1 turbomachinery lecture 5a - airfoil, cascade nomenclature - frames of reference - velocity...
TRANSCRIPT
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Turbomachinery Lecture 5a
- Airfoil, Cascade Nomenclature- Frames of Reference- Velocity Triangles - Euler’s Equation
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Airfoil Nomenclature• Chord: c or b = xTE-xLE; straight line connecting leading edge and
trailing edge• Camber line: locus of points halfway between upper and lower
surface, as measured perpendicular to mean camber line itself• Camber: maximum distance between mean camber line and chord
line• Angle of attack: , angle between freestream velocity and chord line
• Thickness t(x), tmax
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RV V U
C W U
������������������������������������������
Frame of Reference Definitions
1
1 vx
y u
Velocity Components
c u axial component
c c c circumferential component
Variable Stationary or
Absolute Relative V of moving
frame Velocity V, (u,v) VR U=r Velocity C, (Cx, Cu) W, (Wx,Wu) U=r Angle
,
If stationary
C W V ������������������������������������������
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Cascade Geometry Nomenclature
bbx
s pitch, spacing laterally from blade to blade solidity, c/s = b/s stagger angle; angle between chord line and axial
1 inlet flow angle to axial (absolute)2 exit flow angle to axial (absolute)
’1 inlet metal angle to axial (absolute)’2 exit metal angle to axial (absolute)
camber angle ’1 - ’2 turning 1 - 2
Note: flow exit angle does not equalexit metal angle
Note: PW angles referenced to normalnot axial
Concave Side-high V, low p- suction surface
Convex Side-high p, low V- pressure surface
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Compressor Airfoil/Cascade Design
• Compressor Cascade Nomenclature:
Camber - "metal" turning
Incidence +i more turning
Deviation + less turning
Spacing or Solidity
*2
*1
*
*11 i
*22
#chord Airfoils c
pitch D
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Velocity Diagrams• Apply mass conservation across stage
UxA = constant, but in 2D sense– Area change can be accomplished only through change in radius, not solidity.– In real machine, as temperature rises to rear, so does density, therefore normally
keep Ux constant and then trade increase with A decrease– same component in absolute or relative frame
• Rotational speed is added to rotor and then subtracted• If stage airfoils are identical in geometry, then turning is the same and
– V1 = V3
1 1 1 2 2 3 3
1 2 3
cos cos cosx
x x x x
C C C C
or
C C C C constant
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Velocity Diagrams• Velocity Diagram Convention
– Objectives: One set of equationsClear relation to the math
– Conclusion: Angles measured from +X AxisU defines +Y direction
Cx defines +X direction
Velocity Scales• For axial machines
• Vx = u >> Vr
• For radial machines• Vx << Vr at outer radius but Vx may be << or >> Vr at inner radius
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Velocity Diagrams: W C U ������������������������������������������
Compressor and turbine mounted on same shaftSpinning speed magnitude and direction same on both sides of combustorSuction [convex] side of turbine rotor leads in direction of rotationPressure [concave] side of compressor rotor leads in direction of rotation
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Frames of Reference
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Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
C W U ������������������������������������������
: , ,
cos
sin
0
tan / 0
x x
u
u u
u x
Given C U
C C W
C C
W C U
W W
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Velocity Diagrams: W C U ������������������������������������������
Another commonly seen view
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Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
Axial Compressor Velocity Diagram: W C U ������������������������������������������
12
3N
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Turbine Stage Geometry Nomenclature
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Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1
2 1
u uu
u u
C C C
C C
work on rotor
����������������������������
1
2
3
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Analysis of Cascade Forces
Fy
Fx
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Analysis of Cascade Forces
• Conservation mass, momentum
1 1 2 2
2 1
1 2 1 1 2 2
21 2
2 20 2 1 1 2
1 2
cos cos
sin sin
tan tan
[ ]
1tan
2
tan 0.5 tan tan
x
x
y x y y
x
x y m
m
c c c
F p p s
F c s c c m c c
c s
Bernoulli inviscid
p p p c cF F
s
where
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Analysis of Cascade Forces
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2 2
1 22
1/
2
1 12 2
2 tan tan12
tan
x
s xp
x x
yT
x
p T m
Define loss coefficient p c
p FC
c sc
FC
sc
C C
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2
3
sin cos
cos sin
sin 2cos
1 22
cos
/ cos
x m y m
x m y m
mL T m
m
D m
m x m
L F F
D F F
L sC C
lc l
sC
l
where c c
Analysis of Cascade Forces
• L, D are forces exerted by blade on fluid: , / /m mL D
Fy
Fx L
D
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Another View of Turbine Stage
