sharpe2004

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WIND ENERGY
Wind Energ. 2004; 7:177–188 (DOI: 10.1002/we.118)
Research
Article
A General Momentum Theory
Applied to an Energy-extracting
Actuator Disc
D. J. Sharpe*, CREST, Department of Electronic and Electrical Engineering, Loughborough University,
Loughborough, UK
Key words:
wind turbine;
general momentum
theory;
wake rotation
A general momentum theory for an energy-extracting actuator disc modelling a rotor with
a multiplicity of blades having radially uniform circulation is presented that includes the
effects of wake rotation and expansion. A parallel theory directed at the propeller has been
published elsewhere, but not one intended for the wind turbine.The rotation of the wake is
shown to be accompanied by a fall in static pressure that is additional to that which occurs
across the actuator disc and accounts for the energy extraction from the wind. Energy
extraction is recognized in the fully developed wake by the flow regaining the static pressure of the undisturbed wind at the expense of the kinetic energy in the wake. Because the
wake is still rotating in the fully developed wake, the additional fall in static pressure
appears there also and so does not augment the energy extraction.However, the additional
fall in pressure would cause a discontinuity in pressure across the fully developed wake but
is prevented by a further slowing down of the wake. The additional slowing down extracts
a little more energy from the flow than that predicted by the simple momentum theory,
which does not include wake rotation.The overall effect of wake rotation on energy extraction is very small for wind turbines operating at high tip speed ratios but can significantly
increase the predicted power output for turbines operating at low tip speed ratios. Copyright © 2004 John Wiley & Sons, Ltd.
Introduction
Simple actuator disc momentum theory, which is the basis of most blade element/momentum theory codes for
wind turbine performance prediction, is a simplification of a general momentum theory developed first by
Lanchester,1 Joukowski,2 Betz3 and Glauert4 for application to propellers. The simple momentum theory that
regards only the rate of change of axial momentum predicts that the maximum possible power that can be
extracted from a flow is 16/27 times the power available; this is known as the Lanchester–Betz limit. Glauert
found that, for propellers, the general momentum theory, which accounts for wake rotation and expansion,
could be reduced to the simple theory without any significant loss of accuracy by ignoring the effects of wake
rotation in the determination of the rate of change of axial momentum, and, in his treatment of windmills,
Glauert5 assumed that the same simplification applied. It was common at the time Glauert was writing to assume
that the kinetic energy associated with wake rotation is drawn from the translational kinetic energy of the
freestream, Goldstein6 specifically states ‘that the kinetic energy contained in the rotation of the wake has to
* Correspondence to: D. J. Sharpe, CREST, Department of Electronic and Electrical Engineering, Loughborough University,
Loughborough, UK
E-mail: [email protected]
Published online 7 July 2004
Copyright © 2004 John Wiley & Sons, Ltd.
Received 29 August 2003
Revised 13 January 2004
Accepted 23 March 2004
178
D. J. Sharpe
be provided by the motor driving an airscrew’. Accordingly, Glauert incorporated the assumed wake energy
in his development of the simple momentum theory, and this has continued to be the case for latter-day authors
such as Wilson and Lissaman7 and Eggleston and Stoddard.8 The general momentum theory, however, demonstrates that the assumption is not correct; the kinetic energy of wake rotation is acquired by a reduction in the
static pressure in the wake. The rotation of the wake requires a pressure gradient to balance the centrifugal
force on a rotating mass of air, causing a fall in pressure with decreasing radius. The fall in static pressure
exactly matches the kinetic energy of rotation. De Vries9 questions the validity of ignoring the static pressure
drop caused by wake rotation, but it still remains unrecognized.
Wilson and Lissaman7 and Eggleston and Stoddard8 adapt Glauert’s results for the general momentum theory
for the wind turbine and produce an expression for the power coefficient but then follow Glauert in assuming
that the pressure in the far wake is uniformly atmospheric, which does not reflect the loss of pressure associated with wake rotation. Wilson and Lissaman make the same specific statement concerning the kinetic energy
of the wake rotation.
This article reiterates the full analysis of the general momentum theory, which, it is hoped, will facilitate an
improved understanding of the physics of the energy-extracting actuator disc, enabling some fresh conclusions
to be drawn.
