Simple Performance Prediction Methods Module 2 Momentum Theory.

Simple Performance Prediction Methods

Module 2

Momentum Theory

© L. Sankar Wind Engineering, 2009

2

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


3

Continuity• 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).


4

Continuity

Area A1

Density 1

Velocity V1

Area A2

Density 2

Velocity V2

Rate at which mass enters=1A1V1

Rate at which mass leaves=2A2V2


5

Continuity

In compressible flow through a “tube”

AV= constant

In incompressible flow does not change. Thus,

AV = constant


6

Continuity (Continued..)

AV = constant

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

High VelocityLow Velocity


7

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


8

Venturi Tube is a Devicefor

Measuring Flow Ratewe will study later.

Low velocityHigh velocity


9

Continuity

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


10

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


11

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.


12

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)


13

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


14

Momentum EquationFrom 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


15

Bernoulli’s Equation

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


16

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.


17

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.


18

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.


19

Control VolumeV

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


20

Conservation of Mass

44

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


21

Conservation of Mass through the Rotor Disk

44

32

v

vv

VA

VAVAm

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


22

Global Conservation of Momentum

4444

42

42

4

44

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

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


23

Conservation of Momentum at the Rotor Disk

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)


24

Conservation of EnergyConsider 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

44

32

24

23

222

v2

v

v2

1v

2

12

1v

2

1

Vpp

VpVp

VpVp


25

44

32

44

23

v2

v

v2

v

, slide previous theFrom

VAppAD

Vpp

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

4vv VAD

Thus, v = v4/2


26

Induced Velocities

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.


27

Power Produced by the Rotor

limit. Betz called is This

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

27

16

2

1 Pmax

1/3 a :result get the We

0a

Pset

value,maximum its reachespower when determine To

v/Va where,

142

VA

vv14

2

VA vv2

vv2

2vV2

1V

2

1

out flowEnergy -in flowEnergy

3

22

222

22

AV

aa

VVVA

Vm

mm

P


28

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.– We will add these later.

Date post:	22-Dec-2015
Category:	Documents
Upload:	nelson-lambert
View:	218 times
Download:	0 times

Simple Performance Prediction Methods Module 2 Momentum Theory.

Documents