Impianti Eolici
Transcript of Impianti Eolici
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Corso di
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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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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1D Annular Stream Tube Theory
Axial thrust:
Bernoulli’s Th.:
Thrust:
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1D Annular Stream Tube Theory
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:
1D Annular Stream Tube Theory with Wake Swirl
Rotor plane
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1D Annular Stream Tube Theory with Wake Swirl
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:
1D Annular Stream Tube Theory with Wake Swirl
Thrust from axial flow
Thrust from swirl flow
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Optimum design condition:
Solve to get
then set optimality condition
Optimal solution:
or
1D Annular Stream Tube Theory with Wake Swirl
𝜆𝑟𝑜𝑝𝑡2 =
(1 − 𝑎) (4𝑎 − 1)2
(1 − 3𝑎)
𝑎𝑜𝑝𝑡′ =
(1 − 3𝑎)
(4𝑎 − 1)
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Optimum design condition:
Optimal solution:
1D Annular Stream Tube Theory with Wake Swirl
𝜆𝑟𝑜𝑝𝑡2 =
(1 − 𝑎) (4𝑎 − 1)2
(1 − 3𝑎)
𝑎𝑜𝑝𝑡′ =
(1 − 3𝑎)
(4𝑎 − 1)
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Optimum design condition:
Power coefficient:
with optimal solution:
we find:
1D Annular Stream Tube Theory with Wake Swirl
𝐶𝑃,𝑚𝑎𝑥 =24
𝜆2න
Τ1 4
𝑎2 (1 − 𝑎)(1 − 2𝑎)(1 − 4𝑎)
(1 − 3𝑎)
2
𝑑𝑎
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Optimum design condition:
Optimal solution
solved wrt
(for a fixed l)
1D Annular Stream Tube Theory with Wake Swirl
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
Blade Element Momentum Theory
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
Blade Element Momentum Theory
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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:
Blade Element Momentum Theory
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:
Blade Element Momentum Theory
𝜎′ 𝐶𝐿 1 − 𝑎 =4𝑎 sin𝜑2
cos𝜑
𝐶𝑃 =2
𝜆2න𝜆ℎ𝑢𝑏
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).
Blade Element Momentum Theory
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:
Blade Element Momentum Theory
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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.
Blade Element Momentum Theory
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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
Optimal Blade Design
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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)
Optimal Blade Design
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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:
Optimal Blade Design
𝐶𝑃 =2
𝜆2න𝜆ℎ𝑢𝑏
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
Optimal Blade Design
Blade with optimal (max E) chord distribution
Constant chord blade of same solidity
Optimal span-wise twist distributionsOptimal span-wise chord distributions
Blade with optimal (max E) chord distribution
Constant chord blade of same solidity
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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
Optimal Blade Design
Blade with optimal (max E) chord distribution
Constant chord blade of 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?
Optimal Blade Design
Blade with optimal (max E) chord distribution
Constant chord blade of same solidity
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):
Optimal Blade Design
Blade with optimal (max E) chord distribution
Constant chord blade of same solidity
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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