axial flow compressors. elementary theory axial flow compressors

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Axial Flow Compressors

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Page 1: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Page 2: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors• Elementary theory

Page 3: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Page 4: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Comparison of typical forms of turbine and compressor rotor blades

Page 5: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsAxial Flow CompressorsStage= S+RS: stator (stationary blade)R: rotor (rotating blade)First row of the stationary blades is called guide vanes

** Basic operation

*Axial flow compressors: 1) series of stages2) each stage has a row of rotor blades followed

by a row of stator blades.3) fluid is accelerated by rotor blades.

Page 6: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsIn stator, fluid is then decelerated causing change in the kinetic energy to static pressure.

Due to adverse pressure gradient, the pressure rise for each stage is small. Therefore, it is known that a single turbine stage can drive a large number of compressor stages.

Inlet guide vanes are used to guide the flow into the first stage.

Elementary Theory:Assume mid plane is constant r1=r2, u1=u2 assume Ca=const, in the direction of u.

12 www CCC

, in the direction of u.12 www CCC

Page 7: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsInside the rotor, all power is consumed. Stator only changes K.E.P static, To2=To3Increase in stagnation pressure is done in the rotor. Stagnation pressure drops due to friction loss in the stator:C1: velocity of air approaching the rotor.

1 : angle of approach of rotor.u: blade speed.V1: the velocity relative t the rotor at inlet at an angle 1 from the axial direction.V2: relative velocity at exit rotor at angle 2 determined from the rotor blade outlet angle.2: angle of exit of rotor.Ca: axial velocity.

Page 8: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsTwo dimensional analysis: Only axial ( Ca) and tangential (Cw). no radial component

13

13

22

222

a

2

11

C also

stagesimilar a togo air to prepare to

Cget u triangle & V

cosV ontained be V

C assuming

exit.at blade tangnt to

2

21

C

normally

then

Ccanthis

CC

V

VuC

a

aa

)(12

.

oop TTcmW

Page 9: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Page 10: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

)tan-(tan)tan-(tan 2112

21 aaa CCC

(a) tantan/,tantanu/Ca 2211 aCu

from velocity triangles assuming

the power input to stage )(mW12

'ww CCu

rotors. theofexit andinlet at components l tangentiaare 21 ww andCC

where

or in terms of the axial velocity

From equation (a)

)tan-(tanmuCW 12a

Page 11: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsEnergy balance

pao

aoooopop

cuCT

uCTTcTTcTc

/)tan(tan

)tan(tan)()(

21

21

5

12135

pressure ratio at a stage

3 5

1 1

1

1

s

1

2

1 where, isentropic efficiency

Ex.

u 180 m/s, 43.9 , 0.85, 0.8,

150 / , 13.5, 288, 1.183 ,

higher due to centrifugal action

o s os

o o

os

a o s centrifugal

p Tp stage

p T

C m s T R R

Page 12: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Degree of reaction

enthalpy rise in rotor

enthalpy rise in the stager

s

hstatic

static h

is the ratio of static enthalpy in rotor to static enthalpy rise in stage

For incompressible isentropic flow Tds=dh-vdp

dh=vdp=dp/ Tds=0h=p/ ( constant )Thus enthalpy rise could be replaced by static pressure rise ( in the definition of )

1obut generally choose =0.5 at mid-plane of the stage.

Page 13: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

=0: all pressure rise only in stator=1: all pressure rise in only in rotor=0.5: half of pressure rise only in rotor and half is in stator. ( recommend design)

3 1 aAssume C ,and C . ( for simplicity)C const

5

1 21 (tan tan ) / 2

o stagnation stage s

a

T T T T

C u

uCa /,tan1

1 2tan (tan tan ) / 2

Page 14: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressorsspecial condition

=0 ( impulse type rotor)from equation 3

1 2(tan tan ) / 2aC u 1=-2 , velocities skewed left, h1=h2, T1=T2=1.0 (impulse type stator from equation 1)=1-Ca(tan1+tan2)/2u, 2=1velocities skewed right, C1=C2, h2=h3T2=T3

1 2

1(tan tan )

2 2

2 1

2 12 1 1 2

3 1

; symmetric angles

V , ; P P

c V cP P

=0.5from 2

Page 15: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsThree dimensional flow2-D1. the effects due to radial movement of the fluid are ignored.2. It is justified for hub-trip ratio>0.83. This occurs at later stages of compressor.

