Wind powerPart 3: Technology
San Jose State UniversityFX Rongère
February 2009
Wind Turbine Aerodynamics
Energy balance over the stream tube
Inlet: index 1Outlet: index 2Turbine: index T
).( AAAA
AA
AA
AOpen
khmWQdt
dE
Betz’s Simplified Approach
The turbine is perfect: there is no internal entropy generation
AA
AA A
A
Open
smT
Q
dt
dS.
0.
AA
A sm
0.
AA
A hm
Then, when air considered as an incompressible fluid:
Betz’s Simplified Approach
Energy balance equation is reduced to:
In addition, we assume that the wind speed stay axial through the turbine:
Balance of Forces:
2122.2
1. vvmWT
TT vAvAvAm ...... 2211
T
A
TT vFAdvFT
.... The power of the torque absorb by the turbine is equal to the power taken from the air flow
Momentum equation on the system:
A
AA
AA
open
vmFdt
vd.
.
Betz’s Simplified Approach
Then:
21. vvmF
And:
21
22
2121
.2
1
.2
1...
vvv
vvmvvvm
T
T
Introducing a the interference factor:
1
1
v
vva T
Betz’s Simplified Approach
Then:
231
211
22
21
22
21
1.....2
.1..4.2
1..1..
.2
1....
2
1.
aaAvW
vaaAavW
vvAvvvmW
TT
TT
TTT
The power absorbed by the turbine is maximum if:
windwindTMaxT WWAv
aaW
.59.0
27
16.
2..1..4
312
,
Betz’s Simplified Approach
31
01.4.3
0
2
a
aa
da
Wd T
Then
Betz’s law
12
1
.31
.32
vv
vvT
Other factors
Other factors limit the performance of wind turbinesWake rotationAerodynamic dragFinite number of blades and
interaction between themBlade tip losses
Wake rotation In fact, the air is rotating downstream the rotor
by reaction to the torque applied to the rotor
Maximum conversion rate with wake rotation: Glauert’s law
To characterize this effect we introduce: Angular velocity of the rotor : ΩT
Angular velocity imparted to the flow: ω Angular induction factor: a’ = ω/2ΩT
Axial interference factor:
Blade tip speed : λ = ΩTR/v1
We can then show that the transfer power is maximum if:
1
1
v
vva T
14
31'
a
aa
Then the maximum conversion rate is given by the following equation:
With a(λ) defined by:
Betz’s law corresponds to: and:
Maximum conversion rate with wake rotation: Glauert’s law
dxx
xxx
W
WCp
a
Wind
MaxT
Max
)(
25.2
,
31
)41()21()1(24
a
aa
31
141 22
3
1a
Optimal Interference FactorGlauert's model
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10
Blade tip ratio
Inte
rfer
ence
Fac
tor
(a)
By drawing CpMax we can show that: Larger is the blade tip ratio higher is the conversion rate For blade tip ratio greater than 4 conversion rate is close
to Betz’s law The optimal axial interference factor is close to 1/3 for
large enough blade tip ratio
Maximum conversion rate with wake rotation: Glauert’s law
λ
Optimal Coversion RateGlauert's model
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10
Blade tip ratio
Co
nve
rsio
n r
ate
λ
Angular induction factor
Aerodynamic drag The aerodynamics of a blade is defined by:
lcv
DC
lcv
LC
D
L
...21
...21
2
2
L: Lift force
v: Wind speed on the blade
c: Chord of the blade
D: Drag force
l: Span of the bladeα: Angle of attack
Lift and Drag
Lift and Drag are related:
• Lift transfers power
• Drag generates losses
•Bearings•Viscous Friction•Noise
• About 10 -15% of aerodynamic inefficiencies
Design optimization
Blade shape may be optimized using laboratory test and numerical modeling See: Wind Energy, Explained by J.F.
