It Blows You Away

Wind Energy (Today & Tomorrow)Collected and Presented By:

Umair N. Mughal (Post Doc. Researcher)

NUC, Narvik-8505, Norway

Email : unm@hin.no

"Coal, gas and oil will not be the three kings of the energy world for ever. It is no longer folly to look up to the sun and wind, down into the sea's waves"

Introduction

Energy is a major input for overall socio-

economic development of any society

The prices of the fossil fuels steeply increasing

So renewables are expected to play a key role

Wind energy is the fastest growing renewable

Wind turbines are up to the task of producing

serious amounts of electricity

History

Wind Power History

800 – 900 years ago in Europe

140 years ago,water-pumpingwind mills

70 years ago, electric power

1400 – 1800 years ago,in the Middle East

Principles

Like most things, wind is powered by the Sun

• Thermosphere

• Mesosphere

• Stratosphere

• Troposphere

The earth’s atmosphere is composed of 4 layers

The earth’s weather takes place in the lowest level called the Troposphere

The Equation of State (Ideal Gas Law)

Describes relationships between pressure, temperature, and density (Start w/ molecular movement in sealed container Pressure proportional to rate of collisions between molecules and walls).

At constant temperatures, an increase in air density will cause a pressure increase (Add more molecules increase density increase rate of collisions raise pressure)

Under constant density, an increase in temperature will lead to an increase in pressure (Raise temperature increase speed of molecules increase rate of collisions raise pressure)

Pressure = density x temperature x 287 J kg-1

K-1

[ p = ρTR]

Pressure, Wind and Weather Systems� WINDS are horizontal flows of air; winds blow from areas of high pressure to areas of low pressure (nature tries to equalise pressure)

� PRESSURE describes the tendency of the air to rise or to sink at any given place or time.

� Air tends to rise or sink as a result of its density.

� Air density varies with altitude but, at the ground level, air density is governed by its temperature.

� Thus, variations in radiation and temperature control pressure and wind.

Air heated by contact with ground expands; becomes less dense and rises

Sun heats up ground

Insolation

LOW PRESSURE

Denser air drawn in at low level to

replace rising, less dense air

Denser air drawn in at low level to replace rising, less

dense air

Air stops rising when it meets air of equal density, then diverges at high level to produce more wind which eventually sinks elsewhere to complete the circulation cell

GLOBAL PRESSURE & WIND

Antarctic circle 66.5°S

Arctic circle 66.5°N

North Pole 90°N

North Pole 90°N

Equator 0°

Tropic of Cancer 23.5°N

Tropic of Capricorn 23.5°N

ZONE of greatest heating produces LOW PRESSURE

ZONE of least heating produces HIGH PRESSURE

ZONE of least heating produces HIGH PRESSURE

HIGH

HIGH

LOW

Rising air diverges at the tropopause, where a permanent temperature inversion results in warmer air above.

GLOBAL PRESSURE & WIND

EQUATORIAL (Inter-tropical convergence zone - ITCZ) LOW

POLAR FRONT (LOW PRESSURE)

TROPICAL HIGH

POLAR HIGH

Global circulation depends on differential heating over the globe. The system is driven by strong equatorial heating, causing LOW PRESSURE. Equatorial air rises, diverges and descends over the tropics, where HIGH PRESSURE dominates; where it diverges at ground level. This tropical air blows towards the equator, completing the equatorial cell, or towards the mid-latitides where it meets cold, dense polar air blown out from the polar HIGH PRESSURE. These contrasting tropical and polar air masses meet at the POLAR FRONT LOW PRESSURE BELT, where the warmer air is forced upwards by the polar air. At high level, this air again diverges towards the pole or to the tropic.

Strong winds also occur in low latitudes due to stronger heating and steeper presure gradients. Hurricanes and tornadoes are both tropical phenomena.

WIND DIRECTION & STRENGTH� Wind strength depends on the difference in pressure between the high and low pressure systems, and the distance between them.� This is called the PRESSURE GRADIENT; it is a similar concept to the physical slope between two places, shown on a contour map. Pressure is shown by ISOBARS on a weather map.� Pressure difference essentially depends on the temperature difference between the two places.

Locally, wind is channelled down streets (wind canyons).

Beach windbreaks reduce windsped by increasing friction

Strong polar winds due to low friction

Farmers plant trees to protect orchards, houses, stock or prevent soil erosion

Hurricane in Florida

A steep pressure gradient results from a large pressure difference or short distance between places and causes strong wind.