3-D are valid due to 1. due to difference in hub-trip ratio from inlet stages to later-stages, the annulus will have a substantial taper. Thus radial velocity occurs.2. due to whirl component, pressure increase with radius.

Page 16: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

)1/(

11

3s

21

1213

12

12

11

2211

]1[R

stageper rise pressure

)tan(tan

)tan(tan

)tan(tan

)(

tantantantan

o

ss

o

o

p

a

ooooos

a

a

ww

a

T

T

p

p

c

UC

TTTTT

UCm

UCm

CCUmW

C

U

Page 17: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors Design Process of an axial compressor

• (1) Choice of rotational speed at design point and annulus dimensions

• (2) Determination of number of stages, using an assumed efficiency at design point

• (3) Calculation of the air angles for each stage at the mean line

• (4) Determination of the variation of the air angles from root to tip

• (5) Selection of compressor blades using experimentally obtained cascade data

• (6) Check on efficiency previously assumed using the cascade data

• (7) Estimation on off-design performance• (8) Rig testing

Page 18: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors Design process:• Requirements:• A suitable design point under sea-level static conditions

(with =1.01 bar and , 12000 N as take off thrust, may emerge as follows:

• Compressor pressure ratio 4.15• Air-mass flow 20 kg/s• Turbine inlet temperature 1100 K• With these data specified, it is now necessary to

investigate the aerodynamic design of the compressor, turbine and other components of the engine. It will be assumed that the compressor has no inlet guide vanes, to keep weight and noise down. The design of the turbine will be considered in Chapter 7.

Page 19: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsRequirements:• choice of rotational speed and annulus dimensions;

• determination of number of stages, using an assumed efficiency;

• calculation of the air angles for each stage at mean radius;

• determination of the variation of the air angles from root to tip;

• investigation of compressibility effects

Page 20: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors Determination of rotational speed and annulus

dimensions:• Assumptions • Guidelines:• Tip speed ut=350 m/s• Axial velocity Ca=150-200 m/s• Hub-tip ratio at entry 0.4-0.6• Calculation of tip and hub radii at inlet• Assumptions Ca=150 m/s• Ut=350 m/s to be corrected to

250 rev/s•

Page 21: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Equations• continuity

thus

rpstt NtU ***2

at

rta C

r

rrACm

2

211 1

1

)(

12

2

11

2 a

r

rC

mr

t

ra

t

trt

rrr

N /&rget tosolve,2

350t

at

rt C

r

rr

2

21 1

Page 22: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

procedure

311

1

111

21

1

1

1

/106.1

8.2762

150C

bar 01.1,288

11

21

1

1

mkgRT

P

PT

T

P

P

c

CTT

C

PPKTT

oo

p

o

a

aoao

Page 23: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

• From equation (a)

0.60.4 /r

2/350

1

03837.0

r

2

2

fromrassume

rtN

r

rr

t

t

r

t

tr rr /N

0.40.2137260.6

0.50.2262246.3

0.60.2449227.5

tr

Page 24: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

• Consider rps250• Thus rr/rt=0.5, rt=0.2262, ut=2rt*rps=355.3 m/s

7.385V 2

1

211t

at CuGet

1RTa

165.111

a

vM t

Is ok. Discussed later. Results r-t=0.2262, r-r=0.1131, r-m=0.1697 m

Page 25: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow CompressorsAt exit of compressor

mmOutletmminlet

mrC

hrr

mh

rthushrhbut

CAmRT

PP

T

T

c

CTT

KT

P

P

T

Tbargiven

P

P

ma

mr

mm

a

op

ao

o

n

n

o

o

o

o

o

o

1491.0r ;1903.0r : ;1131.0r ;2262.0r :

