Aircraft propulsion turbomachine 2 d

Aeropropulsion Unit

Introduction and 2-D Analysis in

Turbomachinery

2005 - 2010 International School of Engineering, Chulalongkorn University Regular Program and International Double Degree Program, Kasetsart University

Assist. Prof. Anurak Atthasit, Ph.D.

Aeropropulsion Unit

2 A. ATTHASIT Kasetsart University

Work input into compressors

Intro : Thermodynamics concept

Compressor Work/Mass

compressor p3 t3 p2 t2w C T C T

compressor t3 t2w h h

Compressor Pressure Ratio

1t3 t3

c

t 2 t 2

P T

P T

1

p t2

compressor c

c

C Tw 1

Compressor Efficiency

Aeropropulsion Unit


General Design Consideration

Intro : Comp. Design

First step of design : Choice of stage loading

(ex. the pressure rise in relation to the

number of stages and the rotational speed)

Decision to have an axial radial compressor

(ex. For aircraft propulsion the high flow rate per

unit area of the axial is a big advantage)

(ex. Radial compressor has a huge cost advantage

over the axial)

Aeropropulsion Unit


Axial Compressor

Preliminary design : at a mean radius (pitchline)

Criteria have to be chosen for satisfactory

• Blade loading

• Pressure rise at the walls

• Maximum Mach number

Aeropropulsion Unit


Blade Loading : de Haller n° Criteria for endwall loading or pressure rise :

De Haller number 2 1V / V 0.72

De Haller (1953)

But lowing value ---> excessive losses

Aeropropulsion Unit


Blade Loading : diffusion factor

Diffusion factor

High fluid deflection = high rate of diffusion

Definition & termology :

Aeropropulsion Unit


Blade Loading : diffusion factor

Diffusion factor

High velocity gradient ---> high boundary layer thicnkness ---> high losses

w1 2

max 2

1 1

C sV V

V V 2 cDV V

w2

1 1

CV sD 1

V 2V c

max 1 w

sV V 0.5( C )

c When

csolidity

s

Aeropropulsion Unit


Diffusion factor

Diffusion factor

w2

1 1

CV sD 1

V 2V c

Wide range of cascade NACA tests

Criterian's limit :

D < 0.6

Advantage :

'D' help to construct

the velocity diagram

Aeropropulsion Unit


Many criterias left for prelim-design

Degree of reaction

Degree of reaction (°Rc) : T1

T2

T3

2 1 2 1c

3 1 3 1

h h T TR

h h T T

One

stag

e of

com

pre

ssor

°Rc desirable is 0.5 (share the burden)

Stage loading

p tt t

2 2 2

c Th h

( r ) U U

0.3 0.35

Aeropropulsion Unit


Many criterias left for preliminary design

Flow Coefficient

a1 a1C C

r U

0.45 0.55

Flow coefficient

Aeropropulsion Unit


3D Flow Field

Typical gas turbine design procedure

Design procedure

Market

research

Specification Customer

requirements

Preliminary studies: Choice of cycle,

Type of turbomachinery, layout

Turmodynamic design point studies

Aerodynamics of compressor,

turbine, Intake, exhaust, etc.

Ex: Take-off-Thrust

(12,000 N)

c 4.15 m 20kg / s

4T 1100K

Axial flow, Turbojet

Rotational Speed,

Annulus Dim,

N° of stages,

Air Angles,

Balde Design, Etc.

Aeropropulsion Unit


3D Flow Field

Typical gas turbine d'sign procedure

Design procedure

Ex: Take-off-Thrust

(12,000 N)

c 4.15 m 20kg / s

4T 1100K

Axial flow, Turbojet

Rotational Speed,

Annulus Dim,

N° of stages,

Air Angles,

Balde Design, Etc.

Desired

Performance

Parameters

Turbomachinery

Design Criterias

Blade Loading,

etc..

Aeropropulsion Unit


Three Dimensional Flow

Sum of 2-D flow:

- Throughflow field

- Cascade field (blade-to-blade)

- Secondary flow field

3-D Flow

Aeropropulsion Unit


Throughflow field

2-D Flow

Aeropropulsion Unit


Throughflow field Mass flow parameter

2-D Flow

Pm PV V PV M

A RT RRT RT T

1

1 2t t 2

t t

T Tm P 1MFP M M 1 M

A P R P R 2T

Mass Flow Parameter (MFP)

Aeropropulsion Unit


Cascade field

2-D Flow

Aeropropulsion Unit


Cascade field

2-D Flow

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Secondary field

2-D Flow

Aeropropulsion Unit


Coordinate systems

2-D Flow

1 2 3

• Absolute coordinate : U,u2,V2

(fixed to the compressor housing)

U( r )

2V

2u2RV

• Relative coordinate : V2R

(fixed to the rotating blades)

Aeropropulsion Unit


Euler's Equation

Designing Tools

Fluid Mech Eq :

Torque & Work :

A e e i im( r v rv )

C AW

1st Law of Thermo Eq :

