Download - 1 Turbomachinery Centrifugal Compressors Class 13.

1

TurbomachineryCentrifugal Compressors

Class 13

2

Centrifugal Compressor Design

• Geometry• Rothalpy• Impeller Design Considerations

• Eye – Inducer• Slip

• Diffuser Design Considerations• Volute Design Considerations

3

Centrifugal Compressor Components

4


Collector scroll and vanes may be replaced with vaneless diffuser

5


Centrifugal compressor= impeller+diffuser

Geometrical Definitions

Eye: L.E. of impellerImpeller: raises energy level of fluid by increasing radiusInducer: portion of impeller from eye to region where flow turns radialDiffuser: converts K.E. from impeller into P.E.. Adding vanes reduces size of diffuserScroll or volute: collects flow from diffuser and delivers to outlet

6


• Compressor’s role is to produce high pressure flow• High pressure is achieved by imparting kinetic energy

to flow by impeller / rotor.• Kinetic energy is converted to high pressure in

diffuser / stator.• As radius increases, pressure increases

– ½ pressure rise in impeller passage– ½ pressure rise in diffuser passage

• Diffuser may have vanes, but stator passages must have increasing area

• Volute cross-section area increases gradually to exit

7

Centrifugal Compressor Design• In class we defined "Rothalpy" for rotors:

• Since across the impeller I1=I2 then the change in swirl velocity U

explains why static enthalpy rise is so large for centrifugal compared to single-stage axial compressors

• If no prewhirl Cu1=0

2 2 2 2

0

2=constant

2 2 2 2U

rel

C UC W U UI h h h

gJ gJ gJ gJ

2 2 2 21 2 2 1

2 1

( ) ( )

2 2

W W U Uh h

gJ gJ

See result shown later is Example 1

0 2 2 [ ]UW h U C gH for pumps

9

Centrifugal Compressor Design: T.E.

Blade TE shape for opt. performance• Forward: against rotation• Radial• Backward: with rotation

10

Back

Forward

Straight


• Radial T.E.– Ideal no-slip view– Increased Cr/Wr for same U

does not change Cu or work– T02/T01 unchanged with

increased Cr

11


• Backward T.E.– Increased Cr/Wr for same U

decreases Cu or work– T02/T01 decreases with

increased Cr– Stable side of compressor map

12


• Forward T.E.– Increased Cr/Wr for same U

increases Cu or work– T02/T01 increases with

increased Cr– Unstable side of compressor

map

13

14


Radial T.E.

15


Backswept Impellor • Radial: dotted triangle• Backswept: solid

• Same radial component, same mass flow

• Relative velocity increased, absolute decreased

• Increases efficiency, but reduces work absorbing capacity [Cw2 lower]

17

• Prewhirl [in the direction of rotation] added from added upstream guide vanes

• In high PR compressors, may be necessary to provide prerotation to reduce high relative inlet velocity. Also reduces incidence / reduces twist lower bending stresses

• Vane radial design impact on inducer– Free vortex Cu 1/r: high incidence at low rh/rt designs

– Forced vortex Cu rn

Centrifugal Compressor Design: L.E.

Wo

Wn

- IGV will allow untwisted impellor inlet- Untwist will reduce rotor root bending stresses

Centrifugal Compressor Design: Tip

• Tip Leakage Issues– Flow from pressure [+] to suction [-] side over tip

– Flow from downstream [+] to upstream [-]

18

19

Review: Axial Compressor Slip

• Slip: flow does not leave impeller at metal angle

• Carter's Rule:

– Blade turning & solidity are important

• Viscosity plays small role within low loss incidence range

• T.E. thickness & shape significant

Constants &

21

nm

mn

20

Centrifugal Compressor Slip

• Slip: flow does not leave impeller at metal angle [even for inviscid flow] – due to less than perfect guidance from blade.