Tornado in USA

Earth’s Rotation and the Coriolis Force

Coriolis force accelerates moving objects to their right in the NH and to their left in the SH because the sense of Earth’s rotation is opposite in the two hemispheres.

In the north, winds blow anti-clockwise into a

low pressure system. In the

south, they blow clockwise.

Actual wind which blows, as diverted by Coriolis Force

Theoretical wind which would result

solely from pressure gradient

CORIOLIS FORCEHigh

Low

� Pressure gradient wind blows from high presure towards low pressure.� The earth’s rotation diverts this wind direction laterally. This force is called the CORIOLIS FORCE.� The Coriolis force diverts wind the the right in the northern hemisphere; to the left in the south.� The effect is stronger at high altitude where ground level friction is less significant.

LOW

HIGH

In the north, winds blow clockwise out from a high pressure. (In the south, they blow anti-

clockwise).

GLOBAL PRESSURE & WIND

INTER-TROPICAL CONVERGENCE ZONE

-LOW PRESSURETROPICAL HIGH

PRESSURE

TROPICAL HIGH PRESSURE

POLAR FRONTMID-LATITUDE LOW PRESSURE

POLAR FRONTMID-LATITUDE LOW PRESSURE

POLAR HIGH PRESSURE

POLAR HIGH PRESSURE

GLOBAL WIND BELTS (trade winds) are controlled by the major pressure belts, which relate fundamentally to temperature. Regional wind systems (eg the Indian Monsoon) relate to continental heating

effects, and seasonal changes. Local winds relate to smaller scale temperature contrasts (ie Aspect, Albedo, Altitude etc).

What causes wind?

Differences in pressure and temperature• Solar Power

• Uneven heating of the Earth’s surface

• Differences in heating and cooling rates

• Earth’s rotation

Air rises in warm regions where pressure is low (convection)

eg the tropics

Air descends in cold areas where pressure is high (subsidence)

eg the poles

Air rises over warm areas and sinks over cold areas

Force Strength km/h Effect

0 Calm 0-1 Smoke rises vertically

1 Light air 1-5 Smoke drifts slowly

2 Light breeze 6-11 Wind felt on face; leaves rustle

3 Gentle breeze 12-19 Twigs move; light flag unfurls

4 Moderate breeze 20-29 Dust and paper blown about; small branches move

5 Fresh breeze 30-39 Wavelets on inland water; small trees move

6 Strong breeze 40-50 Large branches sway; umbrellas turn inside out

7 Near gale 51-61 Whole trees sway; difficult to walk against wind

8 Gale 62-74 Twigs break off trees; walking very hard

9 Strong gale 75-87 Chimney pots, roof tiles and branches blown down

10 Storm 88-101 Widespread damage to buildings

11 Violent Storm 102-117 Widespread damage to buildings

12 Hurricane Over 119 Devastation

Beaufort Scale

Average Wind Speed is the Key

o Most important variablePower ~ V3

o Double speed and power increases 8 times

o 73% more power in a 12 mph wind than a 10 mph wind

o This is why it is so important to carefully assess

Measured vs Truewind Wind Speeds (m/s)(.23 m/s average difference)

012345678

How

ards Knob

Hilliards K

nob

Turkey R

idge

Wind Springs B

ald

Cow

ee Bald

Cham

bers Mt.

Phoenix Mt

Truewind

Anemometer

Factors Affecting Wind

o Elevationo Obstructionso Surface Roughnesso Perpendicular

Ridgeso Time of dayo Time of year

Velocity with Height

Season and Time of Day Higher winds found late in the evening and early

morning Higher winds seen during the winter

0.000%

0.100%

0.200%

0.300%

0.400%

0.500%

0.600%

0.700%

0.800%

0.900%

1.000%

0 0 0 0 0 0 0 0 0 0 0 0

Hrl

y G

en a

s %

of

An

nu

al T

ota

l

Step 1: What do you want to do?

