wecs

PS 9155 WIND ENERGY CONVERSION SYSTEMS

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Components of WECS-WECS schemes-Power obtained from wind-simple momentum theory-Power coefficient-Sabinin‟s theory-Aerodynamics of Wind turbine

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1. simple momentum theory2. Aerodynamics of Wind turbine3. Bernoulli Equation4. Betz Law5. Power coefficient6. Sabinin’s theory

Padmanathan K Research Scholar

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Aerodynamics-Basics• Newton’s Laws of motion:

A. Law 1 – A body at rest will remain at rest. A body in motion will remain in motion

B. Law 2 – F=MA Force is equal to mass times acceleration

C. Law 3 – For ever action there is an equal and opposite reaction

Bernoulli’s principle of Pressure:An increase in the speed of movement or flow will cause a decrease in the fluid’s pressure.

- Example: the Venturi tube

Low Pressure


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Wind turbine “farms” are being constructed all over the world to extract kinetic energy from the wind and convert it to electrical energy. The mass, energy, momentum, and angular momentum balances are utilized in the design of a wind turbine. The Bernoulli equation is also useful in the preliminary design stage.

MASS, BERNOULLI AND ENERGY EQUATIONS

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Objectives• Apply the conservation of mass equation to

balance the incoming and outgoing flow rates in a flow system.

• Recognize various forms of mechanical energy, and work with energy conversion efficiencies.

• Understand the use and limitations of the Bernoulli equation, and apply it to solve a variety of fluid flow problems.

• Work with the energy equation expressed in terms of heads, and use it to determine turbine power output and pumping power requirements.

Momentum Theory -Overview

• In this module, we will study the simplest representation of the wind turbine as a disk across which mass is conserved, momentum and energy are lost.

• Towards this study, we will first develop some basic 1-D equations of motion.– Streamlines– Conservation of mass– Conservation of momentum– Conservation of energy


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Motion of fluid energykinetic

Pressure in fluid energypotential

KE + PE is constant

Bernoulli’s Equation: derivationConsider a volume V of mass M of incompressible fluid,

KE 1

2Mv2

2 1

2Mv1

2

1

2Vv2

2 1

2Vv1

2

PE Mgy2 Mgy1Vgy2 Vgy1

W F1x1 F2x2

P1A1x1 P2A2x2

P1V P2V

P1 gh1 1

2v1

2 P2 gh2 1

2v2

2


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• Consider a stream tube, i.e. a collection of streamlines that form a tube-like shape.

• Within this tube mass can not be created or destroyed.

• The mass that enters the stream tube from the left (e.g. at the rate of 1 kg/sec) must leave on the right at the same rate (1 kg/sec).


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Area A1

Density 1

Velocity V1

Area A2

Density 2

Velocity V2

Rate at which mass enters=1A1V1

Rate at which mass leaves=2A2V2


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Continuity


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In compressible flow through a “tube”

AV= constant

In incompressible flow does not change. Thus,

AV = constant


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AV = constant

If Area between streamlines is high, the velocity is lowand vice versa.

High VelocityLow Velocity


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Continuity (Continued..)

AV = constant

If Area between streamlines is high, the velocity is lowand vice versa.

In regions where the streamlines squeeze together,velocity is high.

High Velocity

Low Velocity


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Venturi Tube is a Device for Measuring Flow Rate


21Low velocity

High velocity

Bernoulli’s Principle - Lift

“As the velocity of a fluid increases, its internal pressure decreases.” From Newton’s 2nd (F=ma) Shown by Venturi tube

Low Pressure

High Pressure

A1V1=A2V2

Bernoulli’s PrincipleBernoulli’s Principle• Air is a gas and a fluidAir is a gas and a fluid• Air pressure is due to the motion of its

particles• Pressure in a moving stream exerts less

pressure than the air surrounding the moving stream

Quick stream = low air pressureQuick stream = low air pressure

Slow stream = High air pressureSlow stream = High air pressure


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Station 1Density 1

Velocity V1

Area A1

Station 2Density 2

Velocity V2

Area A2

Mass Flow Rate In = Mass Flow Rate Out1 V1 A1 = 2 V2 A2


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Momentum Equation (Contd..)

Density velocity VArea =A

Density dvelocity V+dVArea =A+dA

Momentum rate in=Mass flow rate times velocity= V2A

Momentum Rate out=Mass flow rate times velocity= VA (V+dV)

Rate of change of momentum within this element = Momentum rate out - Momentum rate in

= VA (V+dV) - V2A = VA dV


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Momentum Equation (Contd..)

Density velocity VArea =A

Density dvelocity V+dVArea =A+dA

Rate of change of momentum as fluid particlesflow through this element= VA dV

By Newton’s law, this momentum change must be caused byforces acting on this stream tube.

