Generalised Energy Extraction Limits
Peter Jamieson
April 2008
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Background
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Definitions
Open uniform flow
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Constrained flow
No energy extraction
V0
V0
V0


V0 1  a0
SYSTEM
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Optimum energy extraction
CT
V0
CP
CP
SYSTEM
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8
9
16
27

1
V
3 0

16
1  a0
27
Reference plane
V (1-b)
Rotor
Plane
Induction
Freestream
Far wake
System
V
V (1-2b)
Extraction plane
V (1-a)
 AV ( 1  a )
Conservation of mass flow
When no energy extraction,
a  a0 , b  0
=  Aref V ( 1  b )
 AV ( 1  a0 ) =  Aref V
Aref = A( 1  a0 )
b
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

a  a0

= 

 1  a0 
Reference plane
V (1-b)
Generalised
Equations
Freestream
Far wake
System
V
V (1-2b)
Extraction plane
V (1-a)
Thrust
T = pA
and
T =
Hence
p =
Everywhere
CT
1
 V02 ACT
2
1
 V02 CT
2
= 4 b( 1  b )
=
4 ( a  a0 ) ( 1  a )
( 1  a0 )2
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b


a  a0

= 

 1  a0 
System Equations
GENERAL OPERATION
Open Flow
Constrained
Flow
OPTIMUM OPERATION
Open Flow
Constrained
Flow
a
a
1
3
1  2a0
3
Axial induction in far wake
2a
1  2a  a0
1  a0
2
3
2
3
Performance coefficient, C p
4 a 1  a 
16
27
16
1  a0 
27
Thrust coefficient, CT
4a 1  a 
8
9
8
9
4
p V02
9
4
p V02 1  a0 2
9
Axial induction at rotor plane
Pressure difference across rotor
2
2 a ( 1  a )  V02
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4 a  a0 1  a 2
1  a0 2
4 a  a0 1  a 
1  a0 
2
2 a  a0 1  a 2
1  a0 
2
 V02
Primary purpose of a wind turbine?
• To extract energy from natural air flow
• To extract kinetic energy from natural air flow
• To extract potential energy from natural air flow
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Optimum operation
with same upstream
wind speed
9 m/s
6 m/s
3 m/s
P = 1.000
9 m/s
6.75 m/s
P = 1.125
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3 m/s
9 m/s
18 m/s
P = 2.000
3 m/s
Optimum operation
with same speed local
to turbine
9 m/s
6 m/s
3 m/s
P = 1.00
8 m/s
6 m/s
P = 0.79
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2.67 m/s
3 m/s
6 m/s
P = 0.11
1 m/s
Ideal Rotors and Ideal Systems
A rotor can operate
with fixed geometry in
variable speed to
preserve optimal flow
geometry.
A diffuser or a hill
cannot change
geometry to suit rotor
loading!
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Validation of the Generalised Limit Theory
In open flow, Cp may be expressed in terms of Ct as;
Cp 

1
Ct 1  1  Ct
2

In generalised flow, considering system losses, Cp may be
expressed in terms of Ct as;

1 
C p  Ct 1  a0  S   1  a0  S 2  1  a0 2 Ct
2 
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
0.5 


power performance coefficient
Comparison of New Limit Theory and CFD
1.0
0.8
0.6
CFD
Equation (22)
0.4
0.2
0.0
0.0
0.2
0.4
0.6
thrust coefficient
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0.8
1.0
Optimum State
Torque tracking curve
C
1

Q   p  R 5 pm 3   2
 
2
Optimum chord distribution
cCL

R
B
 1  a
2 Ct
2
2 2
 2  2 1 
m    x 1  am
Optimum twist distribution

1  am 
  0




x
1

a
m 

 x   tan1 
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1


C p 1  am  

 
CL  x 1  am
Optimum Twist Distribution
-1
θ(x) = tan (1-am/ λ x (1+a`
α0
m)) - 
1

a
1
m
 tan)/3

  0
am =x (1+2a
o


x
(
1

a
m

α0
6 deg
1  2 a0
, am  0 ,
a`mwhere am  0
3
λ
9
x
0.2
0.4
0.6
0.8
1.0
0
14.3
4.5
1.0
-0.7
-1.8
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-0.1
16.2
5.5
1.7
-0.2
-1.3
a0
-0.2
18.0
6.5
2.4
0.3
-0.9
a0
am
0.0
0.333
-0.1
0.267
-0.2
0.200

  9 and
-0.5 0  60.000
-2.0
-1.000
-0.5
23.1
9.5
4.5
1.9
0.3
-2
42.0
23.1
14.3
9.5
6.5
Optimum State with a0 ≠ 0
• Torque tracking curve (optimal mode gain) changes
• Optimum chord changes but very little
• Optimum twist changes significantly – pitch change can approximate
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System Issues
• Pressure recovery
• Optimal loading
• Turbine control
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V0 = 9 m/s
p  4 9  V 2  49 N/m2
2
3% p  1.5 N/m
2
p0  105 N/m
Optimal loading
•
Only an ideal system realises Ct = 8/9
•
Ideal systems are variable geometry and don’t exist in reality
•
If there is not full pressure recovery and wind turbines are coupled
in series then they have to share proportions of 8/9
•
There may be compensation in entrainment of wakes by flows
that do not pass through the wind turbines (Opt Ct > 8/9 possible
in a well designed diffuser)
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Turbine Control
•
Considering wind direction, atmospheric characteristics and terrain
characteristics, it will probably be impossible to modify control
deterministically.
•
Can a slow controller vary below rated pitch and optimal mode gain to tune
the turbine for specific operational conditions?
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Conclusions
• Lanchester-Betz limit is special case for open flow
• It appears that whilst 16/27 is not a universal number in actuator
ffffdisc theory, 8/9 may be
• The theory has obvious applicability to wind turbines in diffuser
ffffand ducted turbines in all types of energy systems and may also
ffffhave significant implications for wind farm operation
• The theory is derived analytically without any requirement to use
ffffempirical information. A very useful validation is obtained
ffffcomparing with CFD results.
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