Step 2: Resource Assessment

o Average annual wind speed at the proposed hub height is critical

o Wind mapso Existing data

o Extrapolate wind speed from a lower height to hub height

o Measureo Anemometer, wind vane, temp,

logger

o Flagging, interview locals

Rule 1: Minimize Turbulence

Rule 2: Minimum Tower Height

30’ (9m) above obstructions within 300 – 500’

Visualizing the Turbulence

Use a kite or balloon with streamers

Tower Height Depends on Terrain

Historical Trends of Wind Turbine Sizes

Progression of rotor sizes over time

Rotor diameter vs rated power of MW Series

0

20

40

60

80

100

120

140

0 1 2 3 4 5

Rated Power (MW)

Ro

tor

Dia

met

er (

m)

Square root variation

Best-fit variation

Hub height vs rated power of MW Series

0

20

40

60

80

100

120

140

0 1 2 3 4 5

Rated Power (MW)

Hu

b H

eig

ht

(m)

Minimum height for a given turbine

Maximum height for a given turbine

Minimum hub height vs rotor diameter

0

20

40

60

80

100

120

20 40 60 80 100 120

Rotor Diameter (MW)

Hu

b H

eig

ht

(m)

Minimum hub height

Maximum hub height

Hub height = rotor diameter

What All To Measure

39

 

 

Which quantities do we need to measure?• air temperature• wind speed and direction• pressure• humidity• visibility• cloud distribution• cloud type• type and amount of precipitation

How do we want to measure them?• in as many places as possible• as continuously as possible• as reproducible as possible

we need cheap, simple, and automated measurements

40

 

 

Measurements of air temperature I

• liquid filled / metallic thermometers• effect: T-dependence of volume

• use: volume change ΔV = V0(α1- α2) ΔT

Δl = ΔV / A

where A = area of tube

α1 = coefficient of expansion of liquid

α2 = coefficient of expansion of reservoir

• resistance thermometer• effect: T-dependence of electrical resistance of platinum or nickel

(e.g.: Pt100 with 100 Ω at 0 °C )

• use: R = R20 (1 + α · ΔT)

T = 20 °C + (R/R20-1) / α

the temperature coefficient α is constant in first approximation but tabulated for higher accuracy • thermistor thermometer

• effect: (negative) T-dependence of semiconductor resistance

41

 

 

Measurements of air temperature II

• energy budget of thermometer• sensible heat transfer• radiative heat transfer:

• short wave (gain)• long wave (loss or gain, depending on surroundings)

• (latent heat transfer if wet)

=> generally overestimation of T during the day

=> underestimation of T during night underestimation of T if wet The temperature of atmospheric air must be carefully measured in the shade, in a

ventilated enclosure protected from all radiation.

• response time of thermometer• finite time lag between temperature change and change in measured value• depends on thermal mass of thermometer• depends strongly on wind speed

42

Reminder: water vapour in the atmosphere

The amount of water in a given air volume is crucial for its ability to transfer energy.

Common moisture parameters are:

mass mixing ratio:

where mv is the mass of water vapour and md the mass of dry air

saturation vapour pressure: the vapour pressure that is reached in equilibrium above a plane surface of pure water es or over ice esi. Note that es and esi depend only on temperature and that es > esi at all temperatures.

relative humidity:

dew point: Temperature at which water vapour in a given air volume would start to condensate

frost point: Temperature at which water vapour in a given volume would start to freeze

• water saturation pressure is an exponential function of temperature

• small changes in temperature have a large effect on the amount of water that can be present as water vapour

Every day’s examples:• dry air in heated rooms• “fogging” of glasses• white plumes above chimneys

sw

wRH 100

d

v

m

mw

43

Measurements of air humidity I• hair hygrometer

• effect: detection of change of length of a human (or horse) hair in response to relative humidity changes

• hair length changes as in keratin hydrogen bonds are broken in the presence of water vapour

• slow response

• capacity hygrometer• effect: hygroscopic polymer is placed between two electrodes.

In the presence of water vapour, the volume of the polymer increases, decreasing the capacity of the device

• are easily contaminated• absorption hygrometer

• absorption spectroscopy on H2O can also be used to measure water vapour concentration

44

 

 

Measurements of air humidity II

• dew point hygrometer

• effect: detection of dew on temperature controlled mirror by observation of change in reflectance

• very accurate

• psychrometer • effect: T-difference between two ventilated

thermometers, one of which is covered by a wet wick (wet bulb temperature). T-difference is proportional to relative humidity

• use:

e = esat wet – c (Tdry - Twet)

water vapour partial pressure

water vapour saturation pressure at Twet

45

 

Measurements of air pressure

• mercury barometer• effect: air weight is balanced by mercury weight in a tube

which is open on one end• use:

Δp = p2 – p1 = ρgh

• aneroid barometer • effect: sealed metal box with reduced internal air pressure is

contracting and expanding in response to pressure changes

density of mercury

gravitational acceleration

46

Measurements of wind speed and direction(Wind is a vector quantity comprised of direction and speed)

• wind vane• effect: vane aligns in air flow

• windsock • effect: sock aligns in wind flow and changes shape depending on wind speed (qualitatively)