• If absolute flow enters impeller with no swirl, =0.• If impeller has swirl (wheel speed) , relative to the impeller the

flow has an angular velocity - called the relative eddy [from Helmholtz theorem].

• Effect of superimposing relative eddy and through flow at exit is one basis for concept of slip.

Relative eddy Relative eddy with throughflow

22

Axial Compressor Slip

23

Axial Compressor Slip

24


• "Slip" (Deviation) Reduces Swirl & Work

• Slip Factor

2 2

2 2

s u u i

u i u

V slip velocity W W

C C

2*

2tantan

UUir

WWC

2/1 UVs

2 2/U U idealC C *2 is the ideal relative frame exit flow angle

Vs

25


• Several Correlations for Centrifugal Impeller Slip Factor

Weisner

Stodola

Stanitz

1/2 *2

0.7 * 0.72 2

cos 11 1

(1 tan )Z Z

*2

*2 2

cos1 1

(1 tan )Z Z

*2 2

0.63 0.631 1

(1 tan )Z Z

2 2 2

2 2 2

*2 2

# /

: / /

1 tan 1

r

r x

where Z of impeller vanes and C U

Note C U is like C U for axials

Also for many problems

1

as Z

26

Centrifugal Compressor Design• In general, with possible prewhirl Cu10

• Introduce work done-input factor – In turbines [work done] < 1, due to boundary layer effects– In compressors [work input] > 1, need more power to

account for boundary layer effects

02 01 2 2 1 1 2 1 1U U UW Cp T T U C U C U U C

02 01 2 1 1

103

01 01

1 1

U

c

Cp T T U U C

pTR

p T

27


• Impeller Performance Effects:

03 01 0

03 01 2 2

( ) /

( )ideal

adU

h h pideal work to fluid

actual work to fluid h h U C

1103 03 03

01 01 01

122

01

1 1

1

idealc

cp

p T T

p T T

U

c T

2 2 1

02 01

0c u u

p

W C U no prewhirl C

c T T

28


Splitter impeller vane

Reduce effect of slip by using splitter vane to reduce diffusion

Backward sweep

29


01 01

1:

288 1

0.125 0.250

5.5 / 16,500

H eye T eye

Example Centrifugal compressor

T K p bar

D m D m

m kg s N rpm

• Calculate inlet blade angle at root and tip• Calculate Mach number at eye tip• Assume no whirl at inlet

• Axial velocity can be determined from continuity but density needs trail and error iteration

30


2 2 2 2 21

5 31 01 01

1 11 1

2 21

1 01

/ 1

1 1 11 1

01 01 1

0.125 0.0625 0.038

/ 1 10 / 287 288 1.21 /

5.5119.6

1.21 0.038

119.6288 280.9

2 2 1005

92 1.

T H

x

p

A r r m

Assume p RT kg m

mC C mps

A

CT T K

c

p T pp kPa

p T RT

313 /kg m

Assume no more iteration is needed

31


1 0 1 0

1 1

2 2 2 21 1 1

11

1

2 2216 108

60 60

tan 59.56 tan 40.39

129.96 216 250

0.747

T HT eye H eye

T eye H eyeT eye H eye

x x

T T

Trelative

r N r NU mps U mps

U U

C C

W C U mps

WM

RT

32


• Example #2: Dixon 7.1– A radial vaned centrifugal impeller is required to provide a

supply of pressurized air to a furnace. The specification requires that the fan produce a p0 rise equal to 7.5 cm of water at a volumetric flow rate (Q) of 0.2 m3/s.

– The fan impeller is made from [Z=30] thin sheet metal vanes, the ratio of the passage width to exit height=2 and r=0.1.

