Environmental Engineering Reference
In-Depth Information
cos
2
q
sin
2
f
B
2 p
c
r
(1 -
a
)
2
C
t
=
(
C
L
cos f +
C
D
sin f)
(5-32a)
where
q = coning angle, measured from the plane of rotation (rad)
Turning our attention to the tangential direction, examination of the fluid torque about
the axis of rotation for the streamtube and the increment of rotor torque from lift and drag
forces on the blade segment within the streamtube, it can be shown that Equation (5-22) is
applicable, with a minor change to account for coning, as follows:
a
(1 +
a
)
x
2
=
a
(1 -
a
) cos
2
q
(5-32b)
Thus, Equations (5-32) are the relations that determine the dimensionless induced velocities
a
and
a
¢. Before these equations can be used, however, the local thrust coefficient,
C
t
, must
be modified to account for two effects: the departure of the local thrust coefficient from the
momentum relation (as discussed in the previous section), and the non-uniformity of the in-
duced velocities in the flow, particularly near its outer edges. These so-called
tip losses
will
be discussed first.
The Tip-Loss Factor
Strip theory, as previously developed, does not account for the interaction of shed vortic-
ity with the blade's
bound vorticity
. This effect is usually greatest near the blade tip, although
strong vortex interaction can occur at deployed control surfaces. Denoting the
bound circula-
tion
of all the blades by G, a
tip-loss factor F
is defined as
/
F
= G G
¥
(5-33)
where
G
¥
= bound circulation of a rotor with
B
® ¥ and
c
® 0 (m
2
/s)
Recalling that the induced rotational velocity is directly proportional to the local vorticity in-
tensity, one can give a useful physical interpretation to
F
, which is that it is equal to the ratio
of mean induced velocity in the flow annulus to the induced velocity at the blades.
The flow in the blade tip region significantly affects rotor torque and thrust. Focusing on
these rather than the flow field, we can account for the diminished thrust and torque output of
the tip regions by defining an
effective radius
of the rotor, which is about 3 percent smaller
than the tip radius. In this empirical approach, we set
F
= 1
i f
0 <
r
<
r
e
F
= 0
i f
r
e
£
r
£
R
(5-34)
r
e
» 0.97
R
The interaction between shed and bound vorticity is like a light switch: either on or off.
Propeller development in the 1920s called for better flow models, and the behavior of
the wake of the
optimum propeller
resulted in the development of two tip-loss models. The
optimum propeller, as conceived by Betz [1919], has a
shed vortex sheet
that appears to move
like a rigid body as it is convected away from the rotor. Prandtl [1919] noted that the flow
at the edges of the Betz wake appeared to be like the flow over an infinite stack of round,
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