• cup anemometer• effect: pressure differences produce force on cups which rotate proportional to wind speed• problems: only wind speed in one plane, slow response, overshooting

• propeller anemometer• (ultra)sonic anemometer

• effect: measurement of sound velocity • all 3 wind components, fast, no inertia, simultaneous virtual temperature measurement, measure

turbulence and gusts.o No moving partso Measure by time difference between transmission & reception of a sonic pulse

• hot wire anemometer• effect: energy loss of a heated wire• very fast but fragile

47

Cup anemometer measurements of wind speed

• force balance for cup anemometer:

F1=1/2 Cd1ρA(U – Ux)2

F2=1/2 Cd2ρA(U + Ux)2

Cd1ρA(U – Ux)2 = Cd2ρA(U + Ux)2

where

Cd1 and Cd2 are the drag coefficients for the concave and convex side of the cup

A is the area of the cup

U is the wind speed

Ux is the tangential speed of the cups

ρ is the density of air

UCC

CCUU

dd

ddx 3

1

21

21

• 3 cup anemometers have larger torque and react faster to changes in wind speed

• conical cups are better

• rings for turbulence suppression help

=> angular velocity of the cup anemometer is proportional to the wind speed

• Advantages• Low price• Flexible

• Designs have been developed for all climates.• Simple Installation.• Common instrument, most technicians understand operating principles and

necessary connections• Accuracies of 1% can be achieved with calibration of higher quality devices.• They remain accurate when the wind has a significant vertical component, even

up to 30°.

• Disadvantages• Moving parts wear out.

• Cheap versions are not very accurate.

• Electronic output requires a motor generator, or some type of counting circuit.

• This isn’t very expensive anymore.

• Without provisions for heating, they don’t work well in snow or freezing rain.

• They don’t work well in rapidly fluctuating winds.

Advantages & Disadvantages of Cup Anemometers

49

Measurements of precipitation

• rain gauge

• effect: precipitation is collected and the amount measured e.g. by a tipping bucket. Precipitation collector is heated to convert hail and snow to water

• optical rain gauge

• effect: particles passing through a light beam cause scintillations

Problems in measurements of precipitation

• gauge may alter air flow and thus precipitation locally

• wind shields are necessary

• optical measurement relies on assumptions on droplet size

http://www.usatoday.com/weather/wtipgage.htm

50

Measurements of upper air weather

• radio sonde• small instrument package (temperature, pressure, relative humidity) connected to a

balloon filled e.g. with helium. The balloons usually burst at about 30 km. Data is sent to ground via radio transmission

• ozone sonde• radio sonde which also contains an ozone monitor

• rawinsonde• radiosonde that tracks its position in space and time allowing determination of wind

speed and direction

• dropsonde• sonde that doesn’t ascend with a balloon but is falling on a parachute after being

dropped from an airplane

Perspective – Why are We Measuring the Wind Speed?

o In the wind energy field, there are two primary reasons we might want to measure wind speed.

1. To determine feasibility of wind power development at a site.

2. As part of a wind turbine control system.

o Essentially, to answer the question: “Is it worthwhile to turn the turbine into the wind and start it?”

o The accuracy needed for application 1 is much greater than that needed for application 2.

o Knowledge required for the design of wind turbines includes:

o Wind speed and direction

o Inclination to the ground

o Turbulence levels

o Other weather conditions e.g. Temperature, Pressure, Humidity, Precipitation etc

Wind Data Collection

o Data is most often collected using Meteorological Towers (called MET towers) that have:o Anemometers to measure wind speed,

o Cup anemometer rotates in response to pressure differences

o Wind Vanes to measure wind directiono Typically a pointer in front and fins in the backo The vane rotates until forces are balancedo Wind vane points into the wind

Anemometers and wind vanes mounted at 10 feet above the ground

Also they often have o Thermometers to measure air temperatureo Barometers to measure barometric pressure

to refine air density calculations

o MET towers are either tubular steel 6-8 inches in diameter or for taller towers, are a steel lattice structure (similar to radio towers). Guy wires secure these towers at connections from the tower at several heights to anchors in the ground.

o The anemometers and wind vanes are usually positioned at two or three heights on the tower, with two anemometers and one vane at each height. Three six-foot long booms, one for each anemometer and wind vane, are attached to the tower extending out horizontally. Each sensor is mounted at the end of the boom to minimize the effects of the tower’s wind wake. The booms for the anemometers are positioned in opposite directions.

o The sensors produce data that give the wind speed and direction averaged over 10-minute intervals. This data is recorded and stored by a data logging system at the bottom of the tower. The sensors on the tower are connected to the logger by cable running down the tower.

o Wind data is collected at several heights to determine the local wind shear.

