Wind Turbines N09_087/ 1A

Lecture 3

Aerodynamic Wind Turbine Design - An Introduction -

Th. Carolus

Th. Carolus 08/2009

Historic Windmill in Holland Wind Turbines N09_087/ 2A

- Blades twisted - Adjustable “flaps” - Retractable cloth covering of the blades - For wheat grinding Th. Carolus 08/2009

1 Historic Wind Turbine in Cleveland, Ohio Wind Turbines N09_087/ 3A

Charles F. Brush built what is today believed to be the first automatically operating wind turbine for electricity generation. It was a giant - the World's largest - 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 batteries in the cellar of his mansion. Despite the size of the turbine, the generator was only a 12 kW model. This is due to the fact that slowly rotating wind turbines of the American wind rose type do not have a particularly high average efficiency.

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

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Multi-bladed Wind Pump, U.S.A Mid-West Wind Turbines N09_087/ 4A

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

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2 Modern Three-bladed Horizontal Axis Wind Turbine (HAWT) (Vestas 1.5 MW, 2002) Wind Turbines N09_087/ 5A

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Contents Wind Turbines N09_087/ 6A

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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3 Wind Turbines N09_087/ 7A

..... recall from lecture 1 ......

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1.6 EULER‘s Equation of Turbomachinery (Angular Momentum Analysis) Wind Turbines N09_087/ 8A

control volume

r

δ Fu

Mshaft

δ r

δ Fu Th. Carolus 08/2009

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Apply to angular momentum conservation to CV „blade element“ (BE) GG (r ×∂c) ρ GGGG + ρ ()()rcc× ⋅ ndA= Mshaft ∫V ∫A ∑  ∂t   (1-1) angular momentum external unsteady term torques

With incremental mass flow through BE

 δ mcAc==ρδmm ρ2 πδ r (1-2)

one gets the incremental torque at shaft

δ mrc = δ M  ()22ushaft change of angular momentum from entrance to exit of blade channel

Note: Incremental tangential force is δ Fu δ Fcm=−δ  uu 2 (1-3) change of momentum from entrance to exit Th. Carolus 08/2009 of blade channel

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With the incremental power

δ PMBE= δΩ shaft (1-4)

and

ur= Ω (1-5) SEGNER‘s waterwheel analyzed by one eventually obtains the specific work L. EULER δ P Yuc≡=BE δ m 22u (1-6a)

or, if cu1 ≠ 0

Yucuc=−22uu 11 (1-6b)

EULER equation of turbomachinery (1754) LEONHARD EULER 1707 - 1783

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5 EULER‘s Equation in Terms of c, w, and u Wind Turbines N09_087/ 11A

Take generalized velocity triangles such as:

22 2 cwucmu1111=−−( ) 22 2 =−−wuucc11()2 111uu +

22 2 cccum11=− 1

Eliminate cm1

1 222 uc11u =−() w 1 + u 1 + c 1 2

and insert in Eq. (1-9b)

ww222−−− uu222 cc ⇒ Yucuc()=− =12 +212 + 1 (1-6c) 2 uu211 222

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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

22 11cc21− Yppppth=−=−+() t21 t () 21 (1-7) ρρ   2 Change of Change of Change of total static dynamic pressure pressure pressure (Conservation of energy)

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

ww222−−− uu22 cc2 Y =++1 2 212 1 222

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:

ρ 22ρ 22 pp21−= w 1 − w 2 + u 2 − u 1 (1-8) 22( ) ( ) Th. Carolus 08/2009

6 2. BETZ‘s Theory (1D Axial Momentum Conservation) Wind Turbines N09_087/ 13A

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

1 2 Pm=  c (2-1a) wind 2 0 With the mass flow rate through area A  m =⋅ρ c0 ⋅A (2-2)

1 ⇒ PAc= ρ 3 (2-1b) wind 2 0

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

a) Power provided by the wind is proportional to the throughflow area A b) Power is proportional to third power of wind speed

2005: Rotor diameter 114 m (4,5 MW) Th. Carolus 08/2009

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• Wind turbine retards incoming air • Stream tube is expanding

• Wind exerts thrust force Fax on rotor and thus mast Th. Carolus 08/2009

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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 Th. Carolus 08/2009

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

 PmcccSth,03= ⋅−⋅() D

PPSth, = as in Eq. (2-3) 1 cc22− 2 ( 03) 1 ⇒ cccD ==+()03 (2-6) Power extracted from the streamtube cc03− 2 1 Pm=⋅ () c22 − c 2 03 (2-3) ⇒ Mass flow rate through rotor disc

Theoretical* shaft power of turbine 1 π mcAccD =⋅ρρ ⋅=⋅() + ⋅ 2 D 2403 PFcSth, =⋅ ax D (2-4) Inserting everything in Eq. (2-4) Thrust force Fax (acting on the mast)? ⇒ Theoretical shaft power of turbine Velocity in disc cD ? 1 PFcmcccc=⋅=⋅−⋅ ()() + Axial momentum conservation Sth,0303 ax D 2 11 Fmcc=⋅ () − =ρA()cc+⋅−⋅ ( cc ) () cc + ax 03 (2-5) 2203 03 03