– Assume ad=0.75, m=0.95, slip can be estimated from Stanitz correlation

– Assume R=287 J/(kg-C), p01=101.3 bar, T01=288K

– Assume

5 31 01 01/ ( ) 10 / (287 293) 1.189 /p RT kg m

33

Centrifugal Compressor Design1- Determine the impeller vane exit speed

2- Determine the volumetric flow rate

0 02

2 2 2

2

2

3 20

02

/

1.98 1.981 1 0.934

30

1.2 10 9.81 0.075 882.96 /

882.9647.66 /

1.189 0.934 0.75

adU

U

p p

U C U

C

U Z

p g H x N m

pU m s

2 2 2

1/

2 2 /0.1 2 /

2 2 0.2 302 0.505

0.2 0.2 47.66

rannulus r r r

CQ A C C rb U rb U rb

Ub b

b r Zs r ZQ U r r Z

QZD r m

U

Stannitz21.98

34

Centrifugal Compressor Design2- Determine the volumetric flow rate cont’d

3- Power required if mechanical efficiency is 0.95

4- Determine specific speed

2 / 2 47.66 / 0.505 188.75 / sec

601802 / min

2

Rotational speed

U D rad

N rev

2 22 1.189 0.2 0.934 47.66

531.10.95 0.95m

Q Um WP W

1/2 1/2 1/2

3/4 3/4 3/40

1/2

3/4

188.75 0.20.593

( ) ( / ) (882.96 /1.189)

60 600.593 5.267 / min 0.84 / sec

( ) 2 2

s

s s

Q Qrad

gH p

NQN rev rad

gH

No swirl

35

Axial vs. Radial Machines

36

Diffuser Design: Sta. 23

• Rotation Effect on Diffuser Pressure Rise

– Rotation reduces boundary layer thickness and limits pressure rise in radial portion of impeller

– Johnston & Rothe varied area ratio & rotational speed in 2D diffuser test

– Rotated Diffuser with flow axis radial

37


Centrifugal Compressor & Diffuser

Given

D1 hub 3.00 in R 53.349 ft.lbf/lbm/RD1 shroud 6.00 in Cp 0.24 BTU/lbm/RD2 12.00 in alpha 1 0.00 degreesD3 diffuser 20.52 in back sweep 30.00 degreesN 27,500 RPM No. blades 32.00Flow 3.50 lbm/sec b2 0.350 in.Pt1 14.70 psia Slip Factor WeisnerTt1 519.00 Imp Eff 90.00% Poly, T-Tgamma 1.40 Diff Cp 0.40

Finda) Velocity diagram & flow properties at impeller inletb) Velocity diagram & flow properties at impeller exitc) Velocity diagram & flow properties at diffuser exitd) Adiabatic total to static efficiency at rotor & diffuser e) Adiabatic total to total efficiency at rotor & diffuser

Example #3: Will work through design of each component separately

Know U2, slip, not p02, T02,

38

Centrifugal Compressor Design: Sta. 1

Station 1

A1 sq in 21.2058Dm1 in 4.5000U1 ft/sec 539.9612FPt 0.2784guess M 0.2929Calc M 0.2930error 0.0000Tt/T 1.0172Pt/P 1.0614T 510.2420P 13.8500C1 = Cx1 324.3767U1 = Wu1 539.9612W1 629.9035Beta 1 59.0050

2 21 1

1/ 2

011 1 0

01 1

1

1

2( 1)* 21 0 1

*1 1

2 10 1 0 0

1 1 1 1

4 2

720

11

2

1/ 1 / ( / )

2

0

T HT H m

m

x

U

D DA D D D

RTN mU D FP

p A g

Guess M and calculate

M FP M

Check until M M

T T M p p T T

Since C C M gRT

W

2 21 1 1 1 1 1 1

11 1 1

180tan ( / )

x x x U

U x

U W C W W W

W W

39

Centrifugal Compressor Design: Sta. 2Station 2

U2 1439.8966Slip Factor 0.9177Vs 118.4382A2 13.1947Poly Exp't 0.3175Guess Pt2 56.1893Tt2 794.3942delta ho 66.0946Cu2 1149.2347Wu2 -290.6620Wu2i -172.2238Cr 298.3004C 1187.3177T 677.0886P2 32.1208rho 0.1280flow 3.5000error 0.0000

