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    Workshop on Renewable Energies

    November 14-25, 2005

    Nadi, Republic of the Fiji Islands

    Module 2.1Module 2.1--22

    WIND ENERGYWIND ENERGY -- BASIC PRINCIPLESBASIC PRINCIPLESGerhard J. Gerdes

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    ContentsContents

    J calculate the power in the wind

    J power coefficient

    J efficiency of wind wheel

    J drag devices

    J lift devices

    J comparison of lift and drag devices

    J utilizing wind energyJ maximum power extraction according to Betz

    J thrust according to Betz

    J airfoil theory:

    X drag

    X lift

    X laminar vs. turbulent flow

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    Utilizing wind energyUtilizing wind energy

    J conversion of kinetic energy into mechanical or electricenergy

    32

    wind Av2

    1vm

    2

    1EP === && Pmech = M = M 2 n

    - mass flow rate M - torque

    v - wind speed n - rotational number - air density - angular velocityA - control area

    m&

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    Theoretical maximum powerTheoretical maximum power

    J by reducing the speed of an air mass, wind power is

    converted into mechanical energy

    J to reduce the wind speed to zero, obviously would

    mean maximum power (hypothesis!)

    J not possible - since stopping an air flow completely at

    one given spot (rotor) violates the law of energy

    continuity

    J question: at what speed reduction the maximum wind

    power can be extracted?

    J answer was given by Betz and Glauert 1926 by a

    simple calculation

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    Mass flow m through an area AMass flow m through an area A

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    J Rotor 54m diameter

    J Rotor swept area 1500m

    J Disk of 1 m thickness:

    X Weight of air: 2.8to

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    Betz maximumBetz maximum

    J if energy is extracted, wind speed v2 behind the rotor

    must be lower than v1

    A1 v1 A2 v2

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    ContinuityContinuity

    J assumption: homogeneous air flow

    J product of density, velocity and area is constant

    according to continuity

    332211 AvAvAv ==

    air density, v wind speed, A area

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    Maximum Power Extraction by BetzMaximum Power Extraction by Betz

    J the theoretical maximum power coefficient cp accordingto Betz is

    J this means that a theoretical maximum of 59.3 % of the

    power in the wind can be extracted

    J in practice, modern electricity producing wind turbines

    reach efficiencies of more than 50 % in maximum

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    cpcp-- valuevalue

    J The cp-value is the efficiency of power production by

    wind turbines

    J cp gives the relation between power in the wind (Pwind)

    to the mechanical power extracted by the turbine rotor

    (Pturbine):

    J Usually cp electric is used to indicated the total efficiency

    of power production by a wind turbine, referring to the

    electrical output of the wind turbine generator (Pelectric):

    Wind

    Turbinep

    P

    Pc =

    Wind

    electricelectricp

    P

    Pc =

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    Power in the wind 1Power in the wind 1

    J power in the wind with the velocity v and an area A

    J Pwind is proportional to the air density , area A and the

    third power of the wind velocity v

    J the power output of a WTG follows up to a wind speed

    of approx. 12 m/s wind speed the v energy curve

    J then the power limitation comes into action, to preventthe generator from overload

    air density, v wind speed, A area

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    Power in the windPower in the wind power outputpower outputcharacteristicscharacteristics

    0

    500

    1000

    1500

    2000

    2500

    3000

    3500

    4000

    4500

    0 2 4 6 8 10 12 14Wind speed [m/s]

    Powe

    r[kW]

    Pwind

    PBetz

    PWT

    Pwind

    PBetz

    PWT

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    Practical consequencesPractical consequences

    J first of all, the power in the wind is proportional to the

    third power of wind speedJ this has important practical implications

    X calculating or estimating the power output of turbines

    from wind speed measurements has a high error margin

    X doubling the wind speed results in 8 times the power

    output

    X ability to withstand gusts is of primary importance for

    turbines

    J with around 60 % maximum efficiency, wind turbine

    compare favourably with other energy conversion

    systemsX diesel generator: max. ca. 35 %

    X large steam turbine: max. ca. 40 %

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    Effect of air densityEffect of air density

    J the power output of a wind system is proportional to the

    air density thus the power varies

    X with temperature: low temperatures result in more power

    X with altitude: high altitudes reduce power output

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    J all early wind power systems were drag devices