Wind Data Collection

Wind Data Collection

o The alternate equipment to a MET tower is the SOnic Detection And Ranging system (SODAR).

o An ultrasonic signal is directed upwards into the air. The signal bounces back and is recorded.

o The signal’s transit time gives the height.

o The signal’s frequency shift (Doppler shift) gives the wind speed at height.

Output from Anemometers

o Signal conditioning is usually done within the instrument.

o The output can be an electrical signal to a datalogger or readout device:o Pulse signalo Voltage signal

o For example, 0-10 V corresponds to the velocity measurement range of the instrument.

o Current signalo Typically, 4-20 mA corresponding to the instrument range.o Eliminates voltage drop when the signal is transmitted over larger

distances.

o In addition, the output can be internally digitized and directly sent to a digital data acquisition system.

Importance of Accurate Wind Speed Measurements

o Remember – the power obtained from the wind goes with the cube of the wind speed.

o A small error in the measurement results in a much larger error in the predicted wind power.

o For example, a 5% error at a wind speed of 10 meters/sec leads to a 16% error in predicted wind power.

o 10% anemometer error leads to 33% errors in power prediction.o This could be disastrous if you are monitoring a site for feasibility of wind

power development!

o This also leads to large errors in efficiency calculations, loads predictions, and so on.

The Wind Rose

o Wind speed frequency diagrams do not provide information on the direction that the wind is blowing.

o This is often done by a graph called a “wind rose.”o A wind rose is a polar plot giving the direction, magnitude, and cube of the magnitude of the

wind.o The data is usually averaged over a year.o In some cases, shorter time periods, such as months, may be appropriate.o The percentage time that the wind blows in a particular direction is plotted as a wedge.

Each wedge is divided into color-coded segments. The width of the segments corresponds to the percentage time over the year that the wind has a velocity in the range defined by the color used for the wedge segment.

o Useso The wind rose gives a graphical interpretation of much of the data that is considered

when evaluating a wind power site.o It shows the wind and most of the wind energy comes from a prevailing direction.

o In this case, the site can be designed with this in mind, and turbine technology with limited ability to rotate into the wind can be considered.

o Link to Wind Rose Applet:o www.windpower.org/en/tour/wres/roseplot.htm

Description of the Wind Rose Plot

o The polar plot is usually broken up into 12 30° sectors, and average data is presented over each sector.

o The first rose “petal” in a given direction is the fraction of the time, normalized to 100%, that the wind is blowing from that direction.

o The second petal is the fraction of time wind blows from a particular direction multiplied by the average wind speed in that direction, and normalized to 100%.

o The third petal is the fraction of time wind blows from a particular direction multiplied by the average cube of the wind speed from that direction, and normalized to 100%.

o The last two petals give information about how “useful” the wind is from a given direction.o For example, if the wind usually blows from a particular direction, but not very

hard.

Wind Rose Examples

DirectionSpeedEnergy

Each wedge shows the percentage of time that the wind is from that direction. (usually along one of 16 compass directions)

The radial length of the wedge indicates the total percentage of time for wind from that direction.

The relative radial lengths of each color indicate the percentage time in a particular wind speed range.

Example:In slightly North of NW direction the wind blew 1.5 % of the time (21.5% - 20%) at speeds of 25-30 knots (red).

A turbine sited at this location would need unobstructed terrain in the prevailing wind direction of North to NW.

Blade Element Theory

Lift and drag forces

2Ta VA

2

1F

Maximum Theoretical Torque

Flow Around Infinite Span Cylinder

Streamline Pattern

Pressure Distribution

Velocity Vectors

Lift Distribution

Flow Around Jowkowski/Ideal Airfoil

Streamline Pattern

Pressure Distribution

Velocity Vectors

Lift Distribution

KE to Usable Energy

21

2 a TT A V R

Torque

212

rT

a T

TC

A V R

The torque developed by the rotor shaft is less than the maximum theoretical torque and given in terms of coefficient of torque as

Loads

High Winds

Stochastic Winds

Vertical Wind Shear and Cross Winds

Inertia and Gravity

Tower Interference

Wind Turbulence and Gusts

66

Source: E Hau

Available Wind Power

The kinetic energy of a stream of air:

2mV2

1E

The kinetic energy of the air stream available for the turbine

2a V

2

1E

V A

= Volume of air parcel

available to the rotor

The air parcel interacting with the rotor per unit time has a cross-sectional area equal to that of the rotor (AT) and thickness equal to the wind velocity (V). Power = Enery/Time

3Ta VA

2

1P

• Major Factors: Air density, area of wind rotor and wind velocity

• The most important factor is Wind Speed (Power varies cubic power of velocity) - As the velocity doubles, the power is increased by 8 times. - The rotor area is reduced by a factor of 8.

