Axial compressor - variation of rotor and stator angles from root to tip - 4th March 2010

Axial Compressor

TheoryTheoryVariation of rotor and stator angles

from root to tip

4th March 2010

Prepared by: Cheah CangTo

TURBO GROUP – Axial compressor theory - Variation of rotor and stator angles from root to tip

tip2β

tip1β

Previous discussion covers the theory behind the

calculation of rotor and stator angles at mean

radius.

Further study on previous theory enables

compressor designer to evaluate the change of

angles from root section up to tip section of rotor

and stator (covering all stages in axial compressor).

root1β

root2β

2Axial compressor theory - Variation of rotor and stator angles from root to tip - Cheah CangTo

Variation of rotor angles from root to tip section


Cwdr

( )[ ]

( )[ ] ( )d

rdrrd

Area

c

c

−+××=

≈θ

ππ

θ

2

22

2

2

dr

r ( )[ ] ( )drrdrdrrdrr

d cdrc

×× →−×++× ≈ θθ 0222

2

22

arNote: unit width element

r

CA

r

CV

r

CmFforcelCentrifuga www

cw

×××=

××=

×==

ρρ222

1_

( )drdC

r

drrdC

rrr

c

w

c

w ×××=××××

θρθρ 2

2

ar



dPP +

( ) ( )c

drdrdPPF

θπ ××××+×+= 12

dPP +2

dPP +

( ) ( )

( ) ( ) c

topradial

drdrdPP

drdrdPPF

θ

π

θπ

×+×+

××

×××+×+=

a

12

2,

2P +2

P +

P

c

bottomradial drPF θ××=,

c

sideradial

ddr

dPPF

θ

××

+×=

2sin

22

,

cc

dPddPθ

θ××

+=××

+×

22

Since dθθθθ very small, 22sin

ccdd θθ

≈

cddr

dPP

ddr

dPP θ

θ××

+=××

+×

2222a



dPP +

dPP +2

dPP +

2P +2

P +

P

( ) ( ) cccddr

dPPdrPdrdrdPPF θθθ ××

+−××−×+×+= ( ) ( )

( )cc

ccc

netradial

drdPddrdP

ddrdP

PdrPdrdrdPPF

θθ

θθθ

××+

××

××

+−××−×+×+=

2,

a ( )cdrdP θ××+

2

a



netradialcw FF,

=

( )drdPddrdP

drdCc

cc

w2

××+××

=××× θθ

θρ ( )

rdr

ddPdrdC

drdPdrdC

cc

w

w

2

2

2

+××=×××

××+=×××

θθρ

θθρ

a

rdr

d

Cd

dr

dP

c

w

c 2

2

+×

××=

θ

θρa

r

C

dr

Cd

dr

dP

rd

w

dr

w

c 20

2

21

2

→

×=×

+×

≈θ

ρ

θ

ar

rdr

ddr c

2

→

+×

=×

θρ

a

r

C

dr

dP w

21

=×ρ

Radial equilibrium equation:



PdvvdPdudh

Pvuh

++=

+=

a

Enthalpy Gibbs equation

( ) PdvPdvvdPdhTds

PdvduTds

+−−=

+=

a

PdvvdPdhdu

PdvvdPdudh

−−=

++=

a

a ( )

dPdhTds

vdPdhTds

PdvPdvvdPdhTds

ρ−=

−=

+−−=

a

a

a

ddPdTdsdh

dPTdsdh

ρ

ρ

ρ

11

+=a

dPdr

d

dr

dP

dr

dTds

dr

dsT

dr

dh ρ

ρρ 2

11−++=a

dPdsdh 1Dropping second order terms ��

dr

dP

dr

dsT

dr

dh

ρ

1+=



Stagnation Enthalpy

Chh +=

2

02

[ ]CChh

hh

wa +×+=

+=

22

0

0

2

1

2

a

dr

dCC

dr

dCC

dr

dh

dr

dh ww

aa ++=0

2

a

dPdsdh 1 CdP2

1

dCdCdPdsdh 1

dr

dP

dr

dsT

dr

dh

ρ

1+=Knowing:

r

C

dr

dP w

21

=×ρ

and

dr

dCC

dr

dCC

dr

dP

dr

dsT

dr

dh ww

aa +++=0

1a

ρDropping entropy gradient yields:

dr

dCC

dr

dCC

r

C

dr

dh ww

aa

w ++=2

0a

Vortex energy equation:


drdrrdr


Apart from regions near the walls of the annulus^, the stagnation enthalpy (and temperature) will be uniform across the annulus at entry to the compressor.temperature) will be uniform across the annulus at entry to the compressor.