    J using the force acting on an area perpendicular to thewind direction

    J drag is proportional to the area A, air density and the

    square of wind speed v:

    J the drag coefficient cD depends on the aerodynamic

    quality of the body

    Drag devicesDrag devices

    2

    2AvcD D

    =

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    Drag coefficient depends on shapeDrag coefficient depends on shapeof bodyof body

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    Vertical axis wind wheel 1Vertical axis wind wheel 1

    J simplification

    J wind velocity at the plate: w = v - u

    J blade tip speed u = Rm (rotational speed mean

    radius)

    principle simplified model

    v velocity of wind

    u rotor blade tip-speed

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    Tip speed ratioTip speed ratio

    J blade tip speed ratio is the ratio between the speed ofthe tip of a rotor blade related to the affecting wind

    speed

    J where: angular velocity, Rm blade tip radius, .

    Rm- rotational speed at rotor tip, tip speed ratio

    v

    Rm=

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    Power coefficient of wind wheelPower coefficient of wind wheel

    blade tip speed ratio = Rm / v

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    InterpretationInterpretation

    J obviously, at complete standstill ( = 0) the power is

    zero

    J also, when the the blade-tip speed is equal to wind

    speed ( = 1)

    J the maximum power coefficient of cp max = 0.16 occurs

    at

    opt

    = 0.33,

    when the blade tip moves at 1/3 of the wind speed

    J meaning that only 16 % of the wind power can be

    extracted

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    Drag devicesDrag devices

    J all early wind power systems were drag devices

    J using the force acing on an area perpendicular to thewind direction

    J drag is proportional to the area a, air density and the

    square of rotor speed

    J the drag coefficient cD depends on the aerodynamic

    quality of the body

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    Summary drag devicesSummary drag devices

    J drag devices operate at tip speed ratios < 1, i. e. at tip

    speeds below wind speeds

    J the power coefficient cp of drag devices is below 0.2, i.

    e. the maximum efficiency is 20 %

    J with only 1/3 of the theoretical maximum (Betz: 59.3

    %), drag devices are not suitable for electricity

    generation

    J due to their simple construction, drag devices can be

    used for low power pumping needs etc.

    J however, contemporary use of drag devices is

    practically non-existent

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    Lift devicesLift devices

    J many bodies have not only a drag force component,

    but also a lift forceL

    , perpendicular to the flow

    J L attacks at about of the chord length c, as long as

    the angel of attack is small (< 10 ), thus: cL = f()

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    Lift and drag forces L, DLift and drag forces L, D

    angle of attack

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    Comparison of lift and drag devicesComparison of lift and drag devices

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    Airfoil theory: drag vs. liftAirfoil theory: drag vs. lift

    vWind vWind

    draglift

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    Airfoil theory: lift forceAirfoil theory: lift force

    J lift force =v

    21AcL 21L

    Source: DEWI

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    Airfoil theory: lift forceAirfoil theory: lift force

    Source: DEWI

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    Airfoil theory: streamlines andAirfoil theory: streamlines and

    pressure distributionpressure distribution

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    ( ) ( )bcw2

    cL 2AL

    = ( ) ( )bcw2cD2

    AD

    =

    Airfoil theory: lift and drag at theAirfoil theory: lift and drag at thebladeblade

    X L - lift

    X D - drag

    X c - chord lengthX b - width

    X w - velocity

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    Airfoil theory: laminar vs. turbulentAirfoil theory: laminar vs. turbulent

    flowflow

    Source: DEWI

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    Airfoil theory: laminar vs. turbulentAirfoil theory: laminar vs. turbulentflowflow

    Source: Schlichting / Truckenbrodt

    laminar flow (attached flow) turbulent f low (stall)

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    Wind and energyWind and energy

    J Wind speed is the essential measure influencing thepower output of a wind turbine

    J The power in the wind has to be limited with increasing

    wind speed

    X E.g. 20 m/s on a 40 m rotor of a 600kW turbine equalsroughly 6 MW wind power

    3

    21 AvcP pWind =

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    cp--Diagram

    0 4 8 12 16 20 24

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    Tip speed ratio

    Rotorpo

    wercoefficientcp

    Rated power

    Grid connection

    Constant power

    StartupCut off

    Power < rated power