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Impianti Eolici

Alessandro Croce Dipartimento di Scienze e Tecnologie Aerospaziali

Politecnico di Milano Milano

Anno Accademico 2013-14

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Aerodinamica del rotore

Argomenti trattati:

HAWT

Teoria impulsiva assiale (Betz);

Teoria impulsiva vorticosa;

Azioni aerodinamiche sulle pale;

BEM

Correzioni (tip/hub losses, Glauert, stallo dinamico,…);

Progetto aerodinamico elementare della pala.

VAWT

Turbina a resistenza e portanza;

Aerodinamica (single e multiple stream tube).

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Riferimenti

Rodolfo Pallabazzer , “Sistemi Eolici”, Ed. Rubettino, 2004, ISBN 978-8849810677

Martin O. L. Hansen, “Aerodynamics of Wind Turbines” , Earthscan Publications Ltd., January 2001, ISBN 978-1902916064

J. F. Manwell, J. G. McGowan, A. L. Rogers, “Wind Energy Explained: Theory, Design and Application”, John Wiley & Sons, Ltd, April 2002, ISBN 978-0471499725

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Short Course on Wind Energy- Introduction to Wind Turbine

Aerodynamics -

Carlo L. Bottasso, Alessandro CrocePolitecnico di Milano

April 2016

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2 POLI-Wind Research Lab

Outline

• One-Dimensional annular stream tube theory with wake swirl (A. Betz 1919, H. Glauert 1926, 1935)

• Blade element momentum theory

• Optimum design conditions

• Examples:

- Optimum blade for constant efficiency

- Optimum blade for constant chord

• Recapitulation and conclusions

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1D Annular Stream Tube Theory

Hypotheses:

• Stationary flow

• Constant mass flow rate along stream tube (no interaction among annuli and no mixing at stream tube boundary)

• Incompressible and inviscid flow

• Actuator disk (infinite number of blades)

a = axial induction factor

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Axial thrust:

Bernoulli’s Th.:

Thrust:

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Assuming uniform inflow across rotor disk:

Thrust:

Power:

Betz limit

Turbulent wake state

Glauert empirical relation

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1D Annular Stream Tube Theory with Wake Swirl

a’=angular induction factor

Bernoulli’s Th.:

Pressure jump across rotor:

Thrust:

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Tangential force (rate of change of tangential momentum):

Torque:

Power:


Rotor plane

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Integrating over the rotor disk:

where

Power coefficient:

Variable speed wind turbine: maximize for best possible CP

given a design λ

Local speed ratio

Tip speed ratio (TSR)

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Optimum design condition:


Thrust from axial flow

Thrust from swirl flow

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Solve to get

then set optimality condition

Optimal solution:

or


𝜆𝑟𝑜𝑝𝑡2 =

(1 − 𝑎) (4𝑎 − 1)2

(1 − 3𝑎)

𝑎𝑜𝑝𝑡′ =

(1 − 3𝑎)

(4𝑎 − 1)

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Optimal solution:


𝜆𝑟𝑜𝑝𝑡2 =

(1 − 𝑎) (4𝑎 − 1)2

(1 − 3𝑎)

𝑎𝑜𝑝𝑡′ =

(1 − 3𝑎)

(4𝑎 − 1)

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Power coefficient:

with optimal solution:

we find:


𝐶𝑃,𝑚𝑎𝑥 =24

𝜆2න

Τ1 4

𝑎2 (1 − 𝑎)(1 − 2𝑎)(1 − 4𝑎)

(1 − 3𝑎)

2

𝑑𝑎

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Optimal solution

solved wrt

(for a fixed l)


𝜆 = 7

𝜆𝑟 = 𝜆𝑟

𝑅= 𝜆𝜂

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Blade section: consider velocity components from momentum theory

Blade Element Momentum Theory

Rotor plane

Speed resultant Inflow angle

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Blade section: consider aerodynamic force components


Rotor plane

B = number of blades

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Using the previous relationships, one can compute a and a’ at a generic TSR and blade pitch with an iterative procedure:

• On each annulus, guess a and a’, and iteratively solve:1. Compute φ:

2. Compute α, then CL and CD from polars;3. Compute thrust and torque:

4. Compute a and a’ from thrust and torque:

5. Go to 1 until converged


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Once computed a and a’ one can compute power coefficient:

from 2nd equation of blade element:

with the local solidity s’:

and from the velocity components:

one get:


𝑃 = 𝑅ℎ𝑢𝑏𝑅

𝑑𝑃 = 𝑅ℎ𝑢𝑏𝑅

Ω𝑑𝑄

𝑑𝑄 = 𝐵1

2𝜌𝑉𝑟

2𝑐𝑟𝑑𝑟(𝐶𝐿 sin𝜑 − 𝐶𝐷 cos𝜑)

𝑉𝑟2=

1 − 𝑎 2𝑉2

sin𝜑2

𝜎′ =𝐵𝑐

2𝜋𝑟

𝐶𝑃 =2

𝜆2න𝜆ℎ𝑢𝑏

𝜆

𝜎′𝐶𝐿 1 − 𝑎 2𝜆𝑟

2

sin𝜑1 −

cot 𝜑

𝐸𝑑𝜆𝑟

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Combining the expressions from momentum theory and at the blade element dropping drag in the computation of a and a’ we get:

and

which, included in the previous Cp, and upon simplification, get:


𝜎′ 𝐶𝐿 1 − 𝑎 =4𝑎 sin𝜑2

cos𝜑

𝐶𝑃 =2


𝜆

𝑎′ 1 − 𝑎 𝜆𝑟3

1 −cot𝜑

𝐸𝑑𝜆𝑟

tan𝜑 =𝑎′

𝑎𝜆𝑟

New term function of the aerodynamic efficiency

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Momentum theory is no longer valid at axial induction factors greater than 0.5. The flow patterns through the wind turbine become much more complex /large expansion of the slipstream, turbulence and recirculation behind the rotor).


Momentum theory

Turbulent wake state

Empirical relations Empirical relationships

between the axial induction factor and the thrust coefficient are used.

AeroDyn: " P.J. Moriarty, A.C. Hanse n, AeroDyn Theory Manual, NREL/TP-500-36881 - January 2005

BLADED: Bladed Version 4.0 - Theory Manual - 2010 - GarradHassan & Partners Ltd.

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Theory need to account for the fact that axial induction is not azimuthally uniform

Prandtl’s approximation at the tip:

Similarly at the root:

Combined tip/root loss correction factor:


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Other corrections:

• Skewed wake correction;• Unsteady BEM;• Dynamic stall;• Dynamic inflow model.

References:• Schepers, J. G., and H. Snel. "Joint Investigations of Dynamic Inflow Effects and Implementation of an

Engineering Method." ECN-C 107 (1995).• Hansen M.O.L., “Aerodynamics of Wind Turbines”, Routledge, 2015, ISBN 1317671031.• Leishman, J. Gordon, and T. S. Beddoes. "A Semi‐Empirical Model for Dynamic Stall." Journal of the

American Helicopter society 34.3 (1989): 3-17.• Larsen, Jesper Winther, Søren RK Nielsen, and Steen Krenk. "Dynamic stall model for wind turbine

airfoils." Journal of Fluids and Structures 23.7 (2007): 959-982.• Peters, David A., and Cheng J. He. "Finite state induced flow models. II-Three-dimensional rotor disk."

Journal of Aircraft 32.2 (1995): 323-333.


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Combining the expressions from momentum theory and at the blade element dropping drag for simplicity – but see also Wilson & Lissaman1974):

and, upon simplification, we have the optimal local loading:

Optimal Blade Design

Chord solidity

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From the optimal local loading for maximum CP :

and from the velocity triangle

we find:

Hence the optimal inflow and twist angles are


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Optimal design conditions:

Possible strategies:

A) Assume constant CL (and hence α), e.g. for maximum efficiency E=CL/CD, and compute optimal chord from (1) and twist from (2)

B) Assume constant chord, and compute CL from (1), then its corresponding α, and finally twist from (2)


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Remark:

Since optimal inductions are only functions of inflow angle

power P and thrust T will be the same for strategies A and B when the two rotors have the same solidity (assuming no drag in the equations). But in the more general case:


𝐶𝑃 =2


𝜆

𝑎′ 1 − 𝑎 𝜆𝑟3

1 −cot𝜑

𝐸𝑑𝜆𝑟

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Example: Design TSR = 5Airfoil NACA4412: CL=0.0981 αdeg+0.4033• Constant CL blade: CL max E=0.8• Constant chord blade: chord value selected to have same solidity


Blade with optimal (max E) chord distribution

Constant chord blade of same solidity

Optimal span-wise twist distributionsOptimal span-wise chord distributions



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Example: Design TSR = 5Airfoil NACA4412: CL=0.0981 αdeg+0.4033• Constant CL blade: CL max E=0.8• Constant chord blade: chord value selected to have same solidity




Induction factors (same for both designs) Inflow angle (same for both designs)

Swirl effects are predominant at blade root

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Remark: Apparently the two blades have the same performance (same a and a’, hence same CP), but consider CL span-wise distribution:

What will happen approaching stall?




Aerodynamic efficiency distributionLift coefficient distribution

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Remark: According to this model (no drag), same a and a’ means same CP, but when considering also CD, the local power coefficient distribution shows an higher CP for the optimal chord shape (i.e. higher E):




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Using the previous relationships, one can compute power and thrustat a generic TSR and blade pitch:

• Given a TSR λ and blade pitch β• On each annulus, guess a and a’, and iteratively solve:

1. Compute φ, then α, then CL and CD

2. Compute thrust and torque:

3. Compute a and a’ from thrust and torque:

4. Go to 1 until converged

Performance Off the Design Point

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Effect of Solidity

Effect of changing solidity

Optimal blade shape of the previous example:

• design TSR = 5

• constant CL at max E

• 3 blades

Desirable broad peak for low solidity

Good max CP and broad flat curve

High torque, even at low TSR, good for water pumps

Undesirable sharp peak for high solidity

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Effect of number of blades

Maximum achievable power coefficients as a function of number of blades, no drag

Source: J.F. Manwell, J.G.McGowan, A.L. Rogers, “Wind Energy Explained, Theory, Design and Application”, 2nd Edition, Wiley, 2009

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Effect of Solidity

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Typical behavior of power and thrust coefficient curves:

Performances computed with an aeroelastic code including drag, Glauert correction, tip/hub losses and different airfoils along the blade span.

Real Performance Off the Design Point

Increasing pitchIncreasing pitch

Max CP

a close to 1/3AoA close to max efficiency

Towards low TSRa smallAoA large (towards stall)

Towards high TSRa largeAoA low

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V

Variable Speed Regulation

P

Vr

Pr

Region II - constant TSR strategy

Region III - constant power strategy

Vcut out

Vcut in

Reg

ion I

Weibulldistribution

Annual energy yield:

2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

TSR

CP

Regulation trajectory(see later on in this notes under “Control”)

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V

Initial Rotor Sizing ConsiderationsP

Vr

Pr

Increasing A

Decreasing Vr

Low wind sites, larger rotors(but careful with loads)

On the other hand:

Noise (on shore):

but also:

Hence the rotor size can be estimated as:

See later for regulation strategies that include tip speed limits

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Further Reading

Comprehensive treatment of this material:

T. Burton, D. Sharpe, N. Jenkins, E. Bossanyi, Wind Energy Handbook, John Wiley and Sons, Chichester, England, 2001

If you are interested in historical references:

H. Glauerts, “Windmills and Fans”, Aerodynamic Theory (W.F. Durand, Ed.), Springer, Berlin, Germany, 1935

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Homework Assignment

- Write a computer program (MATLAB, Excel, …) according to these notes

- Choose a design TSR

- Compute the optimal solution with span-wise varying chord (max E)

- Compute performance off the design TSR

- Can you find an approximation which is easier/less expensive to manufacture (e.g. linear taper, linear twist)? Quantify the performance loss wrt the optimal solution

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