selfstudy-346-1
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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
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