1 turbomachinery class 11. 2 axial flow compressors: efficiency loss: centrifugal compressors...
TRANSCRIPT
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Turbomachinery
Class 11
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• Axial Flow Compressors: • Efficiency Loss:
• Centrifugal Compressors • Efficiency Loss:
• Axial Flow turbines: • Efficiency Loss:
1.4h
4b
1.75h
3.63[ .294] 1 0.586
[ .360] 1 10 ...cos
Tw tip
m
tip
Baskharone
rTurbine p K K Z
h r
ECompressor p
h E h
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Configuration Selection & Multidisciplinary Decisions
• Turbomachinery Design Requires Balance Between:
Performance
Weight
Cost
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Optimization Approach
• A Strategy:– Find feasible solution(s) within each discipline– Use each as starting points for multi-disciplined optimization
• Single vs. Multi-Disciplinary Optimization– A discipline’s potential vs. a balanced design– Trading away potential in one discipline to improve another (often
to find feasible design space)
• Pointers– Design variable count: less is more– Initially utilize large scale perturbations to identify gradients– Variable side constraints: consult with other disciplines for input
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Turbomachinery Design
• Consider Turbine Efficiency & Stress
• Performance - Smith Correlation for simplicity– "A Simple Correlation of Turbine Efficiency" S. F. Smith, Journal
of Royal Aeronautical Society, Vol 69, July 1965– Correlation of Rolls Royce data for 70 Turbines– Shows shape of velocity diagram is important for turbine efficiency– Correlation conditions
- Cx approximately constant
- Mach number - low enough
- Reaction - high enough
- Zero swirl at nozzle inlet
- "Good" airfoil shapes
- Corrected to zero clearance
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Smith Turbine Efficiency Correlation
94% 92% 90% 88%
0.8
1.2
1.6
2.0
2.4
2.8
0.4 0.6 0.8 1.0 1.2 1.4
Cx/u
E
Increasing
Note: The sign of E should be negative
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Turbomachinery Design
• Efficiency Variation on Smith Curve
– Increasing E from 1.33 to 2.4 [more negative] (at Cx/U=0.6):
• Higher turning increasing profile loss faster than work.
– Raising Cx/U from 0.76 to 1.13 (at E=1.2):• Higher velocity causes higher profile loss with no
additional work
– Remember - Mach number will also matter!
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Smith Turbine Efficiency Correlation
94% 92% 90% 88%
0.8
1.2
1.6
2.0
2.4
2.8
0.4 0.6 0.8 1.0 1.2 1.4
Cx/u
E
Increasing
Note: The sign of E should be negative
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Typical Optimization Formulations
Aero Structures
Efficiency Weight, Pull
Design Variables Objective Function(s) Thickness distributionChord distributionCG offsets (stacking)
Design Constraints
Design point flow & pressure profile Off-design lapse StabilityCasing clearance
Material propertiesStressTuningFlutter
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Airfoil Structural Overview
• Tools• Hand calculations, finite element analysis
• Design responses: stress, deflection, frequencies, mode shapes
• Design constraints• Strength, life• Tuning• Aero-elastic stability (flutter)• HCF [High Cycle Fatigue] margin
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Low Cycle Fatigue [LCF] Considerations
• Life Limited Parts Vs Limited Useful Life– Disks & high pressure cases – removed at end of certified life– Blades – removed for cause / wear out modes, such as airfoil
erosion• Assessment
– Attachment fillet Kt’s available via Peterson’s or FEA– Nominal stress– S-N curve
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Blade Vibration
• Cantilevered structures attain various modes: bending, torsion, coupled bending / torsion
• Each mode has its own natural frequency• Effect of rotation [shaft] is to stiffen structure and raise natural
frequency• Structural design should be resonance free operating condition
at: design speed, idle speed and other key operating points• Campbell diagram shows possible matches [Excitation] between