• The selection of site is very critical for the success of a wind power

• The power coefficient or the power picked up by the wind turbine rotor is influenced by many factors:

- profile of the rotor blade - number of blades - blade arrangement

Wind Turbine Power and Efficiency

• A wind turbine converts a fraction of the wind energy into mechanical energy

- A part is transferred to the rotor of the wind turbine ( PT ) - Rest is carried away by passing air

• The efficiency is the ratio of actual power developed by wind turbine rotor to the available wind power - defined as power coefficient and expressed as

3Ta

Tp

VA2

1P

C

Rotor Tip Relative Speed

• The rotor power at given wind speed depends on the relative speed between the rotor tip and the wind.

• Higher relative speed between the rotor tip and the wind leads to poor interaction the rotor and the wind.

- For high speed wind approaching a slower moving rotor, a portion of the wind passes the rotor without transferring energy.

- For low speed wind approaching a faster moving rotor, the wind deflects from the rotor and energy is lost due to turbulence and vortex shedding.

Relative speed is defined as velocity of rotor tip and wind speed as

2

rw

R NRV

V V

N = Rotor rotational speed,rpm

= Angular velocity

Also, it can be shown that power coefficient and torque coefficient is related by relative speed:

Pr

T

C R

C V

Solidity and Tip speed ratio

Wind Turbines: Number of Blades

Most common design is the three-bladed turbine. The most important reason is the stability of the turbine. A rotor with an odd number of rotor blades (and at least three blades) can be considered to be similar to a disc when calculating the dynamic properties of the machine.

A rotor with an even number of blades will give stability problems for a machine with a stiff structure. The reason is that at the very moment when the uppermost blade bends backwards, because it gets the maximum power from the wind, the lowermost blade passes into the wind shade in front of the tower.

Blade Element Theory

o We look at a reference blade from a multi-blade system (typically 2 or 3).o Blade, geometry, wind speed, blade RPM, and blade pitch angle are assumed to

be known or chosen.

o We divide the blade into strips, aka blade elements.

o On each strip/elemento We find the local section angle of attack.o We look up the corresponding lift and drag coefficients from a table of airfoil

characteristics.o We correct these for tip losses, root losses, stall delay, swirl losses as needed.o We find lift and drag forces.o We find the propulsive force (in the plane of rotation)o We find the torque contribution of that strip.

o Sum up the toque contribution over all strips to find torque for one blade.

o Multiply by the number of blades, B.

o Vary the wind speed and compute the entire performance map.

Calculation of Angle of Attack

o The figure on the right is from theory

o b is pitch angle (known from from blade geometry)

o U∞ is wind speed

o a is the axial induction factor (axial induced velocity v divided by wind speed), (to be discussed later)

o a’ is a tangential induction factor (tangential induced velocity divided by wr), discussed later.

o lr is the local speed ratio R/ U∞

Lift and Drag are functions of Angle of Attack,

o Once is known, we can look up lift and drag coefficients: Cl and Cd .

o Unfortunately, these quantities influence the axial induction factor a and the tangential induction factor a’.

o An iteration is needed as discussed later.

o If Cl and Cd are known, we can find sectional forces and torques as shown on the right side.

Thrust generated by the Strip of width dr and chord c, for all the blades, B:

Torque generated by the Strip of width dr and chord c, for all the blades, B:

It shows the maximum possible energy — known as the Betz limit — that may be derived by means of an infinitely thin rotor from a fluid flowing at a certain speed peed.

Betz’ law

In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a wind mill) one imagines it to be replaced by a disc that withdraws energy from the fluid passing through it. At a certain distance behind this disc the fluid that has passed through flows with a reduced velocity.

Schematic of fluid flow through a disk-shaped actuator.

Assumptions1. The rotor does not possess a hub, this is an ideal rotor, with an infinite number of blades which have 0 drag. Any resulting drag would only lower this idealized value.2. The flow into and out of the rotor is axial. This is a control volume analysis, and to construct a solution the control volume must contain all flow going in and out, failure to account for that flow would violate the conservation equations.3. This is incompressible flow. The density remains constant, and there is no heat transfer from the rotor to the flow or vice versa.

