Axial flow turbine introduction, construction, types design and calculations.

1. Blade Nomenclature

2. Blade Nomenclature

3. Axial and Radial Flow Turbines
Differences between turbine and compressor:
Long Short
Blade 1 Last blade
Compressor Turbine
Work as nozzle
►
Work as diffuser
►
Direction of rotation is same
as
Life
►
Direction of rotation is
opposite
to lift direction
►
Number of stages is small
<3
►
Number of stages are many
►
Temperature is high,
sometimes
blade cooling is required
►
Temperatures are relative
low
►

4. Axial and Radial Flow Turbines
Differences between Radial and Axial Types.
Axial
Radial
(Centrifugal)
Used for large engines
►
Used for small engines
►
Large mass flow rates
►
Small mass flow rates
►
Better efficiencies
►
Lower efficiencies
►
Expensive
►
Cheap
►
Difficult to manufacture
►
Easy to manufacture
►

5. Axial Flow Turbines
Most of the gas turbines employ the axial
flow turbines.
The chapter is concerned with axial flow
turbines.
The radial turbine can handle low mass
flows more efficiently than the axial flow
machines.

6. Axial Flow Turbine
Elementary Theory of Axial Flow Turbine
► Velocity Triangles.
■ The velocity triangles for one axial flow turbine stage and
the nomenclature employed are shown. The gas enters the
row of nozzle blades with a static pressure and temperature
P1, T1, and a velocity C1, is expanded to P2, T2, with an
increased velocity C2 at an angle α2.
■ The rotor blade angle will be chosen to suit the
direction β2 of the gas velocity V2 relative to the blade at
inlet.
■ V2 and β2 are obtained from the velocity diagram of
known C2, α2, and U.

7. Axial Flow Turbine
• Elementary Theory
 The gas leaves the rotor at β3, T3, with relative velocity
V3 at an angle β3.
 C3 and α3 can be obtained from the velocity diagram.

8. Axial Flow Turbine
► Single Stage Turbine
■ C1 is axial → α1 = 0, and C1 = Cα1. For
similar stages (same black shapes) C1 = C3, and
α1 = α3, called repeating stage.
■ Due to change of U with radius, velocity
triangles vary from root to tip of the blade.

9. Axial Flow Turbine
► Assumptions
■ Consider conditions at the mean diameter of the
annulus will represent the average picture of what
happen to total mass flow.
■ This is valid for low ratio of tip radius to root radius.
■ For high radii ratio, 3-D effects have to be
considered.
■ The change of tangential (whirl) mass is . This
amount produces useful torque.
■ The change in axial component produces the axial
thrust on the rotor.
■ Also there is an axial thrust due to P2 – P3.
■ These forces (net thrust on turbine rotor) are
normally balanced by the thrust on the compressor rotor.

10. Axial Flow Turbine

11. Axial Flow Turbine
► Calculation of Work
Assume Ca= constant
2 3
Ca Ca
Ca


2 2
3 3
tan tan
tan tan
U
Ca
 
 

  


  
2 2 3
tan tan tan tan
e
   
   
2 2
tan tan
U Ca Ca
 
 
(1)

12. Axial Flow Turbine
Applying principle of angular momentum
 
2 3
2 3
(
( )(tan tan )
s
W U C C
U Ca
 
 
 
 
From Equation (1)
2 3
(tan tan )
s
W U Ca  
 
Steady-state energy equation: s
p o
W C T
 
Thus:
2 3
(tan tan )/
1.148, 1.333 and 4
1
s
o p
p
T U Ca C
C
 



  
  


13. Axial Flow Turbine
Elementary theory of axial flow turbine
 
 
1 3,
3,
1
1
1
1 3
1
1
1
1
/
s isent
isent
isent
o s o
s o o
o
s o
o
s o
o o
T T
T T
T
T
T
T
P P







  

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

14. Axial Flow Turbine
ηs is the isentropic stage efficiency based on
stagnation (total) temperature.
1 3
1 3
o o
s
o o
T T
T T





1 3
1 3
( )
o o
s
o
T T
total to static
T T





(used for land-based gas turbines).
Defining
ψ = blade loading coefficient (temperature drop
coefficient)
2
2 s
p o
C T
U




15. Axial Flow Turbine
Thus,
2 3
2 (tan tan )/
a
C U
  
 
Degree of reaction: 0 ≤ Λ ≤ 1


2,3 2,3 2 3
1,3 1 3
1,3
rotor
total
h T T T
T T T
h
  
   
 

For, Ca = const. and C3 = C1
 
1 3
1 3
2 3
( )
(tan tan )
p p o o
a
C T T C T T
U C  
  
 
and relative to rotor blades no work, thus
(a)