1697.0 ;150 ;3.355u rps; 250N

results

m 1491.02

;19303.02

r ;0413.0)2(A

;044.0A , ;kg/m 03.3 bar; 84.3

P

P K; 3.441

2

;5.452 ;9.0 assume ,3174.1

4.01

n

1-n where

];19.4P [ 15.4

rtrt

t

t2

2223

2

222

12

o

22

2

2

1

o

22

2

1

2

1

2

2

1

2

Page 26: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

No. of stages To =overall = 452.5-288=164.5K• rise over a stage 10-30 K for subsonic• 4.5 for transonic• for rise over as stage=25• thus no. of stages =164.5/25 stages 7

- normally To5 is small at first stage

de haller criterion V2/V1 > 0.72

- work factor can be taken as 0.98, 0.93, 0.88 for 1st, 2nd, 3 rd stage and 0.83 for rest of the stages.

Page 27: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Stage by stage design;

• Consider middle plane

• stage 1

• for no vane at inlet wo CuT cp

0 ,/ 9.76 1 smCw

smCC ww /9.76,0 21

m/s 266u thus,r2u mm

Page 28: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

• Angles

o

a

w

a

w

a

thus

C

C

C

Cu

C

u

98.8

bladesrotor in deflection the

14.27tan

67.51tan

64.60tan

21

22

2

22

2

11

check de Haller

0.72 than less is which 79.0cos

cos

cos

cos/

2

1

1/

2

1

2

a

a

C

C

v

v

Page 29: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

856.02

1

)tan(tan2u

C-1

308

249.11 assume

esefficienci cpoly tropi pressures

12

21a

513

3

1

1

5

1

3s

u

CC

equation

KTTT

pT

T

p

p

w

ooo

oo

o

o

o

Page 30: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors• Second stage

05.41 ;06.11

tantan ;tantan

7.42&7.57(b) and (a)

)( 488.2tantan

7.0 take);tan(tan2

)2(

)( 6756.0tantan

)tan(tanc )1(

93.0,25

21

222

1

021

21

21

21

215p

5

aa

a

ao

o

C

u

C

u

solve

bu

C

a

uCT

KT

Page 31: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

0.28 ;24.51

)tan(tan2

);tan(tan

5.0 ,25T 0.88,

3

907.006.11cos

15.27cos

cos

cos

721.0cos

cos

V

V ; stage secondfor

bar 599.1308

251

33325308

21

21215

03

1

2

2

3

2

1

1

2

3

5.3

3

1

3

solvingu

CuCTc

K

stage

C

C

Hallerde

pP

P

T

aaop

os

o

o

o

Page 32: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

Kp

eperformanc

givingthus

take

o 35724333T bar; 992.1246.1599.1)(

246.1333

249.01

p

p

stage rd 3 of *

0.718 ofnumber Haller de ;65.28,92.50

685.0tantan24T

709.028cos

24.51cos

cos

cos

is no.Haller de

3o33

5.3

3o1

o

21

21o5

2

1

3

Page 33: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

:below summerized becan stages three theof eperformanc

).(71.27 38.51

7773.1150

6.26625.0)tan(tan

7267.01506.26683.0

10005.124tantan ;1

)tan(tan2

)tan(tanc

5,6 and 4 stages

/7.18492.50tan150 ;/9.8163.28tan150C

by given are s velocitie whirlthe.92.50

63.28 diagram velocity theofsymmetry From

120

1

21

3

21

21215p

21

012

21

1

5

1

3

the

andyielding

T

T

p

p

u

CuCT

smCsm

and

o

o

o

o

aao

Page 34: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

1oT

1

3

o

o

p

p

13 oo pp 3oT

3op

1oPStage456

1.9922.4472.968

357381405

1.2281.2131.199

2.4472.9683.560

381405429

0.4550.5210.592

Page 35: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors• Stage 7• At entry to the final stage the pressure and

temperature are 3.56 bar and 429 K. the required compressor delivery pressure is 4.15*1.01=4.192 bar. The pressure ratio of the seventh stage is thus given by

KTgiving

th

p

p

os

o

o

8.22

177.1429

T0.901

from detrmined becan ratio pressure the

give torequired rise re temperatue

177.156.3

192.4

5.3

os

71

3

Page 36: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Axial Flow Compressors

