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Velocity Triangle for Turbo-machinery

BY

 P M V Subbarao

Professor

Mechanical Engineering Department

I I T Delhi

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U

Vri

Vre

Vai

UVri

Vai

Inlet Velocity Triangle

U

VreVae

Exit Velocity Triangle

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U

VriVai

Vre

Vae

βiαiαe   βe

Vai: Inlet Absolute Velocity

Vri: Inlet Relative Velocity

Vre: Exit Relative Velocity

Vae:Exit Absolute Velocity

αi: Inlet Nozzle Angle.βi: Inlet Blade Angle.βe: Exit Blade Angle.

αi: Exit Nozzle Angle.

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Va

Vr

Vrc

Vw

U

Vf 

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U

U

U

Vri

Vre

Vai

VaeVni

lo! t"roug" Blades

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luid #yna$ics o% Blades

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& T"e strea$ is delivered to t"e !"eel at an angle αi and velocity Vai.& T"e selection o% angle αi is a co$'ro$ise.

& An increase in αi( reduces t"e value o% use%ul co$'onent )Absolutecircu$%erential *o$'onent+.

& T"is is also called Inlet ,"irl Velocity( V!i - Vai cos)αi+.& An increase in αi( increases t"e value o% axial co$'onent( also called as %lo!

co$'onent.

& T"is is res'onsible %or de%inite $ass %lo! rate bet!een to successive blade.

& lo! co$'onent V%i - Vai sin)αi+ - Vri sin)βi+.& T"e absolute inlet velocity can be considered as a resultant o% blade velocity

and inlet relative velocity.

& T"e t!o 'oints o% interest are t"ose at t"e inlet and exit o% t"e blade.

U

VriVai

Vre

Vae

βiαiαe   βe

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& I% t"e strea$ is to enter and leave t"e blades !it"out s"oc or $uc"losses( t"en relative velocity s"ould be tangential to t"e blade inlet ti'.

& Vri s"ould enter at an angle βi( t"e inlet blade angle.& /i$ilarly( Vre s"ould leave at βe( t"e exit blade angle.& A blade is said to be sy$$etric i% βi - βe.& T"e %lo! velocities bet!een t!o successive blade at inlet and exit are

V%i 0 V%e.

& T"e axial )basic use%ul+ co$'onents or !"irl velocities at inlet andexit are V!i 0 V!e.

U

VriVai

Vre

Vae

βiαiαe   βe

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I$'ulse Turbine

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 Ne!ton1s /econd 2a! %or an I$'ulse Blade:

T"e tangential %orce acting o% t"e 3et is:

- $ass %lo! rate 4 *"ange o% velocity in t"e tangential direction

Tangential relative velocity at blade Inlet : Vri cos)βi+.

Tangential relative velocity at blade exit : 5Vre cos)βe+.

*"ange in velocity in tangential direction: 5Vre cos)βe+ 5 Vri cos)βi+.

5)Vre cos)βe+ 6 Vri cos)βi++.

U

VriVai

Vre

Vae

βiαiαe   βe

( )iriere A   V V m F    β β  coscos   +−=  •

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T"e reaction to t"is %orce 'rovides t"e driving t"rust on t"e !"eel.

T"e driving %orce on !"eel ( )iriere R   V V m F    β β  coscos   +=

  •

7o!er 8ut'ut o% t"e blade(

( )iriereb   V V U m P    β β  coscos   +=  •

#iagra$ E%%iciency or Blade e%%iciency:

stea$inleto% 7o!er9inetic

ou'ut7o!er

=d η ( )coscos

ai

riered 

V m

iV V U m•

•

+=

  β β η 

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( )

coscos

ai

rierid 

V 

iV kV U    β β η 

  +=

( )

coscos

ai

erid 

V 

ik UV    β β η    +=

U

VriVai

Vre

Vae

βiαiαe   βeiriiai   V U V    β α    coscos   +=

i

iairi

U V V 

β 

α 

cos

cos   −=

( )

;cos

coscos

ai

eiai

d V 

ik U V U    

  

   +−

=  β 

β α 

η 

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       +

−=;

cos

coscos:

:

eai

i

ai

d i

k V 

U 

V 

U 

β β α η 

( )

;cos

coscos

ai

eiai

d  V 

ik U V U    

 

  

 +−

=

  β 

β α 

η 

#e%ine Blade /'eed Ratio(

φ 

{ }

   

  

 +−= ;

cos

coscos

ik    eid 

β 

β φ α φ η 

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or a given s"a'e o% t"e blade( t"e e%%iciency is a strong %unction o% φ.

or $axi$u$ e%%iciency:  

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 += ;

cos

coscos::$ax(

i

k    eid 

β 

β α φ η 

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 Impulse-Reaction turbine

& T"is utilizes t"e 'rinci'le o% i$'ulse and reaction.

& T"ere are a nu$ber o% ro!s o% $oving blades attac"ed to t"e rotor ande=ual nu$ber o% %ixed blades attac"ed to t"e casing.

& T"e %ixed blades are set in a reversed $anner co$'ared to t"e $oving

 blades( and act as nozzles.

& T"e %ixed blade c"annels are o% nozzle s"a'e and t"ere is a

co$'aratively s$all dro' in 'ressure acco$'anied by an increase invelocity.

& T"e %luid t"en 'asses over t"e $oving blades and( as in t"e 'ure

i$'ulse turbine( a %orce is exerted on t"e blades by t"e %luid.

& T"ere is %urt"er dro' in 'ressure as t"e %luid 'asses t"roug" t"e

$oving blades( since $oving blade c"annels are also o% nozzle s"a'e.