If the frequently used design condition of constant specific work at all radii is applied, then although h0 will increase progressively through the compressor in applied, then although h0 will increase progressively through the compressor in the axial direction, its radial distribution will remain uniform. Thus dh0/dr = 0 in any plane between pair of blade rows.

dr

dCC

dr

dCC

r

C ww

aa

w ++=2

0Constant specific work at all radii:

drdrr

^ due to the adverse pressure gradient in compressors, the boundary layers along the annulus walls thicken

as the flow progresses.



dr

dCC

dr

dCC

r

C ww

aa

w ++=2

0

When considering possible sets of design conditions, it is usually desirable to retain the constant

specific work-input condition to provide constant stage pressure ratio up to the blade height. It would be

possible, however, to choose a variation of one of the other variables, say Cw, and determine the variation

of Ca. The radial equilibrium requirement would still be satisfied.

In this note, we use the normal design condition:

(a) Constant specific work input at all radii

(b) An arbitrary whirl velocity distribution which is compatible with (a)

To obtain constant work input, U(C - C ) must remain constant across the annulus. Let us consider To obtain constant work input, U(Cw2 - Cw1) must remain constant across the annulus. Let us consider

distributions of whirl velocity at inlet and outlet from the rotor blade given by:

R

baRC

n

w −=1 _2

:

r

rRwhere

R

baRC

n

w =+= Kand

Check whether Cw1 and Cw2 satisfy Uλλλλ(Cw2 - Cw1)Check whether Cw1 and Cw2 satisfy Uλλλλ(Cw2 - Cw1)

R

bCC

R

baR

R

baRCC

ww

nn

ww

_

12

12

2

λλ

=−

−−

+=−

a==

==

Nr

U

srevNwhererNU

π

π

2

/:,2

a constant “It means metal speed at any

specified radius divided by its

radius is a constant value.”

( )r

rUb

R

UbCCU ww

_

12

22 λλλ ==−∴

radius is a constant value.”

UrU

r

U

r

U

__

_

_

=

=

a

( )_

Conclusion“This is independent of radius,


rrU =a

( )_

122 UbCCU ww λλ =−

“This is independent of radius, means the two design conditions (a) and (b) are therefore compatible.”


dr

dCC

dr

dCC

r

C ww

aa

w ++=2

0Constant specific work at all radii:

drdrrwaConstant specific work at all radii:

drC

dCCdCC w

2

0=++

drC

dCCdCC

drr

CdCCdCC

w

wwwaa

2

0

+=−

=++

a

Times both side by “dr”��

drr

CdCCdCC w

wwaa +=−a

In terms of dimensionless R

Re-arranging��

dRR

CdCCdCC w

wwaa

2

+=−Note: _

r

rR =

R



When n = 1 (first power condition)

R

baRC

R

baRC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor exit:

[ ] [ ] dRR

abR

bRa

CC

RR

w

R

a

2

2

1

2

1

1

2

2

22

1

2

1

2

2

++

+=− ∫

dRR

ab

R

bRaab

R

bRaCC

RR

aa

22

2

1

2

1

1

3

2

2

1

2

2

22

2_

2

2

2

+++

++=

−−

∫a

babRab

RabR

bRaabbaab

R

bRaCC

R

aa

11

ln222

222

1

2

1

2222222

1

2

222

22

2

2

22

2_

2

2

2

11

+−+

−−−++=

−−

a

( )RabaRaCC

baRab

R

bRaba

R

bRaCC aa

ln22

022

ln2222

1

2

1

222

2_2

22

2

222

22

2

2

22

2_

2

2

2

+−−=−∴

−+−+−+

−−+=

−−a


( )RabaRaCC aa ln22222

2

2

2+−−=−∴


When n = 1 (first power condition)

R

baRC

R

baRC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor inlet:

[ ] [ ] dRR

abR

bRa

CC

RR

w

R

a

2

2

1

2

1

1

2

2

22

1

2

1

2

1

−+

+=− ∫

dRR

ab

R

bRaab

R

bRaCC

RR

aa

22

2

1

2

1

1

3

2

2

1

2

2

22

2_

1

2

1

−++

−+=

−−

∫a

babRab

RabR

bRaabbaab

R

bRaCC

R

aa

11

ln222

222

1

2

1

2222222

1

2

222

22

2

2

22

2_

1

2

1

11

−−+

+−−−+=

−−

a

( )RabaRaCC

baRab

R

bRaba

R

bRaCC aa

ln22

022

ln2222

1

2

1

222

2_2

22

2

222

22

2

2

22

2_

1

2

1

−−−=−∴

++−−−+

−−+=

−−a



1

2

1−−−=−∴


When n = 0 (exponential condition)

R

baC

R

baC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor exit:

+++

++=

−− ∫ dR

R

ab

R

b

R

a

R

ab

R

baCC

RR

aa

22

2

1

2

1

1

23

22

1

2

2

2

2_

2

2

2

−−+

−−−++=

−−

∫

R

ab

R

bRaabba

R

ab

R

baCC

RRRRR

R

aa

2

2ln2

2

2

1

2

1

22

1

2

2

222

2

2

2

2_

2

2

2

11

a

++−−−+

−−+=

−−

abb

R

ab

R

bRaabb

R

ab

R

bCC

RRRR

aa 22

02

2ln2

2

2

1

2

1

222

2

2

2

22

2

22_

2

2

2

1

a

−+−=−∴

R

abRaabCC aa ln2

2

2_

2

2

2



When n = 0 (exponential condition)

R

baC

R

baC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor inlet:

−++

−+=

−− ∫ dR

R

ab

R

b

R

a

R

ab

R

baCC

RR

aa

22

2

1

2

1

1

23

22

2

2

2

2_

1

2

1

+−+

+−−−+=

−−

∫

R

ab

R

bRaabba

R

ab

R

baCC

RRRRR

R

aa

2

2ln2

2

2

1

2

1

22

1

2

2

222

2

2

2

2_

1

2

1

11

a

−+−+−+

+−−=

−−

abb

R

ab

R

bRaabb

R

ab

R

bCC

RRRR

aa 22

02

2ln2

2

2

1

2

1

222

2

2

2

22

2

22_

1

2

1

1

a

++−−=−∴

R

abRaabCC aa ln2

2

2_

1

2

1



When n = -1 (free vortex condition)

R

b

R

aC

R

b

R

aC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor exit:

33

2

3

2

22

2

2

22_

2

2

2

2211RR

aa dRabbaabba

CC

+++

++=

−− ∫

22

2

2

2

22

22

2

2

22_

2

2

2

1

333

1

22222

2

2

222

2

2

1

2

1

22

R

aa

aa

R

ab

R

b

R

aabba

R

ab

R

b

R

aCC

dRRRRRRR

CC

−−−+

−−−++=

−−

+++

++=

−− ∫

a

22

22

2

2

2

22

22

2

2

22_

2

2

2

1

222

2

222

2

2

1

2

1

22222

aa abba

R

ab

R

b

R

aabba

R

ab

R

b

R

aCC

RRRRRR

+++−−−+

−−−++=

−−

a

2_

2

2

2 aa CC =∴



When n = -1 (free vortex condition)

R

b

R

aC

R

b

R

aC ww +=−=

21,

dRR

CdCCdCC w

wwaa

2

+=−

For rotor inlet:

33

2

3

2

22

2

2

22_

1

2

1

2211RR

aa dRabbaabba

CC

−++

−+=

−− ∫

22

2

2

2

22

22

2

2

22_

1

2

1

1

333

1

22211

2

2

222

2

2

1

2

1

22

R

aa

aa

R

ab

R

b

R

aabba

R

ab

R

b

R

aCC

dRRRRRRR

CC

+−−+

+−−−+=

−−

−++

−+=

−− ∫

a

22

22

2

2

2

22

22

2

2

22_

1

2

1

1

222

2

222

2

2

1

2

1

22222

aa abba

R

ab

R

b

R

aabba

R

ab

R

b

R

aCC

RRRRRR

−+++−−+

+−−−+=

−−

a

2_

1

2

1 aa CC =∴



Degree of Reaction (DOR), ΛΛΛΛ provides a measure of the extent to which the rotor contributes to the Degree of Reaction (DOR), ΛΛΛΛ provides a measure of the extent to which the rotor contributes to the overall static pressure rise in the stage. It is defined as:

ΛΛΛΛ =Static enthalpy rise in the rotor

Static enthalpy rise in the stage

Steady flow energy equation: ( ) ( )1 2222−+−+−

∆=Λ

CCUCCCC

W

TC rotorp

( )12 ww CCUW −=

Since all the work input to the stage takes

Steady flow energy equation: ( ) ( )

( )

( )1

2

1

2

2

2

1

2

2

2

1

12

12

2

2

2

1

2

2

2

1

+−+−

=Λ

−

−+−+−=Λ

CCCC

CCU

CCUCCCC

awwa

ww

wwawwa

aSince all the work input to the stage takes

place in the rotor, the steady flow energy

equation yields:

( )( )

( )( )

( )( )

122

12

2

2

2

1

2

2

2

1

12

2121

+−

−+

−

−=Λ

+−

−+−=Λ

CCU

CC

CCU

CC

CCU

CCCC

wwaa

ww

awwa

a

a

( )221CCTCW −+∆= ( ) ( )

( )( )

( )( )( )

( ) ( )( )

122

22

12

2121

12

2

2

2

1

1212

+−

−++

−

−=Λ

−−

CCU

CCCC

CCU

CC

CCUCCU

ww

wwww

ww

aa

wwww

a

( )

( ) ( )

( ) ( )22

12

2

1

2

2

2

1

2

2

1

2

1:

2

wwrotorp

rotorp

CCUCCTC

CCUCCTCHence

CCTCW

−+−=∆

−=−+∆

−+∆=

( )( )

( )( )( )

( ) ( )

122

22

12

1221

12

2

2

2

1

++

−−

=Λ∴

+−

−+−

−

−=Λ

CCCC

CCU

CCCC

CCU

CC

ww

wwww

ww

aaa

( ) ( )

( ) ( )12

2

2

2

2

2

1

2

1

12

2

2

2

1

2

1

2

1

wwwawarotorp

wwrotorp

CCUCCCCTC

CCUCCTC

−+−−+=∆

−+−=∆

a

a


( )( )

( )1

22

21

12

21 ++

−−

−=Λ∴

U

CC

CCU

CC ww

ww

aa


First power, n = 1First power, n = 1


2_

1

2

1−−−=

( )RabaRaCC ln22222

2_2

+−−=

( ) ( )RabaRaCRabaRaCCC aaaa ln22ln222222

2

_2222

1

_2

2

2

1+−+−−−−=−


2

2

2+−−=

2_

2

2_

1 aa CC =

RabCC aa ln82

2

2

1=−∴

baRC −=

R

baRCw −=

1

R

baRCw +=

2

R

b

R

baR

R

baRCC ww 2

12=

−−

+=− aR

R

baR

R

baRCC ww 2

12=

−+

+=+and

ln22ln8

aRRaRaRRabR

1:

1ln2

12

2

22

ln8

=

+

−

=+

−

=Λ

RWhen

U

aR

U

RaR

U

aR

R

bU

Rab

( )( )

( )1

22

21

12

2

2

2

1 ++

−−

−=Λ

U

CC

CCU

CC ww

ww

aa

111

1:

____

___

__

_

Λ−=⇒Λ−=⇒−=Λ

=

Ua

U

a

U

a

RWhen( ) 2212

− UCCU ww

U

R

U

RUU

r

U

r

UNote =⇒=⇒=

_

_

_

_

1:

11ln121

1ln12__

_

__

_

__

+

Λ−−

Λ−⇒+

Λ−

−

Λ−

=Λ R

RU

RU

RU

RRU


UUr

r

( ) 11ln21_

+−

Λ−=Λ∴

R


Exponential, n = 0Exponential, n = 0

++−−=

R

abRaabCC aa ln2

2

2_

1

2

1

−+−=ab

RaabCC ln22

2_2

−++−

++−−=−

R

abRaabC

R

abRaabCCC aaaa ln2ln2

222

_22

1

_2

2

2

1

−+−=

R

abRaabCC aa ln2

2

2

2

2

2_

2

2_

1 aa CC =

−=−∴

R

ababCC aa 4

2

2

2

1

b

andR

baCw −=

1

R

baCw +=

2

R

b

R

ba

R

baCC ww 2

12=

−−

+=− a

R

ba

R

baCC ww 2

12=

−+

+=+

24

−

−

−aababRa

Rab

aba

ababR

12

1

1112

2

22

4

+

−

=+

−

−

=Λ

+

−

−=+

−

−

=+

−

−

=Λ

aaRaaaR

U

a

Ub

ababR

U

a

Ub

RR

ab

U

a

R

bU

Rab

a

( )( )