Applying conservation of mass to this control volume, the mass flow rate (the mass of fluid flowing per unit time) is given by:                                                

(1)

where v1 is the speed in the front of the rotor and v2 is the speed downstream of the rotor, and v is the speed at the fluid power device. ρ is the fluid density, and the area of the

turbine is given by S.

Betz’ law

The force exerted on the wind by the rotor may be written as

21 vvvSF

212 vvvSFvP

22

21

22

21 2

1

2

1vvvSvvmP

(2)

and the power content in the wind is

(3)

However, power can be computed another way, by using the kinetic energy. Applying the conservation of energy equation to the control volume yields

(4)

Both of these expressions for power are completely valid, one was derived by examining the incremental work done and the other by the conservation of energy. Equating these two expressions yields

2122

2212

1vvvSvvvSP 212

1vvv (5)

Betz’ law

3

1

2

2

1

2

1

231

32

2212

21

31

22

2121

22

21

14

1

4

14

1

2

1

v

v

v

v

v

vvS

vvvvvvS

vvvvSvvvSP

310 2

1vSP

(6)

The work rate obtainable from a cylinder of fluid with area S and velocity v1 is:

(7)

hence

3

1

2

2

1

2

1

2

0

12

1

v

v

v

v

v

v

P

P

(8)

0321

2

12

1

2

1

2

12

0

v

v

v

v

vvd

PPd

3

1

1

2

v

v

01231

2

2

1

2

v

v

v

v

59.027

16

0

max P

P

(9)

(10)

Blade Twist

5-station design as seen from the tip

The blade twists to keep effective angle of attack constant

Blade size and shape

Wind Speed Power Density

o Not all wind power can be extracted or the wind would stopo The Betz Limit of 59.3% is the theoretical maximumo Turbines approach 40% from the rotor, but the mechanical and electrical losses may take

20% of the rotor output

http://www.windpower.dk/tour/wres/powdensi.htm

Grey = total power Blue = useable power Red = turbine power output 0 to 25 m/s on abscissa

Wind Turbine Blade Analysis using the BladeElement Momentum Method

Axial Stream tube around a Wind Turbine

Four stations are shown in the diagram:

1, some way upstream of the turbine, 2 just before the blades, 3 just after the blades and 4 some way downstream of the blades.

Between 2 and 3 energy is extracted from the wind and there is a change in pressure as a result.

Assume p1 = p4 and that V2 = V3. We can also assume that between 1 and 2 and between 3

and 4 the flow is frictionless so we can apply Bernoulli’s equation.

22

2224

4

23

3

22

2

21

1

Up

Up

Up

Up

(11)

Assuming also

UUUandppp at 3241

yields

24

2132 2

1UUpp

(12)

(13)

Noting that force is pressure times area we find that:

dAUUdAppdFx24

2132 2

1

1

21

U

UUa

aUU

aUU

21

1

14

12

(12)

Define the axial induction factor as:

It can also be shown that:

(13)

(14)

Substituting yields: drraaUdFx 2142

1 21 (15)

Consider the rotating annular stream tube shown in Figure 2. Four stations are shown in the diagram 1, some way upstream of the turbine, 2 just before the blades, 3 just after the blades and 4 some way downstream of the blades. Between 2 and 3 the rotation of the turbine imparts a rotation onto the blade wake.

2rmI

IL

dt

dLT

Consider the conservation of angular momentum in this annular stream tube.The blade wake rotates with an angular velocity w and the blades rotate with an angular velocity of W. Recall from basic physics that:

Moment of Inertia of an annulus, (16)

Angular moment, (17)

Torque,(18)

22

rdt

dm

dt

rmd

dt

IdT

(19)

Rotating Annular Stream tube: notation. The Blade Element Model

2rmddT

22 2 rUrdAUmd

drrrUdT 222

So for a small element the corresponding torque will be:

(20)

For the rotating annular element

(21)

(22)

Define angular induction factor :

2

a (23)

Recall that aUU 112

drrUaadT 3114

(24)

Blade element theory relies on two key assumptions:•There are no aerodynamic interactions between different blade elements•The forces on the blade elements are solely determined by the lift and dragcoefficients

Consider a blade divided up into N elements. Each of the blade elements will

experience a slightly different flow as they have a different rotational speed (Ωr), a

different chord length (c) and a different twist angle (γ). Blade element theory involves

dividing up the blade into a sufficient number (usually between ten and twenty) of

elements and calculating the flow at each one.

