Prerequisite - Thermodynamics/ Propulsion Review of Fundamentals/ Ideal Turbojet Performance/ Component Performance/ Non Ideal Cycle Analysis/ Ideal Turbofan Performance/ Non Ideal Turbofan cycle Analysis/ Matching and Off Design/ Turbomachine 2D/ Turbomachine 3D/ Non Ideal Turbomachine 2D/ Axial Flow Compressors Off Design Performance/ Fuel Injection

1. Aeropropulsion
Unit
Performance of Propellers
2005 - 2010
International School of Engineering, Chulalongkorn University
Regular Program and International Double Degree Program, Kasetsart University
Assist. Prof. Anurak Atthasit, Ph.D.

2. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 2
Momentum Theory for Propellers
1. 1-Dimensional analysis and disk is
essentially a discontinuity moving through
the fluid
2. Infinitesimally thin disk of area A which
offers no resistance to fluid passing through
it as frictional forces are negligible
compared with momentum flux and pressure
changes (hence can make assumption 5)
3. Thrust loading and velocity is uniform over
disk
4. Far-field is at free-stream pressure but far up
and downstream
5. Inviscid (thus irrotational), incompressible
and isentropic flow
Thrust
V0 Ve
streamtube
Actuator disc

3. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 3
Control Volume for Actuator Disc
F
V0
Ve
p+Dp
p
V0
r
R
Disc of area A
Control
volume
Inflow along horizontal boundaries
Conservation of Mass in C.V.
Conservation of Momentum
Conservation of Energy
Thrust from Propeller in
function of inlet and exit
velocity
Propulsive Efficiency

4. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 4
Conservation of Mass in the C.V.
 
 
2 2 2 2
0 0
2
0
ˆ 0
0
S
e
e
V ndS
V R V R r V r Q
Q r V V

      

 
     
 

F
V0
Ve
p+Dp
p
V0
r
R
Disc of area A
Control
volume
Inflow along horizontal boundaries
Volume flow into the C.V. through the
horizontal boundaries of the C.V.
Outflow

5. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 5
Momentum Conservation in the C.V.
 
 
 
 
2 2 2 2 2 2 2
0 0 0
2 2 2 2
0 0 0
2
0
ˆ fluid
S
e fluid
fluid e e disc
disc e e
V n VdS F
V R V Q V R r V r F
F r V V V V V F
F r V V V

      


 
     
      
 

F
V0
Ve
p+Dp
p
V0
r
R
Disc of area A
Control
volume
Inflow along horizontal boundaries
Momentum in to C.V. Momentum out of C.V.
Force on the fluid
Continuity equation
help us to eliminate
inflow along
horizontal
boundaries (Q)

6. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 6
Momentum Conservation in the C.V.
2 2
0 0 1
2 2
1
1 1
2 2
1 1
2 2 e e
p V p V
p V p p V
 
 
  
   D 
Bernoulli’s Eq.- up and downstream
Up Down
Pressure Jump
     2 2
0 0 0
1 1
2 2 e e e Dp   V V   V V V V
Continuity requires that  r2Ve=AV1
disc
V1
p p+Dp
  1 0
1
2 e V  V V

7. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 7
Momentum Conservation in the C.V.
 
 
 
 
2 2 2 2 2 2 2
0 0 0
2 2 2 2
0 0 0
2
0
ˆ fluid
S
e fluid
fluid e e disc
disc e e
V n VdS F
V R V Q V R r V r F
F r V V V V V F
F r V V V

      


 
     
      
 

F
V0
Ve
p+Dp
p
V0
r
R
Disc of area A
Control
volume
Inflow along horizontal boundaries
Momentum in to C.V. Momentum out of C.V.
Force on the fluid
  2
disc e e 0 F   r V V V
1. This thrust is an ideal number that does not account for many losses that
occur in practical, high speed propellers, like tip losses.
2. The losses must be determined by a more detailed propeller theory, which
is beyond the scope of this class.
3. The complex theory also provides the magnitude of the pressure jump for a
given geometry.
4. The simple momentum theory, however, provides a good first cut at the
answer and could be used for a preliminary design.

8. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 8
Velocity induced by actuator disc
   
 
2
0 1 1 0
0
2
2
e e F r V V V AV V V
F AV V V
 

     
   
V’~V1-V0 is called the induced velocity and
the thrust is
Thrust = (mass through disc)
(overall change in velocity)

9. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 9
Conservation of energy in the C.V.
F
V0
Ve
p+Dp
p
V0
r
R
Disc of area A
Control
volume
Inflow along horizontal boundaries
 
2 2
2 3 3 3 2
0 2 0 2 2 0
1
ˆ
2
1
1
2
fluid
S
e fluid
V n V VdS F V P
r r Q
R V V V V P
R R R



    
   
        
   

The power absorbed by the fluid is
      2 2
1 0 1 0 0
1 1
2 2 fluid e e e
fluid avg
P AV V V AV V V V V
P FV
 
 
        
 

Outflow
Thrust

10. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 10
Ideal Propulsive Efficiency
In terms of the induced velocity the power absorbed is
1 0
0
1
V
P FV FV
V
  
    
 
power required to keep V=V0
The ideal propulsive efficiency is then
0
1
1
i
i
P
P V
V
  


0 0.1 0.2 0.3 0.4 V’/V0
1.0
0
i
0.8
0.2
0.4
V1=Vavg=V0+V’
Operating range

11. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 11
Thrust and Power from
Momentum Theory: Remark
   
 
2
0 1 1 0
0
1 0
0
2
2
1
e e F r V V V AV V V
F AV V V
V
P FV FV
V
 

     
   
  
    
 
By using only the momentum theory, it is
difficult to evaluate a field of induced
velocity around the rotor or propeller.
Momentum Theory is a global analysis
which gives useful results but can not be
used as a stand-alone tool to design the
rotor.

12. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 12
Thrust Coefficient
  2
0 0
0 0
1
2 4 1
2
V V
F AV V V V A
V V
 
    
        
  
2 0 0
0
4 1
1
2
T
F
A V V C
V V
V
   
    
 
 
0
1
1 1
2 T
V
C
V

   
   
 
2
0 1 1 0
0
2
2
e e F r V V V AV V V
F AV V V
 

     
   
V’=V1-V0 is called the induced velocity and
the thrust is
The thrust coefficient, which is dimensionless,
is defined as
‘F/A’ =Disc
Loading

13. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 13
Power coefficient
  ,
2 0
0 0
1
1 1 1
1 2
2
P i T T T
P V
C C C C
V
V A V
  
       
   
 
 
 
1 0
0
2
0 0
0 0
1
1
2 4 1
2
V
P FV FV
V
V V
F AV V V V A
V V
 
  
    
 
    
        
  
Where
And
For a statically thrusting propeller
V0=0 and the non-dimensional
coefficients don’t apply. Instead,
and Vs’=V1 3
1
2
2
s
s
s
s s s s
F
V
A
F
P FV FV
A


 
   

14. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 14
Thrust Variation with Flight Speed
Equating the power to the propeller for
the moving and static cases yields:
3
0
1 2
1
2 2
s F F
FV
 A  A
 
     
 
This equation may be put in the
following form:
2
0 0
2
1
1 4
2 s s s s
F V V F
F V V F
 
    
     
Power coefficient
  ,
2 0
0 0
1
1 1 1
1 2
2
P i T T T
P V
C C C C
V
V A V
  
       
   
 
 
 
1 0
0
2
0 0
0 0
1
1
2 4 1
2
V
P FV FV
V
V V
F AV V V V A
V V
 
  
    
 
    
        
  
Where
And
For a statically thrusting propeller
V0=0 and the non-dimensional
coefficients don’t apply. Instead,
3
2
2
s
s
s
s
F
V
A
F
P
A


 

1.0
0
F/Fstatic
0 V 1 0/V’static
V’static=(Fstatic/2A)1/2
thrust drops as
speed increases

15. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 15
Force and Velocity on Blade Element
dL
dD wr
V0
V V’ e
VR
ai
f
f ai
f
Axis of rotation
Chord line
cos  sin  i i dF  dL f a  dD f a
dF
g Induced angle
The induced angle
ai
depends on the
induced velocity
V’

16. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 16
The Blade Element for a Propeller
dr
r
Blade element
Axis of rotation
C(r)
      2 1
cos sin
2 L e i i dF  Bc cdr V  f a  f a 
dL
dD wr
V0
V V’ e
VR
ai
f
f ai
f
Axis of rotation
Chord line
cos  sin  i i dF  dL f a  dD f a
dF
g Induced angle
‘B’ Number of Blades
Drag/Lift Coef. Ratio

17. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 17
Thrust and Power for a Propeller
dr
r
Blade element
Axis of rotation
C(r)
      2 1
cos sin
2 L e i i dF  Bc cdr V  f a  f a 
dL
dD wr
V0
V V’ e
VR
ai
f
f ai
f
Axis of rotation
Chord line
cos  sin  i i dF  dL f a  dD f a
dF
g Induced angle
‘B’ Number of Blades
Drag/Lift Coef. Ratio
   
