1 Turbomachinery Lecture 5a - Airfoil, Cascade Nomenclature - Frames of Reference - Velocity Triangles - Euler’s Equation

1

Turbomachinery Lecture 5a

- Airfoil, Cascade Nomenclature- Frames of Reference- Velocity Triangles - Euler’s Equation

2

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

3

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 ��

4

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

5

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

6

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

7

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

8

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

Frames of Reference

9

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C C

across rotor

Rotor (Blade)

Stator (Vane)


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

10

Velocity Diagrams: W C U ��

Another commonly seen view

11

Rotor (Blade)

Stator (Vane)


2 1u u uC C C

across rotor

Rotor (Blade)

Stator (Vane)


2 1u u uC C C

across rotor

Axial Compressor Velocity Diagram: W C U ��

12

3N

13

Turbine Stage Geometry Nomenclature

14

Rotor (Blade)

Stator (Vane)


2 1

2 1

u uu

u u

C C C

C C

work on rotor

��

1

2

3

16

Analysis of Cascade Forces

Fy

Fx

17


• 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

18


20

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

19

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


• L, D are forces exerted by blade on fluid: , / /m mL D

Fy

Fx L

D

20

Another View of Turbine Stage

W U

a x

V W

C C

C C

21

Rotor (Blade)

Stator (Vane)


2 1

2 1

u uu

u u

C C C

C C

work on rotor

��

1

2

3

22

Combined Velocity Diagram of Turbine Stage

Across turbine rotor

Work across turbine rotor

Effect on increased m

Reason for including IGVs

25

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

26

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

27

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

28

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

29

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

30

Compressor Stage Thermodynamic and Kinematic View

31


Variable behavior - P0, T0, K.E.

32


• 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)


2 1u u uC C C

across rotor

Rotor (Blade)

Stator (Vane)


2 1u u uC C C

across rotor

W C U ��

33

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

34


• 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

35


• 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

36

Turbomachinery Lecture 5b

- Flow, Head, Work, Power Coefficients- Specific Speed

37

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

38

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

39

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

40

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

41

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

42

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

43


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

44


Pr

Flow, Wc

E

A

B

A,B

Pr

Flow, Wc

E

Nc1

Nc2

B1

B2

A1 A2

B1

B2A1

A2

45

UA

W1A

C1ACx1

11


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

46

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

47

Nondimensional Parameters

48

Dimensional Analysis of Turbomachines

49

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

50

Power Coefficient

3 2 5

Power

N D

QHPower

QHPower

51

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

52

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

53

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

54

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 &

Documents

1 Turbomachinery Lecture 5a - Airfoil, Cascade Nomenclature - Frames of Reference - Velocity Triangles - Euler’s Equation