Turbomachinery

Summary Equations

2

Important Equations

1 /1 /2 2

1 1

T P or T CPT P

2

0 0 0

1 /

2 2

1 1

1 /

.

.2

=

.

0

pP v

v

p RT Eq of state

p Vh e pv e h Enthalpy def

cc c R Kinetic theory

c

T P or Adiabatic eq of stateT P

T CP if s

Important Equations

2

2

2

2

2

2

2

0.24

. .32.174 778.16 .24.sec

6008.8sec

1716 1.4 / 0.4 6008.8sec

1287 1.4 / 0.4 1004.5

sec

BTUCplbm R

ft lbm ft lbf BTUlbf BTU lbm R

ftR

ftR RCp

mK

4

Gibbs Equation

02 022 1

01 01

0

022 1

01

02 2 1

01

1ln ln

,

1 ln

exp 1

p

p

Apply at stagnation state

T Ps sc T P

For adiabatic processes T constant

Ps sc P

P s sP R

1 /

1 /

11

pT T

Tad

T

Turbines

1 /

1 / 11

pc c

cad

c

Compressors

Efficiency

00 1

02 12

0

0

0max

0

1cos11

21.0888

287cos 14.3181.4

0.5787

0.5787 /1.0888 0.531cos0.5787 /14.318 0.0404

m TM RgFPp A g

M

Engm T

p A SI

if choked

Engp Am

SIT

7

Similarity – Compressible Flow

0102 02

01 01 01 01

,Re, ,m Tp T Nf

p T p T

01

01

m Tp

02

01

pp

01

NT

03 2

0 0

00 022 2

0 01 0

m m RTFlow coefficientND p D

Tp T NDHead coefficientN D T ND T

02 01 01 02 01 02( , , , , , , , , )p f D N m p T T

8

Total Pressure Mass Flow Parameter

• Defines common flow parameters.

• Corrected flow to standard day [eliminate effect of outside ambient conditions].

0

0 cosm RTP A

0

0

519

14.7

Tm mP

0 0

0 0STD STD

T pT p

9

RV V U

C W U

Frame of Reference Definitions

1

1 vx

y u

Velocity Componentsc u axial componentc 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

10

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

11

Compressor Airfoil/Cascade Design

• Compressor Cascade Nomenclature:

Camber - "metal" turning

Incidence

Deviation

Spacing or Solidity

*2

*1

*

*11 i

*22

#chord Airfoils cpitch D

12

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Axial Compressor Velocity Diagram: W C U

12

3N

Frames of Reference

13

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

C W U

: , ,cossin

0tan / 0

x x

u

u u

u x

Given C UC C WC CW C U

W W

14

Analysis of Stage Performance – Compressor Rotor

02 01 2 2 1 1

2 1 2 2 1 1

0 1 1 22 1

01 01 1

12 1 02 01

2 1

tan tan

1 tan tan

tan tan

p u u

U U U x x

stage x x x

p x

px

x x

c T T U C U C

C C C U C C

T UC C CT c T U C

cC T TC UC

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

Rotor (Blade)

Stator (Vane)

Relative = Absolute - Wheel Speed

2 1u u uC C Cacross rotor

W C U

15

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

N

16

Relative Flow Conditions

• T0R - changes with wheel speed across a rotor• T0 - no change with radius across a stator• No work done by a stationary object!

• Using Isentropic relation between P & T, Ideal Exit Relative Total Pressure is:

• P02Ri is ideal, assuming 100% efficiency

2 202 2 1

01 01

12

R

R p R

T U UT gJC T

/ 12 202 2 1

01 01

12

Ri

R p R

P U UP gJC T

17

Reaction• Definition: 1=rotor inlet 2=rotor exit 3=stator exit

• For axial machines

• For Cx constant

2 1

3 1 0

change across rotor transfer across stage

stage

h h hRh h h

12

21

22

2 uu CCUWW

R

12

21

22

2 uu CCUWW

R

18

Relationships Between Work Coefficient, Flow Coefficient, Reaction and Flow Angles for Constant Cx, U Machines

22tan 1RE

222tan 1RE

22tan 2RE

222tan 2RE

02

xC hEU U

Losses in Compressors and Turbines

• Compressors

• Turbines

• Alternative form

19

02 022

1 / 2Ri RP PZW

02 02

01 1

Ri R

R

P PP P

02 02

02 2

Ri R

R

P PYP P

0 02

01 01

1R R

R R

p pp p

20

Impact of Deviation on Airfoil Shape

• Carter’s Rule for compressor cascades

• Carter’s Rule for turbine cascades

• Metal angle decreased to achieve design exit angle goals

8i e

Tbwheres

/

/44 4 1

c e e

i e

i e e ic e

exit deviationairfoil camber

bor wheres

21

Compressor Design for Lift, Min Loss, Max Range & Choke Margin

Avoid flow reversal

IdealAvoid separation

Avoid leadingedge sep. bubble

Can also view this in terms of ps/p0

22

Compressor Loss Analysis - Lieblein's Dfactor• Correlation of cascade data [Velocities in Relative Frame]

or de Haller [ 0.72<W2/W1<1]

• Momentum thickness [ ] correlated to Dfactor [0 < Dfactor < .7]

• Loss coefficient related to cascade wake momentum thickness

• Efficiency related to loss coefficient

1 22

1 1

12u u

factor

V VVDV V

7.50.006 0.0002 fDec

2

01 02 2 1

01 1 2 1 2

cos2cos cos

x

x

p p cp p s c

12

22

2

coscos211

RSx

UC

E

23

Turbine - Zweifel Coefficient

Area Fideal

Area F

Solidity play important role in turbine efficiency: (1) spacing small, fluid getsmaximum turning force with large wall friction forces; (2) spacing large, fluidgets small turning force with small wall friction losses.

[ ]zideal

F Z MattinglyF

F is the tangential force per unit airfoil lengthF can be found by integrating the airfoil static pressure distribution

Component Matching Criteria

24

022

022 2 4 4 4 2

04

2 4 04 022 4 2 4

2 4 02

1 /1

_________________________________________________

/ /

C T T C f C pref

C c C T c C C C

C TC c C C T C C T

TRN N m f m f m m cT

TN N N N N N N N

T

m m p Tm m m m m m

p T

04

1 1

03 0504

02 04

_________________________________________________. 1

______________________________________________

1 1 1

C T

C T

m C m T

pCm T pT

C

mech efficiency w f w

C p pf C T

p p