Environmental Engineering Reference
In-Depth Information
3.1.1 Double-multiple-streamtube analysis of curved-blade VAWTs
The double-multiple-streamtube method described above considered a blade element
located at a given radius from the rotor axis. This radius would be constant over the
length of each blade for a straight-bladed VAWT. It is a relatively simple matter to
adapt the equations above to other VAWTs such as curved-blade Darrieus machines
where the blade radius is a function of elevation. Many curved-blade Darrieus
machines have been constructed with a variant of the Troposkien blade shape where
the ends of each blade comprise straight sections and the middle section has a con-
stant radius. Whatever the actual shape of the blade may be the resultant velocity at a
particular blade element is a function of the elevation from the mid-plane of the rotor
(Fig. 15) and we can defi ne
z
(
z
) as the angle of the blade element to the vertical.
Since only the wind velocity component normal (not spanwise) to the blade
results in lift and drag forces, we require the magnitude of the local resultant veloc-
ity normal to the plane of the blade element,
W
, as illustrated in Fig. 15.
W
is then
given by:
2
2
W
=Ω+
(
r
Ua
(1) in)
−
q
+
(
Ua
(1) s s)
−
q
z
(11 )
∞
∞
where
r
(
z
) is the local radius of the blade element. The aerodynamic forces acting
on the blade element in the horizontal plane can be determined by modifying eqn
(5) so as to account for angle
z
as follows [23]:
2
r
W
F
=
c C
(
cos
qz
cos
−
C
sin
q
)
(12)
n
t
2
When analysing a curved-blade VAWT using the double-multiple-streamtube
model, eqns (6) - (9) must also be modifi ed so that
C
n
is replaced by
C
n
cos
z
,
sec
q
is replaced by sec
q
sec
z
and the maximum rotor radius,
R
, is replaced by
the local radius,
r
.
Ω
r
+
U
∞
(1−
a
) sin
q
a
z
z
U
∞
(1
−
a
)cos
z
r(z)
r(z)
U
∞
(1
−
a
)
U
∞
(1
−
a
) cos
q
cos
z
R
Figure 15: Schematic of kinematics of blade and wind velocity on a curved VAWT
blade: (a) elevation of top half of blade showing component of wind
acting normal to the plane of a blade element; (b) resultant velocities
acting normal to the blade.
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