361
10.3 Wind Turbine Performance Analysis
From a = 1 3 follows:
A completely tight, plane plate, placed perpendicularly to the wind has a drag coefficient of 1.28. So, the working wind turbine takes up an axial force of about 70 %
of the force exerted on a tight, plane plate. This puts a very large load on the rotor
and the supporting tower.
It subsequently appears with (10.7) that not all energy that passes uninhibitedly
through the frontal surface of a wind energy system can be captured. As the flow
retards when capturing its kinetic energy, the mass flow passing through the rotor
disc decreases. There is thus an optimal retardation, found to be a = 1 3 according to
the preceding derivation.
The preceding theory is generally known as ‘Momentum Theory’ or ‘RankineFroude Actuator Disc Theory’ (Rankine 1865; Froude 1889; Betz 1920).
10.3.2 Multiple Streamtube Analysis
The analysis in the preceding section may be refined by considering a series of
concentric annular streamtubes with an infinitesimal thickness and by writing the
momentum and work relations for each streamtube. With this analysis, the rotation
effect behind the rotor may be taken into account. A wind turbine example was
already studied in Chap. 2, Exercise 2.5.8. Figure 10.10 represents an infinitesimal
streamtube. The figure also renders the velocity triangles immediately upstream
and downstream of the rotor disc. At the position of the rotor disc, the axial velocity
component is represented by (
)
1
0
− a v . There is no tangential velocity component
immediately upstream of the rotor. The tangential velocity component immediately
downstream of the rotor is represented by −2bu.
The pressure variation in the streamtube may be assumed symmetrical. The momentum balance in axial direction stays then the same as in the preceding analysis:
with dA = 2πr dr.
With v
av
a
3
0
1 2
= −
(
) , it follows that
(10.9)
The work equation in the relative system between the positions just upstream and
downstream of the rotor is
C T = =
8
9
0 889
.
.
1
2
0
0
3
(
)
(1 ) (
),
a
dT dA p p
v
a dA v v
r
=
−
=
−
−
2
1
2
0 (1 ) 2 .
p p
v
a a
r
−
=
−
10.3 Wind Turbine Performance Analysis
From a = 1 3 follows:
A completely tight, plane plate, placed perpendicularly to the wind has a drag coefficient of 1.28. So, the working wind turbine takes up an axial force of about 70 %
of the force exerted on a tight, plane plate. This puts a very large load on the rotor
and the supporting tower.
It subsequently appears with (10.7) that not all energy that passes uninhibitedly
through the frontal surface of a wind energy system can be captured. As the flow
retards when capturing its kinetic energy, the mass flow passing through the rotor
disc decreases. There is thus an optimal retardation, found to be a = 1 3 according to
the preceding derivation.
The preceding theory is generally known as ‘Momentum Theory’ or ‘RankineFroude Actuator Disc Theory’ (Rankine 1865; Froude 1889; Betz 1920).
10.3.2 Multiple Streamtube Analysis
The analysis in the preceding section may be refined by considering a series of
concentric annular streamtubes with an infinitesimal thickness and by writing the
momentum and work relations for each streamtube. With this analysis, the rotation
effect behind the rotor may be taken into account. A wind turbine example was
already studied in Chap. 2, Exercise 2.5.8. Figure 10.10 represents an infinitesimal
streamtube. The figure also renders the velocity triangles immediately upstream
and downstream of the rotor disc. At the position of the rotor disc, the axial velocity
component is represented by (
)
1
0
− a v . There is no tangential velocity component
immediately upstream of the rotor. The tangential velocity component immediately
downstream of the rotor is represented by −2bu.
The pressure variation in the streamtube may be assumed symmetrical. The momentum balance in axial direction stays then the same as in the preceding analysis:
with dA = 2πr dr.
With v
av
a
3
0
1 2
= −
(
) , it follows that
(10.9)
The work equation in the relative system between the positions just upstream and
downstream of the rotor is
C T = =
8
9
0 889
.
.
1
2
0
0
3
(
)
(1 ) (
),
a
dT dA p p
v
a dA v v
r
=
−
=
−
−
2
1
2
0 (1 ) 2 .
p p
v
a a
r
−
=
−