Environmental Engineering Reference
In-Depth Information
induction factors [Lock
et al.
1926, Wilson
et al.
1976]. A
thrust coefficient
, which can also
be used to characterize the different flow states of a rotor, is shown in Figure 5-11(b) as a
function of the axial induction factor, and is defined as
T
0.5 r
AU
2
C
T
=
(5-30a)
By Momentum Theory:
C
T
= 4
a
| 1 -
a
|
(5-30b)
where
C
T
= rotor thrust coefficient
Wind turbines will normally operate in the
windmill state
, with 0 £
a
£ 0.5. For nega-
tive inductions (
a
< 0) it is simple to continue the analysis to show that the device will act
as a propulsor producing an upwind force (
i.e.
C
T
< 0) that adds energy to the wake. This is
typical of the
propeller state
.
When a wind turbine rotor operates at tip-speed ratios appreciably above its design
value, blade tips may be driven into the
turbulent wake state
. As illustrated by the data in
Figure 5-11(b) obtained on wind turbines, autogiros, and helicopters, rotor thrust increases
with increasing induction in the turbulent wake state, instead of decreasing as predicted by
Equation (5-30b). Thus, momentum theory is considered to be invalid for induction factors
larger than about 0.4. Glauert's empirical formula [1926] and, more recently, Equation (2-16)
[Spera 2008] are acceptable models of rotor thrust behavior for induction factors from 0.4 to
1.0, or, equivalently, 0.96 <
C
T
< 2.0 [Dugundji
et al.
1978].
When the induction factor is somewhat over unity, the flow regime is called the
vortex
ring state
, a condition which is experienced by helicopters during powered (slow) descent.
A particularly interesting case occurs for axial inductions greater than unity, where the rotor
reverses the direction of flow. This may be physically modeled by considering a powered
propeller with blades pitched so that they induce a forward flow. This is termed the
propel-
ler brake state
, with power being added to the flow to create downwind thrust on the rotor.
Further discussion of wind rotor states is given by Eggleston and Stoddard [1987].
Summary Comments on Actuator Disk Theory
The actuator disk models discussed in this section provide limiting values of rotor
performance and a general understanding of the rotor configuration for a particular operat-
ing condition. For example, there is a maximum power coefficient of 16/27 for HAWTs,
and this limit (called the
Betz
or
Lanchester-Betz
limit) can be approached when the wake
rotation is low. Further, we noted that wind turbines operating at high values of induction
develop forces that are considerably different from the axial forces predicted by the mo-
mentum relation, Equation (5-30). The assumptions of actuator disk theory, particularly
the assumption of an infinite number of blades, restrict our understanding of the effect of
rotor geometry (
i.e.
blade airfoil section, chord, and twist) on HAWT operation. Addition-
ally, we find that the Betz limit is higher than the power coefficients achieved in practice,
because actual rotors (1) have a finite number of blades and (2) are acted upon by drag
forces.
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