Pr t-t 3.8224Tr 1.5306Eta - ad 87.98%

Pr t-s 2.1851Eta - ad 47.16%

Alpha2 75.4492a2 1275.5646M2 0.9308

2 2

1/ 2 *2

*20.7

2 2 2 2

02

1

02 02 0 02 01

2 0 2 2 2 2

*2 2 2 2 2

2

/ 720

1 cos [ ]180 ;

[1 ]

519( /14.7) ( )

/

/ tan[ ]180

poly

s

p

U U U

U ideal U s r U ideal

U D N

Weisner backsweepZ

V U A D b

Guess p then calculate

T p h c T T

C gJ h U W C U

W W V C W

T

2

1202 2 02 2 02

2 2 2

2 2 2 2 1

1

02 01 02 01

12 2

( / )2

/( )

1/ /

1180

sin [ / ]

p

r

ad

U

CT p p T T

gJc

p RT

m A C check with m

PrPr p p Tr T T

Tr

C C

40


2 2 2 21 2 2 1

2 1 4.46 35.562 2

W W U Uh h

gJ gJ

41


• Impeller Performance:

– Efficiency up to 90 to 95% Polytropic

– Small size reduces efficiency

– Thickness, finish affect performance

– Clearance

– Multi-stage usually shrouded because axial location (and therefore, ) difficult to control

2

3.b

42


• Basic Parameters

h = height of 2D diffuser

Y = entrance width

L = Length

B = entrance boundary layer blockage

W = inlet velocity

• Rotation Number

1 2

1 2 2

To

h r UR

W W W

h

43


44

Radial Diffuser Performance: Sta. 23

• Pressure coefficient [compressor-type definition]

• Effectiveness, Diffuser Efficiency

3 2 3 2

202 22 2

12

p

P P P PC for low speed

P P C

22 22 3 2

22 3

Incompressibly: 1ppi

pi

C C C AC

C C A

2

2

3

for constant b: 1p diff i

rC

r

45

Rotation Reduces Diffuser Pressure Rise

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25

Area Ratio

PressureCoefficient

Ideal Cp

Ro=.000

Ro=.006

Ro=.029

Ro=.039

Diffuser Issues

46

Appreciable Stall

Jet Flow

No Stall

47

Radial Diffuser Design: Sta. 23

• Diffusers for Centrifugal Impellers:

Rotor Exit Flow Often Contains:

High Swirl up to 80

High Velocity M>1 often

Distortion Impeller Separation

Unsteady Flow Rotating Distortion

48

Radial Diffuser Design: Sta. 23

• Diffuser reduces velocity & swirl

• Diffuser Configurations:

Vaneless inexpensive, limited effectiveness,

work over wide range of operation, large, ok for industrial use

Vaned more expensive, better performance, smaller, sensitive to incidence effects

Pipe most expensive, best performance, compact, complete angle control

49

Vaneless Diffuser Performance: Sta. 23

• Vaneless Diffuser: removes swirl of fluid by increasing radius

• Continuity:

• Inviscid Tangential Momentum, (Friction Important for quantitative calculations!).

For constant angular momentum:

2 2 2 2

2

1r

rr

m C rb

C r bC

rb r

2 2

1u u

u

r C r C

Cr

2

2

tan

tan tan

u

r

Cconstant

C

50


• Swirl angle

Increases - compressible flow

Constant - incompressible flow

Reduced by friction

51


• Flow trace for vaneless, incompressible diffuser

• Ex: Radius ratio=2, =77.6: =180 ! In other words, radius will grow by factor of 2.26 as flow revolves 1800 around the diffuser

• Lot's of friction in Vaneless Diffusers

22 lntan

r

r

r

dr

C

Cd

C

rd

C

drdt

r

u

ur

Flow path in vaneless, incompressible diffuser is log spiral outward radially and circumferentially

52


• Diffuser Flow not Simple

– 1D Inviscid Analysis with Momentum and Continuity Gives Incorrect Pressure Rise in Diffuser (Too High)

– Need More Physics for Valid Solution – Friction !

