Velocity Triangle for Turbo-machinery

BY

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

U

Vri

Vre

Vai

UVri

Vai

Inlet Velocity Triangle

U

VreVae

Exit Velocity Triangle

U

VriVai

Vre

Vae

iie e

Vai: Inlet Absolute VelocityVri: Inlet Relative VelocityVre: Exit Relative VelocityVae:Exit Absolute Velocity

i: Inlet Nozzle Angle.i: Inlet Blade Angle.e: Exit Blade Angle.i: Exit Nozzle Angle.

Va

Vr

Vrc

Vw

U

Vf

U

U

U

Vri

Vre

Vai

VaeVni

Flow through Blades

Fluid Dynamics of Blades

• The stream is delivered to the wheel at an angle i and velocity Vai.

• The selection of angle i is a compromise.

• An increase in i, reduces the value of useful component (Absolute circumferential Component).

• This is also called Inlet Whirl Velocity, Vwi = Vai cos(i).

• An increase in i, 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(i) = Vri sin(i).

• 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.

U

VriVai

Vre

Vae

iie e

• 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 i, the inlet blade angle.• Similarly, Vre should leave at e, the exit blade angle.• A blade is said to be symmetric if i = e.• 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.

U

VriVai

Vre

Vae

iie e

Impulse Turbine

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(i).

Tangential relative velocity at blade exit : -Vre cos(e).

Change in velocity in tangential direction: -Vre cos(e) - Vri cos(i).

-(Vre cos(e) + Vri cos(i)).

Tanential Force,

U

VriVai

Vre

Vae

iie e

iriereA VVmF coscos

The reaction to this force provides the driving thrust on the wheel.

The driving force on wheel iriereR VVmF coscos

Power Output of the blade,

iriereb VVUmP coscos

Diagram Efficiency or Blade efficiency:

steaminlet ofPower Kinetic

ouputPower d

2

coscos

2ai

riered

Vm

iVVUm

2

coscos2

ai

rierid V

iVkVU

2

coscos2

ai

erid V

ikUV

U

VriVai

Vre

Vae

iie e

iriiai VUV coscos

i

iairi

UVV

cos

cos

2

1coscos

cos2

ai

eiai

d V

ikUVU

1

cos

coscos2

2

eaii

aid i

kV

U

V

U

2

1coscos

cos2

ai

eiai

d V

ikUVU

Define Blade Speed Ratio,

1

cos

coscos2

ik e

id

For a given shape of the blade, the efficiency is a strong function of

For maximum efficiency: 0

d

d d

01cos

cos2cos2

i

k ei

2

cos02cos i

i

1

cos

cos

2

coscoscos2max, i

k eiiid

1

cos

coscos22

max, ik e

id

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.

U

VriVai

VreVae

iie e

pva vr

The reaction effect is an addition to impulse effect.

The degree of reaction

stage in the dropenthalpy The

blades moving in the dropenthalpy The

First law for fixed blades:

0 1 22

20

2

101VV

hh

First law for moving blades:

2

21

2

212 rVV

hh r

• 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 1, the inlet blade angle.

• Similarly, Vr2 should leave at 2, the exit blade angle.

• In an impulse reaction blade, Vr2 > Vr1.

• 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.

U

Vr1Va1

Vr2Va2

112 2

2

21

2

212 rVV

hh r

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(2).

Tangential relative velocity at blade exit : -Vr2 cos(2).

Change in velocity in tangential direction: -Vr2 cos(2) – Vr1 cos(1).

-(Vr2 cos(2) + Vr1 cos(1)).

Tangential Force, 1122 coscos rrA VVmF

U

Vr1Va1

Vr2Va2

112 2

The reaction to this force provides the driving thrust on the wheel.

The driving force on wheel 1122 coscos rrR VVmF

Power Output of the blade,

1122 coscos rrb VVUmP

Diagram Efficiency or Blade efficiency:

steaminlet ofPower Kinetic

ouputPower d

2

coscos

21

1122

a

rrd

Vm

VVUm

First law for fixed blades:

2

20

2

101VV

hh

First law for moving blades:

2

21

2

212 rVV

hh r

22

21

220

2

2021 rVVVV

hh r

22

21

220

21

202 raVVVV

hh r

stage in the dropenthalpy The

blades moving in the dropenthalpy The

20

21

hh

hh

21

220

21

21

2

20

21

2

2

ra

r

VVVV

VV

hh

hh

r

r

20

21

22

112VVVV arr

20

21

2

112VVVV arr

1111 coscos ra VUV 1

111 cos

cos

UV

V ar

Losses in nozzle, Nozzle blade loss factor,

ison

a

V

V

,

1

exit nozzleat Velocity cIsoentropi

cityinlet velo absolute Actual

2,

1122 coscos2

ison

rrstage V

VVU

20

21

2

112VVVV arr

21

1122 coscos2

a

rrd V

VVU

21

1122

021

22 coscos1

21

a

ra

stage V

VVVVUr

1

111 cos

cos

UV

V ar

21

11

112

20

21

2

1

112 coscoscos

cos1cos

cos2

a

aa

a

stage V

UVVV

UVU

For a given shape of the blade, the efficiency is a strong function of U/Va1

For maximum efficiency: 0

1

a

stage

VU

d

d

21

11

11

2

2

1

0

2

1

11

2

1

coscos

coscos1

1cos

cos2

a

a

a

a

a

stage V

VU

VVV

U

VU

21

11

11

2

2

1

0

2

1

11

2

1

coscos

coscos1

1cos

cos2

a

a

a

a

a

stage V

VU

VVV

U

VU