* i.e. for inviscid flow Th. Carolus 08/2009

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23 P 1 cc c ⇒ S,th 22 2 (2-7) ==...1 + − − ≡ CP,th (theoretical power coefficient) Pccc2 Wind 11 1

Optimum:

cPth,

c3

c0

¾ Max. power for c3/c0 = 1/3; i.e. wind turbine must retard incoming flow to 33% of it original velocity

¾ CP,th,opt = 16/27 = 0.593 or roughly 60%

¾ CP,th,opt = 16/27 is the BETZ limit; maximum possible power coefficient of any wind turbine! (Albert BETZ, 1926)

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Hints

1 • Turbine shaft power including various losses: PC =⋅ ηρ ⋅ Ac 3 (2-8) SP,th 2 0 ≡CP

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

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9 Example: A 500 kW Wind Turbine Wind Turbines N09_087/ 19A

max. power from the wind 1 PAc= ρ 3 Wind 2 0 shaft power of Power [kW] he wind turbine

Enercon E40

Wind velocity at hub level [m/s]

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3. Blade Element Analysis and Optimal Blade Design Wind Turbines 3.1 Kinematics N09_087/ 20A

So far: BETZ optimum (without wake rotation) 1 cc= 303 12 cccc=+=() D 2303 0 Introducing the „axial induction (retardation) factor“ a at rotor disc

cac30=−(12) (3-1)

cacD =−(1 ) 0 (3-2a) 1 yields for the BETZ optimum a value of a = 3 Th. Carolus 08/2009

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

Compare propeller: Wake downstream of rotor is rotating

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

At disc: cccDm⇒ 1, m2, cu2

Far wake: cc3 ⇒ m3 , cu3

Employing the previous axial induction factor

ccmm12==−(1 ac) 0 (3-2b) and defining a‘ as the „tangential induction factor“ (3-3) cacu20=−2 '

What are now optimum values?

a = ? a'?= Th. Carolus 08/2009

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Thrust from axial momentum conservation on CV „coaxial strip“

 2 δ Fccma(a)crrax=−( 03 m )δρπδ =−14 0 (3-4)

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Thrust exerted on blade element (BE)

δ FppAax =−( 21)δ

In axial cascade with incompressible flowuu12= , cccmm12= = D

ρ 22ρ 22 ⇒ pp−= w − w + u − u (1-11) 2122( 1 2) ( 2 1) ρ 2 =+−−+ uc22() uc c 2 2 ()DuD( 2 ) ρ =−2uc c2 2 uu22 =−21ρua'2 () + a'

2 ⇒ δ Fax =+ρπδ u a'(14 a') r r (3-5)

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

22 ⇒ aaca'a'u(11−=+) 0 ( ) (3-6a) i.e. a and a‘ are not independent from each other Th. Carolus 08/2009

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Define design tip-speed ratio

raΩ λ = (3-7a) c0

and local tip-speed ratio

ru λλr == (3-7b) rca 0

Then Eq. (3-6a) becomes

2 aaa'a'(11−=) ( +)λr (3-6b)

with a solution

11 aa(1− ) a'=− +1 +2 = f ()λr (3-8) 22 λr

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Now define BE power coefficient δ P C = BE P,BE (3-9) δ Pwind

From EULER‘s equation (1-9a)  2 δ Pucmua'acrrBE==−22 u δ ρ (14) 0πδ

and with 0.6 ρ 3 δ Pcrrwind = 0 2πδ 2 0.5 λ and Eq. (3-8) 0.4 r

2 0.3 P,BE ⇒ Ca'afP,BE=−(1 )λ r =(λ r ) (3-10) C 0.2 Strategy for optimization: 0.1 Given λr find a and a‘ for maximum CP,BE 0 0 0.1 0.2 0.3 0.4ρ 3 0.5 δ PcrrTh. Carolus= 08/20092πδ a wind 2 0

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Result

Conclusion:

¾ (Only) for λr →∞theBETZvaluesareobained: a = 0.33 a‘ = 0

CP,BE = 0.59

ρ 3 ¾ For smaller values of λr the wake rotation reduces a, CP,BE,max and δincreasesPcrrTh. Carolus= 08/2009a‘2πδ wind 2 0

13 3.2 Optimum blade design Wind Turbines N09_087/ 27A

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

lz 4a'λ σ ≡= r 2πr 2 cD 11++aa'' (3-11) cL ()11− a −+λλr r 1 cL 11−−a a

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

cL, cD from airfoil selected

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

1 ε = 0 σ

L 0.5 c

0 -1 0 1 10 10 10 r λr = λ ra

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Low and high tip-speed ratio wind 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!) Wind Turbines N09_087/ 30A

α = local angle between chord and relative velocity = angle of attack β∞ = local angle between plane of rotation and relative velocity γ = local angle between the chord and plane of rotation = pitch (angle)

γ = βα∞ −

twist = γ - γ(tip)

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15 References Wind Turbines N09_087/ 31A

• Gasch, R., Twele, J.: Plants. Solarpraxis AG, , 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 : A State-of-the-Art. Lecture Series 2007-5, von Karman Institute for , Belgium

• Mathew, S.: Wind Energy, 2006

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

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