The Actuator Disc
Most horizontal axis wind energy converters employ a rotor comprising a number of rotating blades. The blades
sweep out a disc and by virtue of their aerodynamic design acquire a bound circulation, which develops a pressure difference across the disc and a driving torque about the rotor axis. To simplify the analysis, it is assumed
that the number of blades is infinite, providing circumferential uniformity to the flow, but with a total circulation that is finite. In the wake of the rotor the air rotates in reaction to the torque, attaining angular momentum and changing the static pressure in the wake. By losing some of its kinetic energy, the wind must slow
down, but only that mass of air which actually passes through the rotor disc is assumed to be affected and that
mass remains separate from the air which does not pass through the rotor disc. A boundary can be drawn containing the affected air mass, and this boundary can be extended upstream as well as downstream, forming the
so-called stream-tube which is of circular cross-section. By definition, no air flows across this boundary and
so the mass flow rate of the air flowing within the stream-tube will be the same at all streamwise positions.
Because the air slows down and is assumed to be incompressible, the cross-sectional area of the stream-tube
must expand to accommodate the slower-moving air.
Although, ultimately, kinetic energy is extracted from the airflow, a sudden step change in streamwise velocity across the rotor disc is not possible because of the unlimited accelerations and forces that this would require.
Pressure energy, however, can be extracted in a step-like manner using the actuator disc as a permeable
diaphragm between the two pressure regions. The disc first causes the approaching air to slow down gradually from the freestream speed U• to U•(1 - a), which results in a steady rise in the static pressure above the
freestream level p•. Across the disc there is a drop in static pressure such that, having passed through the disc,
the air is below p•. As the air proceeds downstream, the static pressure rises to match the freestream level at
the wake boundary, causing a further slowing down of the wind to U•(1 - b) in the fully developed wake. The
induced velocity factor b in the developed wake is greater than the corresponding factor a at the disc.
Wake Rotation and Expansion
The exertion of a torque on the rotor disc by the air passing through it requires an equal and opposite torque
to be imposed upon the air. The consequence of the reaction torque is to cause the air to rotate in a direction
opposite to that of the rotor; the air gains angular momentum and so in the wake of the rotor disc the air particles have a velocity component in a direction which is tangential to the rotation as well as having an axial
velocity component.
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
Energy-extracting Actuator Disc
179
The change in tangential velocity is conventionally expressed in terms of a tangential flow induction
factor a¢. Upstream of the disc the tangential velocity is zero. Immediately downstream of the disc, a
radial distance r from the axis of rotation, the tangential velocity is 2Wra¢, W being the angular velocity of
the rotor. At the mid-plane of the disc, however, the induced tangential velocity is assumed to be Wra¢. If
the rotor disc is assumed to be an infinitesimally thin disc of circumferential bound vorticity, then the
induced tangential velocity on the disc is exactly Wra¢, whereas immediately downstream of the disc the
tangential velocity is 2Wra¢. The velocities and pressures at the different stages of the flow are shown
Figure 1.
Static pressure within the wake
The angular velocity of an air particle in the wake will give rise to a centrifugal force, which has to be balanced by a radial pressure difference across the particle, causing a radial pressure gradient that is a feature of
the entire wake.
The angular momentum of a particle of air remains constant as it travels along the wake, but expansion of
the flow from radius r to radius rw will cause the angular velocity of the particle to fall from 2Wa¢ immediately downstream of the disc to 2Wb¢ in the fully developed wake, where b¢ is less than a¢.
Therefore the angular momentum per unit volume in the wake is
2 rWa ¢r 2 = 2 rWb ¢rw2
(1)
Application of Bernoulli’s theorem
The total energy of the flow upstream of the rotor disc is constant everywhere at the freestream level, ignoring potential energy. Downstream of the disc the total head of a particle remains constant but at a lower level
than that of the freestream because of the extraction of energy by the rotor.
As a fluid particle passes through the rotor disc, it will have a radial velocity w because of the flow expansion. In the fully developed wake the radial velocity is zero but the tangential velocity remains, together with
the necessary radial pressure gradient, which means that, except at the very edge of the fully developed wake,
the static pressure pw will be less than the freestream level p•.
Figure 1. Velocities and pressures in the expanding stream-tube
Copyright © 2004 John Wiley & Sons, Ltd.
