ENGINEERING TRIPOS PART IB PAPER 8 – ELECTIVE (2)

Mechanical Engineering for Renewable Energy Systems

Lectures 4, 5 and 6 Dr. Digby Symons

Design of Wind Turbines – Blade aerodynamics, Loads & Structure

Lecturer’s Master Copy

CONTENTS

4 Wind Turbine Blade Aerodynamics ....................................................................... 3 4.1 Introduction......................................................................................................... 3

4.2 Aerofoil Aerodynamics....................................................................................... 4

4.3 Wind Turbine Blade Kinematics ........................................................................ 7

5 Blade Element Momentum Theory ....................................................................... 12 5.1 Momentum changes .......................................................................................... 12

5.2 Blade forces ...................................................................................................... 13

5.3 Induction factors ............................................................................................... 14

5.4 Iterative procedure ............................................................................................ 14

6 Blade Loading ......................................................................................................... 19 6.1 Aerodynamic Loading ...................................................................................... 19

6.2 Centrifugal Loading .......................................................................................... 21

6.3 Self Weight loading .......................................................................................... 22

6.4 Combined Loading............................................................................................ 23

6.5 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

4 WIND TURBINE BLADE AERODYNAMICS

4.1 Introduction

4.1.1 Aim

Preliminary design of a wind turbine:

• Aerodynamics of blade

• Loadings on blade

• Structure of blade

4.1.2 Wind turbine type

Horizontal axis wind turbine (HAWT) with 3 blade upwind rotor – the “Danish concept”:

4.1.3 Load cases

We will consider two load cases:

1) Normal operation – continuous loading

• Aerodynamic, centrifugal and self-weight loading

• Deflection - blades must not hit tower

• Cyclic stresses – fatigue

2) Extreme wind loading – storm loading with rotor stopped

• Strength calculation

3

4.2 Aerofoil Aerodynamics

4.2.1 Lift, drag and angle of attack

LF

V DFα

c

4.2.2 Lift and drag coefficients

Define non-dimensional lift and drag coefficients

cV

FC LL

221 ρ

=

cV

FC DD

221 ρ

=

4

4.2.3 Variation of lift and drag coefficients with angle of attack

How does lift and drag vary with angle of attack α ?

DL CC ,

LC

DC

α

πα2≈LC

Stall:

4.2.4 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

4.2.5 Example aerofoil shape used in wind turbines

Lift and drag coefficients for the NACA 0012 symmetric aerofoil (Miley, 1982)

6

4.3 Wind Turbine Blade Kinematics

4.3.1 Blade rotation

θα

Relative wind

V

rϖθ

7

4.3.2 Wake rotation

rδ

R r

N T

)21(0 aV −

Axial velocity

0V )1(0 aV −

0

Tangential velocity

ar ′ϖar ′ϖ2

0 Rotor plane

8

4.3.3 Annular control volume

Wake rotates in the opposite sense to the blade rotation ω

4.3.4 Wind and blade velocities

Induced wind velocities seen by blade + blade motion

Local twist angle of blade = θ

ar ′ϖ2

)21(0 aV − )1(0 aV −

Rr

ϖ

ar ′ϖ

0V

)1(0 aV −

rϖ

ar ′ϖ

9

4.3.5 Blade relative motion and lift and drag forces

)1(0 aV −

Local angle of attack = α

Relative wind speed V has direction rel θαφ +=

where

)1(sin 0 aVVrel −=φ

and

)1(cos arVrel ′+= ωφ

raVaω

φ)1()1(

tan 0′+

−=

LF and are aligned to the direction of V DF rel

Obtain and for LC DC θφα −= from table or graph for aerofoil used

θLF

DF

α)1( ar ′+ϖ

relV

10

4.3.6 Resolve forces into normal and tangential directions

We can resolve lift and drag forces into forces normal and tangential to the rotor plane:

φφ sincos DLN FFF +=

φφ cossin DLT FFF −=

We can normalize these forces to obtain force coefficients:

cV

FC

rel

NN

221 ρ

= cV

FCrel

TT

221 ρ

=

Hence:

φφ sincos DLN CCC +=

φφ cossin DLT CCC −=

LF

DF

NF

TFrelV

φ

11

5 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)

5.1 Momentum changes

Thrust from the rotor plane on the annular control volume is Nδ

ruVuruVmN δρπδ )(2)( 1010 −=−=

= rVaVVar δρπ ))21(()1(2 000 −−− = raaVr δρπ )1(4 20 −

Torque from rotor plane on this control volume is Tδ

θδ rCmT = = arm ′22 ϖ

= = raur δωρπ ′34 raaVr δωρπ ′− )1(4 03

ar ′ϖ2

)21(0 aV − )1(0 aV −

Rr

ϖ

ar ′ϖ

0V

12

5.2 Blade forces

Now equate the momentum changes in the flow to the forces on the blades:

5.2.1 Normal forces

rBFN Nδδ =

raaVr δρπ )1(4 20 − = rcCVB Nrel δρ 2

21

= rcCaVB Nδφ

ρ 2

220

sin)1(

21 −

Therefore: raπ4 = NcCaBφ2sin)1(

21 −

Define the rotor solidity: rBrcr

πσ

2)()( =

Hence: aaCN

−= 1sin4 2

σφ

5.2.2 Tangential forces

rrBFT Tδδ =

raaVr δωρπ ′− )1(4 03 = rcCVrB Trel δρ 2

21

= rcCaraVrB Tδφφω

ρcossin

)1()1(21 0 ′+−

Therefore: ar ′π4 = TcCaBφφ cossin

)1(21 ′+

Use the rotor solidity σ : aaCT

′+=′ 1cossin4σ

φφ

13

5.3 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: 1sin4

12

+

=

NCσφ

a

Angular induction factor: 1cossin4

1'−

=

TC

a

σφφ

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 a these equations must be solved iteratively.

a '

5.4 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 V and rotor angular velocity 0 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 C and C for L D α from table or graph for the aerofoil used.

5. Calculate C and N 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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5.4.1 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 = 8.25

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

15

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

)

5.4.2 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

16

Angular induction factor a’

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 2 4 6 8 10 12 14 16 18 20 2

r (m)

Ang

ular

indu

ctio

n fa

ctor

a'

2

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)

17

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) P = 184 kW

Coefficient of performance23

03

0 21

21 RV

P

AV

PC pπρρ

== = 0.42

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

(%)

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6 BLADE LOADING

6.1 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 ′+−

=

6.1.1 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:

22max,b

IMd

IM

NN

To

TT

Naero +=σ

19

6.1.2 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.

20

6.2 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 −+ ++

21

6.3 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:

2max,b

IM

NN

swsw =σ

Simplified method: split blade into elements, assume each element has uniform self weight.

22

6.4 Combined Loading

22max,b

IMd

IM

NN

To

TT

Naero +=σ

)()(

rArFc

c =σ

2max,b

IM

NN

swsw =σ

Operational maximum stress: =maxσ aeromax,σ + cσ swmax,σ+

Minimum stress at same location: =minσ aeromax,σ + cσ swmax,σ−

23

6.5 Storm Loading

6.5.1 Drag force on blade

Blades parked. Extreme wind speed

)(21)( 2

max rcCVrF DD ρ= load per unit length

DF

maxV c

maxV = 50 m/s, c = 1.3m

Re =µ

ρ cVmax = 1.225 ×50× 1.3 / 1.79 × 10-5 = 4.4 × 106

Hence = 1.5 DC

24

6.5.2 Bending moment

Find bending moment at root of blade

∫=Rr DdrrFMmin

κσ EI

My

==

2od

IM

=σ

12)( 33

io ddbI −=

6.5.3 Shear stress

∫=Rr DdrFSmin

IySAq c=

wtq τ=

Note:

High solidity rotor (multi bladed) gives excessive forces on tower during extreme wind speeds. Therefore use fewer blades.

25