W U
a x
V W
C C
C C
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Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1
2 1
u uu
u u
C C C
C C
work on rotor
����������������������������
1
2
3
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Combined Velocity Diagram of Turbine Stage
Across turbine rotor
Work across turbine rotor
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Effect on increased m
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Reason for including IGVs
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Euler’s Compressor / Turbine Equation• Work = Torque X Angular Velocity
– Angular Velocity of the Rotor– Torque About the Axis of the Rotor
21 21
Periodicity @ B & D, integer # of blades pitches apart Identical flow conditions along B & D
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Euler’s Equation• Only tangential force produces the torque on the
rotor. By the momentum equation:
• Since flow is periodic on B & D the pressure integral vanishes :
dmCPdAF U
DB
U
&
2 1U U UF m C C
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Euler’s Equation• Moment of rate of Tangential Momentum is Torque []:
– rate of work = F x dU = F x rd = [angular momentum][]– torque vector along axis of rotation
• Work rate or energy transfer rate or power:
• Power / unit mass = H = head
• 1st Law:
2 2 1 1U U Ur F m r C rC
2 2 1 1
1 1 2 2
c U U
t U U
Rate of work done on fluid m r C rC Pump equation
Rate of work done on rotor m rC r C Turbine equation
0W Q H
2 2 1 1 2 2 1 1U U U U
W pVdA dpH dh r C rC U C U C
m VdA
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Euler’s Equation• Euler's Equation Valid for:
– Steady Flow– Periodic Flow– Adiabatic Flow– Rotor produces all tangential forces
• Euler's Equation applies to pitch-wise averaged flow conditions, either along streamline or integrated from hub to tip.
2 22 2 1 1
0
2
/ secUnits
32.174 778.16sec
U U
fm m
f
U C U C ft BTUh
ft lbft lbgJ lblb BTU
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Euler’s Equation• Euler Equation applies directly for
incompressible flow, just omit “J” to use work instead of enthalpy:
2 2 1 1U UQ U C U CW
g
3
3
2
/ sec /
/ sec , / sec
/ sec
f m
U
m f
W ft lb lb ft
Q ft U C ft
g ft lb lb
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Compressor Stage Thermodynamic and Kinematic View
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Compressor Stage Thermodynamic and Kinematic View
Variable behavior - P0, T0, K.E.
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Compressor Stage Thermodynamic and Kinematic View
• Across rotor, power input is
• Across stator, power input is
• From mass conservation, and if cx = constant, then
• Euler’s equation
02 01pW mc T T
03 020W T T
1 11 1
1 1
1 11 1 2 2
1 1
tan tan
tan tan tan tan
u u
x x
u u
x x
C Wand
C C
C W Uor
C C
2 1 2 1
1 2
tan tan
tan tan
U U x
x
W mU C C mUC
or in terms of rotor blade angles
W mUC
Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
Rotor (Blade)
Stator (Vane)
Relative = Absolute - Wheel Speed
2 1u u uC C C
across rotor
W C U ������������������������������������������
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Analysis of Stage Performance
• Geometry = velocity triangles• Flow = isentropic relations [CD]• Thermodynamics =Euler eqn., etc.
– All static properties independent of frame of reference– All stagnation properties not constant in relative frame
0
211
2
TT
M
0/( 1)
211
2
pp
M
/( 1)
00
Tp p
T
02 01 2 2 1 1
12 1 02 01
2 1
tan tan
p u u
px
x x
c T T r C rCg
gcCT T
C rC
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Compressor Stage Thermodynamic and Kinematic View
• Euler’s equation continued
Large turning (1 - 2) within rotor leads to high work per stage, but this is in reality limited by boundary layer effects
for constant U, the work per pound of air decreases linearly
with increasing mass flow rate. Thus slight increases in m leads to decreased W, decreased pressure ratio leading to lower m
002 012 1
01 01
0 03 01 02 01 1 2tan tan
stage UU U
p
xstage
p
Th h U CW mU C C
m T c T
UCT T T T T
c
2 1 2 2 1 1
0 1 22 1
01 01 1
tan tan
1 tan tan
U U U x x
stage x x
p x
C C C U C c
T C CU
T c T U C
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Compressor Stage Thermodynamic and Kinematic View
• Stage pressure ratio is
03
01
03 01
03 01
1 103 03 0
01 01 01
Pr
1
s
sisen ad
sad
p
p
T T
T T
p T T
p T T
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Turbomachinery Lecture 5b
- Flow, Head, Work, Power Coefficients- Specific Speed
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Work Coefficient
• Define Work Coefficient:
• Applying Euler's Equation to E
02
hE
U
0 2 2 1 12 2
u uh U C U CE
U U
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Work Coefficient
2 1
2 1
2 1
1 2
2 1
for axial machines (const. r)
Remembering
Letting
1
u u
u u
u u
x x
x u u
x x
U U
C CE
UW C U
U W CE
UC C
C W CE
U C C
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Work Coefficient
• This equation relates 2 terms to velocity diagrams and applies to both compressors & turbines. The physics, represented by Euler’s Equation, matches the implications of Dimensional Analysis.
