Environmental Engineering Reference
In-Depth Information
2.2 Optimum blade shape
Neglecting drag all relevant forces can be derived from lift via (compare to
Fig. 9 )
2
dT
rw
(27)
=
Bc
C
L
·
cos
j
dr
2
and torque
2
dQ
rw
( 28 )
=
Bc
C
L
·
sin
j
· .
r
dr
2
Adding to eqn (10) the momentum balance for thrust, equation (17), and torque
equation (19) solution for
a
and
a
′
is possible. The following fi nal equations are
obtained:
a
a
s
Ccos
·
j
L
=
,
( 29 )
2
2
(1
−
)
4
sin
(
j
)
a
′
s
C
L
=
( 30 )
1
+
a
′
4
cos
j
With
s
=
BC
/2
p
r
solution for
c
·
C
L
which is proportional to the circulation:
Bc
w
C
4(2
12
sin
j
cos
j
j
−
1
)
L
=C=
ls
,
(31 )
r
L
2
p
v
+
cos
1
can be performed. Division by
r
/
R = l
r
/ l
R
results in the desired ratio
c
/
r
against
r
/
R
(Fig. 13). Together with Glauert's theory the more extended approach
of Wilson [18] and De Vries [17] , which includes also the lift to drag ratio and
tip losses:
(1
−
aF aF
a
)
s
Ccos
·
j
L
=
,
( 32 )
2
2
(1
−
)
4
sin
(
j
)
aF
a
′
s
C
L
=
.
( 33 )
1
+
′
4
cos
j
with
F
from eqn 12 is obtained. It can be seen that for
C
L
/
C
D
= 100 and
B
= 3,
there are only small differences from the Glauert theory. Recently [40] it was
found that the widely used optimization approach by Betz may be overcome
by the older one of Joukowsky [36] thereby stating that a constantly loaded
rotor with a fi nite number of blades may be superior to that with a Betz-type
load distribution.
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