53


Diffuser

P3 41.7482Cu3 672.0671

Guess Al3 77.4467Cr3 149.6501C3 688.5269T3 754.9461rho3 0.1493Area 3 22.5629Flow 3.5000Error 0.0000

Po3 49.8952Pr t-t 3.3942Eta ad t-t 0.7875

Pr t-s 2.8400Eta ad t-s 0.6548

3 2 02 2

23 2 2 3

3

3 2 3 3

22 2 3

3 3 3 3 02

3 3 3 3 3 2

3 3 3 3 1

103 3 02 3

1

03 01 02 01

[ ]

/ tan[ ]180

2

/( )

( / )

1/ /

1

diff

U Udiff

r U

U rp

diff

r

33 ad

p p Cp p p

DC C assuming

D

Guess C C

CC C C T T

gJc

p RT A D b

check m C A m

p p T T

PrPr p p Tr T T

Tr

54

Vaned Diffuser Design: Sta. 23

• Vaned Diffusers

– Vanes added to remove swirl of fluid at a higher rate than done by increasing radius

– Better Performance BUT matching is critical

– Steifel 1972

34% increase in throat area, same vane shape

15% greater rotor flow

55

Vaned Diffuser Design: Sta. 23• Diffuser Performance

Rundstadler & Deane 2D & Conical diffusers

Large Range of:

Area Ratio

Length/Height

Aspect Ratio

Mach No.

Reynolds No.

Blockage

• Effect of increasing diffuser angle: range of steady and transient flows

• Blockage drives effectiveness!

56


• Diffuser Geometry

• Throat area is critical

• Pipe Diffuser better with M>1

57


• Throat Blockage Drives Effectiveness

58


• Throat Blockage Driven by diffusion between blade exit and throat

59

Volute and Vaneless Diffuser: Sta. 34

• Impeller, Vaneless Diffuser, and Volute

60

Volute: Sta. 34

Radius characteristicof volute is Rv

Rv is function of angular position, usually linear from entrance to exit

61

Volute Design

• Volute Design Considerations

– Effects overall performance

– Tongue aligned at only one Cx/U

– Pressure Loss

– Space requirements

– Flow Uniformity

62

Volute Design

• Volute designed for uniform flow around circumference at diffuser exit

• Angular momentum sets swirl velocity

• Volute area set to match uniform flow from diffuser by angular momentum and continuity

63

Volute Design

• Common design practice is to maintain simple conservation of angular momentum along mean streamline in volute [r=rV]

1

1.0

1.0

2 [ ]

3

ref ref

V V

C r if no diffusionc

if some diffusionC r

reference exit of impeller no vanes

exit of diffuser

64

Volute Design

• The Best Volute Design is Not Perfect

Stagnation point on tongue creates local high pressure

Curvature stops at end of volute changing radial

pressure gradient

Tongue incidence at off-design 3 creates general distortion

Vanes or dual volutes used to reduce distortion

65

Volute Design

3 3 3

2

/ 360u vol u vol

v total

vol vol

v

-Constant angular momentum : r C r r C

-Since r varies, volute flow varies. At volute flow is : m m

-Circular volute area : A r

-r can be found at any by combining co

3 3

3

2 2 3 33 3

3

2

3 3

3 3 3 3

33

.

. :

: 2360

: 2360

2 cot 1360

u

vol

ur vol u vol vol

vol

vol u

vol r

ntinuity & angular momentum

r CAng momentum

r r

r CContinuity m r b C r C r

r r

r r CbFor incompressible flows

r r r r C

bor

r

2

3 3

vol volr r

r r

.