Manwell, J.G. McGowan and A.L.Rogers, John Wiley, 2002, for detailed explanations
See also: Riso DTU – National Laboratory for Renewable Energy http://www.risoe.dtu.dk/
Interaction between blades
Blade tip speed is limited by the interaction between the blades. Typically, if the rotation is too fast (λ>λMax) then the flow for each is perturbed by the previous one
Conversion rate WMax/WWind
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6
Blade Tip Speed
Con
vers
ion r
ate
Multiblade Wind pump
Three Blade Wind Turbine
Two Blade Wind Turbine
Vertical axis Wind Turbine (Darrieus)
λ
Blade tip losses Local rotating airflow at the tip of the blades due to the
difference of pressure between the faces of the blades Similar to aircraft wings
Thesis at DTU has shown a gain of 1.65% but with an increase in thrust of 0.65%
Performance glider with wingletsWingtip loss of Boeing 737
Source: Mads Døssing Vortex Lattice Modelling of Winglets on Wind Turbine Blades Risø-R-1621(EN) Aug. 2007
Actual Wind Turbine Conversion Rate
Example: Liberty of Clipper Windpower
Power curve
Actual Turbine Conversion rate
Power curve and Conversion rate
GE 1.5S
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
1400.00
1600.00
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Wind Speed (m/s)
Po
wer
(kW
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Series2
Series1
Optimization Parameters
Theoretical Power CurveConversion rate: .35
-100
0
100
200
300
400
500
0 5 10 15 20 25 30
Wind speed (m/s)
Pow
er (W
/m2)
Cut in
Cut out
Efficiency
Max Power
Generated Energy Generated Energy = Wind Speed Distribution x Turbine Power
CurveCapacity Factor
0
100
200
300
400
500
0 5 10 15 20 25
Wind Speed (m/s)
Win
d T
urbi
ne P
ower
(k
W/m
2)
0%
2%
4%
6%
8%
10%
Win
d P
roba
bilit
y
Capacity Factor
Lower Capacity factor allows higher energy capture Generally preferred CF is between 30% and 40%
5 12% 33% 4% 0% 89%5.5 16% 33% 3% 0% 82%6 20% 32% 2% 0% 76%
6.5 24% 30% 1% 0% 68%7 28% 29% 1% 0% 60%
7.5 32% 27% 1% 0% 50%8 36% 25% 1% 0% 45%
8.5 40% 23% 0% 0% 38%9 44% 22% 0% 0% 33%
Energy after Cut out
Energy at max eff
Average Wind Speed
Capacity Factor
Captured Energy/ Wind
Energy before cut-in
Eff 0.35Max Power 400 W/ m2Cut-in 4 m/ sCut-out 25 m/ s
Characteristics of the tested turbine
hoursPowerMax
EnergyGeneratedAnnualCF
760,8..
Wind Turbine Optimization
0%
10%
20%
30%
40%
50%
5 6 7 8 9
Average Wind Speed (m/s)
0%
20%
40%
60%
80%
100%
Capacity Factor
Captured Energy/ Wind Energy
Energy at max eff
Wind Turbine Optimization
Conv. Rate 0.35Max Power 400 W/ m2Cut-in 4 m/ sCut-out 25 m/ s
0
50
100
150
200
250
0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24
00.010.020.030.040.050.060.070.080.090.1
Annual Wind Turbine Energy
Annual Wind Energy (kWh/ m2)
Wind and Power Distribution
Wind Turbine Optimization
0
20
40
60
80
100
120
140
0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24
Wind Speed (m/s)
Ann
ual E
nerg
y (K
Wh/
m2)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Annual Wind Turbine Energy
Annual Wind Energy (kWh/ m2)
vm = 6 m/ s
CF=24%Captured Energy =33%
Conv. Rate 0.35Max Power 400 W/ m2Cut-in 4 m/ sCut-out 25 m/ s
0
50
100
150
200
250
0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24
00.010.020.030.040.050.060.070.080.090.1
Annual Wind Turbine Energy
Annual Wind Energy (kWh/ m2)
Wind and Power Distribution
Wind Turbine Optimization
0
50
100
150
200
250
300
350
0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24
Wind Speed (m/s)
Ann
ual E
nerg
y (K
Wh/
m2)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Annual Wind TurbineEnergy
Annual Wind Energy(kWh/ m2)
vm = 9m/s
CF=44%Captured Energy =22%
Conv. Rate 0.35Max Power 400 W/ m2Cut-in 4 m/ sCut-out 25 m/ s
0
50
100
150
200
250
0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24
00.010.020.030.040.050.060.070.080.090.1
Annual Wind Turbine Energy
Annual Wind Energy (kWh/ m2)
Companies to follow GE: www.ge.com Clipper Windpower: www.clipperwind.com Vestas: www.vestas.com Repower: www.repower.com Gamesa: www.gamesa.com Suzlon: www.suzlon.com Mitsubishi: www.mitsubishi.com