Environmental Engineering Reference
In-Depth Information
Since this integral for the power involves two dependent variables, another relation is
required. This is the
momentum equation
, as follows:
a
(1 +
a
)
x
2
=
a
(1 -
a
)
(5-22)
A unique way of illustrating this relation is to consider the velocities at the rotor
plane of
rotation
of a HAWT, as shown in Figure 5-10. The flow is assumed to be uniform in annular
streamtubes with no circumferential variations. Under these conditions, two-dimensional
flow may be assumed. In the absence of drag, the velocity
induced
at the rotor must be
caused by lift and, hence, be perpendicular to the relative velocity. Two expressions for the
tangent of the angle from the plane of rotation to the relative wind vector may be developed
under the condition that the total induced velocity is perpendicular to the relative velocity.
These are
(1 -
a
)
U
(1 +
a
)
r
W
(1 -
a
)
(1 +
a
)
x
tan f =
=
(5-23a)
a r
W
a U
a
a
x
tan f =
=
(5-23b)
Equating the right-hand-sides of Equations (5-23) also produces Equation (5-22). The
Calculus of Variations
can now be used to solve Equation (5-21) with the constraint of Equa-
tion (5-22), obtaining the following relationship between the rotational and axial induction
factors:
1 - 3
a
4
a
- 1
a
=
(5-24)
so that
1 -
a
1 - 3
a
(5-25)
x
= (4
a
- 1)
Hence,
1/3
>
a
>
1/4
. A tabulation of variations in the parameters
x
,
a
,
a¢
, f, and
C
P
is
given in Table 5-2. Since high-speed rotors easily reach tip-speed ratios of 7 or more, it
can be seen that most of an ideal rotor will operate with
a
= 1/3 and a rotational velocity
distribution in the form of an
irrotational vortex
. The power coefficient for various tip-
speed ratios is also given in Table 5-2, by equating l to
x
. At low tip-speed ratios, power
coefficients are low because of the large rotational kinetic energy in the wake. At high
tip-speed ratios, the power coefficient approaches 0.593 and the wake rotation approaches
zero.
Further information may be obtained from the Glauert optimum disk model using the
blade-element theory
. Blade-element theory equates the thrust on a radial increment of blade
of length
dr
to the momentum change in a flow annulus (streamtube) of area 2p
rdr
. The
blade torque is then equated to the moment of momentum in a similar fashion. As the quanti-
ties
a
and
a
¢ are known for each radial position, the relative velocity and the angle f may be
determined. Figure 5-10 illustrates the velocities and forces in relation to the blade configu-
ration. Of course, since we have assumed that the drag is zero, the only force that acts on the
blade is lift.
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