Simple Performance
Prediction Methods
Module 2
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
© L. Sankar
Wind Engineering, 2009
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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
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Continuity
Rate at which mass enters=r1A1V1
Rate at which mass leaves=r2A2V2
Area A1
Density r1
Velocity V1
© L. Sankar
Wind Engineering, 2009
Area A2
Density r2
Velocity V2
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Continuity
In compressible flow through a “tube”
rAV= constant
In incompressible flow, r 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 low
and vice versa.
Low Velocity
High Velocity
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Wind Engineering, 2009
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Continuity (Continued..)
High Velocity
AV = constant
If Area between streamlines
is high, the velocity is low
and vice versa.
In regions where the
streamlines squeeze together,
velocity is high.
Low Velocity
© L. Sankar
Wind Engineering, 2009
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Venturi Tube is a Device
for
Measuring Flow Rate
we will study later.
Low velocity
High velocity
© L. Sankar
Wind Engineering, 2009
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Continuity
Mass Flow Rate In = Mass Flow Rate Out
r1 V1 A1 = r2 V2 A2
Station 1
Density r1
Velocity V1
Area A1
Station 2
Density r2
Velocity V2
Area A2
© L. Sankar
Wind Engineering, 2009
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Momentum Equation (Contd..)
Density r
velocity V
Area =A
Density r+dr
velocity V+dV
Area =A+dA
Momentum rate in=
Mass flow rate times velocity
= rV2A
Momentum Rate out=
Mass flow rate times velocity
= r VA (V+dV)
Rate of change of momentum within this element =
Momentum rate out - Momentum rate in
= r VA (V+dV) - rV2A = r VA dV
© L. Sankar
Wind Engineering, 2009
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Momentum Equation (Contd..)
Density r
velocity V
Area =A
Density r+dr
velocity V+dV
Area =A+dA
Rate of change of momentum as fluid particles
flow through this element= r VA dV
By Newton’s law, this momentum change must be caused by
forces acting on this stream tube.
© L. Sankar
Wind Engineering, 2009
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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
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Forces acting on the Stream tube
Horizontal Force = Pressure times area of the ring=(p+dp/2)dA
(p+dp)(A+dA)
Pressure
times
area=pA
Area of this ring = dA
Net force = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp • dA/2-Adp
© L. Sankar
Wind Engineering, 2009
Product of two
small numbers13
Momentum Equation
From the previous slides,
Rate of change of momentum when fluid particles flow
through the stream tube = rAVdV
Forces acting on the stream tube = -Adp
We have neglected all other forces - viscous, gravity, electrical
and magnetic forces.
Equating the two factors, we get:
rVdV+dp=0
This equation is called the Euler’s Equation
© L. Sankar
Wind Engineering, 2009
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Bernoulli’s Equation
Euler equation:
rVdV + dp = 0
For incompressible flows, this equation may be integrated:
r  VdV +  dp  0
Or,
1
rV 2 + p  Const
2
Bernoulli’s
Equation
Kinetic Energy + Pressure Energy = Constant
© L. Sankar
Wind Engineering, 2009
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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
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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.
© L. Sankar
Wind Engineering, 2009
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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.
© L. Sankar
Wind Engineering, 2009
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Control Volume
Station1
V
Disk area is A
Station 2
Station 3
V- v2
V-v3
Stream tube area is A4
Velocity is V-v4
Station 4
Total area S
© L. Sankar
Wind Engineering, 2009
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Conservation of Mass
Inflowthrough he
t top ρVS
Out flow through he
t bottom ρV S - A 4  + ρ(V  v 4 )A4
1
Ouflow through he
t side  m
 Inflowat thetop Out flowat thebottom
 ρv 4 A 4
© L. Sankar
Wind Engineering, 2009
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Conservation of Mass through the
Rotor Disk
V-v2
V-v3
m  rAV  v 2   rAV  v3 
 rA4 V  v 4 
Thus v2=v3=v
There is no velocity jump across the rotor disk
The quantity v is called velocity deficit at the rotor disk
© L. Sankar
Wind Engineering, 2009
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Global Conservation of Momentum
Momentuminflow through op
t  rV 2 S
 1V
Momentumout flow through he
t side  m
 rA 4 v 4V
Momentumout flow throughbot t om
r S - A 4 V 2 + r V  v 4 2 A4
P ressure is at mospheric on all
t hefar field boundaries.
Drag on t herot or, D  Momentumrat ein MomentumRate out
D  rA 4 (V  v 4 ) v 4  m v 4
Mass flow rate through the rotor disk times
velocity loss between stations 1 and 4
© L. Sankar
Wind Engineering, 2009
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Conservation of Momentum at the
Rotor Disk
p2
V-v
Due to conservation of mass across the
Rotor disk, there is no velocity jump.
Momentum inflow rate = Momentum outflow rate
p3
V-v
Thus, drag D = A(p2-p3)
© L. Sankar
Wind Engineering, 2009
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Conservation of Energy
Consider a particle that traverses from station 1 to
station 4
1
2
V-v
3
4
V-v4
We can apply Bernoulli equation between
Stations 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
1
2
p2 + r V  v   p + rV 2
2
2
1
1
2
2
p3 + r V  v   p + r V  v 4 
2
2
v 

p2  p3  r V  4  v 4
2 

© L. Sankar
Wind Engineering, 2009
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From thepreviousslide ,
v4 

p3  p2  r V   v 4
2

v4 

D  A p2  p3   rAV   v 4
2

From an earlier slide, drag equals mass flow rate through the
rotor disk times velocity deficit between stations 1 and 4
D  rAV  vv4
Thus, v = v4/2
© L. Sankar
Wind Engineering, 2009
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Induced Velocities
V
V-v
The velocity deficit in the
Far wake is twice the deficit
Velocity at the rotor disk.
To accommodate this excess
Velocity, the stream tube
has to expand.
V-2v
© L. Sankar
Wind Engineering, 2009
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Power Produced by the Rotor
P  Energyflow in - Energyflow out
1
1
2
m V 2  m V  2v
2
2
 2m vV  v 

V2
 2 rAV  v  v  rA
2
2

V2
2
 rA
41  a  a
2
where, a  v/V
  v 2 v 
41   
  V  V 

T o determinewhen power reachesits maximumvalue,
P
0
a
We get theresult :
set
a  1/3
1
 16 
rAV 3  
2
 27 
T husat best only16/27of theinflowingenergy may be convertedinto power.
P max
T hisis called Betz limit.
© L. Sankar
Wind Engineering, 2009
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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, nonuniform flows, etc.
– We will add these later.
© L. Sankar
Wind Engineering, 2009
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