Environmental Engineering Reference
In-Depth Information
The incremental thrust,
dT
, and torque,
dQ
, acting on an annulus are given by
dT
= 0.5 r
V
r
C
L
c B
cos f
dr
(5-26a)
dQ
= 0.5
r
r
V
r
C
L
c B
sin f
dr
(5-26b)
where
V
r
= relative wind velocity (m/s)
c
= blade chord length at radius r (m)
B
= number of rotor blades
Assuming that the far-wake axial induction is twice the axial induction at the rotor, momen-
tum expressions yield
dT
= 4 p
r
r
U
2
(1 -
a
)
a dr
(5-27a)
dQ
= 4 p
r
3
r
U
W (1 -
a
)
a dr
(5-27b)
Combining Equations (5-26) and (5-27) then gives
a
1 -
a
B c C
L
cos f
8 p
r
sin
2
f
(5-28a)
=
a
1 +
a
B c C
L
8 p
r
cos f
=
(5-28b)
Now,
a
,
a
¢, and f are known as functions of
x
, so a dimensionless blade parameter equal to
(
Bc
W
C
L
/2p
U
) may be determined. Results are listed in the right-hand column of Table 5-2.
Note that an optimum blade for a given tip-speed ratio and constant
C
L
will have a chord that
approaches a maximum at
x
approximately equal to 0.75.
To illustrate the use of Table 5-2 to determine an optimum blade shape, consider a two-
bladed rotor designed to operate at l = 6 with
C
L
= 0.8. The left-hand column of the table may
be used to determine radial position along the blade, since
r
R
=
r
W
U
R
W
U
=
x
l
=
x
(5-29a)
6
The required blade chord can be obtained from the right-hand column as follows:
c
R
B c
W
C
L
2 p
U
2 p
B C
L
U
R
W
= 0.654
B c
W
C
L
2 p
U
(5-29b)
=
The required twist angle, b, of the
zero-lift line
(Fig. 5-10) can be obtained from the relation-
ship b = f - a, where a is the angle of attack. Assuming
ideal lift
, for which
C
L
= 2p sin a,
the angle of attack is 7.3 degrees for
C
L
= 0.8. Thus
b = f - 7.3
°
(5-29c)
Thus, although the effects of finite blade number are not included, Table 5-3 gives an indica-
tion of the rotor geometry required to achieve maximum
C
P
at a tip-speed ratio of 6. The
blade is noted to have
nonlinear taper and twist
. In order to determine the effects of finite
blade number, drag, and operation at other tip-speed ratios, another approach is required.
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