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Part B of wind turbine lectures
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SAILING ACROSS THE WIND
• Boat speed added to true wind
• Apparent wind exceeds true wind
• Boat speed can exceed wind speed Apparent wind
True wind
Boat speed
EARLY AERODYNAMIC LIFT-TYPE WIND MILL
• Old Portugese windmill • Cloth strung between wood
masts
Torrey, 1976
AERODYNAMICS • Sail, airplane wings,
airplane propellers, wind turbine blades, are all airfoils
• Definitions – Angle of attack – Chord length
• Air passing over airfoil generates lift and drag
Vrelative
α
Lift
Drag
Angle of attack
Chord
ANGLE OF ATTACK • Low angle of attack
– Lift exceeds drag • Moderate angle of attack
– Both increase but lift more than drag
• Steep angle of attack – Stall – More drag than lift
Lift Drag
LIFT AND DRAG COEFFICIENTS • Lift coefficient for lift force F
𝐶𝐿 =𝐿
12 𝜌𝜌𝑉𝑟𝑟𝑟2
• Drag coefficient for drag force D
𝐶𝐷 =𝐷
12 𝜌𝜌𝑉𝑟𝑟𝑟2
where A is the blade area (surface area wind flows over)
A = chord x blade length
Blade length
EFFECT OF ANGLE OF ATTACK
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20
Lift
coef
ficie
nt
Angle of attack (degrees)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 5 10 15 20
Drag
coe
ffic
ient
Angle of attack (degrees) CL and CD values calculated for SD7062 airfoil by Huahui Tan
COMBINED EFFECT OF LIFT AND DRAG
• Ratio of CL/CD illustrates importance of angle of attack
0
20
40
60
80
100
120
140
0 5 10 15 20
C lift
/Cdr
ag
Angle of attack (degrees) CL/CD values calculated for SD7062 airfoil by Huahui Tan
BLADE SECTION GEOMETRY
r Wind
V0
v
V0
V1 Vrel
v
v = 2πrN where N =rotation
speed (rpm)
View of section from blade end Rotor axis Lift
Drag
ANGLE DEFINITIONS
V0
V1 Vrel
v
• Relative wind speed
𝑉𝑟𝑟𝑟 = 𝑉12 + 𝑣2
• Φ = relative wind angle
𝜑 = arctan𝑉1𝑣
• α = angle of attack • β = blade pitch angle
𝛽 = 𝜑 − 𝛼
Rotor axis
Lift Drag
φ β α
FORCES
V0
V1 Vrel
v
• Driving force rotates blade 𝐹𝐷𝑟𝐷𝐷𝐷𝐷𝐷 = 𝐿𝐿𝐿𝐿𝜑 − 𝐷𝐿𝐷𝐷𝜑
• Thrust force loads bearings 𝐹𝑡𝑡𝑟𝑡𝑡𝑡 = 𝐿𝐿𝐷𝐷𝜑 + 𝐷𝐿𝐿𝐿𝜑
𝐿 = 𝐶𝐿 ∙ 12𝜌𝜌𝑉𝑟𝑟𝑟2
𝐷 = 𝐶𝐷 ∙ 12𝜌𝜌𝑉𝑟𝑟𝑟2
View
Rotor axis
Lift Drag
φ β α
LIFT FORCE • Driving force on blade element:
𝐹𝐷𝑟𝐷𝐷𝐷𝐷𝐷 = 12𝜌𝜌𝑉𝑟𝑟𝑟
2 𝐶𝐿 ∙ 𝐿𝐿𝐿𝜑 − 𝐶𝐷 ∙ 𝐿𝐷𝐷𝜑
• Rotor speed v is one component of Vrel 𝑣 = 𝑟 ∙ 𝜔
where 𝜔= blade rotation frequency (radians/s)
𝐹𝐷𝑟𝐷𝐷𝐷𝐷𝐷 = 12𝜌𝜌 𝑉12 + 𝑟𝜔2 ∙ 𝐶𝐿 𝛼 ∙ 𝐿𝐿𝐿𝜑 − 𝐶𝐷 𝛼 ∙ 𝐿𝐷𝐷𝜑
where: 𝜑 = 𝑎𝑟𝐿𝑎𝑎𝐷 𝑉1𝑟𝑟
and 𝛼 = 𝜑 − 𝛽
FORCE DEPENDENCE ON ROTATING SPEED • SD7062 airfoil for illustration
– Blade tip radius R = 3.38m – V1 = 9.33m/s – β=15o – Two radii (1.8m, 3.35m) – Force/unit blade length
• 0
20
40
60
80
100
120
140
0 10 20 30
Forc
e /u
nit b
lade
leng
th
(arb
itrar
y un
its)
Rotation frequency (radians/s)
r = 1.8m
r = 3.35m
β= 15o
FORCE AT FIXED ROTATION SPEED • Often desirable to run rotor at
fixed speed to match generator synchronous speed
• Calculation shows effect of wind speed for fixed rotation speed.
• Large difference in force developed!
• Large wind turbines use blade pitch control – adjust β 0 5 10 15 20
Forc
e fr
om a
irfoi
l sec
tion
(arb
itrar
y un
its)
Wind speed (m/s)
r = 1.8m
β = 15o
Ω = 10
INTEGRATE OVER BLADE LENGTH • Define Power coefficient
𝐶𝑃 =𝑃𝑎𝑎𝑡𝑡𝑎𝑟12𝜌𝜌𝑉0
3
• Indicates the fraction of power available in the wind that is produced as electrical output:
• Depends on wind speed and on rotor speed
Blade length
POWER COEFFICIENT & TIP SPEED RATIO • Characterize rotor speed by tip
speed ratio:
𝑇𝐷𝑇 𝑆𝑇𝑆𝑆𝑆 𝑅𝑎𝑎𝐷𝐿 =2𝜋𝑅𝜋𝑉0
where: R = blade radius N = rotor speed (rpm) V0 = upstream wind speed
• TSR useful for all circumstances,
constant rotor speed or not 0 2 4 6
Pow
er co
effic
ient
Tip speed ratio
β = 15o
WHAT IS THE MAXIMUM POWER THAT CAN BE EXTRACTED FROM THE WIND?
• Betz limit is max power that can be extracted from the wind
𝑃𝑚𝑎𝑚 𝑤𝐷𝐷𝑖 𝑡𝑡𝑟𝑡𝐷𝐷𝑟 =12𝜌𝜌𝑉03
1627
• For Betz limit, CP = 0.593 • Real wind turbines have lower values of CP
SUMMARY - AERODYNAMIC LIFT-TYPE WIND ENERGY TECHNOLOGY
• Air passing over an airfoil shape (e.g. a wind turbine blade) generates lift and drag
• Blade angle of attack determines relative amounts of lift and drag
• Blade sees relative wind velocity, the vector sum of the true wind speed and the rotor speed
• Actual power output characterized by power coefficient CP, which depends on the tip speed ratio, the ratio of rotor tip speed to upstream wind speed.
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