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ENGINEERING TRIPOS PART IB PAPER 8 – ELECTIVE (2) Mechanical Engineering for Renewable Energy Systems Dr. Digby Symons Wind Turbine Blade Design Student Handout
ENGINEERING TRIPOS PART IB PAPER 8 – ELECTIVE (2)
Mechanical Engineering for Renewable Energy Systems
Dr. Digby Symons
Wind Turbine Blade Design
Student Handout
CONTENTS
1 Introduction............................................................................................................... 3
2 Wind Turbine Blade Aerodynamics ....................................................................... 4
2.1 Aerofoil Aerodynamics....................................................................................... 4
2.2 Wind Turbine Blade Kinematics ........................................................................ 7
3 Blade Element Momentum Theory ....................................................................... 12 3.1 Momentum changes .......................................................................................... 12
3.2 Blade forces ...................................................................................................... 13
3.3 Induction factors ............................................................................................... 14
3.4 Iterative procedure ............................................................................................ 14
4 Blade Loading ......................................................................................................... 19
4.1 Aerodynamic Loading ...................................................................................... 19
4.2 Centrifugal Loading.......................................................................................... 21
4.3 Self Weight loading .......................................................................................... 22
4.4 Combined Loading............................................................................................ 23
4.5 Storm Loading .................................................................................................. 24
More detailed coverage of the material in this handout can be found in various books,
e.g. Aerodynamics of Wind Turbines, Hansen M.O.L. 2000
2
1 INTRODUCTION
1.1.1 Aim
Preliminary design of a wind turbine
•
•
•
1.1.2 Wind turbine type
Horizontal axis wind turbine (HAWT) with 3 blade upwind rotor – the “Danish concept”:
1.1.3 Load cases
We will consider two load cases:
1) Normal operation – continuous loading
• Aerodynamic, centrifugal and self-weight loading
•
•
2) Extreme wind loading – storm loading with rotor stopped
•
3
2 WIND TURBINE BLADE AERODYNAMICS
2.1 Aerofoil Aerodynamics
2.1.1 Lift, drag and angle of attack
c
V α
2.1.2 Lift and drag coefficients
Define non-dimensional lift and drag coefficients
4
2.1.3 Variation of lift and drag coefficients with angle of attack
How does lift and drag vary with angle of attack α ?
α
DL CC ,
Stall:
2.1.4 Application of 2D theory to wind turbines
• Tip leakage means flow is not purely two dimensional
• Wind turbine blades are spinning with an angular velocity ω
• The angle of attack depends on the relative wind velocity direction.
5
2.1.5 Example aerofoil shape used in wind turbines
Lift and drag coefficients for the NACA 0012 symmetric aerofoil (Miley, 1982)
6
2.2 Wind Turbine Blade Kinematics
2.2.1 Blade rotation
θ α
Relative wind
V
rϖ θ
7
2.2.2 Wake rotation
)1(0 aV − )21(0 aV −
0V
0
0
R
Axial velocity
Tangential velocity
Rotor plane
a = axial induction factor
a’ = angular induction factor
8
2.2.3 Annular control volume
Wake rotates in the opposite sense to the blade rotation ω
)21(0 aV − 0V )1(0 aV −
ϖ
r R
2.2.4 Wind and blade velocities
Induced wind velocities seen by blade + blade motion
)1(0 aV −
ar ′ϖ
Local twist angle of blade = θ
9
2.2.5 Blade relative motion and lift and drag forces
θ
)1(0 aV −
Local angle of attack = α
Relative wind speed has direction relV θαφ +=
where
)1(sin 0 aVVrel −=φ
and
)1(cos arVrel ′+= ωφ
LF and are aligned to the direction of DF relV
Obtain and for LC DC θφα −= from table or graph for aerofoil used
10
2.2.6 Resolve forces into normal and tangential directions
LF
DF
relV
We can resolve lift and drag forces into forces normal and tangential to the rotor plane:
We can normalize these forces to obtain force coefficients:
Hence:
φφ sincos DLN CCC +=
φφ cossin DLT CCC −=
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3 BLADE ELEMENT MOMENTUM THEORY Split the blade up along its length into elements.
Use momentum theory to equate the momentum changes in the air flowing through the turbine with the forces acting upon the blades.
Pressure distribution along curved streamlines enclosing the wake does not give an axial force component. (For proof see one-dimensional momentum theory, e.g. Hansen)
3.1 Momentum changes
ar ′ϖ2
)21(0 aV − 0V )1(0 aV −
ar ′ϖ ϖ
r R
Thrust from the rotor plane on the annular control volume is Nδ
ruVuruVmN δρπδ )(2)( 1010 −=−= &
Torque from rotor plane on this control volume is Tδ
θδ rumT &= = arm ′22 ϖ&
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3.2 Blade forces
Now equate the momentum changes in the flow to the forces on the blades:
3.2.1 Normal forces =Nδ
= raaVr δρπ )1(4 20 −
= rcCaV
B Nδφ
ρ 2
220
sin)1(
21 −
Therefore: raπ4 = NcCaBφ2sin)1(
21 −
Define the rotor solidity:
Hence:
3.2.2 Tangential forces =Tδ
= raaVr δωρπ ′− )1(4 03
= rcCaraVrB Tδφφω
ρcossin
)1()1(21 0 ′+−
Therefore: ar ′π4 = TcCaBφφ cossin
)1(21 ′+
Use the rotor solidity σ :
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3.3 Induction factors
These equations can be rearranged to give the axial and angular induction factors as a function of the flow angle.
