Transcript Slide 1

Simple Performance
Prediction Methods
Momentum Theory
© L. Sankar
Helicopter Aerodynamics
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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
Helicopter Aerodynamics
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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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Helicopter Aerodynamics
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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
Helicopter Aerodynamics
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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
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Helicopter Aerodynamics
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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
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Helicopter Aerodynamics
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Conservation of Mass through the
Rotor Disk
V-v2
V-v3
m  AV  v 2   AV  v3 
 A4 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
Helicopter Aerodynamics
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Global Conservation of Momentum
Momentuminflow through op
t  V 2 S
 1V
Momentumout flow through he
t side  m
 A 4 v 4V
Momentumout flow throughbot t om
 S - A 4 V 2   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  A 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
Helicopter Aerodynamics
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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
Helicopter Aerodynamics
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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   V  v   p  V 2
2
2
1
1
2
2
p3   V  v   p   V  v 4 
2
2
v 

p2  p3   V  4  v 4
2 

© L. Sankar
Helicopter Aerodynamics
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From thepreviousslide ,
v4 

p3  p2   V   v 4
2

v4 

D  A p2  p3   A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  AV  vv4
Thus, v = v4/2
© L. Sankar
Helicopter Aerodynamics
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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
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Helicopter Aerodynamics
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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 

2
V2   v  v 
 2 AV  v  v  A
41   
2   V  V 
2

V2
2
41  a  a
 A
2
where, a  v/V

T o determinewhen power reachesits maximumvalue,
P
0
set
a
We get theresult : a  1/3
1
 16 
AV 2  
2
 27 
T husat best only16/27of theinflowingenergy may be convertedinto power.
P max
T hisis called Betz limit.
© L. Sankar
Helicopter Aerodynamics
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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.
© L. Sankar
Helicopter Aerodynamics
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