This document outlines the topics and content covered in a unit on 2-D analysis in turbomachinery flow with loss taught from 2005-2010. The unit covered 2-D blade design criteria such as diffusion factor and degree of reaction. It also covered 2-D flow analysis for blades with loss, including isentropic/polytropic loss, loss coefficient, and work done factor. The document provides examples of these concepts and notes they were practiced in class.

1. Aeropropulsion
Unit
2-D Analysis in Turbomachinery Flow with Loss
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
2
A. ATTHASIT
Kasetsart University
Topics
•2D Blade Design Criterions
–Diffusion Factor
–De Haller number
–Degree of Reaction
•2-D Flow Analysis: Blade with Loss
–Isentropic/ Polytropic Loss
–Loss Coefficient
–Work Done Factor
•In-class Practice

3. Aeropropulsion
Unit
3
A. ATTHASIT
Kasetsart University
Cascade field
2-D Flow

4. Aeropropulsion
Unit
4
A. ATTHASIT
Kasetsart University
Cascade field
2-D Flow

5. Aeropropulsion
Unit
5
A. ATTHASIT
Kasetsart University
Blade Loading : diffusion factor
Diffusion factor
High fluid deflection = high rate of diffusion
Definition & termology :

6. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 6
Blade Loading : diffusion factor
Diffusion factor
High velocity gradient ---> high boundary layer thicnkness ---> high losses
w
1 2
max 2
1 1
C s
V V
V V 2 c D
V V

 

 
2 w
1 1
V C s
D 1
V 2V c

  
max 1 w
s
V V 0.5( C )
c
When   
c
solidity
s
 

7. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 7
Diffusion factor
Diffusion factor
2 w
1 1
V C s
D 1
V 2V c

  
Wide range of cascade NACA tests
Criterian's limit :
D < 0.6
Advantage :
'D' help to construct
the velocity diagram

8. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 8
Blade Loading : de Haller n°
Criteria for endwall loading or pressure rise :
De Haller number 2 1 V / V  0.72
De Haller (1953)
But lowing value ---> excessive losses

9. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 9
Many criterias left for prelim-design
Degree of reaction
Degree of reaction (°RcT1 ) :
T2
T3
2 1 2 1
c
3 1 3 1
h h T T
R
h h T T
 
  
 
One stage of compressor
°Rc desirable is 0.5 (share the burden)
Stage loading
t t p t
2 2 2
h h c T
( r ) U U
  


  
0.3   0.35

10. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 10
Many criterias left for preliminary design
Flow Coefficient
a1 a1 C C
r U


 
0.45   0.55
Flow coefficient

11. Aeropropulsion
Unit
11
A. ATTHASIT
Kasetsart University
Velocity Diagrams
Velocity Diagrams

12. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 12
Velocity Diagrams & Euler's Equation
Velocity Diagrams
p t2 t1 2 2 1 1 C (T T ) ( r v r v )
2 r 1 r 
 2 1 2
p t 2 t1 1 2
1
u u
C (T T ) r tan tan
r u
  

 
    
 
 2 1 2
p t 2 t1 2 1
1
u u
C (T T ) r tan tan
r u
  

 
    
 

13. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 13
i i i i i i i
i i i i
i i i i i
m PV cos V cos P P
V cos M
A RT RT RT R T
   
 

   
Velocity Diagrams & Flow Annulus Area
Velocity Diagrams
2 r 1 r 
i
ti
i
ti i M
m T
A
P cos MFP


14. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 14
Flow with loss : Introduction
Flow with loss
Adiabatic Stage Efficiency s 
( 1 ) /
t3s t1 t3s t1 t3 t1
s
t3 t1 t3 t1 t3 t1
h h T T ( P / P ) 1
h h T T T / T 1
 

   
  
  
When t t3 t1 T  T T
/( 1 )
t3 t
s
t1 t1
P T
1
P T
 


  
   
 

15. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 15
Flow with loss : Introduction
Flow with loss
ti ti ti t t t
c
t t t t t t
dh dT dT / T 1 dP / P
e
dh dT dT / T dT / T



   
Adiabatic Polytropic Efficiency c e
t3 t1
c
t3 t1
1 ln( P / P )
e
ln(T / T )



  0.9
(Preliminary design)
When t t3 t1 T  T T
ec /( 1 ) ec /( 1 )
t3 t3 t
t1 t1 t1
P T T
1
P T T
     
    
      
   

16. Aeropropulsion
Unit
16
A. ATTHASIT
Kasetsart University
Flow with loss : Life is still not easy …
Flow with loss
Adiabatic Polytropic Efficiency
Adiabatic Stage Efficiency
When
are
unknown …

17. Aeropropulsion
Unit
17
A. ATTHASIT
Kasetsart University
Flow with loss : Experiment data
Flow with loss
Cascade tests result :
• The optimum angle (minimum loss)
• Profile drag coefficient (cascade efficiency)
(must be increased to account for end losses (e.g., tip leakage, wall boundary layer or cavity leakage)

18. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 18
Flow with loss : Cascade data
Flow with loss
Total pressure loss coefficient
t ,drop ti te
c 2
dynamic i
P P P
P V / 2



