9 Wind Turbines9. Wind Turbines

1

9.1 Wind Energy ‐Wind Power9.1 Wind Energy  Wind Power

A moving air with velocity of     has a kinetic energy of∞Vg y gy∞

][21 2 JmVE ∞=

If the moving air has a density    , then the kinetic 2

ρ

energy per volume of air becomes:

1 ][21

32

mJVEV ∞= ρ

2

The Energy Extracting Stream‐tube of a Wind Turbine 

The volume flow rate per second through A is:

3][

3

SmAVV ∞=φ

3

Power = Energy per Second

Power = Energy per Volume x Volume per secondPower = Energy per Volume x Volume per second

Combining the above equations gives:

AVVPair ∞∞ ×= 21 ρair ∞∞2ρ

][21 3 WAVPair ∞= ρ

4

From the above derived equation:

• The power is proportional to the density .  Density 

varies with height and temperature

• In case of horizontal axis windmills the power is 

proportional to the area (area swept by the blades)proportional to the area  (area swept by the blades) 

and thus to R2.

h h h b f h d b d• The power varies with the cube of the undisturbed 

wind velocity . Note that the power increases eightfold 

if the wind speed doubles.5

9 2 M i P C ffi i9.2 Maximum Power Coefficient

• The act al po er e tracted b the rotor blades is the• The actual power extracted by the rotor blades is the 

difference between upstream and downstream 

powers.

Th i i i h d h h• The maximum power extraction is reached when the 

wind downstream is 1/3 of the undisturbed 

upstream velocity .

6

The axial Stream tube model 

( ) ( )22. 2

1oextr VVrateflowmassP −××= ∞

7

⎟⎞

⎜⎛ −

××= ∞ oVVArateflowmass ρ ⎟

⎠⎜⎝

××=2

Arateflowmass ρ

( )221 o VVVV

AP ×⎟⎞

⎜⎛ −

×××= ∞ρ ( )max 22 oVVAP −×⎟⎠

⎜⎝

×××= ∞ρ

⎟⎞

⎜⎛ V

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−×

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −×××= ∞

∞

∞∞ 2

2max 32

321 V

V

VV

AP ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛−×⎟

⎠

⎞⎜⎝

⎛×××= ∞∞

∞

932

21 2

2.max

VV

VAP ρ

3.max 2

12716

∞⎟⎠⎞

⎜⎝⎛= AVP ρ

227 ⎠⎝

8

9.3 Classification of Windmill Rotors9.3 Classification of Windmill Rotors

9 3 1 H i t l A i R t9.3.1 Horizontal Axis Rotors

Horizontal axis wind turbines (HAWT) haveHorizontal axis wind turbines (HAWT) have

their axis of rotation horizontal to the ground

and almost parallel to the wind stream. Most

of the commercial wind turbines fall under

this categorythis category.

9

Advantages of horizontal axis wind turbines are:

• Low cut‐in wind speed and easy furling

• They show relatively higher power coefficientThey show relatively higher power coefficient

Disadvantage of horizontal axis wind turbines are:

• Generator and gearbox are to be placed over the tower 

making its design more complex and expensivemaking its design more complex and expensive

• They need for tail or yaw drive to orient the turbine 

towards wind.10

Depending on the number of blades, HAWTs are classified 

as single bladed, two bladed, three bladed and multi 

bladed.

11

9.3.2 Vertical Axis Rotor9.3.2 Vertical Axis Rotor

The axis of rotation of vertical axis windmill is vertical to

the ground and almost perpendicular to the wind

di tidirection.

The advantages of these windmills are:

• They can receive wind from any direction.

• Complicated yaw devices are not needed

12

• Generator and gearbox of such systems can be housed 

at the ground level which makes the tower design 

simple and more economicalsimple and more economical

• Maintenance of these windmills can be done at the 

ground level

The major disadvantage of these systems is that theyThe major disadvantage of these systems is that they 

are not self starting.

13

9.4 The Rotor9.4 The Rotor

The  windmill rotates because of forces acting on the 

blades.

The cross sections of these blades have several forms.

Air flow over blades (airfoil) results two forces, Lift and ( ) ,

Drag.

Lift is  the force measured perpendicular to the airflow 

and drag is measured parallel to the flowg p

14

Lift and DragLift and Drag

• The lift force result in a force working in tangential

direction at some distance from the rotor center.

• This force is diminished by the component of the drag• This force is diminished by the component of the drag

in the tangential direction.

15

Lift d D fLift and Drag forces

16

The product of the net tangential force multiplied by 

the corresponding distance from the rotor center givesthe corresponding distance from the rotor center gives 

the contribution of the blade element to the torque Q 

of the rotor.

The rotor rotates at angular speed      ,Ωg p ,

[ ]rad2Ω [ ]sradnπ2=Ω

17

The power such a rotor extracts from the wind is

transformed to mechanical power.

This power is equal to the product of the torque and

the angular speed.

[ ][ ]WQP Ω×=

18

9.5 Rotor Blade Design9.5 Rotor Blade Design

The windmill rotates because of forces acting on theThe  windmill rotates because of forces acting on the 

blades.

The cross sections of these blades have several forms.

Ai fl bl d ( i f il) l f Lif dAir flow over blades (airfoil) results two forces, Lift and 

Drag.

Lift is  the force measured perpendicular to the airflow 

d d i d ll l t th fland drag is measured parallel to the flow

19

Lift and DragLift and Drag

• Chord line: ‐ it connects the leading edge and the 

trailing edge of the airfoil.

• Angle of attack: an angle between the chord line and• Angle of attack: ‐ an angle between the chord line and 

the direction of the airflow.