C te tiW m( h h )

te ti e e i ih h ( r v rv ) 'This relation will be used to obtain total

temperature changes throughout the blade' p te ti e e i iC (T T ) ( r v rv )

Aeropropulsion Unit


Velocity Diagrams Velocity Diagrams

Aeropropulsion Unit


Velocity Diagrams & Euler's Equation Velocity Diagrams

p t2 t1 2 2 1 1C (T T ) ( r v r v )

2r1r

2 1 2

p t2 t1 1 2

1

u uC (T T ) r tan tan

r u

2 1 2

p t2 t1 2 1

1


r u

Aeropropulsion Unit


ii i i i i ii i i i

i i i i i

PPV cos V cos PmV cos M

A RT RRT RT T

Velocity Diagrams & Flow Annulus Area Velocity Diagrams

2r1r

i

ti

i

ti i M

m TA

P cos MFP

Aeropropulsion Unit


Coordinate systems : important!

2-D Flow : Caution!

1 2 3

• Absolute coordinate : U,u2,V2

(fixed to the compressor housing)

• Relative coordinate : V2R

(fixed to the rotating blades)

t1 t1R t1 t1RT ,T ,P ,P

t 2 t2R t2 t2RT ,T ,P ,P

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Mean radius geometry

Constant mean radius

Nomenclature for constant mean radius

Constant Mean Radius

Aeropropulsion Unit


Mean radius stage calculation

Flow without loss

rm

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Stage calculation



&Velocity diagram

t1

1 2

1

TT

1 1 / 2 M

t1

1 /( 1 )2

1

PP

1 1 / 2 M

1t1 t1

1 1

P T

P T

Aeropropulsion Unit



Flow without loss

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Step 1: Find Triangle "V" at Station 1

Properties

Geometry

V_Compo

V_Compo

(relative)

Properties

(relative)

1T 1a1P

1 1 1V M a 1 1 1u V cos 1 1 1v V sin

1

t1

1

t1 1 M

m TA

P cos MFP

U r

1R 1v r v 2 2

1R 1 1RV u v 1R1R

1

VM

a

t1R 1 1RT f (T ,M , ) t1R 1 1 t1RP g( P ,T ,T , )

Aeropropulsion Unit



Flow without loss

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Step 2: Find "P&T" at Station 2

Properties :

t 2 t2T ,P ?

t 2R t2RT ,P ?

t 2 t3 t1 c,stageP P P ( 1 ) /

t 2t 2 t3 t1

t1

PT T T

P

t 2R t1R t2R t1RT T ,P P

Aeropropulsion Unit



Flow without loss

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Step 3: Find Triangle "V" at Station 2 ?

Which Eq can link "V" from station 1-2 ?

2 1 2

p t2 t1 1 2

1


r u

Euler's Equation :

Then 2 is determined

Aeropropulsion Unit



Flow without loss

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK


22 1 2R 2 2

1

uu u ,v u tan

u

2RV

1 22 2R 2

2

vv U v , tan

u 2V

Step 5: Find "T&P" at Station 2

/( 1 )2

2 22 t2 2 t2

P t2

V TT T ,P P

2C T

Aeropropulsion Unit



Flow without loss

t1 t1

m

1 3

t321 3

1 t1

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

PuM M 0.7, 1.1, 1.3

u P

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Step 6: Find the rest at Station 2

2 2 2a ,M ,A


t3 t3R t3 t3RP ,P ,T ,T ?

3 3P ,T

3 3 3 3 3a ,V ,u ,v ,A

Aeropropulsion Unit



Flow without loss

Rotor : Adiabatic in the relative reference frame

Stator : Adiabatic in the absolute reference frame

§ I M P O R T A N T §

Aeropropulsion Unit


Flow with loss : Introduction

Flow with loss

Adiabatic Stage Efficiency s

( 1 ) /

t3s t1 t3s t1 t3 t1s

t3 t1 t3 t1 t3 t1

h h T T ( P / P ) 1

h h T T T / T 1

When t t3 t1T T T

/( 1 )

t3 ts

t1 t1

P T1

P T

Aeropropulsion Unit


Flow with loss : Introduction

Flow with loss

ti ti ti t t tc

t t t t t t

dh dT dT / T dP / P1e

dh dT dT / T dT / T

Adiabatic Polytropic Efficiency ce

t3 t1c

t3 t1

ln( P / P )1e

ln(T / T )

0.9

(Preliminary design)

When t t3 t1T T T

c ce /( 1 ) e /( 1 )

t3 t3 t

t1 t1 t1

P T T1

P T T

Aeropropulsion Unit


Flow with loss : Life is still not easy …

Flow with loss

Adiabatic Polytropic Efficiency

Adiabatic Stage Efficiency When

are

unknown …

Aeropropulsion Unit


Flow with loss : Experiment data

Flow with loss

Cascade tests result :

• The optimum angle

(minimum loss)

• Profile drag coefficient

(cascade efficiency)

(must be increased to account

for end losses (e.g., tip leakage,

wall boundary layer or cavity

leakage)