 
Remark :
• Rotor - relative reference
• Stator - fixed reference

19. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 19
Flow with loss : Total pressure loss coefficient
Flow with loss
t ,drop ti te
c 2
dynamic i
P P P
P V / 2



  Example for Rotor
t1R t 2R
cr 2
1 1R
P P
V / 2




2 2
t 2R 1 1R 1 1R
cr cr
t1R t1R t1R
P V PM
1 1
P 2P 2P
 
   
2
t 2R 1R
cr /( 1 )
t1R 2
1R
P M / 2
1
P 1
1 M
2
 



  
  
  
 

20. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 20
Flow with loss : Total pressure loss coefficient
Flow with loss
For Rotor
2
t 2R 1R
cr /( 1 )
t1R 2
1R
P M / 2
1
P 1
1 M
2
 



  
  
  
 
For Stator
2
t3 2
cs /( 1 )
t 2 2
2
P M / 2
1
P 1
1 M
2
 



  
  
  
 
How can we evaluate the total
pressure ratio of a stage ?
t3
t1
P
P

cs , 2 2 2 R cr , 1R 1R 1
t3 t 2 2 t 2R t1R 1
t 2 M 2 M t 2R M t1R M 1 M t1 M
P P P P P P
P P P P P P
 
           
           
           

21. Aeropropulsion
Unit
21
A. ATTHASIT
Kasetsart University
Blockage in the Compressor Annulus
Axial velocity distribution
(a)At first stage
(b)At fourth stage
1.The change in axial velocity affects the work-absorbing capacity of the stage.
2.The reduction in work capacity can be accounted for by use of the work-done factor λ which is a number less than unity
Variation of mean work- done factor with number of stages
ptetieeiiC(TT)(rvrv)

22. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 22
In-class Practice: Ch10P01
Mean radius stage calculation
Flow with loss
Stage calculation
t1 t1R t1 t1R T ,T ,P ,P
t2 t2R t2 t2R T ,T ,P ,P
&Velocity diagram
t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  
t T
Prove
• Obj: Able to
use the
fundamental
equation
under the
correct
assumptions
Analysis
• Obj:
Understand
the physical
meaning of
each
parameters
Calculation
• Obj: Able to
solve the
relations
under the
constraints of
corrected unit,
constant, …
etc.

23. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 23
Mean radius stage calculation
Flow with loss
t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  
Step 1: Find Triangle "V" at Station 1
Properties
Geometry
V_Compo
V_Compo
(relative)
Properties
(relative)
1 T 1 a 1 P
1 1 1 V  M a 1 1 1 u V cos 1 1 1 v V sin
1
t1
1
t1 1 M
m T
A
P cos MFP

U r
1R 1 v r v 2 2
1R 1 1R V  u v 1R
1R
1
V
M
a

t1R 1 1R T  f (T ,M , ) t1R 1 1 t1R P  g( P ,T ,T , )

24. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 24
Mean radius stage calculation
Flow with loss
Step 2: Find "P&T" at Station 2
Properties :
t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  
cr , 1R
t 2R
t 2R t1R
t1R M
P
P P
P

 
  
 
t2R t1R T T  290.07K
t2 t1 t T  T  T
t 2 Still don't know P ,let's keep it later!

25. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 25
Mean radius stage calculation
Flow with loss
? Step 3: Euler's Equation :
 2 1 2
p t 2 t1 1 2
1
u u
C (T T ) r tan tan
r u
  

 
    
 
Euler's Equation :
Then 2 is determined
t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  

26. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 26
Mean radius stage calculation
Flow with loss
Step 4: Find Triangle "V" at Station 2
2
2 1 2R 2 2
1
u
u u ,v u tan
u
   2R V
1 2
2 2R 2
2
v
v U v , tan
u
     2 V
Step 5: Find "T&P" at Station 2 t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  
/( 1 )
2
2 2
2 t 2 2 t 2R
P t 2R
V T
T T ,P P
2C T
  
 
    
 

27. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 27
Mean radius stage calculation
Step 6: Find the rest at Station 2
2 2 2 a ,M ,A
Step 7: Find the rest at Station 3
t3 t3R t3 t3R P ,P ,T ,T ?
3 3 P ,T
3 3 3 3 3 a ,V ,u ,v ,A
Flow with loss
t1 t1
m
1 3
2
1 3 t
1
cr cs
T 288.16K,P 101.3kPa
1000rad / s,r 0.3048m
40 , 1,m 22.68kg / s
u
M M 0.7, 1.1, T 22.43K
u
0.09, 0.03
1.4,Cp 1.004kJ / kgK,R 0.287kJ / kgK

  
 

 
 
    
    
 
  

28. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 28
t2R t2R T ,P
cr ,M1R
Euler Eq.
2
Mean radius stage calculation - Summary
Flow with loss
t1 t1 m
1 3
1 3 2 1 t
cr cs
T ,P , ,r
, , ,m
M ,M ,u ,u , T
,

  
 

t1 t1R t1 t1R T ,T ,P ,P
t2 t1 t T  T  T
cs 2  ,M

29. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 29
e d
1 u
g
Design Constrain
(Performance control)
Design Overview
Design constrain
Where is the starting point… and the next step …?
Blade Profile Determination
b
c t1 T
t1 P
 1 3  
m
1 M
cr 
cs 
Fixed Parameters
(input data)
a m r

3 M
2 u
t T
f
Variable Parameters
Obtained Flow Properties

30. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 30
Discusion
Design constrain
Preliminary design parameters ….?
a
b
c
d
t1 T
t1 P

e
m r
1 
3 

m 1 M 3 M
2 u
1 u
t T
cr 
cs 
f
g
Nobody can help me ….?!?!

31. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 31
Engineering Aproach
Repeating-Stage, Repeating-Row, Mean-Line Design
R-S, R-R, M-L
Input data

1 M
g
D
ec
t1 1 T 
Variables
Flow Properties in each station
Repeating-Stage (Exit condition = Inlet condition)
Repeating-Row (Mirror-image of each row)
Mean-Line Design
Assumption :
- 1=2=3,1=2=3
- u1=u2=u3
- Constant mean radius
- Polytropic efficiency representing stage losses
- Two-dimensional flow (extimate annulus area)
c
t1 P
1 t u f T
Controled parameter
Simplified
Your life !

32. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 32
Repeating-Stage, Repeating-Row, Mean-Line Design
R-S, R-R, M-L
Repeating-row constraint : 1=2=3,1=2=3
2R 1 2 v  v r v
or
1 2 v  v r
3 2 3R 2 3 1     v  v ,v  v
And also

33. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 33
Repeating-Stage, Repeating-Row, Mean-Line Design
Diffusion Factor
R-S, R-R, M-L
Diffusion factor : High velocity gradient ---> high boundary layer thicnkness ---> high losses
2R 1R 1R
1R 1R
V v v s
D 1
V 2V c

  
c
solidity
s
 
2R 1R 1R 3 2 3 2 2 1
2
1R 1R 2 2 1
V v v V v v cos tan tan
D 1 1 1 cos ...
V 2V V 2V cos 2
  

   
  
         
2 1   f (D, , )

34. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 34
Repeating-Stage, Repeating-Row, Mean-Line Design
Stage Total Temperature/Pressure Ratio
R-S, R-R, M-L
 
2 2
t3 1 1
s 2 2
t1 1 2
T ( 1)M cos
1 1
T 1 ( 1) / 2 M cos
 

 
  
     
   
c
c
e /( 1 )
t3 t3 e /( 1 )
s s
t1 t1
P T
P T
 
   

  
    
 
s 1 1 2
s s c
f (M , , , )
f ( , ,e )
   
  


From Euler Eq: p t2 t1 p t3 t1 2 1 C (T T ) C (T T ) r( v v )
2 1 r  v v & Diffusion Factor ?
H
O
w

35. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 35
Repeating-Stage, Repeating-Row, Mean-Line Design
Degree of Reaction and Stage Efficiency
R-S, R-R, M-L
( 1 ) /
s
s
s
1
1
  


 


s s s c   f ( , , ,e )
  2 2
2 1 2 1 3 2 2 3 p
c
3 1 3 1 3 1 t3 t1
h h T T T T V V / 2C
R 1 1
h h T T T T T T
   
      
   
 
 
2 2
2 3 p
c 2 2
2 3 p
V V / 2C 1
R 1
V V / C 2

   

Euler Eq :
Stage Efficiency :

36. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 36
Repeating-Stage, Repeating-Row, Mean-Line Design
Stage Exit Mach Number
R-S, R-R, M-L
1
1 2
V
f ( , )
r
 


     
3 3 3 1
2 2
1 1 1 3 s 1 1
M V / RT T 1
1
M V / RT T 1 ( 1) / 2 M ( 1) / 2 M

   
   
   
1 1 1 1 1
1 2 1 1 2 1 1 2
V u / cos u / cos 1
r v v u (tan tan ) cos (tan tan )
 
     
  
  
Inlet velocity/Wheel speed ratio :

37. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 37
Repeating-Stage, Repeating-Row, Mean-Line Design
Stage Loading and Flow Coefficient
R-S, R-R, M-L
p t 2 1
2
2 1
C T tan tan
( r ) tan tan
 

  
 
 

1
1 2
u 1
r tan tan

  
 

Flow Coefficient : Axial velocity/rotor speed
Stage Loading : stage work/rotor speed squared
0.3   0.35
0.45   0.55

38. Aeropropulsion
Unit
38
A. ATTHASIT
Kasetsart University
Repeating-Stage, Repeating-Row, Mean-Line Design General Solution
R-S, R-R, M-L c(D0.5,1,e0.9)

39. Aeropropulsion
Unit
Kasetsart University A. ATTHASIT 39
Repeating-Stage, Repeating-Row, Mean-Line Design
General Solution
R-S, R-R, M-L
c (D  0.5,  1,e  0.9 )

40. Aeropropulsion
Unit
40
A. ATTHASIT
Kasetsart University
Conclusion