20

To describe the performance of an airfoil independent 

of size and velocity, Lift L and drag D are divided by         

where21 where,AV 2

21 ρ

⎥⎦⎤

⎢⎣⎡= 3kgDensityAirρ [ ]s

mVelocityFlowV =⎥⎦⎢⎣ 3myρ [ ]s

[ ]2)( mLengthBladechordAreaBladeA ×==

21

The results of these divisions are called lift coefficient           

and drag coefficientC Cand drag coefficient    .lC dC

AVLCl 21

=AV

DCd 21=

AV 2

21 ρ AV 2

21 ρ

The amount of lift and drag depends on the angle of 

attack. This dependence is a given characteristic of an p g

airfoil is always presented in           and           graphs.α−lC α−dC

22

For the design of a windmill it is important to find from

such graphs the and values that correspond with a

minimum ratio.

lC α

l

dC

C

23

Drag lift ratio, angle of attack and lift coefficient

for different airfoils24

The mechanical power can be expressed as the power

in air multiplied by a factor .PC

airpmech PCP ×=

is called power coefficient and is a measure for the

aipmech

PC

success we have in extracting power from the wind.

PC h23

21 RV

PC mechP πρ ∞

=

25

The local speed ratio is the speed U of the rotor at

radius r by the wind speed.

Ω

The speed‐ratio of the element of the rotor blade at∞∞

Ω==

Vr

Vu

rλ

p

radius R is called tip‐speed ratio:

∞

Ω=

VR

oλ

26

Calculation of blade chords and blade settingg

• Design of the rotor consists in finding both values of

the chord and the setting angle ,

Th i l i h l b t th h d d

β

• The setting angle is the angle between the chord and

the plane of rotation.

27

The following parameters must be found before

making the calculation of the chords and the settingmaking the calculation of the chords and the setting

angles:

Rotor R: the radius

: the design tip speed ratiodλ

B: number of blades

Airfoil : design lift coefficientldC

: Corresponding angle of attackdα

28

The choice of and B are more or less related as the

following table suggests

dλ

following table suggests.

λ B

1 6 – 20

2 4 – 12

dλ

2 4 12

3 3 – 6

4 2 – 4

5‐8 2 – 3

8‐15 1 ‐ 2

29

Th f l d d iThe type of load determines :

• Water pumping windmills driving piston pumps have   1 <    < 2.

dλ

dλ

• Electricity generating wind turbines usually have 4 <      < 10.dλ

The radius of a rotor can be fixed by a formula,y

( ) 3

21

∞×=

VCPRp πρ

Where:              can be approximated to be equal to 0.1 for wind 

pump and it could be changed to 0.15 to 0.2 for electric 

( )pCρ21

generators.30

The airfoil data are selected from Table 1 FourThe airfoil data are selected from Table 1. Fourformulas describe the required information about ,and C.

β

Chord:   ( )rBCrC φπ cos18

−=

Blade setting angle: 

ldBC

αφ −= rrBg g

Flow angle: ⎟⎟⎠

⎞⎜⎜⎝

⎛=r λ

φ 1arctan32Flow angle: 

Design Speed:

⎟⎠

⎜⎝ rλ3

rdd ×= λλDesign Speed:  Rdrd ×λλ

31

Example: Find the chord C and setting angle of the blade

( )for a curved plate profile (10 % curvature) with the

given parameters:

⎪⎨

⎧ =6

37.1B

mRRotor ⎨

⎧ =lCAirfoil

1.1

⎪⎩

⎨==

26

d

BRotorλ ⎩

⎨= o

d

Airfoil4α

32

Soln.

To keep the lift coefficient at a constant value of , a

varying chord C and varying setting angle will result.

ldC

To keep the blade with a constant chord (for ease of

d i ) h h lif ffi i ill l hproduction) then the lift coefficient will vary along the

blade.

33

Constant Lift CoefficientConstant Lift Coefficient

By dividing the radius of the rotor at four points andl i th b f l th f ll i lapplying the above formulas the following values are

found

position r/R r(m) C (m)

1 0.25 0.34 0.5 42.3o 4 o 38.3 o 0.337

rλ rφ dα β

2 0.5 0.68 1 30.0 o 4 o 26.0 o 0.347

3 0 75 1 03 1 5 22 5 o 4 o 18 5 o 0 2983 0.75 1.03 1.5 22.5 o 4 o 18.5 o 0.298

4 1 1.37 2 17.7 o 4 o 13.7 o 0.247

34

h f b l h h h d f h bl d hThe figure below shows the chord of the blade at thefour division points.

35

Constant ChordConstant Chord

• The constant lift coefficient approach has a difficulty of

manufacturing as the twist varies discontinuously along

th bl d T id th t t t h i dthe blade. To avoid that a constant approach is used.

• To have a constant chord the lift coefficient at different

positions along the blade will vary.

( )rl BCrC φπ cos18

−=

36

h l f k l h l fThe angle of attack also varies with variation in lift

coefficient. Therefore graph is needed to determineα−lC

values at different positions.

Choosing a chord of 0 324 m and three positions alongChoosing a chord of 0.324 m and three positions along

the blade, and applying the above formula the following

data are found.

position r(m) C (m) chosen rλ rφ lC α β1 0.5 0.324 0.73 35.9 1.23 6.4 29.5 27

2 0.86 0.324 1.26 25.7 1.10 3.6 22.1 23

3 1 22 0 324 1 78 19 5 0 91 0 2 19 3 19

r l

3 1.22 0.324 1.78 19.5 0.91 0.2 19.3 19

37

The blade shape and setting angles of the blade are

shown below.

38