Environmental Engineering Reference
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and tangential coeffi cients,
C
n
and
C
t
, as a function of the angle of attack,
a
, as
follows:
a a a a
(4)
The force acting on a blade in the direction of the local resultant air fl ow,
W
, is then:
CC
=
cos
+
C
sin
,
CC
=
sin
−
C
cos
n
L
D
t
L
D
2
r
W
F
=
c C
( s
q
−
C
in)
q
(5)
n
t
2
where
r
is the density of the air,
W
is the local resultant air velocity,
c
is the chord
length and
q
is the angle between the streamtube and the local radius to the rotor
axis (see Fig. 7). Using the impulse-momentum principle the forces on the blade
at the upstream and downstream portions of the cylinder may then be computed
and related to the resulting deceleration of the fl ow to give expressions for the
upstream and downstream induction factors:
2
u
Nc
W
a
(1
−=
a
)
sec
q
(
C
cos
q
+
C
sin
q
),
u
u
nu
tu
2
8
p
R U
W
∞
(6)
2
d
Nc
a
(1
−=
a
)
sec
q
(
C
cos
q
+
C
sin
q
)
d
d
nd
td
2
8
p
R
U
∞
where
N
is the number of blades and
R
is the radius of the rotor. Equation (6) is
solved iteratively together with the following auxiliary eqns (7) and (8) so as to
fi nd the unknown parameters in the problem. The angles of attack on the blade at
the upstream and downstream locations are given by:
Ua
(1
−
) cos
q
Ua
(1
−
) cos
q
∞
u
a
d
tan
a
=
,
tan
a
=
(7)
u
d
Ω+
RU
(1
−
a
) sin
q
Ω+
RU
(1
−
a
) sin
q
∞
u
a
d
where
= d
b
/d
t
is the angular velocity of the rotor. The local resultant relative
velocities are then given by:
Ω
2
2
WR
=Ω+
(
U
(1) sin)
−
a
q
+
(
U
(1) c s
−
a
q
) ,
u
∞
u
∞
u
(8 )
2
2
W
=Ω+−
(
RUa
(1
) sin
q
)
+−
Ua
(1
) cos
q
)
d
a
d
a
d
The torque,
Q
, generated by the blade for each of the streamtubes may then be
estimated from:
2
u
r
W
Nc
C
⎛
⎞
nu
Q
=
A
sec
q
C
cos
q
+
,
⎜
⎟
u
u
tu
⎝
⎠
22
p
R
4
(9 )
2
d
r
W
C
Nc
⎛
⎞
nd
Q
=
A
sec
q
C
cos
q
+
⎜
⎟
d
d
td
⎝
⎠
22
p
R
4
and hence the total torque and shaft power from the rotor may be determined by
integration of eqn (9) around the circumference of the rotor.
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