66

Volute Design1

3 33 3 3

2

3 33

3

cot 1 1 2 cot360 360

1

:

vol

u rr

uV vol u vol

vol

r b bSolving the quadratic

r r r

CP CRadial static pressure gradient in volute : C

r r r

r CLetting r r r C

r r

From mass

3 3

1/ 22 2

2 2

33 3 3

3 3

2

3 33

3

/ 2 / 2

/ 21 1

r

rvol r r r

vol Vol rvol u vol

rvol u

v v v v

u volvol

vol

conservation (rC = constant),

C C C C

r r C C C

Cp p r Cthen

r r r r r r

r C rp p

r r r

2

3

3 2r

vol

C

r

Volute exit radial velocity is arbitrary and can be changed to optimize diffusion.

67

Volute Design

1

3 33 3 3

3 3

cot 1 1 2 cot 0.1821360 360

594.21 / sec

0vol o-vol v

vol

uu vol

V

rvol

The equations for p , p , r all apply for all from 0 to 360 .

Volute Exit : Incompressible calculations.

r b b

r r r

r CC ft

r

C

3

2 23

/ 2 74.81 / sec

43.07

r

volvol u vol rV

V

C ft

rp p C C psia

r

68

Volute DesignVolute Exit

"C" 0.0076rv/r3 0.1311rv 1.3449rV 11.6049Cuv 594.1787Crv 74.8250Cv 598.8716Pv 42.8861Pov 48.6634

Pr t-s 2.9174Eta t-s 0.6744

Pr t-t 3.3104Eta t-t 0.7685

a3 1346.9071M3 0.4446

3 3

3 3 3

03 02 3 3

3

3

3 3

3

3

3

323

,

/( )

, ,

2

/

/

/ 2 ???

/

U r

diff exit

vol Vol vol

UU vol

Vol

r vol r

volU vol

p T conditions at exit of diffuser

p RT

T T C C from diffuser exit

Dr

b width of diffuser

r r r r r

CC from angular momentum

r r

C C from

r rp C

g

2

3

1

/ r volvol

Cr r

69

Volute DesignVolute Exit

"C" 0.0076rv/r3 0.1311rv 1.3449rV 11.6049Cuv 594.1787Crv 74.8250Cv 598.8716Pv 42.8861Pov 48.6634

Pr t-s 2.9174Eta t-s 0.6744

Pr t-t 3.3104Eta t-t 0.7685

a3 1346.9071M3 0.4446

Compressible Solution in Volute

P3 41.7482 Cr3 149.6501T3 754.9461 Cu3 672.0671rho 3 0.1493 r3 10.2600To3 794.3942 rCu3 6895.4079rCu3 6895.4079 b3 0.3500

rV/r3 rV Cuv Crv delta P1.0000 10.3 672.0671 74.8250 -0.18041.0100 10.4 665.4129 74.8250 -0.03911.0200 10.5 658.8893 74.8250 0.09391.0300 10.6 652.4923 74.8250 0.21911.0400 10.7 646.2183 74.8250 0.33711.0500 10.8 640.0639 74.8250 0.44811.0600 10.9 634.0255 74.8250 0.55271.0700 11.0 628.1000 74.8250 0.65111.0800 11.1 622.2843 74.8250 0.74381.0900 11.2 616.5753 74.8250 0.83091.1000 11.3 610.9700 74.8250 0.91291.1100 11.4 605.4658 74.8250 0.99001.1200 11.5 600.0599 74.8250 1.06251.1295 11.6 594.9876 74.8250 1.1277

70

Volute DesignVolute Radius vs Circumference

10.010.511.011.512.0

0 90 180 270 360

Circumference - Degrees

Vol

ute

Rad

ius -

in.

Pressure vs CircumferenceVolute Mean Flow

404244

464850

0 90 180 270 360

Circumference - Degrees

Pre

ss

ure

- p

sia

Ptv

Psv