Overall performance characteristics are determined by numerical integration along

the blade span.

Relative flow

Flow onto the turbine blade

arr

r 12

Lift and drag coefficient data area available for a variety of aerofoils from wind tunnel data. Since most

wind tunnel testing is done with the aerofoil stationary we need to relate the flow over the moving

aerofoil to that of the stationary test. To do this we use the relative velocity over the aerofoil. More

details on the aerodynamics of wind turbines and aerofoil selection can be found in Hansen and

Butterfield (1993). In practice the flow is turned slightly as it passes over the aerofoil so in order to

obtain a more accurate estimate of aerofoil performance an average of inlet and exit flow conditions is

used to estimate performance.

The flow around the blades starts at station 2 and ends at station 3. At inlet to the blade the flow is not

rotating, at exit from the blade row the flow rotates at rotational speedω. That is over the blade row

wake rotation has been introduced. The average rotational flow over the blade due to wake rotation is

therefore ω/2. The blade is rotating with speed Ω. The average tangential velocity that the blade

experiences is therefore Ωr+ 1/2ωr.

aU

ar

1

1tan

1

(25)

The value of β will vary from blade element to blade element

The local tip speed ratio is defined as

1U

rr

(26)

Forces on the turbine blade

a

ar

1

1tan

cos

11 aUW

cossin

sincos

dDdLdF

dDdLdF

x

So the expression for tanβ can be further simplified:

(27)

And hence the relative velocity is

(28)

note that by definition the lift and drag forces are perpendicular and parallel to the incoming flow. For each blade element one can see:

(29)

where dL and dD are the lift and drag forces on the blade element respectively. dL and dD can be found from the definition of the lift and drag coefficients as follows:

rcdWcdDrdcWcdL DL22

2

1

2

1 (30)

Lift and Drag Coefficients for a NACA 0012 Aerofoil

This graph shows that for low values of incidence the aerofoil successfully produces a large amount of lift with little drag. At around i = 14º a phenomenon known as stall occurs where there is a massive increase in drag and a sharp reduction in lift.

rcdccWBdF

rcdccWBdF

DL

DLx

sincos2

1

cossin2

1

2

2

rdrcccWBdT DL sincos2

1 2

rdrccaU

dT

rrdccaU

dF

DL

DLx

22

221

2

221

sincoscos

1

cossincos

1

If there are B blades the forces are calculated as

(31)

The Torque on an element, dT is simply the tangential force multiplied by the radius.

(32)

The effect of the drag force is clearly seen in the equations, an increase in thrustforce on the machine and a decrease in torque (and power output)

These equations can be made more useful by noting that b and W can be expressed in terms of induction factors

where σ’ is called the local solidity and is defined as: r

Bc

2

(33)

(34)

Tip Loss Correction

Blade tip vortices remain close to the rotor and to the following blades for several rotor revolutions =>a strongly three-dimensional induced velocity field =>fluctuating air loads on the blade =>affecting the rotor performance & source of vibration and noise

At the tip of the turbine blade losses are introduced in a similar manner to those found in wind tip vorticies on turbine blades. These can be accounted for in BEM theory by means of a correction factor. This correction factor Q varies from 0 to 1 and characterises the reduction in forces along the blade.

Tip Loss Correction

cos/

/12/expcos

2 1

Rr

RrBQ (35)

The results from cos-1 must be in radians.

drrUaaQdT

drraaUQdFx

31

21

14

142

1

rdrccaU

dT

rrdccaU

dF

DL

DLx

22

22

2

22

sincoscos

1

cossincos

1

Tip Loss Correction

The tip loss correction is applied to the axial force and torque as

(36)

Blade Element Momentum EquationsWe now have four equations, two dervied from momentum theory which express the axial thrust and the torque in terms of flow parameters Eq.36

(33)

To calculate rotor performance Equations 36 from a momentum balance are equated with Equations 33. Once this is done the following useful relationships arise:

22 cos4

sincos

1,

cos4

cossin

1 r

DLDL

Q

cc

a

a

Q

cc

a

a

(34)

dTdP

R

r

R

rh h

dTrdPdP

32

2

1UR

dT

P

PC

R

r

windP

h

Power Output

The contribution to the total power from each annulus is:

The total power from the rotor is:

(35)

(36)

Where rh is the hub radius. The power coefficient CP is given by:

(37)

Using Equation 33 it is possible to develop an integral for the power coefficient directly. After some algebra:

rL

DrP d

c

caaQC

h

tan11

8 32 (38)

THANK YOU