Page 70 - Wind Energy Handbook

44 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

tions are necessary because the total energy is different upstream and downstream.
Bernoulli’s equation states that, under steady conditions, the total energy in the
flow, comprising kinetic energy, static pressure energy and gravitational potential
energy, remains constant provided no work is done on or by the fluid. Thus, for a
unit volume of air,
1 2
rU þ p þ rgh ¼ constant: (3:5)
2

Upstream, therefore, we have

1 2 1 2 þ
r 1 U þ r 1 gh 1 ¼ r d U þ p þ r d gh d (3:6)
1
d
d
2 2
Assuming the flow to be incompressible (r 1 ¼ r d ) and horizontal (h 1 ¼ h d ) then,
1 2 1 2 þ
rU þ p 1 ¼ rU þ p (3:6a)
2 1 2 d d
Similarly, downstream,

1 2 1 2
rU þ p 1 ¼ rU þ p d (3:6b)
d
w
2 2
Subtracting these equations we obtain
1 2 2
þ

( p p ) ¼ r(U U )
d d 2 1 w
Equation (3.4) then gives
1 2 2
r(U U )A d ¼ (U 1 U w )rA d U 1 (1 a) (3:7)
1
w
2
and so

U w ¼ (1 2a)U 1 (3:8)
That is, half the axial speed loss in the stream-tube takes place upstream of the
actuator disc and half downstream.

3.2.2 Power coefficient

The force on the air becomes, from Equation (3.4)

2
þ

F ¼ ( p p )A d ¼ 2rA d U a(1 a) (3:9)
d d 1