16. Axial Flow Turbine
 
 
 
2 2
2 3 3 2
2 2 2
3 2
2 2 2
3 2
1
( )
2
1
sec sec
2
1
tan tan
2
p
a
a
C T T V V
C
C
 
 
  
 
 
 
2 2
1
3 2
2
2 3
1 3 2 3
tan tan
(tan tan
a
a
C
T T
T T U C
 
 



 
1
3 2
(tan tan )
2
C
U
 
  
Substitute in (a):


17. Axial Flow Turbine
   
   
3 2
3 2
2 2
2 2 2 2
3 2
2 2
2 2 2 2
3 2
tan tan
a w a w
w w
a a
V V C C u C C u
C u C u
C C
 
   
      
 
   
 
   
 
2 3 3 2
and
   
 
3 2 3 2
and
V V  
 
3 2 3 2
,
C C  
 
a
C
U
 
Λ = 0.5 → Symm. velocity triangles
● Λ = 0 : Impulse turbine
● Λ = 1 :
Defining flow coefficient:

18. Axial Flow Turbine
2 1
3 2
2 (tan tan )
(tan tan )
2
   

 
 
  
Adding:
3
1 1
tan 2
2 2
 

 
  
 
 
2
1 1
tan 2
2 2
 

 
  
 
 
From:
2 2
3 3
3 3
2 2
(tan tan )
(tan tan )
1
tan tan
1
tan tan
a
a
U C
U C
 
 
 

 

 
 
 
 

19. Axial Flow Turbine
If , Λ, and  are assumed, blade angles can be determined.
● For aircraft applications:
3 < ψ < s, 0.8 <  < 1
● For industrial applications:
 is less (more stages)
 is less (larger engine size)
α3 < 20 (to min. losses in nozzle)
● Loss coefficient:

1 2
1
2 2
2
( )
2
2
/ 2
n nozzle stator
p
o o
N
o
T T
C C
P P
Y
P P







Λ and Y: The proportion of the leaving energy
which is degraded by friction.

20. Axial Flow Turbine
Example (Mean diameter design)
Given:
1
1 3
1 3
1
Single-stage turbine
= 20 kg/s
= 0.9
= 1100 K
Temperature drop, = 145 K
Pressure ratio, / = 1.873
Inlet pressure, = 4 bar
t
o
o o
o o
o
m
T
T T
P P
P


Assumptions:
Rotational speed fixed by compressor: N = 250 rps
Mean blade speed: 340 m/s
Nozzle loss coefficient:
2 2
2
2 / 2
N
p
T T
C C





21. Axial Flow Turbine
/
t r
r r
2 3 1 3
1
,
0
a a
C C C C

 

Calculation:
a)Λ degree of reaction at mean radius
b)Plot velocity diagrams
c)Blade height h, tip/root radius,
Assume:
3
2 2
2 2 1.148 145 10
2.88
340
s
p o
C T
U

   
  
flow coefficient 0.8
a
C
U
   
The temperature drop coefficient:
Assume (try):

22. Axial Flow Turbine
3 3
1
tan tan
 

  3
tan 1.25

 
3
1 1
tan 2
2 2
0.28
 

 
  
 
 
  
■ To get Λ use
This is low as a mean radius value because Λ will
be low or negative at the root.
This introduce a value for α3.
Take α3 = 10°
* To calculate degree of reaction Λ:
■ Get β3:
α3 = 0

23. Axial Flow Turbine
3 3 3
3
1
tan tan tan 1.426
1 1
tan 2
2 2
0.421 (Acceptable)
  

 

   
 
  
 
 
  
Reaction at root should be checked.
Thus α3 = 10°, β3 = tan-1 1.426 = 54.96
2
2
1 1
tan 2 0.374
2 2
0.421
2.88
0.8
20.49
 




 
   
 
 
 


 

24. Axial Flow Turbine
2 2
2
1
tan tan 1.624
58.38
 


  
  
3 3 2 2
, , , , U
   
/
t r
r r
1
1 :axial a
C C

With knowledge of
plot velocity diagrams.
* Determine blade height h and tip/root radius ratio,
.
Assumption:
Calculation of area at Section 2 (exit of nozzle)

25. Axial Flow Turbine
2
2
2 1
2
2 2 2
2
2
2 2
340 0.8 272 m/s
cos 519 m/s
1100 K
5.9 K
2
a
a
o o
o
p
C U
C C C
T T
C
T T T
C


   
  
 
   
2
2
2 2
2
0.05 117.3 5.9 K
2
976.8 K
N
p
C
T T
C
T


    

 
1 1 1
2
/ 1
4
2
2
2.49 bar
o
P
o o
o
P T
P
P T
  

 
 
 
 

 

