Proof
of Betz' Law
This page gives a proof of Betz' law. Before reading this
page you should have read the pages in the Guided Tour on how
the wind turbine deflects the
wind and Betz' Law. If
you do not follow the argument in detail, just glance through
the rest of this page, which uses Betz' own reasoning from his
book Wind-Energie from 1926 to explain the law.
Proof
of Betz' Theorem
Let
us make the reasonable assumption that the average wind speed
through the rotor area is the average of the undisturbed wind
speed before the wind turbine, v1, and the wind speed
after the passage through the rotor plane, v2, i.e.
(v1+v2)/2. (Betz offers a proof of this).
The
mass of the air streaming through the rotor during one second
is
m = F (v1+v2)/2
where m is the mass per second,
is the density of air, F is the swept rotor area and [(v1+v2)/2]
is the average wind speed through the rotor area. The power
extracted from the wind by the rotor is equal to the mass times
the drop in the wind speed squared (according to Newton's second
law):
P = (1/2) m (v12 - v22)
Substituting m into this expression from the first equation
we get the following expression for the power extracted from
the wind:
P = (/4) (v12 - v22) (v1+v2) F
Now, let us compare our result with the total power in the
undisturbed wind streaming through exactly the same area F, with
no rotor blocking the wind. We call this power P0:
P0 = (/2) v13 F
The ratio between the power we extract from the wind and the
power in the undisturbed wind is then:
(P/P0) = (1/2) (1 - (v2 / v1)2) (1 + (v2 / v1))
We may plot P/P0 as a function of v2/v1:
We can see that the function reaches its maximum for v2/v1 = 1/3,
and that the maximum value for the power extracted from the wind
is 0,59 or 16/27 of the total power in the wind.
Click here to go back the Guided Tour page on Betz'
Law.
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