Environmental Engineering Reference
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a
¼
1
U
1
=
U
0
ð
3
:
3
Þ
so that the larger the value of a the greater the reduction in speed as the wind
passes through the blades.
3.3.3 Conservation of Angular Momentum
In vector form, dQ, the torque acting on the blade elements within the streamtube
can be obtained from Eq.
2.7
. From this can be deduced the scalar equation for the
contribution to Q, the torque acting about the axis of rotation:
dQ
¼
qr
1
W
1
U
1
2pr
1
dr
1
ð
3
:
4
Þ
assuming that there is no swirl upstream of the blades. (This is an important
assumption that will be examined in more detail in
Chap. 4
.) Downstream of the
blades, the angular momentum of the streamtube is conserved so rW
2
= r
?
W
?
.
Using this relationship and conservation of mass
Þ
W
2
r
2
dr
¼
4pqU
0
1
a
Þ
a
0
Xr
3
dr
dQ
¼
2pqU
0
1
a
ð
ð
ð
3
:
5
Þ
where W
2
= 2a
0
Xr defines (twice) the rotational interference factor. The geo-
metric significance of a and a
0
are discussed in the next section. Note that the
average W seen by the blades is
Þ=
2
¼
W
2
=
2
¼
a
0
Xr
W
¼
W
0
þ
W
2
ð
ð
3
:
6
Þ
where a
0
is the rotational interference factor.
3.4 The Forces Acting on a Blade Element
The analysis of the previous two sections gives the velocity components for each
blade element at radius r. The situation is summarised in Fig.
3.2
. The velocity in
the wind direction is U
1
and the circumferential velocity is the sum of Xr and W as
defined in Eq.
3.6
. Adding these velocities vectorially, and ignoring any radial
velocity, gives the non-dimensional velocity U
T
:
2
Þ
2
þ
1
þ
a
0
U
T
¼
1
a
ð
½
ð
Þ
k
r
ð
3
:
7a
Þ
where k
r
is the local speed ratio (of the blade element)
k
r
¼
rX
=
U
0
¼
kr
=
R
ð
3
:
7b
Þ
U
T
is usually called the ''total'' or ''effective'' velocity as seen by the blade element.
a is the angle of attack, which is sometimes called the angle of incidence. This is
one of the three important angles defined in Fig.
3.2
: h
P
the twist, is the angle
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