Environmental Engineering Reference
In-Depth Information
Finally, an expression for the radial gradient in axial velocity may be obtained from
Euler's equation:
U
2
-
V
1
2
d
dr
1
d
dr
1
w
1
r
1
= ( W + w
1
)
(5-17d)
These four equations may be used to obtain the relationships among thrust, torque, and flow
in the wake. Closure cannot be obtained without specification of one of the variables, say w.
When this is done, the flow is found to have the following features:
--
The pressure varies across the wake due to the rotational velocity.
--
The rotor and wake axial velocities vary radially.
--
The angular velocity of the fluid, which is opposite in direction to the rotation of the
rotor, changes discontinuously at the rotor.
In the expression for the wake radial velocity gradient, Equation (5-17d), let us assume
that
r
2
w is constant (
i.e.
, the rotor wake is an
irrotational vortex
). Then, the wake axial veloc-
ity is constant along a radius, because this equation's right-hand-side is zero. Defining
V
=
U
(1 -
a
)
(5-18a)
V
1
=
U
(1 -
b
)
(5-18b)
where
a
= axial induction (retardation) factor at the disk
b
= axial induction (retardation) factor at the far-wake
We may obtain from Equation (5-17c)
( 1
-
a
)
b
2
4 l
2
(
b
-
a
)
a
=
b
2
(5-19a)
1 -
where, as before, l is the tip-speed ratio. The power coefficient is given by
b
2
(1 -
a
)
2
b
-
a
C
P
=
Examination of Equation (5-19a) shows that the axial velocity reduction at the disk is
always approximately one-half the reduction in the far-wake for tip-speed ratios above 2,
which is the same result reached when wake rotation was neglected. The above equation for
the power coefficient requires some modification, since the assumption that
r
2
w is constant
produces infinite velocities near the axis. In lieu of an irrotational vortex wake, we may
substitute a
Rankine vortex wake
, which contains a rotational core with a constant angular
velocity equal to the maximum specified for the rotor. This leads to
b
(1 -
a
)
2
b
-
a
(5-19b)
C
P
=
b
+ (2
a
-
b
) W w
max
where
w
max
= angular velocity of the wake core (rad/s)
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