26. Axial Flow Turbine
For the nozzle:
1
1
2
1 2
1 1
1
1
/(2 ) 1 1
1
2 2
1 4
2.16
2 1.853
o p
o
c
c
M
T T C C
M
T T
P
P
P


 
 

  
  

 
   
 
 
P2 > Pc, the nozzle is not choked. 2
, 2.49
throat
Thus P P
 
3
2
2 2
2
2
2 2 2 2
2
2
2 2
2
2 2 2 2 2 2 2
0.833 /
, , m , 0.0833
throat area of nozzles; A
, m 0.0437 , also A cos
a
a
P
kg m
RT
m
A or C A A m
C
m
N
C
or C A N A N m A N
 



 
  
  

   

27. Axial Flow Turbine
Calculate areas at section (1) inlet nozzle and (3) exit rotor.
3
1 1
1
1 1
1
1 1 3 3
3
2
1
1 1
1
1 1
1
3
1
1 1
1
2
1 1 1
, but C C , 276.4 /
cos
1067
2
3.54
1.155 /
0.626
a
a a
o
p
o o
a
C
C C C and C m s
C
T T T K
c
P T
P bar
P T
P
kg m
RT
m C A A m



 


    
   
 
  
 
 
 
  
  

28. Axial Flow Turbine
3 1 5
3
3 3
o
2
3
3 3
1
3 3
3
3
3
3 5
5
Similarly at outlet of stage ( rotor)
T 1100 145 955 ,
922
2
1.856
0.702 /
o o
o
p
o o
T T K given
C
T T T K
c
P T
P bar
P T
P
kg m
RT


 

     
   
 
  
 
 
 
  
3
2
3 5 5
2
3 3 3
3/ 0.702 /
0.1047
Blade height and annulus radius ratio
a
P RT kg m
m C A A m
 

  
  

29. Axial Flow Turbine
Mean radius
m
340
2 0.216
2 (250)
for known (A); A 2 r
m m m
u Nr r m
also h



   

t
r ,
2 2 2
m r m
m
A h h
h then r r r
r

     
using areas at stations 1,2,3 thus
2
1m
A
m
h1
/
t r
r r
3
2
1
Location
0.1047
0.0833
0.0626
0.077
0.0612
0.04
1.43
1.33
1.24

30. Axial Flow Turbine
Blade with width W
Normally taken as W=h/3
Spacing s between axial blades
t
r
a t
space
0.25, should not be less than 0.2 W
width
r
* should be 1.2 1.4
r
unsatisfactory values such as 0.43 can be reduced by
changing axial velocity through .
increasing C reduce r check has to
s
w
will

 

v
be made for mach number M .

31. Axial Flow Turbine
Vortex Theory
The blade speed ( u=r) changes from root to tip, thus
velocity triangles must vary from root to tip.
Free Vortex design
axial velocity is constant over the annulus.
Whirl velocity is inversely proportional to annulus.
,
C
,
tan
tan
C
,
tan
3
3
2
2
const
r
t
cons
C
t
cons
r
t
cons
C
a
a






Along the radius.
 
2 3 2 3
( ) tan
s
W u C C C r C r cons t
   

    

32. Axial Flow Turbine
For variable density, m is given by



t
r
r
r
a
a
rdr
C
m
C
r
r
m
2
2
2
2
2
)
2
(






   
2 2
2
2
a 2
2 2
2
3 3
3
tan tan
C cosntant, thus changes as
tan tan (a)
tan tan (b)
a
m
m
m
m
C r cons t r C
but is
r
r
similarly
r
r
 

 
 
 
 
  
 
 
  
 

33. Axial Flow Turbine
2 2
2
2
s 3
3
2 2 2 2
m
2
2
a 3 3
3 3
3 3
tan tan , , tan tan
r
tan (c)
r
for exit of rotor u C tan tan
tan tan (d)
a a
a
m
m a
a
m
m
m a
u
u C C thus
C
u
r
m
r C
C
r r u
thus
r r C
   

 
 
   
 
 
  
 
   
 
 
 
  
 
   
Ex: Free vortex
Results from mean diameter calculations
2 2m 3
3 2
3
58.38, 20.49, 10 ,
54.96, 0.0612, 0.216,
0.077,
2
o
m m
m m
r m
h r
h
h r r
  

  
  
  

34. Axial Flow Turbine
58.33
8.52
0
54.93
Tip
51.13
12.12
39.32
62.15
Root
54.96
10
20.49
58.38
mean
3
 3

2 3
2
3
2 3
m
a
1.164,( ) 0.877, 1.217, 0.849
u 1
also 1.25, Results are
C
m m m m
t t r t
m
a
r r r r
r r r r
u
C

   
 
   
   
 
 
   
  