vibrational mode frequencies and multiples of shaft rotation [N]• Multiples of N caused by stators, blades, struts in neighboring
rows
• Examples: – Forced spring – mass damping– Chinook helicopter
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Motion of a damped spring-mass system
14
/ / /0
220
220
220
0
: 1 02
2 02
3 02
kmy cy ky where natural frequency of system
m
cCases signal decays overdamped
m
ccritically damped
m
cunderdamped
m
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Forced motion: Damped spring-mass system
15
/ / / cos[ ]my cy ky A t
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CHINOOK HELICOPTER
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Airfoil Tuning Represented on Campbell Diagram
• Airfoil frequency vs. rpm• Excitation orders
– Static flow disturbance relative to the rotating frame
– Source = inlet distortions– Freq = EO*RPM/60
• Project Requirements– 1st bending @ RL > 20%– 2nd & 3rd modes @ RL > 5%
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HCF Strength Assessed with Goodman Diagram
Steady Stress (ksi)Steady Stress (ksi)
Vib
rato
ry S
tres
s (k
si,
0-p
eak)
Vib
rato
ry S
tres
s (k
si,
0-p
eak)
UltimateUltimateStrengthStrength
AlternatingAlternatingStrengthStrength
AMS4928 R=-1 Goodman DiagramAMS4928 R=-1 Goodman DiagramSmooth, Minimum PropertiesSmooth, Minimum Properties
Vibratory LimitVibratory Limit
Steady LimitSteady Limit
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Stresses
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Secondary Air Systems
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S S RR
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Turbomachinery Design Structural Considerations
Centrifugal stresses in rotating components• Rotor airfoil stresses
– Centrifugal due to blade rotation [cent]• Rim web thickness
– Rotating airfoil inserted into solid annulus (disk rim). – Airfoil hub tensile stress smeared out over rim
• Disk stress [disk]– Torsional: Tangential disk stress required to transfer
shaft horsepower to the airfoils– Thermal: Stresses arising from radial thermal
gradients• Cyclic effect called low-cycle fatigue (LCF)
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Turbomachinery DesignStructural Considerations
• Blade pitch [s] at Rmean chosen for performance s/b, h/b values• Need to check if [s] too small for disc rim attachment
• number of blades have an upper limit• Fir tree holds blade from radial movement, cover plates for axial
• slight movement allowed to damp unwanted vibrations• manufacturing tolerances critical in fir tree region
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Structural Design Considerations• Airfoil Centrifugal Stress
Blade of constant cross section has mass:
2BMPull r
g
2RT DD
h
4
T Rm
D Dr
2 2
2.
222.
2 2 2.
[ ]
sec
12
0.5 2 2
T
H
centrifugal m
centrifugalcent
m m
R
cent T H
m TR
cent m T H
dF Rdm R AdR
dFdRdR
A
for constant blade cross tional area
U RRdR
R
N r r
An
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Turbomachinery DesignStructural Considerations
Centrifugal stress is limited by blade material properties
2
2
3
[ ][ ]
[ ]
2[ / ]
2 2 12 2 2 12 60 30
0.28 / [ ]
[ ]2
ccs
B
T H T Hmean
metal metal cs
T Hblade
ccs
Blade Pull P lbfStress psi
Blade cross section area A in
MPull r
g
D D D D N NR rad s
M mass L A lbm in for steel
D DL in
PStress
A
2
2
2 2 12 900 2 790,000metal anT H T H A ND D D D
Ng
Aan
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Turbomachinery DesignStructural Considerations
Centrifugal stress is limited by blade material properties
Gas bending
Cent. bending
L
From Rear
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Mechanical Design – Minimizing Root Moments
Blade is balanced about rim to minimizeBlade is balanced about rim to minimize
Bearing stress maldistributionBearing stress maldistribution
Bending stress on disk webBending stress on disk web
Disk rim rollingDisk rim rolling
Blade airfoil is tilted to offset root bending stressesBlade airfoil is tilted to offset root bending stresses
Axial & tangential tiltsAxial & tangential tilts
CGCG
Air pressureAir pressure
