Environmental Engineering Reference
In-Depth Information
The distance between the blades is the circumference divided by the number of blades.
As the flow traverses the path shown, two distinct effects occur to change the flow pattern
from that of the actuator disk which was used to develop strip theory: (1) An increase in
tangential velocity occurs between the leading and trailing edges of the airfoil, so that the
flow follows a curved path, and (2) the axial component of the flow increases because the
airfoil thickness
reduces the size of the flow path. We will treat these two effects separately,
developing the equation
Da
c
= Da
1
+ Da
2
(5-38a)
where
Da
1
= effect of finite airfoil thickness (rad)
Da
2
= effect of finite airfoil width (rad)
Because of
blockage
by the blade thickness, axial velocity between the blades is in-
creased, which in turn increases the wind angle, f. This causes flow streamlines to be dis-
placed in a direction normal to the blade chord by an amount
w
, as shown in Figure 5-14 (a).
The circumferential spacing between the blades in this figure is greatly reduced for purposes
of illustration. The actual spacing-to-chord ratio is large, so we may employ a one-dimen-
sional flow model. Using the requirement for flow continuity, an increment of the displace-
ment,
dw
, is expressed in terms of the blade thickness as follows:
B
cos f
o
2 p
r
t
cos a
d
x =
B
cos f
o
2 p
r
dw
=
t dx
where
f
o
= flow angle prior to rotational induction (
i.e.
,
a¢
= 0) (rad)
t
= local thickness of airfoil, normal to chord line (m)
x = coordinate in the direction of the relative wind (m)
x
= airfoil chordwise coordinate, measured from leading edge (m)
The flow displacement
w
causes an increase in the apparent angle of attack of magnitude
c
c
Da
1
=
w
c
=
1
c
dw
dx
dx
=
B
cos f
o
2 p
r c
t dx
0
0
The latter integral is the airfoil cross-sectional area,
A
a
. Using the thickness distribution func-
tions for the
NACA four-digit series
of airfoils,
A
a
» 0.68
c t
max
where
t
max
= maximum thickness of the airfoil (m)
Therefore, the increase in angle of attack caused by the finite thickness of the blades is
c
R
t
c
max
l
(5-38b)
Da
1
= 0.11
B
r
R
2
(1 -
a
)
2
+
l
2
The curvature of the flow caused by the finite blade width, illustrated in Figure 5-14(b),
results in a change in the circulation developed by the blades. It is assumed that the induced
tangential velocity is developed linearly along the blade, from a value of zero at the leading
edge to its final value of 2
a
¢
r
W at the trailing edge. Note that blade coning is ignored in this
development. The change in f from a leading edge to a trailing edge is
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