2 1 2 Basic Principles

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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

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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

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