lecture 3 aerodynamic wind turbine design - an …n...aerodynamic wind turbine design - an...

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1Wind TurbinesN09_087/ A

Th. Carolus 08/2009

Lecture 3

Aerodynamic Wind Turbine Design - An Introduction -

Th. Carolus


Th. Carolus 08/2009

Historic Windmill in Holland

- Blades twisted- Adjustable “flaps”- Retractable cloth covering of

the blades- For wheat grinding

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Historic Wind Turbine in Cleveland, Ohio

Charles F. Brush built what istoday believed to be the firstautomatically operating wind turbine for electricity generation. It was a giant - the World'slargest - with a rotor diameter of 17 m (50 ft.) and 144 rotor blades made of cedar wood. The turbine ran from 1888 to 1908 and charged the batteriesin the cellar of his mansion. Despite the size of the turbine, the generator was only a 12 kW model. This is due to the factthat slowly rotating wind turbines of the American wind rose type do not have a particularly high averageefficiency.

www.windpower.org/en/pictures/brush.htm


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- Multi-bladed for high torque- Stormproof due to ring in the

outer blade region

Multi-bladed Wind Pump, U.S.A Mid-West

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Modern Three-bladed Horizontal Axis Wind Turbine (HAWT)(Vestas 1.5 MW, 2002)


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Contents

1. Review: A Few Fundamentals of 1D-Turbomachinery Theory

2. BETZ‘s Theory (1D Axial Momentum Conservation)

3. Blade Element Analysis and Optimal Blade Design

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..... recall from lecture 1

......


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1.6 EULER‘s Equation of Turbomachinery (Angular Momentum Analysis)

controlvolume

Mshaft

δ Fu

δ Fu

r

δ r

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Apply to angular momentum conservation to CV „blade element“ (BE)

With incremental mass flow through BE

one gets the incremental torque at shaft

( ) ( )( )× ∂⋅+

∂× =∫∫ ∑

anguunsteady termexternaltorqu

lar momentumes

AV shaftr c c Mc

n dAr

tρ

ρ

( ) =2 2

change of angular momentum from entrance to exit

of blade channel

u shaftm r c Mδ δ

= = 2m mm c A c rδ ρ δ ρ πδ

(1-1)

(1-2)

= −change of momentum from entrance to exit

of blade channel

u uF c mδ δ2

Note:Incremental tangential force is

(1-3)

δ Fu


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With the incremental power

and

one eventually obtains the specific work

or, if

=u rΩ

≡ = 2 2BE

uPY u cm

δδ

=BE shaftP Mδ δ Ω

EULER equation of turbomachinery (1754) LEONHARD EULER

1707 - 1783

(1-4)

(1-5)

(1-6b)

≠1 0uc

= −2 2 1 1u uY u c u c

(1-6a)

SEGNER‘swaterwheelanalyzed byL. EULER

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EULER‘s Equation in Terms of c, w, and u

Take generalized velocity triangles such as:

( )( )

m u

u u

c w u c

w u u c c

= − −

= − − +

22 21 1 1 1

2 2 21 1 1 1 12

u mc c c= −2 2 21 1 1

Eliminate mc 1

( )uu c w u c= − + +2 2 21 1 1 1 1

12

and insert in Eq. (1-9b)

( ) − − −= − = + +

2 2 21 2 2 1 2

2 2 2

2 1 11

2 2 2 2u uw w u u c cY u c u c⇒ (1-6c)


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yields the valuable result that the static pressure difference across the BE (and thus e.g. the axial thrust on the BE) only depends on the change of u und w:

Comparing this Eq. (1-7) with (1-6c)

(1-7)( ) ( ) −= − = − +

2 22 1

2 1 2 11 1

2Change ofChange of Change of dynamic total static pressurepressure pressure

th t tc cY p p p p

ρ ρ

( ) ( )− = − + −2 2 2 22 1 1 2 2 12 2

p p w w u uρ ρ

If fluid friction is neglected, the complete shaft power is equivalent to

- the total pressure drop across the BE in case of a turbine- the total pressure rise across the BE in case of a pump

− − −= + +

2 2 22 1

2 211

22 2

2 2 2w w u u c cY

(1-8)

(Conservation of energy)

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Max. power provided by the wind of velocity c0 through an area A (without turbine!)