PullPull
CG OffsetCG Offset
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Turbomachinery DesignStructural Considerations
• Bending stress on a cantilevered bead under aerodynamic loading [Kerrebrock]
• Centrifugal stress is typically larger than bending stress
31
2
max
12
x Tbending s
T
C rspU c t
c/s=
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Typical Centrifugal Stress Values
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Typical Centrifugal Stress Values
0 0
20 3 2 3 3 2 2
3 2 3 2
3 2
: 1200 4.0
0.75 0.51 10,500 / min
50% 0.7 2.5 [ ]
1 2 3
/ /
/ tan tan
tan tan /
T H
mean
u u u u u u
u u
First stage turbine T K p bar
r m r m N rev
R E
stator inlet stator exit rotor exit
E h U C C U C W U C W U
E W W U
R
3 2
2 3 2 2
2 2
2 68.2 46.98
50%
/ 2 0.315 2 346.4
242.45 / cos 652.86m T H m m
x m x
For R
at r r r U Nr mps
C U mps C C mps
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Typical Centrifugal Stress Values
22 02 2
/ 1
2 22
01 01
2 2 2
3
2 2
96%
/ 2 1016.3
11 1.986
39.1 /
8,000 /
412.32 0.518,000 1 2.437
3 2 0.75
stator
p
x
m
c
Given
T T C c K
p Tp bar
p T
m A C kg s
For tapered blade of material kg m
MPa
Need to determine if blade with this stress level will last 1000hr to rupture
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• Airfoils inserted into slots of otherwise solid annulus [rim]
• Airfoil tensile stress is treated as ‘smeared out’ over rim
• Disk supports rim and connects to shaft
Turbo Design - Structural Considerations
2 [ ]c blades hub
disk bladesrim
n A
r x
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Turbomachinery DesignStructural Considerations
• The average tangential stress due to inertia then is:
• The contribution of the external force to the average tangential stress is
• so that the total average tangential stress becomes:
2
2V
t
F I
A A
A
Frim2
A
F
A
I rimt
2
2
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Turbomachinery DesignStructural Considerations
• For the same speed and pull, the average tangential stress can be reduced by:
– increasing disk cross sectional area
– decreasing disk polar moment of inertia - moving mass to ID of disk
A
F
A
I rimt
2
2
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Turbomachinery DesignStructural Considerations
• Stress and major flow design parameters (, E) relate directly to achievable
• Recalling from Dimensional Analysis:
• Higher stress () at constant N and Dmean occurs on longer blades and lower flow coefficient ()
2
1
1
x
x
C m m N
U AU AN D
C m N
U D
m N
D
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Turbomachinery DesignStructural Considerations
• Also :
• Flow, Density & Work are set by cycle requirements
• Stress (P/A) capability is set by material, temperature, & blade configuration
• Parametric effects– increased N increased (to first order), decreased E (to 2nd
order)– increased D decreased (to first order), decreased E (to 2nd
order)
02 2
1xh C m NE
N D U D
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40Plot shows effect of +20% change in N, D & stress on Cx/U, E, and Efficiency. Stress changes allowable blade height or annulus area.
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Turbomachinery Gaspath Design Problem• Objective: to illustrate interaction of several design parameters
, stress level (cent), x, cost, weight flowpath dimensions
• Design a baseline turbine and 3 alternative configurations
– Dmean or weight and cost on
– Aan or Cx or weight on
– Stress level on • All turbine designs have the following conditions
1 2
01 01
1 2
0
2
1 1 1
50 /
200 28,800 2200
50%
1.0
2 cossin 1.0
cos /
x x
mean mean
x x
x x b xw x
x mean b mean
m lbm s C C
p psia psf T R
D D R
span LAR h same
b b
b b n bZ where
s D n D
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Turbomachinery Gaspath Design Problem• Design: fill in the missing blanks in the table below
• Account for tip clearance losses as a 2% debit in efficiency
• Remember cent AanN2 and cost blade count (nb)
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Turbomachinery Gaspath Design Problem• Base Case: Assume only for this case M1=0.8 is given.