2

2


35. Axial Flow Turbine
'
3
o
1
o
3
o
1
o
s
)
1
/(
1
o
3
o
1
o
s
1
o
'
3
o
1
o
s
3
o
1
o
os
1
2
a
1
2
a
3
2
a
3
2
a
3
o
1
o
p
os
p
3
3
2
2
a
T
T
T
T
where
)
)
p
p
(
1
(
T
)
T
T
1
(
T
T
T
T
)
tan
(tan
UC
m
)
tan
(tan
UC
m
)
tan
(tan
UC
m
)
tan
(tan
UC
m
)
T
T
(
c
m
T
c
m
W
tan
tan
tan
tan
C
U

















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


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






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











36. EES Design Calculations of Axial Flow Turbine
Known Information
To1 = 1100 [K]
Pratio = 1.873
DelTs = 145
Ettaturbine = 0.9
Assumptions
U = 340 [m/s]
Nrps = 250
= 0.8
3 = 10
Loss nozzle = 0.05

37. EES Design Calculations of Axial Flow Turbine
cp = 1148 R = 0.287 = 1.333
DelTs = To1 – To3
Pratio =
Po1
Po3
Ca = C2 · cos ( 2 )
=
Ca
U
Gamr =
– 1
Epsi = 2 · cp ·
DelTs
U
2
Epsi = 2 · · ( tan ( 2 ) + tan ( 3 ) )
Reaction =
2
· ( tan ( 3 ) – tan ( 2 ) )
U = Ca · ( tan ( 2 ) – tan ( 2 ) )
U = Ca · ( tan ( 3 ) – tan ( 3 ) )

38. EES Design Calculations of Axial Flow Turbine
Calculate A2
Loss nozzle =
T2 – T2dash
C2
2
2 · cp
To2 = To1
To2 – T2 =
C2
2
2 · cp
Po1
P2
=
To1
T2dash
Gamr
Po1
Pc
=
+ 1
2
Gamr
Pth = P2
Rho2 =
Pth
R · T2
A2 =
m
Rho2 · Ca
A2 · cos ( 2 ) = A2N

39. EES Design Calculations of Axial Flow Turbine
Calculate A1
To1 – T1 =
C1
2
2 · cp
Po1
P1
=
To1
T1
Gamr
Rho1 =
P1
R · T1
C1 = Ca
A1 =
m
Rho1 · Ca
Calculate A3
To3 – T3 =
C3
2
2 · cp
Po3
P3
=
To3
T3
Gamr
Rho3 =
P3
R · T3
C3 = Ca
A3 =
m
Rho3 · Ca

40. EES Design Calculations of Axial Flow Turbine
Blade height
U = 2 · · Nrps · rm
Blade height at section 1
A1 = 2 · · rm · h1
rt1 = rm +
h1
2
rr1 = rm –
h1
2
rratio1 =
rt1
rr1
Blade height at section 2
A2 = 2 · · rm · h2
rt2 = rm +
h2
2
rr2 = rm –
h2
2
rratio2 =
rt2
rr2
Blade height at section 3
A3 = 2 · · rm · h3
rt3 = rm +
h3
2
rr3 = rm –
h3
2
rratio3 =
rt3
rr3

41. EES Design Calculations of Axial Flow Turbine
A1 = 0.06345 A2 = 0.08336 A2N = 0.04372 A3 = 0.1046 2 = 58.37
3 = 10 2 = 20.49 3 = 54.97 C1 = 272 C2 = 518.7
C3 = 272 Ca = 272 cp = 1148 [J/kgK] DelTs = 145 Epsi = 2.88
Ettaturbine = 0.9 = 1.333 Gamr = 4.003 h1 = 0.04666 h2 = 0.06129
h3 = 0.07692 Lossnozzle = 0.05 m = 20 [kg/s] Nrps = 250 [rev per sec] P1 = 355.1
P2 = 248.8 P3 = 186.1 Pc = 215.9 = 0.8 Po1 = 400 [kPa]
Po3 = 213.6 Pth = 248.8 Pratio = 1.873 R = 0.287 [kJ/kgK] Reaction = 0.4211
Rho1 = 1.159 Rho2 = 0.8821 Rho3 = 0.7029 rratio1 = 1.242 rratio2 = 1.33
rratio3 = 1.432 rm = 0.2165 rr1 = 0.1931 rr2 = 0.1858 rr3 = 0.178
rt1 = 0.2398 rt2 = 0.2471 rt3 = 0.2549 T1 = 1068 T2 = 982.8
T2dash = 977 T3 = 922.8 To1 = 1100 [K] To2 = 1100 [K] To3 = 955
U = 340 [m/s]

42. Axial Flow Turbine

43. Axial Flow Turbine