With the mass flow rate through area A

⇒

= 20

12windP m c

= ⋅ ⋅m c Aρ 0

=windP Acρ 30

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2. BETZ‘s Theory (1D Axial Momentum Conservation)

Important:

a) Power provided by the wind isproportional to the throughflowarea A

b) Power is proportional tothird power of wind speed

1993: Rotor diameter 34 m (0,5 MW)

2005: Rotor diameter 114 m (4,5 MW)

(2-1a)

(2-2)

(2-1b)


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• Wind turbine retards incoming air• Stream tube is expanding• Wind exerts thrust force Fax on rotor and thus mast

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Assumptions(1) Rotor replaced by a thin actuator disc at station D(2) Steady uniform flow upstream of the disc(3) Uniform and steady velocity at the disc(4) No wake rotation produced by the disc(5) Flow passing through the disc is contained both upstream and downstream

by the boundary stream tube(6) Flow is incompressible


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Power extracted from the streamtube

(2-3)

Theoretical* shaft power of turbine

(2-4)

Thrust force Fax (acting on the mast)? Velocity in disc cD ?

Axial momentum conservation

(2-5)

* i.e. for inviscid flow

Combining Eq. (2-5) and (2-4)

as in Eq. (2-3)

⇒

⇒ Mass flow rate through rotor disc

Inserting everything in Eq. (2-4)⇒ Theoretical shaft power of turbine

= ⋅ − ⋅, 0 3( )S th DP m c c c

( )( )

−= = +

−

2 20 3

0 30 3

1122D

c cc c c

c c

( ) πρ ρ= ⋅ ⋅ = ⋅ + ⋅ 20 3

12 4Dm c A c c D

=,S thP P

( )

( ) ( )ρ

= ⋅ = ⋅ − ⋅ +

+ ⋅ − ⋅ +

, 0 3 0 3

0 3 0 3 0 3

1( )2

1 1 = ( )2 2

S th ax DP F c m c c c c

A c c c c c c

= ⋅,S th ax DP F c

= ⋅ −0 3( )axF m c c

= ⋅ −2 20 3

1( )2

P m c c

(2-6)

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⇒

= = + − − ≡

(theoretical power coefficient)S,thP ,th

Wind

P c c c... CP c c c

2 3

2 2 2

1 1 1

1 12

3

0

cc

,P thc

(2-7)

Optimum:

Max. power for c3/c0 = 1/3; i.e. wind turbine must retard incoming flowto 33% of it original velocity

CP,th,opt = 16/27 = 0.593 or roughly 60%

CP,th,opt = 16/27 is the BETZ limit; maximum possible power coefficientof any wind turbine! (Albert BETZ, 1926)


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≡

= ⋅ ⋅P

S P ,th

C

P C Acη ρ 30

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• Turbine shaft power including various losses: (2-8)

• Effect of taking into account wake rotation: see next chapter

Hints

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shaft power of he wind turbine

max. powerfrom thewind

=WindP Acρ 30

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Pow

er [k

W]

Wind velocity at hub level [m/s]

Example: A 500 kW Wind Turbine

Enercon E40


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3. Blade Element Analysis and Optimal Blade Design

=3 013

c c

( )= + =0 3 01 22 3Dc c c c

Introducing the „axial induction (retardation) factor“ a at rotor disc

( )= −3 01 2c a c

( )= − 01Dc a c

=13

a

So far: BETZ optimum (without wake rotation)

yields for the BETZ optimum a value of

(3-1)(3-2a)

3.1 Kinematics

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and defining a‘ as the„tangential induction factor“

What are now optimum values?

( )= = −1 2 01m mc c a c

= ?a

Compare propeller:Wake downstream of rotor is rotating

Take complete flow kinematics including wake rotation into account for optimum wind turbine design

⇒ , , D m umc c c c2 21

= −2 02 'uc a c

=' ?a

⇒ , umc c c3 33

At disc:

Far wake:

Employing the previous axial induction factor

(3-2b)

(3-3)

Link between rotor geometry and performance - Optimum wind turbine design?