1/ 220 01 1 1 1 1
1 1 11 0 1 0 1 0 0
11
0
01 1 2
1 1 1
10.8 ( ) 1
2
0.7532 1731.9
2 2 2 ( 2) 2 0.5tan 1.666 59
2 2 0.9
cos 1731.9 cos(59) 891.0x
a TC C C C CM f M M
a a a a T a a
CC fps
a
E R
C C fps
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Turbomachinery Gaspath Design Problem• Base Case: Assume only for this case M1=0.8 is given.
01 2 21
01 1 1
/ 891.0 / 0.9 990
1202 2 1.2605 15.126
2 / 60 2 15,000
0.3087 44.45cos ( )
/( ) 44.45 /( 15.126) 0.93
/ 0.93
x
mean mean
an
an mean
x
U C fps
U UD R ft in
N
m TA ft in
p MFP M
L A D in
b L AR L in
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Turbomachinery Gaspath Design Problem• Base Case:
2 2 10 2 2
01 1
44.45 15,000 1 10 [ / min ]
2 2 0.5 2tan 0.5555 29.0 [ ]
2 2 0.9
29 ( 59) 88
2 cos59sin88 1.177
cos 29
60.14 60
an
xw
x meanb
x
A N x in
R Eby convention
Z
Dn Number of airfoils
b
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Turbomachinery Gaspath Design Problem• Base Case:
0
2
0 78.28 /
2.0, 0.9 90.7 2.0( ) 88.7
Find h
EUh Btu lbm
gJ
Find from Smith turbine correlation
E tip clearance
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Turbomachinery Gaspath Design Problem
• Baseline Design:
• Account for tip clearance losses as a 2% debit in efficiency
• Remember cent AanN2 and cost blade count (nb)
2
[ ]790,000
anc
A NStress psi
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Turbomachinery Gaspath Design Problem• Alternate Design 1: Given N, Aan1N2, Dmean1
2
102 2
2
15% 1.15 1.15 990 1139.0
15% 1.15 15.126 1.15 17.39
1 10/( ) 0.813
17.39 15,000
base
mean mean base
an
an mean
an mean
U increased by U U fps
D increased by D D in
A N constant, therefore compute new span L
xL A N D N in
A D L
2
02 2
1
17.39 0.813 44.42
/ 0.813
78.28 32.174 7781.511
/( ) 1139
2 2 2 ( 1.511) 2(0.5) 1.255tan
2 2
x
in
b L AR in
hE
U gJ
E R
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Turbomachinery Gaspath Design Problem• Alternate Design 1:
1
010 1
01 1 1 1 1
1/ 221
1 1 1 10
011 1 11 1 1
0 0 0
11
1.0883 1.0883 50 2200 0.2873
cos 200 17.14 0.825cos cos
1( ) 1
2
49.02 2200cos cos 2.018 cos
1139.0
tan
an
x
Guess
m TFP Get M
p A
Cf M M M Get C
a
RTC C C CGet
U a U a a
11 1
0
01 2
(1.255 / )
: , , , 4
: 58.8 / 0.7527x
CUnknowns M with equations set up iteration
a
Solution C U
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Turbomachinery Gaspath Design Problem• Alternate Design 1:
01 2
1 1
0
58.8 / 0.7527
2 1.511 2 0.5tan 18.75
2 2 0.7527
18.75 ( 58.8) 77.55
2 cos( 58.8)sin(77.55) 1.068
cos(18.75)
x
w
xw
x meanb
C U
E R
Determine solidity from Z
Z
Determine the number of airfoils
Dn
b
1.068 17.39
71.76 720.8
[ ] 93.3% 2%[ ] 91.3%
bx
n
Determine turbine efficiency
from Smith chart tip clarance
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Turbomachinery Gaspath Design Problem• Summary