1 turbomachinery centrifugal compressors class 13
TRANSCRIPT
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
Centrifugal Compressor Components
Collector scroll and vanes may be replaced with vaneless diffuser
5
Centrifugal Compressor Components
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
Centrifugal Compressor Design
• 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
8
9
Centrifugal Compressor Design: T.E.
Blade TE shape for opt. performance• Forward: against rotation• Radial• Backward: with rotation
10
Back
Forward
Straight
Centrifugal Compressor Design: T.E.
• 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
Centrifugal Compressor Design: T.E.
• 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
Centrifugal Compressor Design: T.E.
• 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
Centrifugal Compressor Design: T.E.
Radial T.E.
15
Centrifugal Compressor Design: T.E.
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]
Centrifugal Compressor Design: T.E.
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
Centrifugal Compressor Slip
22
Axial Compressor Slip
23
Axial Compressor Slip
24
Centrifugal Compressor Slip
• "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
Centrifugal Compressor Slip
• 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
Centrifugal Compressor Design
• 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
Centrifugal Compressor Design
Splitter impeller vane
Reduce effect of slip by using splitter vane to reduce diffusion
Backward sweep
29
Centrifugal Compressor Design
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
Centrifugal Compressor Design
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
Centrifugal Compressor Design
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
Centrifugal Compressor Design
• 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 Design
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
Centrifugal Compressor Design: Sta. 2
2 2 2 21 2 2 1
2 1 4.46 35.562 2
W W U Uh h
gJ gJ
41
Centrifugal Compressor Design
• 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
Diffuser Design: Sta. 23
• 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
Diffuser Design: Sta. 23
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
Vaneless Diffuser Performance: Sta. 23
• Swirl angle
Increases - compressible flow
Constant - incompressible flow
Reduced by friction
51
Vaneless Diffuser Performance: Sta. 23
• 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
Vaneless Diffuser Performance: Sta. 23
• 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
Centrifugal Compressor Design: Sta. 23
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
Vaned Diffuser Design: Sta. 23
• Diffuser Geometry
• Throat area is critical
• Pipe Diffuser better with M>1
57
Vaned Diffuser Design: Sta. 23
• Throat Blockage Drives Effectiveness
58
Vaned Diffuser Design: Sta. 23
• 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