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D. J. Sharpe
Bernoulli’s theorem can therefore be applied separately to both the upstream and downstream conditions:
1
1
1
2
rU •2 + r • = rU •2 (1 - a) + rw 2 + pdu
2
2
2
(2)
1
1
1
2
2
rU •2 (1 - a) + r(2 Wa ¢r) + rw 2 + pdd
2
2
2
1
1
2
2
= rU •2 (1 - b) + r(2 Wb ¢rw ) + pw (rw )
2
2
(3)
where pdu and pdd are the static pressures immediately upstream and downstream respectively of the rotor disc
and pw(rw) is the static pressure in the fully developed wake at radius rw.
Subtraction of the two equations results in the following expression for the drop in pressure across the rotor
disc:
1
2
2
r U •2 (2 - b)b - 4 W 2 (b ¢rw ) - (a ¢r) + p• - pw (rw )
(4)
2
2
The angular momentum per unit volume of the flow downwind is 2rWa¢r and remains constant, in this inviscid representation, as the wake develops. Therefore 2rWa¢r2 = 2rWb¢rw2 and hence the pressure drop across the
disc is
Dpd = pdu - pdd =
Dpd =
[
{
]}
1
r[U •2 (2 - b)b - 4(b ¢ - a ¢)W 2 a ¢r 2 ] + p• - pw (rw )
2
(5)
The lift forces on the blades forming the rotor disc are normal to the resultant velocity relative to the blades
and so no work is done on or by the fluid. Therefore Bernoulli’s theorem can be applied to the flow across the
disc, relative to the disc spinning at angular velocity W, to give
1
1
1
2
rU •2 (1 - a) + rW 2 r 2 + rw 2 + pdu
2
2
2
1
1
1
2
2
= rU •2 (1 - a) + rW 2 (1 + 2 a ¢) r 2 + rw 2 - pdd
2
2
2
(6)
Consequently,
Dpd = 2 rW 2 (1 + a ¢)a ¢r 2
(7)
From equations (5) and (7) the overall static pressure drop from far upstream of the disc to the fully developed wake is
p• - pw (rw ) =
1
r[ 4 W 2 (1 + b ¢)a ¢r 2 - U •2 (2 - b)b]
2
(8)
Differentiation of (8) with respect to rw and using equation (1) to replace a¢r2 with b¢rw2 results in an expression for the radial pressure gradient in the rotating wake:
dpw
db
d
(b ¢rw2 ) + 4 rW 2 b ¢ 2 rw
= rU •2 (1 - b)
- 2 rW 2 (1 + 2 b ¢)
drw
drw
drw
(9)
However, the radial equilibrium of the rotating wake also provides an expression for the radial pressure
gradient:
dpw
2
= r(2 Wb ¢) rw
drw
(10)
Consequently,
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
Energy-extracting Actuator Disc
181
2 W 2 (1 + 2 b ¢)
d
db
(b ¢rw2 ) = U •2 (1 - b)
drw
drw
(11)
The radial variations of b and b¢ depend upon the radial distribution of blade circulation, which would, in turn,
depend upon the geometrical design of the blades.
Uniform blade circulation strength
It is possible to obtain a solution for equation (11) if it is assumed that the bound circulation is uniform along
the blade span. A direct consequence of uniform circulation is that the angular momentum in the wake will
also be radially uniform.
Radially uniform angular momentum means that
2 Wa ¢r 2 = 2 Wb ¢rw2 = constant
If a¢t is the tangential flow factor at the edge of the disc (‘t’ is for blade tip) and b¢t is the corresponding value
at the edge of the fully developed wake, then
a ¢r 2 = at¢R 2 = b ¢rw2 = bt¢Rw2 = constant
(12)
where Rw is the radius of the fully developed wake.
Inspection of equations (11) and (12) leads directly to the conclusion that the wake flow factor b is radially
uniform.
Assuming that a is also radially uniform, then, for continuity,
rw drw 1 - a Rw2 rw2 a ¢
=
=
=
=
rdr
1 - b R2 r 2 b ¢
(13)
and this forms a link between the axial and tangential flow induction factors.
The assumption of uniformity for a is certainly true at high tip speed ratios, but it has not been rigorously
established that the condition is true at all tip speed ratios. As will be shown later, the uniformity of a is an
equivalent assumption to that of no radial transfer of axial momentum.
Application of the axial momentum theory
For the whole stream-tube the axial force due to pressure will be equal to the rate of change of momentum of
the air. The sides of the stream-tube and the upstream cross-section are subject to atmospheric pressure, but
the static pressure in the fully developed wake is lower than atmospheric because of the rotation of the wake.
