Environmental Engineering Reference
In-Depth Information
vicinity of the turbine to a value
V
t
at the turbine and an even lower value of
V
w
downstream of
the turbine, called the wake region. The turbine power
P
is then the product of the mass flow rate
V
t
A
, and the reduction of kinetic energy of the wind,
V
2
V
2
through the turbine,
ρ
/
2
−
w
/
2, or
V
t
V
2
A
1
2
ρ
V
2
w
P
=
−
(7.10)
As the air slows down both upstream and downstream of the wind turbine, it undergoes a pressure
rise that is very small compared to atmospheric pressure. This provides a pressure drop across the
wind turbine of amount
V
2
V
2
V
2
V
2
ρ(
−
w
)/
2 and a corresponding axial thrust force of
ρ(
−
w
)
A
/
2.
But this force must also equal the reduction in momentum of the wind flow,
ρ
V
t
A
(
V
−
V
w
)
. It then
follows that
V
t
is the average of
V
and
V
w
:
1
2
(
V
t
=
V
+
V
w
)
(7.11)
The power
P
can now be expressed in terms of
V
and
V
w
as
V
3
A
(
2
1
2
ρ
1
+
V
w
/
V
)
(
1
−
V
w
/
V
)
P
=
(7.12)
2
The factor in brackets has a maximum value of 16/27 when
V
w
=
3, in which case 8/9 of the
wind's kinetic energy has been removed by the wind turbine. As a consequence, the wind turbine
power cannot exceed the limit:
V
/
1
2
ρ
V
3
A
16
27
P
≤
(7.13)
The foregoing considerations do not reveal the detailed mechanism whereby the wind flow
exerts a torque on the wind turbine rotor in the direction of its rotation, thereby generating me-
chanical power. Figure 7.16 depicts the amount and direction of the wind flow relative to a section
of the turbine blade at a radius
r
from the turbine axis. In the tangential direction, the velocity
L
D
2
rf
V
t
V
rel
Figure 7.16
The motion of the wind relative to a turbine blade consists of an axial speed
V
t
and a tangential
speed 2
π
rf
that generates a lift force
L
and drag force
D
.