( )1

22

21

12

2

2

2

1 ++

−−

−=Λ

U

CC

CCU

CC ww

ww

aa

111

1:

11

___

__

_

Λ−=⇒Λ−=⇒−=Λ

=

+=+

−=Λ

Ua

U

a

U

a

RWhen

UUUa( ) 22

12− UCCU ww

U

R

U

RUU

r

U

r

UNote =⇒=⇒=

_

_

_

_

1:

1

12

11

121_

_

_

____

+

Λ−

−

Λ−⇒+

Λ−−

Λ−

=Λ

RRU

URU

UU


UUr

r

12

11_

+

−

Λ−=Λ∴

R


Free vortex, n = -1Free vortex, n = -1

2_

1

2

1 aa CC =

2_2

CC = 02

2

2

1=− aa CC

2_

2

2_

1 aa CC =

2

2

2 aa CC = 021

=− aa CC

ba

andR

b

R

aCw −=

1

R

b

R

aCw +=

2

R

b

R

b

a

a

R

b

R

aCC ww 2

12=

−−

+=−

R

a

R

b

R

a

R

b

R

aCC ww 2

12=

−+

+=+

2a

( )( )

( )1

22

21

12

2

2

2

1 ++

−−

−=Λ

U

CC

CCU

CC ww

ww

aa

RR

=

+

−=+

−=+

−

=Λ

1:

11012

2

22

0

RWhen

UR

a

UR

a

U

R

a

R

bU

( ) 2212

− UCCU ww

Λ−=⇒Λ−=⇒−=Λ

=

____

___

__

_

111

1:

U

Ua

U

a

U

a

RWhen

U

R

U

RUU

r

U

r

UNote =⇒=⇒=

_

_

_

_

1:

+

Λ−

−=+

Λ−

−=+

Λ−

−=Λ2

____

1

1

1

1

1

1

RUR

R

U

UR

U


UUr

r

Λ−−=Λ∴

_

21

11

R


( )_

( )_

122 TCUbCCUW spww λλ ∆==−=

_

TCb

p∆=∴

_

2 U

b

λ

=∴

Note: this is applicable for all cases (i.e. n = 1, 0 and -1)Note: this is applicable for all cases (i.e. n = 1, 0 and -1)



− CU

−= −

a

w

C

C

CU1tanβ

Variation of rotor and stator angles

can therefore be calculated as a

function of radius.

= −

a

w

C

C1tanα

function of radius.



Example: RB211-24G

RB211-24G’s axial compressors at ISO conditions:

a. Pressure ratio = 20

b. Mass flow rate = 100 kg/s

c. LP axial compressor = 7 stages


c. LP axial compressor = 7 stagesd. HP axial compressor = 6 stages



ctcheah

Text Box

beta 1

ctcheah

Text Box

beta 2

ctcheah

Text Box

alpha 1

ctcheah

Text Box

alpha 2

ctcheah

Text Box

root

ctcheah

Text Box

tip

ctcheah

Text Box

mean radius

ctcheah

Line

ctcheah

Text Box

a1 = alpha 1 a2 = alpha 2 b1 = beta 1 b2 = beta 2 fv = free vortex (n = -1) exp = exponential (n = 0) fp = first power (n = 1)

ctcheah

Rectangle

ctcheah

Line



ctcheah

Text Box

dhr = de Haller number for rotor dhs = de Haller number for stator fv = free vortex (n = -1) exp = exponential (n = 0) fp = first power (n = 1)

ctcheah

Line

ctcheah

Rectangle

ctcheah

Text Box

root

ctcheah

Text Box

mean radius

ctcheah

Line

ctcheah

Text Box

tip


RB211-24G HP compressor: stage 1RB211-24G HP compressor: stage 1





End of document

Axial compressor - variation of rotor and stator angles from root to tip - 4th March 2010

Documents

Transcript of Axial compressor - variation of rotor and stator angles from root to tip - 4th March 2010