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Thrust from axial momentum conservation on CV „coaxial strip“

( )= − = − 20 3 01 4ax mF c c m a( a )c r rδ δ ρ π δ (3-4)

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Comparing Eqs. (3-4) and (3-5)

i.e. a and a‘ are not independent from each other

( ) ( )( ) ( )( )

( )

− = − + −

= + − − +

= −

= − +

2 2 2 22 1 1 2 2 1

22 2 22

22 2

2

2 2

2

222 1

D u D

u u

p p w w u u

u c u c c

uc c

u a' a'

ρ ρ

ρ

ρ

ρ

(1-11)

( )= −2 1axF p p Aδ δ

Thrust exerted on blade element (BE)

In axial cascade with incompressible flow ,

( )= +2 1 4axF u a' a' r rδ ρ π δ

( ) ( )− = +2 201 1a a c a' a' u

= =1 2m m Dc c c=1 2u u

⇒ (3-5)

⇒

⇒ (3-6a)


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Define design tip-speed ratio

and local tip-speed ratio

=0

arcΩλ

( ) ( )− = + 21 1 ra a a' a' λ

( ) ( )−= − + + =2

11 1 12 2 r

r

a aa' f λ

λ

= =

0r

a

r ur c

λ λ

(3-7a)

(3-7b)

with a solution

Then Eq. (3-6a) becomes

(3-6b)

(3-8)

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2windP c r rρδ π δ

From EULER‘s equation (1-9a)

and with

and Eq. (3-8)

Now define BE power coefficient

= BEP ,BE

wind

PC

Pδδ

( )= = −22 2 01 4BE uP u c m u a' a c r rδ δ ρ π δ

= 30 2


( ) ( )= − =21P ,BE r rC a' a fλ λ

Strategy for optimization:Given λr find a and a‘ for maximum CP,BE

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

a

CP,

BE

λr

⇒

(3-9)

(3-10)


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Result

Conclusion:

(Only) for λr → ∞ the BETZ values are obained:a = 0.33a‘ = 0CP,BE = 0.59

For smaller values of λr the wake rotation reduces a, CP,BE,max and increases a‘

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3.2 Optimum blade design

( )≡ =

+ + − − + − −

'

' 'r

Lr

D

r

L

a

a aa

lz

acc

rc

a

λσπ

λ λ2

42 1 11 1 1

1 1(3-11)

... similar as for fan design (lecture 1):

a, a‘ from optimization with respect to max. power output

cL, cD from airfoil selected


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10-1 100 1010

0.5

1

c Lσ

Optimum design: Dimensionless chord length vs. local tip-speed ratio

=ra

rr

λ λ

ε = 0

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Low and high tip-speed ratiowind turbines.

From the book by Albert BETZ : Windenergie und ihre Ausnutzung durch Windmühlen, 1926


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HInt: Wind turbine jargon (mostly used but sometimes confused!)

α = local angle between chord and relative velocity = angle of attackβ∞ = local angle between plane of rotation and relative velocityγ = local angle between the chord and plane of rotation = pitch (angle)

twist = γ - γ(tip)

∞= −γ β α

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References

• Gasch, R., Twele, J.: Wind Power Plants. Solarpraxis AG, Germany, 2002

• Burton, T., Sharpe, D., Jenkins, N, Bossanyi, E.: Wind Energy Handbook. John Wiley&Sons, Ltd, 2008

• Spera, D. A. (editor): Wind Turbine Technology. ASME Press 1998

• Brouchaert, J.-F. (editor): Wind Turbine Aerodynamics: A State-of-the-Art. Lecture Series 2007-5, von Karman Institute for Fluid Dynamics, Belgium

• Mathew, S.: Wind Energy, 2006

• Hanson, O.L.: Aerodynamics of Wind Turbines. earthscan 2009

lecture 3 aerodynamic wind turbine design - an …n...aerodynamic wind turbine design - an...

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