2p Ú 0RDpd rdr - 2p Ú 0Rw ( p• - pw )rw drw = 2prÚ 0Rw U •2 (1 - b)bdrw
(14)
Substitution of equations (7), (8) and (12) into (14) gives
(1 - a)b 2 = 4 l2 at¢(b - a)
(15)
where l = WR/U• is the tip speed ratio of the rotor.
Equation (15) could have been obtained by taking the differential form of equation (14), which means that
there is no radial transfer of axial momentum, but this relies on the value of a being uniform.
At the far wake boundary with the freestream flow the static pressure will be atmospheric, so, from
equation (8),
1
r[U •2 (2 - b)b - W 2 4(1 + bt¢)at¢R 2 ] = 0
2
(16)
which, combined with equation (15), reduces to
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
182
D. J. Sharpe
Figure 2. The ratio b/a as a function of b and l
at¢ = 1 - 2
a
b
(17)
Eliminating b from equations (15) and (17) results in
a(1 - a) = l2 at¢(1 - at¢ 2 )
(18)
From equations (17) and (18) a relationship between the axial flow factors a and b is found:
a2 1 Ê b2
b
ˆa 1
+
- 12
+ - 2 =0
2
2
Ë
¯
8 l
b 2 8l
b
(19)
For a range of values of b from 0·1 to 0·95 and a range of values of l from 1·0 to 6·0 the value of b/a is
calculated from equation (19) and plotted in Figure 2. For tip speed ratios greater than 2 the ratio b/a is very
close to 2·0 but always greater than this value, asymptotically approaching 2·0 as l approaches infinity. When
b/a exactly equals 2, equation (16) shows that there is no rotation in the wake. At tip speed ratios less than 2
the ratio b/a lies well above 2·0.
Thus at high tip speed ratios the general momentum theory corresponds closely to the simple momentum
theory, but at low tip speed ratios there is a significant difference between the two theories.
The rotational velocity at the edge of the wake is not zero and so there is a commensurate loss of static pressure. In the near wake there is a discontinuity of static pressure across the wake boundary, but in the far wake
there is no discontinuity even though the rotational velocity is non-zero. To achieve the atmospheric pressure
just inside the wake boundary in the far wake, the axial velocity must reduce that much more than if there
were no wake rotation, i.e. in the simple momentum theory case. The reason why a/b is less than 0·5 is because
of the additional fall in axial velocity required to regain the static pressure lost by wake rotation. The additional fall in axial velocity gives rise to an increased power extraction compared with that predicted by the
simple momentum theory.
Power extraction
A certain way to determine the power extraction is from the rate of change of angular momentum. The
angular momentum of the wake does not change as the wake develops and so the rotational conditions
immediately behind the rotor plane can be used. Power is equal to the rotor torque times the rotor angular
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
Energy-extracting Actuator Disc
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velocity W and torque is equal to the rate of change of angular momentum of the fluid passing through the
disc. The rate of change of angular momentum of the fluid passing through an annular ring of the disc of radius
dr is equal to the mass flow rate through the ring times the change in tangential velocity times the radius of
the ring. Integration of the rate of change of angular momentum for the whole disc provides the torque on the
rotor.
Power =
1
2
r 4(WR) at¢U • (1 - a)pR 2
2
(20)
which yields the expression for the power coefficient
CP = 4 l2 at¢(1 - a)
(21)
and, combined with equation (15), gives
CP =
b 2 (1 - a)
b-a
2
(22)
Figure 2 shows that, for high tip speed ratios, a/b approaches 0·5 and equation (22) converges on the result of
the simple momentum theory.
The power extraction can also be derived by considering the rate of loss of total head by the fluid passing
through the disc, and the same results as offered by equations (20)–(22) are achieved.
Wilson and Lissaman7 and de Vries9 have formed equation (22) from Glauert’s propeller theory4 by changing the signs of a and b and show that values of CP that exceed the Lanchester–Betz limit are predicted at low
tip speed ratios. No previous author has established the reason for the high predicted values of CP.
Pressure drop across the disc
The pressure drop across the disc is given by equation (7), which, combined with equations (12) and (15), can
be written as
Dpd = r
U •2 b 2 (1 - a) Ê
R2 ˆ
1 + at¢ 2
2(b - a) Ë
r ¯
(23)
If the rate of change of axial momentum is employed to determine the power extraction, then it is clear, by
comparison with equation (22), that only the left-hand term in the brackets in equation (23) contributes to the
power. The second term is the fall in static pressure in the wake immediately behind the disc caused by the
wake rotation. Because the wake continues to rotate as it develops, the pressure drop caused by the wake rotation means that the pressure field is not uniform; atmospheric pressure is achieved only at the outer wake
boundary. The axial force on the wake caused by the rotational pressure drop is zero. The rotation of the wake
does not constitute a loss of power extraction.
