Environmental Engineering Reference
In-Depth Information
General Momentum Theory with Wake Rotation
Further idealized-flow modeling can be accomplished with the additional consideration
of wake rotation. While the initial free-stream wind is not rotational, interaction with a rotat-
ing wind machine will cause the wake to rotate. In the case of a propeller, the wake rotates
in the same direction as the blades. In the case of an energy-extracting device (such as a
HAWT), the wake rotates in the opposite direction. If there is rotational kinetic energy in
the wake in addition to translational kinetic energy, we may expect (from thermodynamic
considerations) less power extraction than if the wake has translational kinetic energy only.
Joukowski [1918] considered the effect of wake rotation in the analysis of propellers.
Adopting his notation to the analysis of wind turbines, the effect of wake rotation on power
removal may be estimated. Using a streamtube analysis, equations can be written that ex-
press the relation between the wake velocities (both axial and rotational) and the correspond-
ing wind velocities at the rotor disk. In addition, for certain special cases, an expression for
the power coefficient can be obtained. The main outcome of this approach is a measure of
the effects of rotation on the relative values of the induced (retarded) velocities at the rotor
and in the wake.
Figure 5-8 illustrates the geometry of the
streamtube model
of wind flow through a
HAWT. Assuming fluid drag is zero, the resulting equations are
Continuity
:
(5-17a)
V r dr
=
V
1
r
1
dr
1
r
2
w
=
r
1
w
1
Moment of momentum
:
(5-17b)
W + w
1
/2
V
1
-
W + w 2
V
/
0.5(
U
-
V
1
)
2
=
V
1
w
1
r
1
Energy
:
(5-17c)
where
r
,
r
1
= radial coordinates at the rotor and at the far wake, respectively (m)
dr
,
dr
1
= radial thickness of the streamtube at rotor and far wake, respectively (m)
w, w
1
= angular velocity in the fluid at rotor and far wake, respectively (rad/s)
Figure 5-8. Geometry of the streamtube model of flow through a HAWT rotor.
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