& T"e relative velocity increases in t"e $oving blades.

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U

VriVai

VreVae

βiαiαe   βe

 'va

vr 

T"e reaction e%%ect is an addition to i$'ulse e%%ect.

T"e degree o% reaction

stagein t"edro'ent"al'yT"e

 blades$ovingin t"edro'ent"al'yT"e=Λ

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irst la! %or %ixed blades:

< ; :

:

<

:

;<;

V V 

hh

−=−

irst la! %or $oving blades:

:

:

;

:

:;

:   r V V hh  r  −=−

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& I% t"e strea$ is to enter and leave t"e blades !it"out s"oc or $uc"

losses( t"en relative velocity s"ould be tangential to t"e blade inlet ti'.

& Vr; s"ould enter at an angle β;( t"e inlet blade angle.& /i$ilarly( Vr s"ould leave at β( t"e exit blade angle.& In an i$'ulse reaction blade( Vr > Vr;.

& T"e %lo! velocities bet!een t!o successive blade at inlet and exit are

V%; 0 V%.

& T"e axial )basic use%ul+ co$'onents or !"irl velocities at inlet and

exit are V!; 0 V!.

U

Vr;Va;

VrVa

β;α;α   β

:

:

;

:

:;

:   r V V 

hh  r 

−=−

U

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 Ne!ton1s /econd 2a! %or an I$'ulse5reaction Blade:

T"e tangential %orce acting o% t"e 3et is:

- $ass %lo! rate 4 *"ange o% velocity in t"e tangential direction

Tangential relative velocity at blade Inlet : Vr; cos)β+.

Tangential relative velocity at blade exit : 5Vr cos)β+.

*"ange in velocity in tangential direction: 5Vr cos)β+ ? Vr; cos)β;+.

5)Vr cos)β+ 6 Vr; cos)β;++.

( );;   coscos   β β  r r  A   V V m F    +−=•

U

Vr;Va;

VrVa

β;α;α   β

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T"e reaction to t"is %orce 'rovides t"e driving t"rust on t"e !"eel.

T"e driving %orce on !"eel ( );;   coscos   β β  r r  R   V V m F    +=•

7o!er 8ut'ut o% t"e blade(

( );; coscos   β β  r r b   V V U m P    +=  •

#iagra$ E%%iciency or Blade e%%iciency:

stea$inleto% 7o!er9inetic

ou'ut7o!er

=d η ( )coscos

;

;;

a

r r d 

V m

V V U m•

•

+=

  β β η 

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irst la! %or %ixed blades:

:

:

<

:

;<;

V V 

hh

−=−

irst la! %or $oving blades:

:

:

;

:

:;

:   r V V hh  r  −=−

::

:

;

::

<

:

:<

:;   r V V V V 

hh  r 

−+

−=−

::

:

;

::

<

:

;

:<

:   r a  V V V V 

hh  r 

−+

−=−

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stagein t"edro'ent"al'yT"e

 blades$ovingin t"edro'ent"al'yT"e=Λ

<

;

hhhh

−−=Λ

:

;

::

<

:

;

:

;

:

:<

:;

:

:

r a

r 

V V V V 

V V 

hh

hh

r 

r 

−+−

−=

−

−=Λ

{ }:<

:

;

::

;;:

V V V V a

r r 

−   

  

Λ−Λ

+=

{ }:<

:

;

:

;;:

V V V V a

r r 

−   

  

Λ−Λ

+=

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;;;;  coscos

  β α  r a  V U V    +=

;

;;

;cos

cos

β 

α    U V 

V 

  a

r 

−

=

2osses in nozzle( Nozzle blade loss %actor( φ  

ison

a

V 

V 

(

;

exitnozzleatVelocitycIsoentro'i

cityinlet veloabsoluteActual==φ 

( ):(

;;:: coscos:

ison

r r 

 stage V 

V V U    β β η 

  +

=

{ }:<

:

;

:

;;:

V V V V a

r r 

−   

  

Λ−Λ

+=

( )

;

;; coscos

a

r r d 

V 

V V U    β β η 

  +=

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{ }

:

;

;;:

:

<

:

;

:: coscos

;

:;

a

r a

 stageV 

V V V V U r   

 

 

 

 

 +− 

 

 

 

 

Λ−

Λ+

=

β β φ 

η 

;

;;

;

cos

cos

β 

α    U V V 

  a

r 

−=

{ }

;

;

;

;;

<

;

;

;; coscos

coscos

;cos

cos

a

aa

a

 stage V 

U V V V 

U V U 

   

 

 

 

 

   

  

    −+− 

  

  

Λ−Λ

+   

  

    −

=

β β 

α β 

β 

α φ 

η 

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or a given s"a'e o% t"e blade( t"e e%%iciency is a strong %unction o% U/V a1.

or $axi$u$ e%%iciency:   <

;

=

  

 

 

a

 stage

V 

U d 

d η 

;

;

;

;

;

;

<

;

;

;

;

coscos

cos

cos;;cos

cos

a

a

a

a

a

 stageV 

V 

U 

V 

V V 

U 

V 

U 

 

    

 

 

 

 

    

 

 

 

  −+

   

  

 − 

  

  

Λ−Λ+

    

 

 

 

  −

=

β β 

α 

β β 

α 

φ 

η 

;

;

;

;

;

;

<

;

;

;

;coscos

cos

cos;;cos

cos

a

a

a

a

a

 stageV 

V 

U 

V 

V V 

U 

V 

U 

     

 

 

 

 

   

 

 

 

 

  −

+

   

  

 −      Λ−

Λ+

   

 

 

 

 

  −

=

β β 

α 

β β 

α 

φ 

η