Environmental Engineering Reference
In-Depth Information
tan h
p
¼
2
3k
p
r tan a
max
3k
p
r
þ
2 tan a
max
ð
6
:
9a
Þ
so
sin h
p
cos h
p
¼
1
Þ
4 sin 2a
max
ð
Þ
þ
6k
p
r cos 2a
max
ð
Þ
2
sin 2a
max
ð
ð
6
:
9b
Þ
4
þ
9k
p
r
2
4
þ
9k
p
r
2
where, interestingly, the first term depends only on the choice of aerofoil, and the
last two on the design tip speed ratio as well. Integration of (
6.7
) using Eq.
6.8
for
cr, gives the expression for the coefficient of the stationary blade torque, Q
s
:
Q
s
16
9k
p
C
l
;
max
C
Q
;
s
¼
2
qU
s
pR
3
¼
ð
I
1
þ
I
2
þ
I
3
Þ
ð
6
:
10
Þ
1
where U
s
is the starting wind speed. The integrals on the right are, in the same
order as the integrands in (
6.9b
):
I
1
¼
1
r
h
ð
Þ
sin 2a
max
ð
Þ
ð
6
:
11a
Þ
tan
1
I
2
¼
4 sin 2a
max
ð
Þ
3k
p
2
3k
p
r
h
2
tan
1
ð
6
:
11b
Þ
3k
p
and
!
4
þ
9k
p
4
þ
9k
p
r
h
ð
Þ
I
3
¼
2 cos 2a
max
ln
ð
6
:
11c
Þ
3k
p
To estimate the rated torque, assume that the turbine operates at the Betz-
Joukowsky limit at rated wind speed U
p
. It is also assumed that the generator and
drive train have no inefficiencies. Thus
2
qU
p
pR
3
¼
16
Q
p
C
Q
;
p
¼
ð
6
:
12
Þ
1
27k
p
Figure
6.7
shows the ratio of the stationary torque from (
6.10
) to the rated
torque from (
6.12
) for the parameter values given in the caption. Recall from
Fig.
1.12
that the ratio of resistive to rated generator torque typically falls between
0.01 and 0.02, then even the highest curve in Fig.
6.7
should give cause for
concern.
When it is further realised that (
6.8
) allows c to become infinite as r ; 0, and a
practical limit on blade chord is likely to be around 0.1 then the situation gets
worse. This is demonstrated by the calculations for c
max
= 0.1 in the figure with
the twist distribution unaltered. These calculations are somewhat more involved
than for the unconstrained chord and are not described. Further, N now becomes
important, and Fig.
6.7
shows that three blades produce significantly more torque
than two. In summary: Fig.
6.7
shows how much attention the blade designer has
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