Axial induction factor:
Angular induction factor:
However, recall that the flow angleφ is given by: ra
Vaω
φ)1()1(
tan 0′+
−=
Because the flow angle φ depends on the induction factors and these equations must be solved iteratively.
a 'a
3.4 Iterative procedure
Choose blade aerofoil section.
Define blade twist angle θ and chord length c as a function of radius r.
Define operating wind speed and rotor angular velocity 0V ω .
For a particular annular control volume of radius r :
1. Make initial choice for a and a’ , typically a = a’ = 0.
2. Calculate the flow angle φ .
3. Calculate the local angle of attack θφα −= .
4. Find and for LC DC α from table or graph for the aerofoil used.
5. Calculate and NC TC .
6. Calculate a and a’ .
7. If a and a’ have changed by more than a certain tolerance return to step 2.
8. Calculate the local forces on the blades.
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3.4.1 Example wind turbine
Blade element theory has been applied to an example 42 m diameter wind turbine with the parameters below. Each element has a radial thickness rδ = 1m.
Incident wind speed 0V 8 m/s
Angular velocity ϖ 30 rpm
Blade tip radius R 21 m
Tip speed ratio 0/VRωλ =
Number of blades B 3
Air density ρ 1.225 kg/m3
Blade shape (chord c and twist θ ) are based on the Nordtank NTK 500/41 wind turbine (see Hansen, page 62).
Chord c
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12 14 16 18 20 2
Radius r (m)
Cho
rd c
(m)
2
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Blade twist angle θ
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Bla
de tw
ist a
ngle
(deg
)
3.4.2 Results of BEM analysis
Axial induction factor a
0.00
0.05
0.10
0.15
0.20
0.25
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Axi
al in
duct
ion
fact
or a
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Angular induction factor a’
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Ang
ular
indu
ctio
n fa
ctor
a'
Flow angle φ and local angle of attack θφα −=
0
5
10
15
20
25
30
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Ang
le (d
eg)
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Normal and tangential NF TF forces on blade
0
100
200
300
400
500
600
700
800
900
1000
1100
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Forc
es p
er u
nit l
engt
h (N
/m)
FN (N/m)
FT (N/m)
Total power (3 blades)
Coefficient of performance
Contribution of blade elements to total torque (and therefore power)
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12 14 16 18 20 22
r (m)
Con
trib
utio
n to
tota
l tor
que
(%/m
)
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4 BLADE LOADING
4.1 Aerodynamic Loading Once values of a and a’ have converged the blade loads can be calculated:
NN cCaVFφ
ρ 2
220
sin)1(
21 −
=
TT cCaraV
Fφφ
ωρ
cossin)1()1(
21 0 ′+−
=
4.1.1 Stresses at blade root
The normal force causes a “flapwise” bending moment at the root of the blade. NF
∫ −=R
rNN drrrFM
min
)( min
The tangential force causes a tangential bending moment at the root of the blade. TF
∫ −=R
rTT drrrFM
min
)( min
For convenience we will neglect the relatively small twist of the blade cross section and assume that these bending moments are aligned with the principal axes of the blade structural cross section. The maximum tensile stress due to aerodynamic loading is therefore given by:
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4.1.2 Deflection of blade tip
drdSFN =
drdMS =
)]([2
2
rEIM
drvd
=≈− κ
Simplified approach:
• Split blade into elements.
• Assume that for each element the loading and flexural rigidity EI are constant.
NF
• Find the shear force and bending moment transferred between each element.
• Use data book deflection coefficients for each element.
• Find the cumulative rotations along the blade.
• Find the cumulative deflections along the blade.
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4.2 Centrifugal Loading
The large mass of a wind turbine blade and the relatively high angular velocities can give rise to significant centrifugal stresses in the blade.
Consider equilibrium of element of blade:
rrmdr
dFc 2)( ω−= )()(
rArFc
c =σ
Simplified method:
• Split blade up into elements.
• Assume each element has a constant cross-section
2
111,, )(21)( ωnnnnncnc rrrrmFF +−+= +++
= ( ) 22211, 2
1 ωnnnc rrmF −+ ++
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4.3 Self Weight loading
The bending moment at the blade root due to self weight loading can dominate the stresses at the blade root. Because the turbine is rotating the bending moment is a cyclic load with a frequency of πω 2/=f . The maximum self-weight bending moment occurs when a blade is horizontal.
Bending moment at root of blade due to self weight
∫ −=R
rsw drrrgrmM
min
)()( min
where m(r) is the mass of the blade per unit length. This is a tangential (edge-wise) bending moment and therefore the maximum bending stress due to self-weight is given by:
Simplified method: split blade into elements, assume each element has uniform self weight.
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4.4 Combined Loading
22max,b
IMd
IM
NN
To
TT
Naero +=σ
)()(
rArFc
c =σ
2max,b
IM
NN
swsw =σ
Operational maximum stress: =maxσ
Minimum stress at same location: =minσ
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4.5 Storm Loading
4.5.1 Drag force on blade
Blades parked. Extreme wind speed
load per unit length
c maxV DF
maxV = 50 m/s, c = 1.3m
Re =μ
ρ cVmax =
Hence = DC
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4.5.2 Bending moment
Find bending moment at root of blade
∫ −=Rr DdrFrrMmin
)( min
κσ EI
My
==
2od
IM
=σ
12)( 33
io ddbI −=
.5.3 Shear stress
4
∫=Rr DdrFSmin
IySAq c=
wtq τ=
Note:
igh solidity rotor (multi bladed) gives excessive forces on tower during extreme wind eeds. Therefore use fewer blades.
Hsp
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