Simple Performance Prediction Methods Module 2 Momentum Theory.
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Transcript of 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
© L. Sankar Wind Engineering, 2009
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).
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
5
Continuity
In compressible flow through a “tube”
AV= constant
In incompressible flow does not change. Thus,
AV = constant
© L. Sankar Wind Engineering, 2009
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Continuity (Continued..)
AV = constant
If Area between streamlines is high, the velocity is lowand vice versa.
High VelocityLow Velocity
© L. Sankar Wind Engineering, 2009
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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
© L. Sankar Wind Engineering, 2009
8
Venturi Tube is a Devicefor
Measuring Flow Ratewe will study later.
Low velocityHigh velocity
© L. Sankar Wind Engineering, 2009
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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
© L. Sankar Wind Engineering, 2009
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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
© L. Sankar Wind Engineering, 2009
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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.
© L. Sankar Wind Engineering, 2009
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)
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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.
© L. Sankar Wind Engineering, 2009
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.
© L. Sankar Wind Engineering, 2009
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.
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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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
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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)
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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.
© L. Sankar Wind Engineering, 2009
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
© L. Sankar Wind Engineering, 2009
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.