Page 71 - Wind Energy Handbook

THE ACTUATOR DISC CONCEPT 45

As this force is concentrated at the actuator disc the rate of work done by the force
is FU d and hence the power extraction from the air is given by

3
Power ¼ FU d ¼ 2rA d U a(1 a) 2 (3:10)
1
A power coefficient is then defined as

Power
C P ¼ (3:11)
1 3
rU A d
1
2
where the denominator represents the power available in the air, in the absence of
the actuator disc. Therefore,

C P ¼ 4a(1 a) 2 (3:12)

3.2.3 The Betz limit

The maximum value of C P occurs when

dC P
¼ 4(1 a)(1 3a) ¼ 0
da

1
which gives a value of a ¼ .
3
Hence,
16
¼ ¼ 0:593 (3:13)
C P max
27
The maximum achievable value of the power coefficient is known as the Betz
limit after Albert Betz the German aerodynamicist (119) and, to date, no wind
turbine has been designed which is capable of exceeding this limit. The limit is
caused not by any deficiency in design, for, as yet, we have no design, but because
the stream-tube has to expand upstream of the actuator disc and so the cross section
of the tube where the air is at the full, free-stream velocity is smaller than the area
of the disc.
C P could, perhaps, more fairly be defined as

Power extracted Power extracted
C P ¼ ¼ (3:14)
Power available 16 1 3
rU A d
1
27 2
but this not the accepted definition of C P .