Forces acting on the Control Volume

• Surface Forces– Pressure forces which act normal to the surface– Viscous forces which may act normal and tangential to

control volume surfaces

• Body forces– These affect every particle within the control volume.– E.g. gravity, electrical and magnetic forces– Body forces are neglected in our work, but these may be

significant in hydraulic applications (e.g. water turbines)


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Forces acting on the Stream tube

Pressuretimesarea=pA

(p+dp)(A+dA)

Horizontal Force = Pressure times area of the ring=(p+dp/2)dA

Area of this ring = dA

Net force = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp • dA/2-Adp

Product of two small numbers

Momentum Equation


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From the previous slides,

Rate of change of momentum when fluid particles flowthrough the stream tube = AVdV

Forces acting on the stream tube = -Adp

We have neglected all other forces - viscous, gravity, electricaland magnetic forces.

Equating the two factors, we get: VdV+dp=0

This equation is called the Euler’s Equation

Bernoulli’s Equation


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Euler equation: VdV + dp = 0

For incompressible flows, this equation may be integrated:

ConstpV

Or

dpVdV

2

2

1

,

0

Kinetic Energy + Pressure Energy = Constant

Bernoulli’sEquation

Actuator Disk Theory: Background

• Developed for marine propellers by Rankine (1865), Froude (1885).

• Used in propellers by Betz (1920)• This theory can give a first order estimate of HAWT

performance, and the maximum power that can be extracted from a given wind turbine at a given wind speed.

• This theory may also be used with minor changes for helicopter rotors, propellers, etc.


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Assumptions

• Momentum theory concerns itself with the global balance of mass, momentum, and energy.

• It does not concern itself with details of the flow around the blades.

• It gives a good representation of what is happening far away from the rotor.

• This theory makes a number of simplifying assumptions.


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Assumptions (Continued)

• Rotor is modeled as an actuator disk which adds momentum and energy to the flow.

• Flow is incompressible.• Flow is steady, inviscid, irrotational.• Flow is one-dimensional, and uniform through

the rotor disk, and in the far wake.• There is no swirl in the wake.


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Control Volume


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V

Disk area is A

Total area S

Station1

Station 2

Station 3

Station 4

V- v2

V-v3

Stream tube area is A4

Velocity is V-v4

Conservation of Mass

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1

444

Aρv

bottom at the Outflow topat the Inflow

m side he through tOuflow

)Avρ(VA-SρV bottom he through tOutflow

ρVS tophe through tInflow


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Conservation of Mass through the Rotor Disk

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v

vv

VA

VAVAm


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Thus v2=v3=v

There is no velocity jump across the rotor disk

The quantity v is called velocity deficit at the rotor disk

V-v2

V-v3

Global Conservation of Momentum

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4

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1

2

vv)v(A D

out Rate Momentum

-in rate MomentumD,rotor on the Drag

.boundaries fieldfar the

allon catmospheri is Pressure

vA-S

bottom through outflow Momentum

vA

Vm side he through toutflow Momentum

V op through tinflow Momentum

mV

AVV

V

S


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Mass flow rate through the rotor disk times velocity loss between stations 1 and 4

Conservation of Momentum at the Rotor Disk


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V-v

V-v

p2

p3

Due to conservation of mass across theRotor disk, there is no velocity jump.

Momentum inflow rate = Momentum outflow rate

Thus, drag D = A(p2-p3)

Conservation of Energy

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23

222

v2

v

v2

1v

2

12

1v

2

1

Vpp

VpVp

VpVp


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Consider a particle that traverses from station 1 to station 4

We can apply Bernoulli equation betweenStations 1 and 2, and between stations 3 and 4. Not between 2 and 3, since energy is being removed by body forces.Recall assumptions that the flow is steady, irrotational, inviscid.

1

2

3

4

V-v

V-v4

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44

23

v2

v

v2

v

, slide previous theFrom

VAppAD

Vpp

4vv VAD


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From an earlier slide, drag equals mass flow rate through the rotor disk times velocity deficit between stations 1 and 4

Thus, v = v4/2

Induced Velocities


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V

V-v

V-2v

The velocity deficit in theFar wake is twice the deficitVelocity at the rotor disk.

To accommodate this excessVelocity, the stream tube has to expand.

Power Produced by the Rotor

limit. Betz called is This

power. into converted bemay energy inflowing theof 16/27only best at Thus

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16

2

1 Pmax

1/3 a :result get the We

0a

Pset

value,maximum its reachespower when determine To

v/Va where,

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VA

vv14

2

VA vv2

vv2

2vV2

1V

2

1

out flowEnergy -in flowEnergy

3

22

222

22

AV

aa

VVVA

Vm

mm

P


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Summary• According to momentum theory, the velocity

deficit in the far wake is twice the velocity deficit at the rotor disk.

• Momentum theory gives an expression for velocity deficit at the rotor disk.

• It also gives an expression for maximum power produced by a rotor of specified dimensions.

• Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc.


