Environmental Engineering Reference
In-Depth Information
Fig. 7.5
Actuator disc
7.4 Wind Turbine Aerodynamic and Control
In order to best understand the wind turbine control challenges, the fundamental theory
of the wind power extraction process and the upper bound of conversion efficiency
C
pmax
k
;
ð
of the wind power P
wind
to mechanical power P
cap
must first be clarified.
Basically, actuator disc theory is used to derive the P
cap
given in Eq. (
7.1
) and
the maximum C
p
k
;
ð
. The actuator disc is a generic device that has the ability to
extract wind energy when it is immersed in streamlined airflow passing through a
virtual tube (see Fig.
7.5
).
Therefore, due to energy extraction, the downstream wind speed is necessarily
slower than the upstream wind speed. However, the mass flow rate must be the
same everywhere in the tube. Hence,
q
A
wind
v
wind
¼
q
A
v
¼
q
A
2
v
2
ð
7
:
10
Þ
where
A
wind
;
A
, and
A
2
are the upstream, disc, and downstream areas, respec-
tively. The force or thrust (F
d
) exerted by the wind on the actuator disc is given as:
F
d
¼
q
A
v
ð
v
wind
v
2
Þ
ð
7
:
11
Þ
Equivalently, F
d
can also be defined in term of pressure difference before (P
+
)
and after (P
-
) the disc. Based on Bernoulli's equation, the total energy of the flow
remains constant provided no work is done on the fluid. Hence, this equation can
be applied upstream and downstream of the actuator disc as follows:
1
=
2
qv
wind
þ
P
1
¼
1
2
qv
2
þ
P
þ
ð
7
:
12
Þ
;
1
2
qv
2
þ
P
1
¼
1
2
qv
2
þ
P
by taking the difference between expressions in Eq. (
7.12
), F
d
can be rewritten as
follows:
Þ ¼
1
F
d
¼A
P
þ
P
2
q
Að
v
wind
v
2
Þ
ð
ð
7
:
13a
Þ
