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Short Summary of the Mechanics of Wind Turbine

Korn Saran-Yasoontorn

Department of Civil EngineeringUniversity of Texas at Austin

8/7/02

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Summarized from

Wind energy explained: theory, design and application./ Manwell, J. F. / Chichester / 2002

Wind turbine technology: fundamental concepts of wind turbine engineering. / New York / 1994

Wind energy conversion systems/ Freris, L.L./ Prentice Hall/ 1990

Wind turbine engineering design/ Eggleston, D.M. and Stoddard, F.S./ New York/ 1987

Introduction to wind turbine engineering/ Wortman, A.J./ Butterworth Publishers/ 1983

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Yaw system

HubDrive train

Column

RotorSimple model ofMicon 65/13

1D steady wind flow(12m/s)

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Fundamental Concepts

Mass flow rate = Av

Energy per unit volume = 1/2 v2

Power = rate of change of energy

= force * velocity

= (Av) 1/2 v2 = 1/2 Av3

Dynamic pressure = force/area

= power/vA

=1/2 v2

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Actuator Disk Model with no wake rotation

U1U2 U3 U4

rotor disk

Assumptions:

i. Homogeneous, incompressible, steady wind

ii. Uniform flow velocity at disk (uniform thrust)

iii. Homogenous disk

iv. Non-rotating disk

stream tube boundary

upstream

downstream

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)(

)(

41

14

uumT

uumT

dt

dpF

Conservation of Linear Momentum

where T is the thrust acting uniformly on the disk (rotor) which can be written as a function of the change of pressure as follow

)( 32 ppAT

7

244

233

222

211

2

1

2

12

1

2

1

upup

upup

)]1(4[2

1 21 aaAuT

Bernoulli’s Equation (energy conserved)

Relate above equations and define the axial induction factor, a as

1

21 )(

u

uua

we obtain

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Power output of the turbine is defined as the thrust times the velocity at the disk. Hence

])1(4[2

1 231 aaAuP

)1(4)2/1//(

)1(4)2/1/(3

23

aaAuTC

aaAuPC

T

p

)]1(4[2

1 21 aaAuT

Wind turbine rotor performance is usually characterized by its power and thrust coefficients

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Notice that

Wind velocity at the rotor plane is always less than the free-stream velocity when power is being absorbed.

This model assumes no wake rotation, i.e. no energy wasted in kinetic energy of a twirling wake.

The geometry of the blades does not involve the calculations.

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If the axial induction factor of the rotor is founded, one can simply calculate for the thrust and power output.

An ideal turbine generates maximum power. After some manipulations, one can find that the axial induction factor, a, for the ideal turbine is 1/3.

Even with the best rotor design, it is not possible to extract more than about 60 percent of the kinetic energy in the wind

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Wind Velocity

Total Pressure

Dynamic Pressure

Static Pressure

p3

p2

upstream disk downstream

u1

u2

u4

1/2u2

p0 p0

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Actuator Disk Model with wake rotation

U1U2 U3 U4

rotor disk

U

U(1-2a)U(1-a)

dr

r

The thrust distribution is circumferentially uniform.(infinite number of blades)

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Conservation of Linear momentum

rdruaadT

uumrdT

dt

dpF

22

1)1(4

)()(

2

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Conservation of Angular Momentum

rdrruaadQ

rrdrurrmddQ

22

1)1(4

)(2))()((

2

22

14

rdrrdAppdT

rpp

2])2

1([)(

)2

1()(

232

232

2

a

Bernoulli’s Equation (energy conserved)

Define the angular induction factor a’ as

rdruaadT 22

1)1(4 2

Hence,

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uRRrr ,/

2

)1(

)1(raa

aa

Equating the thrust on an annular element derived from the conservation of linear momentum and the Bernoulli’s equation gives

where

For an ideal turbine that produces maximum power output,

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31

a

aa

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In summary

rdruaadT 22

1)1(4 2

rdrruaadQ 22

1)1(4 2

2

)1(

)1(raa

aa

Notice that

the geometry of the blades still does not involve the calculations.

if the turbine is assumed to be ideal generating maximum power, one can find a and a’ in each section.

once a and a’ are founded, the total thrust and rotor torque can be determined by integration along the blade spanwise.

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Blade Element Theory

Blade geometry is considered in this part and we may use this to calculate the induction factors that relates the thrust and rotor torque.

blade elementr

dr

rotor blade

R

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Lift and Drag Forces

u

Ωr

urel

FL

FD

)2

1(

)2

1(

2

2

relDD

relLL

AuCF

AuCF

Note that CL and CD vary with cross section

(top view)

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Typical Variation of Aerofoil Coefficients

Val

ues

of

Co

effi

cien

ts

-10 0 90angle of attack (degrees)

Cl

Cd

1.0

0.5

0.0

flow separation

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Relative Velocity

u(1-a)

Ωr(1+a’)

urel

FL

FD

Wind velocity at the rotor blade is u(1-a) in horizontal direction. Also, the wind rotates with the angular velocity of ω/2 (=Ωa’) while the angular velocity of the rotor is Ω in the opposite direction.

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u(1-a)

Ωr(1+a’)

urel

dFL

dFD

dFN

dFTθp

φ

Blade Geometry

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From blade geometry, one simply obtains

the following relations.

cossin

sincos2

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1

sin

)1(

)1(

1tan

2

2

DLT

DLN

relDD

relLL

rel

r

p

dFdFdF

dFdFdF

cdruCdF

cdruCdF

auu

a

a

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Finally, the total normal force on the section and torque due to the tangential force operating at a distance, r, from the center are

crdrCCuBdQ

cdrCCuBdF

DLrel

DLrelN

)cossin(2

1

)sincos(2

1

2

2

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Since the forces and moments derived from momentum theory (actuator model) and blade element theory must be equal,

crdrCCuBdrruaadQ

cdrCCuBrdraaudT

DLrel

DLrel

)cossin(2

1)1(4

)sincos(2

1)1(4

23

22

from momentum theory blade element theory

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)cos(sin

)sin(cos2sin4

r

rL Bc

rC

One can solve for C and α at each section by using this equation and the empirical C vs α curves. Once both parameters are known, a and a’ at the section can be determined from

)sin8/()1(

)sin8/(cos)1( 2

rL

L

rBcCaa

rBcCaa

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Iterative solution for a and a’

1. Guess values of a and a’2. Calculate φ3. Calculate angle of attack, α4. Calculate Cl and Cd

5. Update a and a’6. Check if a > 0.5

(In the case of turbulent wake this analysis may lead to a lack of convergence to a solution)

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Note that to keep the lift and drag coefficients, and thus the angle of attack, constant over the spanwise of the blade, it is necessary to twist the blade along the length. This however may increase the complexity of their manufacture.

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Tip loss factor

The tip loss factor allows for the velocities and forces not being circumferentially uniform due to the rotor having a finite number of blades. The Prandtl tip loss factor can be express as

)115.0arccos(exp)/2( 2

B

R

rF