362
10 Wind Turbines
or
(10.10)
Combination of (10.9) with (10.10) produces b as a function of a:
(10.11)
with r
l the local speed ratio:
0
/ .
r
u v
l =
Work (power) gained from the streamtube is given by Euler’s formula:
meaning with (10.11):
(10.12)
Expression (10.12) replaces expression (10.7) on an infinitesimal basis. Expression
(10.7) is recovered for b = 0 (no post-swirl) and a = constant over the radius.
The power (10.12) may be optimised for each streamtube, giving
or
(10.13)
2
2
2
2
2
1
2
1
2
(1 2 )
,
2
2
2
2
a
a
v
v
p
p
u
u
b
r
r
+
+
=
+
+ +
2
1
2
2 (1 ) .
p p
b
b u
r
−
=
+
2
(1 )
(1 ),
r
b
b
a
a
l
+
=
−
2
0 (1 )(2
),
dP
dA v
a bu
r
=
−
2
3
0
2 (1 ) .
1
a
a
dP
dA v
b
r
−
=
+
2
2
2
(1 ) 2 (1 )
(1 )
0,
1
(1 )
a
a
a a
a db
b
da
b
−
−
−
−
−
=
+
+
db
da
b
a
a
a
=
+
−
−
(
)(
)
(
)
.
1
1 3
1
Fig. 10.10 Streamtube with infinitesimal thickness through a horizontal-axis wind turbine