2 11 x u u
x x
C W CE
U C C
2 11 tan tanx
x
CE
UC
whereU
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Work vs. Flow Coefficients
-6.0
-4.0
-2.0
0.0
2.0
4.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 = Cx/U
E
Tan2-tan1=21
0
-1
-2
-4
-6-8
-10
Compressors
Turbines
2 11 1 (tan tan )E t
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Work and Flow CoefficientsExample:
1
1 2
201 01
2 1
0.5 30 88%
700 400
519 2116.8 /
:
, Pr, ( ), .
o
x x
Given the following machine data
E
U fps c c fps
T R p lbf ft
Find What kind of machine is it and
Tr turning E vs
Solution:
2
0 02 01 01
0
1 1
0.5 700 7009.787 / ( ) ( 1)
32.174 778
1 9.787 /(0.24 519) 1.0786
0,
Pr 1 1 0.88 1.0786 1 1 1.2637
p p
EU x xh Btu lbm c T T c T Tr
gJ x
Tr
Since E and h stage is a compressor
Tr
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Work and Flow Coefficients
Solution continued:
1 1 1
1 1 1
1 1 1 1
2 1 2 1 2
2 1
tan 400 tan 30 230.9
230.9 700 469.1
tan / 49.5
11 (tan tan ) tan tan 16.6
/
16.6 ( 49.5) 32.9
u x
u u
ou x
o
x
o
c c fps
W c U fps
W c
EE
c U
Rotor turning
For plotting on Work diagra
1
2 1
/ 400 / 700 0.571
tan tan tan( 16.6) tan(30) 0.875
x
m
c U
t
W1
C1
U
Cx1
11
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Work and Flow Coefficients
W1A
C1A
UA
Cx1
11
2 2 1 1 2 12 2 2
2 12
( )
2 ( )
/(2 )
u u u u
u uPW
c
U C U C U C ChE
U U UC Ch
EU g J U
Note:
Similar velocity triangles at different operating conditions will give thesame values of E (work) and (flow) coefficient
UB
Since angles stay the sameand Cx/U ratio stays the same,E is the same
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Work and Flow Coefficients
Pr
Flow, Wc
E
A
B
A,B
Pr
Flow, Wc
E
Nc1
Nc2
B1
B2
A1 A2
B1
B2A1
A2
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UA
W1A
C1ACx1
11
Work and Flow Coefficients
Effect on velocity triangles
Low E High E 2 11 tan tanx
x
CE
UC
whereU
1
W1A
C1ACx1
1
UA
1
1
,
arg
xFor same c U
Low E corresponds to small or low camber
High E corresponds to l e or high camber
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Work and Flow Coefficients• Effect on velocity triangles of varying E = (cu2 - cu1)/U is
design– low E results in low airfoil cambers– high E results in higher cambers
• Effect of varying = cx/U in design– low results in flat velocity triangles, low airfoil
staggers, and low airfoil cambers– high results in steep velocity triangles, higher airfoil
staggers, and higher airfoil cambers
Prove these statements by – sketching compressor stage and – sketching corresponding 3 sets of velocity triangles
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Nondimensional Parameters
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Dimensional Analysis of Turbomachines
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Returning to Head Coefficient• Also "Head" is P/ (Previously shown), 2 can be a
pressure coefficient. Incompressible form:
• Compressible form:
• Remembering compressor efficiency definitions, for incompressible flow:
02 2 2
h PE
U U
1 /01
2 2
1P rgJC T PE
U
/ideal
E Eactual
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Power Coefficient
3 2 5
Power
N D
QHPower
QHPower
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Power Coefficient
• Power Coefficient = Head Coefficient * Flow Coefficient
3 3 5 2 2 3
H Q H Q
N D N D ND
3 2 1
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Returning to Head Coefficient
• Now that has been shown to be corrected speed, return to
2
2 22
2 2
1
/
r
P PPU URT
P
U T
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Flow and Head Coefficients
• Many compressor people use & to represent stage performance scaled to design speed.
where "des" refers to the design point corrected flow etc. for the stage.
2
2
1
/1
/
/
/
desr
des
NP
N
m N
N
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Specific Speed
• Ns is a non-dimensional combination of so that diameter does not appear.
1/ 2 1/ 2 3/ 21/ 213/ 4 3/ 4 3/ 2 3/ 2
2
/
/s
Q N DN
H N D
4/3
2/1
H
QNN s
21 &