Environmental Engineering Reference
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The remaining task is to relate the forces acting on the blade and the torque
generated to the local rotor azimuth angle,
b .
However, independent variables,
such as
W, a
u
and
a
d
, have to this point been calculated only as a function of the
angle
q
between the streamtube and the local radius arm from the rotor axis. The
blade azimuth angle
b
is related to
q
through the degree of expansion of all
the streamtubes passing through the rotor. It is a relatively simple matter to deter-
mine
b
after the local streamwise velocities,
U
u
and
U
d
, have been found as func-
tions of
q
and the corresponding azimuth angles may then be computed as detailed
by Sharpe [23]:
q
p
2
2
U
p
2
U
⎛⎞
( )
( )
∫
u
∫
d
bq
=
d,
q bq
=
b
+
d
q
⎜
⎝⎠
(10 )
d
u
d
UU
+
2
UU
+
u
d
u
d
0
q
The double-multiple-streamtube model described above has been implemented
by the present author and illustrative results are presented in Figs 12-14 for
a straight-bladed VAWT with a rotor radius of 20 m and blades based on the
NACA0012 profi le with a rotor solidity of
s
=
Nc
/
R
= 0.15. Lift and drag data
have been taken at an average blade Reynolds number of
Re
m
= 1.0
10
6
from
the data provided by a key publication on the lift and drag characteristics of
aerofoils for VAWTs [46].
×
30
TSR = 2.0
TSR = 4.0
TSR = 6.0
20
10
0
-150
-100
-50
0
50
100
150 200 250 300
Blade equatorial angle,
b
(º)
-10
-20
-30
Figure 12: Angle of attack of resultant wind velocity,
, as a function of blade
azimuth angle,
b
, and tip speed ratio. Predicted from a double-multiple-
streamtube analysis of a straight-bladed VAWT (NACA0012H blade
profi le,
Re
= 1.0
a
10
6
,
R
= 10 m,
c
= 0.5 m, number of blades
N
= 3,
×
Ω
= 3.14 rad/s).
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