Wind power technology - Wind Power Integration: Connection and System Operational Aspects - page 87

AV V 0

0

Figure 3.31 Turbine air flow

Hence

1
2 mV 2

1
2 mV 0 ¼

1
2

r A V þ V 0
2

½ V 2 V 0 P ¼

" #

1 2

1
4 r AV 3

V 0
V

V 0
V

¼ 1 þ

" #

1 2

1
2 r AV 3

1
2

V 0
V

V 0
V

¼ 1 þ

¼ wind power C p

The Betz Limit

" #

1 2

Þ 1 x 2

1
2

V 0
V

V 0
V

1
2 ð

C p ¼ 1 þ ¼ 1 þ x

x can range from 0 to 1. The maximum value is when

dC p
dx ¼ 0 ¼ 3 x 2

2 x þ 1

Hence

3 x 2 þ 2 x 1 ¼ 0

The roots are 1/3 and 1.
With x ¼ 1/3, C p ¼ 16 = 27 0 : 59, which is the Betz Limit.