Download Velocity Triangles for Turbo

Velocity Triangle for Turbo-machinery
BY
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
U
Vai
Vri
Vai
Vri
Inlet Velocity Triangle
U
U
Vae
Vre
Vre
Exit Velocity Triangle
ae
Vae
ai
U
bi
be
Vai
Vre
Vai: Inlet Absolute Velocity
Vri: Inlet Relative Velocity
Vre: Exit Relative Velocity
Vae:Exit Absolute Velocity
ai: Inlet Nozzle Angle.
bi: Inlet Blade Angle.
be: Exit Blade Angle.
ai: Exit Nozzle Angle.
Vri
Vf
Vrc
Vr
Vw
Va
U
Flow through Blades
Vre
U
U
Vae
Vni
Vri
U
Vai
Fluid Dynamics of Blades
ae
Vae
•
•
•
•
•
•
•
•
•
ai
U
bi
be
Vai
Vre
Vri
The stream is delivered to the wheel at an angle ai and velocity Vai.
The selection of angle ai is a compromise.
An increase in ai, reduces the value of useful component (Absolute
circumferential Component).
This is also called Inlet Whirl Velocity, Vwi = Vai cos(ai).
An increase in ai, increases the value of axial component, also called as flow
component.
This is responsible for definite mass flow rate between to successive blade.
Flow component Vfi = Vai sin(ai) = Vri sin(bi).
The absolute inlet velocity can be considered as a resultant of blade velocity
and inlet relative velocity.
The two points of interest are those at the inlet and exit of the blade.
ae
Vae
ai
U
bi
be
Vai
Vre
Vri
• If the stream is to enter and leave the blades without shock or much
losses, then relative velocity should be tangential to the blade inlet tip.
• Vri should enter at an angle bi, the inlet blade angle.
• Similarly, Vre should leave at be, the exit blade angle.
• A blade is said to be symmetric if bi = be.
• The flow velocities between two successive blade at inlet and exit are
Vfi & Vfe.
• The axial (basic useful) components or whirl velocities at inlet and exit
are Vwi & Vwe.
Impulse Turbine
ae
Vae
ai
U
bi
be
Vai
Vri
Vre
Newton’s Second Law for an Impulse Blade:
The tangential force acting of the jet is:
F = mass flow rate X Change of velocity in the tangential direction
Tangential relative velocity at blade Inlet : Vri cos(bi).
Tangential relative velocity at blade exit : -Vre cos(be).
Change in velocity in tangential direction: -Vre cos(be) - Vri cos(bi).
-(Vre cos(be) + Vri cos(bi)).

Tanential Force, FA   mVre cos b e  Vri cos b i 
The reaction to this force provides the driving thrust on the wheel.

The driving force on wheel FR  mVre cos b e  Vri cos b i 
Power Output of the blade,

Pb  m U Vre cos b e  Vri cos b i 
Diagram Efficiency or Blade efficiency:
Power ouput
d 
Kinetic
Power of inlet steam

m U Vre cos b e  Vri cos bi 
d 

m Vai2
2U kVri cos b e  Vri cos bi 
d 
Vai2
2UVri k cos b e  cos bi 
d 
2
Vai
ae
Vae
ai
U
bi
be
Vai
Vre
Vai cos a i  U  Vri cos bi
Vri
Vai cos a i  U
Vri 
cos b i
 cos b e 
2U Vai cos a i  U  k
 1
cos bi 

d 
2
Vai
 cos b e 
2U Vai cos a i  U  k
 1
cos bi 

d 
2
Vai
2

U  

U
 cos b
d  2 cos a i     k
 1
Vai
Vai   cos bi e 




Define Blade Speed Ratio, f
 cos b e 
d  2fcos a i  f k
 1
 cos bi 
For a given shape of the blade, the efficiency is a strong function of f.
For maximum efficiency:
d d
0
df
 cos b e 
2cos a i  2f k
 1  0
 cos bi 
cos a i
cos a i  2f   0  f 
2
d ,max
cos a i  cos b e 

 2 cos a i cos a i 
 1
 k
2  cos bi 

d ,max
 cos b e 
 f cos a i  k
 1
 cos bi 
2
2
Impulse-Reaction turbine
• This utilizes the principle of impulse and reaction.
• There are a number of rows of moving blades attached to the
rotor and equal number of fixed blades attached to the casing.
• The fixed blades are set in a reversed manner compared to the
moving blades, and act as nozzles.
• The fixed blade channels are of nozzle shape and there is a
comparatively small drop in pressure accompanied by an
increase in velocity.
• The fluid then passes over the moving blades and, as in the
pure impulse turbine, a force is exerted on the blades by the
fluid.
• There is further drop in pressure as the fluid passes through the
moving blades, since moving blade channels are also of nozzle
shape.
• The relative velocity increases in the moving blades.
ae
Vae
ai
U
Vre
bi
be
Vai
Vri
The reaction effect is an addition to impulse effect.
The degree of reaction

