wind Industry Association
Updated 1 June 2003
http://www.windpower.org/en/tour/wres/betz.htm
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