   
2
0
2
0
1
cos 1 tan
2
1
sin 1 cot
2
R
L e i i
R
L e i i
F B cc V dr
P B cc rV dr
 f a  f a
 w f a  f a
     
     


The induced angle ai
depends
on the induced velocity V’

18. Aeropropulsion
Unit
18
A. ATTHASIT
Kasetsart University
Blade Element Theory and Momentum Theory: Remarks
1.When the two theories are combined, it is possible to evaluate a field of induced velocity around the rotor or propeller, and therefore correct the inflow conditions assumed in the basic blade element theory.
2.The induced velocities aren't known until the blade loads are computed. With the loading available one can re-compute the field of induced velocities.
3.This is an iterative method, generally the quantity that is iterated for is the thrust coefficient. The combined Blade Element Momentum Theory is a fairly accurate analytical tool (for lightly loaded rotors or propellers) that can be used by the engineer early in the design of a rotor.

19. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 19
Propeller Characteristics and Non
Dimension Parameter: AF
Blade activity factor (AF)
is a measure of solidity
and therefore power
absorption capability
r
R
c(r)
5 1 3
0
10
16
c r r
AF d
D R R
   
    
    
  3 1.28 10
Bc B AF
R

 
   
Solidity for constant chord blade
Typical range: 100<AF<150

20. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 20
Advance Distance for One
Revolution
2r
2 r tang
g
In-plane distance moved in 1 revolution
Advance Distance

21. Aeropropulsion
Unit
21
A. ATTHASIT
Kasetsart University
Non Dimension Analysis: Advance Ratio
wr
g
Axis of rotation
Chord line
g = blade pitch angle
Advance ratio J= V0 /wD
Advance during 1 revolution 2r
2 r tang
g
In-plane distance moved in 1 revolution

22. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 22
Non Dimension Analysis: Thrust
and Torque
2 4 0
2 2 2
2 4
2 4
( ) ; ;
( ) Re; ;
d e f
tip
t
K V
F Const n D func
D n D n Dn
F Const n D func M J
F K n D





      
         
      
     
  
K= Fluid bulk elasticity modulus (N/m2)
K/ρ = a2 where a is the
speed of sound, this is like
Where KT is called the thrust
coefficient and in general is a
function of propeller design, Re,
Mtip and J.
Since torque is a force multiplied
by a length, it follows that
2 5
torque Torque  K  n D
wr
g
Axis of rotation
Chord line

23. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 23
Propulsive Efficiency
The power supplied to the propeller is Pin where
2 5 2 2 in torque P   nTorque   nK  n D
The useful power output is Pout where
2 4
out 0 0 t P V F V K  n D
Therefore the efficiency is given by
2 4
0 0 0
2 5 2 2 2
out t t t
i
in torque torque torque
P V K n D V K K V
P nK n D nK D K nD


   
  
     
  
0 Advance ratio J

Fine pitch
Coarse pitch
Which is often written using a power coefficient
2 t
Pow torque i
Pow
K
K K J
K
   
Where KX is a function
of propeller design,
Re, Mtip and J.

24. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 24
Pitch Operation
wr
g
Axis of rotation
Chord line
VL 0
D
R F
V0 High J: cruise
Low J
g wr
Axis of rotation
Chord line
V0 F
R
L
D
V0
Low J: T-O
High J
Fine Pitch Operation
Coarse Pitch Operation

25. Aeropropulsion
Unit
25
A. ATTHASIT
Kasetsart University
Propeller Efficiency Curves
Typical propeller efficiency curves as a function of advance ratio
Many propellers contain a mechanism in the hub to change the overall pitch of the blades in response to a servo command from a control system for that the prop efficiency could be very high for a wide range of operating conditions
g is varied to keep w constant at best engine speed
Fine pitch
Coarse pitch

26. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 26
Conclusion
*
2
1
*
2
1
1
*
*
2
*
1
2( 1)
2
*
1
2
1
1
2
1
2
1
1
2
1
2
1
1
2
1
1
1 2
1
2
T
T
M
P
P
M
P
P
T
M
T
P
m AV AM
R T
M
A
A M
g
g
g
g
g
g
g
g
g
g

 g
g

g
g






  
 
 
  
 
  
           
  
 
   
           
 
    
    
    
  
 
 
2
0
0 t
dA d du
A u
udu dP
dh dh udu
dP d dT
P T
a
P






g
  
 
  
 

P dP
T dT
d
A dA
u du
 





P
T
A
u

dx
2
dP
P 
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