The enthalpy drop in the moving blades
The enthalpy drop in the stage
p
va
vr
First law for fixed blades:
h0  h1 
V12  V02
2
0
First law for moving blades:
h1  h2 
V r22  Vr21
2
1
2
a2
Va2
a1
Vr2
U
b1
b2
Va1
Vr1
• If the stream is to enter and leave the blades without shock or much
losses, then relative velocity should be tangential to the blade inlet tip.
• Vr1 should enter at an angle b1, the inlet blade angle.
• Similarly, Vr2 should leave at b2, the exit blade angle.
• In an impulse reaction blade, Vr2 > Vr1.
h1  h2 
V r22  Vr21
2
• The flow velocities between two successive blade at inlet and exit are
Vf1 & Vf2.
• The axial (basic useful) components or whirl velocities at inlet and exit
are Vw1 & Vw2.
a2
Va2
a1
Vr2
U
b1
b2
Va1
Vr1
Newton’s Second Law for an Impulse-reaction Blade:
The tangential force acting of the jet is:
F = mass flow rate X Change of velocity in the tangential direction
Tangential relative velocity at blade Inlet : Vr1 cos(b2).
Tangential relative velocity at blade exit : -Vr2 cos(b2).
Change in velocity in tangential direction: -Vr2 cos(b2) – Vr1 cos(b1).
-(Vr2 cos(b2) + Vr1 cos(b1)).

Tangential Force, FA   mVr 2 cos b 2  Vr1 cos b1 
The reaction to this force provides the driving thrust on the wheel.

The driving force on wheel F  mV cos b  V cos b 
R
r2
2
r1
1
Power Output of the blade,

Pb  m U Vr 2 cos b 2  Vr1 cos b1 
Diagram Efficiency or Blade efficiency:
Power ouput
d 
Kinetic
Power of inlet steam

m U Vr 2 cos b 2  Vr1 cos b1 
d 

m Va21
First law for fixed blades:
h0  h1 
V12  V02
2
First law for moving blades:
h1  h2 
h0  h2 
V r22  Vr21
2
V12  V02
2

V r22  Vr21
2
2
2
Va21  V02 V r 2  Vr1
h0  h2 

2
2

The enthalpy drop in the moving blades
The enthalpy drop in the stage


h1  h2
h0  h2
V r22  Vr21
h1  h2
 2
h0  h2 Va1  V02  V r22  Vr21
Vr 2
   2
2
 V r1  
 Va1  V0
1  
Vr 2
   2
2
 Vr1  
V

V
 a1
0
1  
2

2
2



2U Vr 2 cos b 2  Vr1 cos b1 
d 
Va21
   2
2
V r 2  V r21  
 Va1  V0
1  


Va1 cos a1  U  Vr1 cos b1
Va1 cos a1  U
Vr1 
cos b1
Losses in nozzle, Nozzle blade loss factor, f
Va1
Actual absolute inlet velo city
f

Isoentropi c Velocity at nozzle exit Vn,iso
2U Vr 2 cos b 2  Vr1 cos b1 
 stage 
2
Vn,iso





2
2
2
2Uf  V r 1  
Va1  V0 cos b 2  Vr1 cos b1 



1






 stage 
Va21

2

Va1 cos a1  U
Vr1 
cos b1

2
2Uf 


 stage 
2

 Va1 cos a1  U     2
 Va1 cos a1  U 
2

  
 cos b1 
 Va1  V0 cos b 2  
cos b1
cos b1


 1  





Va21


U 2
2 f 
Va1 


 stage 
U

 cos a1 
Va1

 cos b1




U 2
2 f 
Va1 


 stage 
U

cos
a


1
Va1

 cos b1


2
U


cos
a

2


1
  V  
Va1



 
1   0   cos b 2  
  1     Va1  
 cos b1










 cos b 
1






Va21
2
U


cos
a

2


1
  V  
Va1



0
 



1

cos
b


2
  1     Va1  
 cos b1










 cos b 
1






Va21
For a given shape of the blade, the efficiency is a strong function of U/Va1.
For maximum efficiency:
d stage
U
d
V
 a1




0