It is common in the literature5,7,8 to attribute the whole of the pressure drop, given in equation (23), to the
change of momentum of the flow through the disc, but the above analysis shows this to be incorrect.
Ideal power coefficient
The maximum value of CP can be obtained by differentiating the expression for power (equation (21)) with
respect to a and setting the result to zero to give
dat¢
at¢
=
da 1 - a
(24)
From equation (15) is obtained
1 - 2 a = l2 (1 - 3at¢ 2 )
Copyright © 2004 John Wiley & Sons, Ltd.
dat¢
da
Wind Energ. 2004; 7:177–188
184
D. J. Sharpe
giving
(1 - 2 a)(1 - a) = l2 (1 - 3at¢ 2 )at¢
(25)
From equations (15) and (25) comes the expression
at¢ 2 =
3a - 1
5a - 1
(26)
Eliminating a¢t from equations (18) and (25) gives an expression for the ideal value of a:
125a 5 - 325a 4 + 290 a 3 - 106 a 2 + (17 - 12 l4 )a + 4 l4 - 1 = 0
(27)
For tip speed ratios greater than about 1·5 the optimum value of a is very slightly greater than 1/3, although
the value of b becomes close to 2/3 at tip speed ratios greater than about 3·5 (Figure 3) at low tip speed ratios,
b rises rapidly, signalling that, in practice, the wake would break up and mix with the freestream flow. This
equation does not yield an appropriate real root when the value of l falls below 0·806, but, more significantly,
the value of b rises to unity at l = 0·93, which is an absolute limit on the applicability of the theory.
Wilson and Lissaman7 suggest that wake mixing begins when a rises above about 1/3 and so it may be
assumed in the context of the present theory that mixing becomes significant when b rises above 2/3, providing a limit for the applicability of the theory. Figure 4 shows the variation of the ideal power coefficient with
tip speed ratio, comparing values with and without the restriction on the value of b of 2/3.
Even with the imposed restriction on b the ideal power coefficient is predicted to exceed the Lanchester–Betz
limit at very low tip speed ratios. The results of Figure 4 are significantly in contrast to the ideal power coefficient predicted by Glauert,5 who assumes that kinetic rotational energy of the wake has to be drawn from the
kinetic energy of the freestream rather than from a reduction in the static pressure. Almost all other authors
have adopted Glauert’s reasoning. Wilson and Lissaman7 discuss the vortex structure of the wake but state that,
close to the axis of rotation, tangential velocities would become too large to be sustained and a Rankine vortex
core would result. In both References 5 and 7 the assumption is made that the tangential velocity of the fluid
cannot exceed that of the blades, but there is no physical argument to support this case. Although the theory
predicts infinite velocities at the centre line of the wake, this is no different from all other momentum theo-
Figure 3. Variation of the axial flow factors with tip speed ratio as given by equations (19) and (27)
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
Energy-extracting Actuator Disc
185
Figure 4. Variation of the ideal power coefficient with tip speed ratio
ries that include wake rotation. In practice the problem of infinite velocity is resolved because the blade root
will be at a finite radius.
It is clear, however, that at the operational tip speed ratios adopted for the vast majority of wind turbines
the simple momentum theory result for axial induced velocities can be used without any significant loss of
accuracy. The rotational induced velocities require further examination.
Thrust force on the disc
The normal thrust force on the disc can be determined directly from the pressure drop expressed in equation
(23).
The inner radius of the disc is R0 and the rotor thrust is given by
R
T=
=
1
b 2 (1 - a) Ê
R2 ˆ
rU •2
1 + at¢ 2 2prdr
Ú
2
b - a R0 Ë
r ¯
1
b 2 (1 - a) ÈÊ
R02 ˆ
a
R ˘
rU •2 pR 2
1 - 2 + 2Ê 1 - 2 ˆ lnÊ ˆ ˙
Í
Ë
b - a ÎË
b ¯ Ë R0 ¯ ˚
2
R ¯
If the thrust coefficient is defined as
CT =
Thrust
1
rU •2 pR 2
2
then
CT =
b 2 (1 - a) ÈÊ
R02 ˆ
a
R ˘
1 - 2 + 2Ê 1 - 2 ˆ lnÊ ˆ ˙
Í
Ë
b - a ÎË
b ¯ Ë R0 ¯ ˚
R ¯
(28)
Clearly, if the inner radius is zero, the thrust force will be infinite, but this would not occur in practice
because the blade roots would have a finite radius. At low tip speed ratios the thrust force will be greater than
the simple theory predicts, as illustrated in Figure 5, which shows the variation of the thrust coefficient as a
function of a and l assuming a root radius of 10% of the tip radius.