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Bernoulli’s Principle Again

Courtesy of FAA: Pilot’s Handbook of Aeronautical Knowledge, AC 61-23B

Winds over a mountain

Below is a NASA illustration of whatBelow is a NASA illustration of whatsurface pressure is exerted on what surface pressure is exerted on what surface areas of the wing. surface areas of the wing.

Bernoulli’s Principle Again


Lift Vector


Blades• Aerofoil design

•Air flow over the blade develops lift force and causes the blade to rotate

Drag• Induced drag is a by-product of

lift and is greatly affected by changes of airspeed.

Drag Types Induced drag is the unavoidable by-product of lift and

increases as the angle of attack increasesParasite drag is caused by any aircraft surface that

deflects or interferes with smooth airflow around airplaneSkin-friction drag - between the outer surfaces of

the aircraft and the air through which it moves. Reduced by using glossy, flat finishes on surfaces

Form drag - resistance of air to the shape of the aircraft. Form drag can be reduced by streamlining the aircraft shape.

Aerodynamics - Stalls • When does an airplane stall?

– When it exceeds the critical angle of attack.

Chord line=the line from the leading edge of the wing to the trailing edge

Relative wind=perpendicular to lift, relative to the airfoil

What is angle of attack?Angle of attack is the angle between the chord line and the relative wind

Basic airfoil terminology

Camber = distance between mean camber line (mid-point of airfoil) and the chord line (straight line from leading edge to trailing edge)Thickness = distance between upper and lower surfaces (measured perpendicular to chord line)Span = length of airfoil normal to the cross-section

Camber

Thickness

Torque / P-factor (Left-Turning Tendencies)

• Newton’s 3rd law: “For every action there is an equal and opposite reaction.”– Propeller rotates CW when

viewed from pilot’s seat.– Torque reaction rotates the

airplane CCW about longitudinal axis

• P-factor (asymmetrical thrust) caused by descending blade taking a greater “bite” of air than ascending blade at high angle of attack

Four Aerodynamic Forces1.Lift 2.Thrust 3.Drag 4.Weight

•The engines provide The engines provide THRUST.THRUST.•The wings provide The wings provide LIFT.LIFT.•Gravity provides the Gravity provides the ‘G force. (weight)‘G force. (weight)•And, fluid friction provides the And, fluid friction provides the DRAG.DRAG.

LIFTLIFT

GRAVITYGRAVITY

THRUSTTHRUSTDRAGDRAG


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Lift briefly exceeds weight.Rearward component of the Lift adds to dragUpward component of Thrust is called the Lift of Thrust

Second law of motion says that a force results whenever a mass is accelerated F = maThird law states for every action there is an equal and opposite reaction

Four Forces of Flight Lift opposes Weight Thrust opposes Drag In straight, unaccelerated flight, L = W & T = D

Lift created by pressure differential around wing. High pressure on lower surface and low pressure on the upper surface – low pressure caused by increased airflow velocity over top of airfoil.

Weight – downward force of gravity Drag – rearward retarding force Thrust – forward force propelling airplane through air

Airfoils

• What is NACA?– National Advisory

Committee for Aeronautics

– Chartered in 1915, operational from 1917-1958

– The National Aeronautics and Space Act of 1958 created NASA from NACA

Airfoils - Nomenclature

• Chord line - straight line connecting the leading and trailing edges of an airfoil• Camber line – locus of all points equidistant from top and bottom of airfoil• Camber – distance between chord line and camber line • Thickness – maximum distance between top and bottom surfaces of wing• Leading Edge• Trailing Edge• Wingspan (b)• Aspect Ratio (AR = b2/S)

Low p

High p

Angle of Attack

• Angle between wing chord line and relative wind• The angle of attack at which airplane stalls does

not change

Aerodynamics-Basics

• Because air is a fluid, it utilizes the properties of the Coanda effect: the tendency for a fluid to follow the object along its flow path.

• http://www.youtube.com/watch?v=AvLwqRCbGKY

• http://www.youtube.com/watch?v=S-SAQtODAQw

http://www.youtube.com/watch?v=AvLwqRCbGKY

http://www.youtube.com/watch?v=AvLwqRCbGKY

http://www.youtube.com/watch?v=S-SAQtODAQw

http://www.youtube.com/watch?v=S-SAQtODAQw

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The power in the wind than can be extracted by a wind turbine is proportional to the cube of the wind speed and is given in watts by:

where represents the aerodynamic efficiency of the rotor.ρ -air density, A- rotor swept area, U-wind speed C p power coefficient

C p - Power Coefficient

C p

POWER OUTPUT FROM THE WIND MACHINE

POWER CONTAINED IN WIND

= ________________________________________

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Sabinin’s theoryThe main merit of prof. G.H.Sabinin in wind power for ever remain the presence proved to him so-called « the affixed weight » as a result of which the greatest possible part of energy which can be taken from an ideal rotor makes 68,6 % (instead of 59,3 % on A.Betz). As appeared, almost all world does not know about it and counts aerodynamics of rotors using the formulas A.Betz. A limit of 59,3 % name a limit and even law A.Betz

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