• the corresponding air angles, assuming 50 per cent reaction, are then 1=50.98,

0.717. ofnumber Haller de

ry satisfacto a )(52.28 10

2 with

Page 37: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EES– "Determination of the rotational speed and annulus dimensions"

– "Known Information"

– To_1=288 [K]; Po_1=101 [kPa]; m_dot=20[kg/s]; U_t=350 [m/s]

– $ifnot ParametricTable

– Ca_1=150[m/s];r_r/r_t=0.5;cp=1005;R=0.287;Gamma=1.4

– $endif

– Gamr=Gamma/(Gamma-1)

– m_dot=Rho_1*Ca_1*A_1 "mass balance"

– A_1=pi*(r_t^2-r_r^2) "relation between Area and eye dimensions"

– U_t=2*pi*r_t*N_rps

– C_1=Ca_1

– T_1=To_1-C_1^2/(2*cp)

– P_1/Po_1=(T_1/To_1)^Gamr

– Rho_1=P_1/(R*T_1)

– $TabStops 0.5 2 in

Page 38: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EESDetermination of the rotational speed and annulus dimensions

Known Information

To1 = 288 [K] Po1 = 101 [kPa] m = 20 [kg/s] U t = 350 [m/s]

Ca1 = 150 [m/s] r r

r t = 0.5

cp = 1005 R = 0.287 = 1.4

Gamr =

– 1

m = 1 · Ca1 · A1 mass balance

A1 = · ( r t2

– r r2

) relation between Area and eye dimensions

U t = 2 · · r t · N rps

C1 = Ca1

T1 = To1 – C1

2

2 · cp

P1

Po1 =

T1

To1

Gamr

1 = P1

R · T1

Page 39: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EESCalculate radii at exit section

Choose (round) rotational speed as 250 rps

N rps = 250

Thus calc new value for tip speed

rt1 = 0.2262

U t = 2 · · rt1 · N rps

rm = 0.1697

Known Information

To1 = 288 [K]

P ratio = 4.15

Assumptions

Etta inf = 0.9

Ca2 = Ca1

Ca1 = 150 [m/s]

Gamr =

– 1

Page 40: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EESnratio =

1

Etta inf · Gamrnratio=(n-1)/n=(1/etta inf )/gamr

P ratio = Po2

Po1

To2

To1 =

Po2

Po1

nratio

m = 2 · Ca2 · A2

A2 = 2 · · h · rm

C2 = Ca2

T2 = To2 – C2

2

2 · cp

P2

Po2 =

T2

To2

Gamr

2 = P2

R · T2

r t = rm + h

2

r r = rm – h

2

Page 41: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EES

A2 = 0.04398 Ca1 = 150 [m/s] Ca2 = 150 [m/s] cp = 1005 [J/kgK] C2 = 150 [m/s] Ettainf = 0.9

= 1.4 Gamr = 3.5 h = 0.041 [m] m = 20 [kg/s] nratio = 0.3175 Nrps = 250 [rev per sec]

Po1 = 101 [kPa] Po2 = 419.2 P2 = 384 [kPa] Pratio = 4.15 R = 0.287 [kJ/kgK] 2 = 3.032

rt1 = 0.2262 [m] rm = 0.1697 [m] rr = 0.1491 [m] rt = 0.1903 [m] To1 = 288 [K] To2 = 452.5 [K]

T2 = 441.3 [C] Ut = 355.3 [m/s]

Page 42: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EESCalculate number of stages

Known Information

To1 = 288 [K] Po1 = 101 [kPa] m = 20 [kg/s]

P ratio = 4.15 Tooutlet = 452.5

Assumptions

delTstage = 25

Ca1 = 150 [m/s] cp = 1005 R = 0.287 = 1.4

Gamr =

– 1

delTov = Tooutlet – To1

Nstages = delTov

delTstage

Page 43: Axial Flow Compressors. Elementary theory Axial Flow Compressors

Design calculations using EES

Ca1 = 150 [m/s] cp = 1005 [J/kgK] delTov = 164.5 delTstage = 25 = 1.4 Gamr = 3.5 m = 20 [kg/s] Nstages = 6.58

Po1 = 101 [kPa] Pratio = 4.15 R = 0.287 [kJ/kgK] To1 = 288 [K] Tooutlet = 452.5