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
186
D. J. Sharpe
Figure 5. The thrust coefficient as a function of a and l for R0 = 0·1R
The Simple Momentum Theory
The simple momentum theory dissociates the axial momentum change from the angular momentum change
and so b = 2a. The power developed by the flow through an annular ring of radius r and radial width dr from
the rate of change of axial momentum is
2
dP = rU •2 (1 - a) 2 aU • 2prdr
(29)
whereas the power developed from the rate of change of angular momentum is
dP = rU • (1 - a)2 a ¢W 2 r 2 2prdr
(30)
Equating the two expressions for power gives
U •2 a(1 - a) = a ¢(Wr)
2
Thus
a¢ =
a(1 - a)
l2 m 2
(31)
where m = r/R. Because a is radially uniform, we have
a(1 - a) = l2 at¢
(32)
Equation (32) is a simplified version of equation (18) as would be achieved if a¢t were to be considered as being
negligibly small compared with unity.
When the axial and tangential induced velocities are superimposed, the pressure drop across the disc is
Dpd = r[U •2 2 a(1 - a) + 2 W 2 a ¢ 2 r 2 ]
(33)
Equation (33) is relevant when the simple momentum theory, is used in conjunction with the blade element
theory, because the blade element force, derived from two-dimensional aerofoil data, must balance the whole
of the pressure force and not just the force arising from the rate of change of axial momentum.
In the fully developed wake there will be a pressure difference across the wake boundary because the wake
is rotating and there will be a consequent pressure drop: b = 2a and so there is no additional slowing down of
Copyright © 2004 John Wiley & Sons, Ltd.
Wind Energ. 2004; 7:177–188
Energy-extracting Actuator Disc
187
the wake to equalize the pressure across the wake boundary. In the near wake it would appear that there is a
pressure difference predicted across the wake boundary, as indeed there is in the simple momentum theory,
but the momentum theories consider only the conditions far upstream, at the actuator disc and in the fully
developed wake: no predictions are made about the conditions elsewhere in the flow field. At the edge of the
disc the pressure difference is also non-zero across the wake boundary, and this problem has been addressed
by van Kuik.10
For optimal operation the Lanchester–Betz limit applies at all tip speed ratios.
High-solidity, Low-tip-speed-ratio Wind Turbines
It would appear that wind turbines, designed according to the above theory, intended to operate at very low
tip speed ratios could achieve a maximum power coefficient that exceeds the Lanchester–Betz limit. Such turbines would have to be of very high solidity, with a large number of blades. To complete the design, the blade
element theory would need to be employed, but the only available blade element data are for isolated, twodimensional aerofoils. The mutual interactions between the closely spaced, probably overlapping, blades would
be large and so the available aerofoil data would be useless for the purpose of designing high-solidity turbines.
Conclusions
A general momentum theory for an energy-extracting actuator disc modelling a rotor with blades having radially uniform circulation that includes the effects of wake rotation and expansion has been presented. The
general momentum theory is well known but the fall in pressure caused by the rotation of the wake that the
theory predicts is not usually recognized. Accounting for the wake rotational pressure drop changes some of
the established conclusions of the momentum theory that appear in the literature. Principally, the theory establishes that there is no loss of efficiency associated with the rotating wake and that, ignoring any losses associated with the high tangential velocities close to the rotor axis, it is possible, theoretically, to exceed the
Lanchester–Betz limit.
In practical terms the changes determined by the theory are not significant for wind turbine rotors designed
to operate at high tip speed ratios. For turbines that operate at low tip speed ratios, however, the effect of the
theory upon the blade design may well result in an improved performance compared with designs that have
been undertaken ignoring the pressure drop caused by wake rotation. Unfortunately, the necessary blade
element data are not available.
Acknowledgement
Dr David Infield, Director of CREST, is thanked for his encouragement, advice, suggestions and
proofreading.
References
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