Aeropropulsion Unit


Flow with loss : Cascade data

Flow with loss

Total pressure loss coefficient

t ,drop ti tec 2

dynamic i

P P P

P V / 2

Remark :

• Rotor - relative reference

• Stator - fixed reference

Aeropropulsion Unit


Flow with loss : Total pressure loss coefficient

Flow with loss

t ,drop ti tec 2

dynamic i

P P P

P V / 2

Example for Rotor

t1R t2Rcr 2

1 1R

P P

V / 2

2 2

t2R 1 1R 1 1Rcr cr

t1R t1R t1R

P V PM1 1

P 2P 2P

2

t2R 1Rcr /( 1 )

t1R 2

1R

P M / 21

P 11 M

2

Aeropropulsion Unit


Flow with loss : Total pressure loss coefficient

Flow with loss

For Rotor

2

t2R 1Rcr /( 1 )

t1R 2

1R

P M / 21

P 11 M

2

For Stator

2

t3 2cs /( 1 )

t 2 2

2

P M / 21

P 11 M

2

How can we evaluate the total

pressure ratio of a stage ?

t 3

t1

P

P

2 1Rcs , 2 2 R cr , 1R 1

t3 t 2 t 2R t1R2 1

t2 2 t2R t1R 1 t1M MM M M M

P P P PP P

P P P P P P

Aeropropulsion Unit



Flow with loss

Stage calculation



&Velocity diagram

t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

tT

Aeropropulsion Unit



Flow with loss

t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK


Properties

Geometry

V_Compo

V_Compo

(relative)

Properties

(relative)

1T 1a1P

1 1 1V M a 1 1 1u V cos 1 1 1v V sin

1

t1

1

t1 1 M

m TA

P cos MFP

U r

1R 1v r v 2 2

1R 1 1RV u v 1R1R

1

VM

a

t1R 1 1RT f (T ,M , ) t1R 1 1 t1RP g( P ,T ,T , )

Aeropropulsion Unit



Flow with loss

Step 2: Find "P&T" at Station 2

Properties :

t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

cr , 1R

t 2Rt2R t1R

t1R M

PP P

P

t 2R t1RT T 290.07K

t 2 t1 tT T T

t 2PStill don't know ,let's keep it later!

Aeropropulsion Unit



Flow with loss

Step 3: Euler's Equation : ?

2 1 2

p t2 t1 1 2

1


r u

Euler's Equation :

Then 2 is determined t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Aeropropulsion Unit



Flow with loss


22 1 2R 2 2

1

uu u ,v u tan

u

2RV

1 22 2R 2

2

vv U v , tan

u 2V

Step 5: Find "T&P" at Station 2 t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

/( 1 )2

2 22 t2 2 t2R

P t2R

V TT T ,P P

2C T

Aeropropulsion Unit




2 2 2a ,M ,A


t3 t3R t3 t3RP ,P ,T ,T ?

3 3P ,T

3 3 3 3 3a ,V ,u ,v ,A

Flow with loss

t1 t1

m

1 3

21 3 t

1

cr cs

T 288.16K ,P 101.3kPa

1000rad / s,r 0.3048m

40 , 1,m 22.68kg / s

uM M 0.7, 1.1, T 22.43K

u

0.09, 0.03

1.4,Cp 1.004kJ / kgK ,R 0.287kJ / kgK

Aeropropulsion Unit


t 2R t2RT ,P

cr , 1RM

Euler Eq.

2

Mean radius stage calculation - Summary

Flow with loss

t1 t1 m

1 3

1 3 2 1 t

cr cs

T ,P , ,r

, , ,m

M ,M ,u ,u , T

,

t1 t1R t1 t1RT ,T ,P ,Pt 2 t1 tT T T

cs 2,M

Aeropropulsion Unit


d e 1ug

Design Constrain (Performance control)

Design Overview

Design constrain

Where is the starting point… and the next step …?

Blade Profile Determination

b

c t1Tt1P

13

m1M

cr

cs

Fixed Parameters

(input data)

a mr

3M2u

tT

f

Variable Parameters

Obtained Flow Properties

Aeropropulsion Unit


Discusion

Design constrain

Preliminary design parameters ….?

a

b

c

d

t1T

t1P

e

mr1

3

m1M 3M

2u

1u

tT

cr

cs

f

g

Nobody can help me ….?!?!

Aeropropulsion Unit


Engineering Aproach Repeating-Stage, Repeating-Row, Mean-Line Design

R-S, R-R, M-L

Input data

1Mg

D

ec

t1T1

Variables

Flow Properties in each station

Repeating-Stage (Exit condition = Inlet condition)

Repeating-Row (Mirror-image of each row)

Mean-Line Design

Assumption :

- 1=2=3,1=2=3

- u1=u2=u3

- Constant mean radius

- Polytropic efficiency representing stage losses

- Two-dimensional flow (extimate annulus area)

c t1P1utTf

Controled parameter

Simplified

Your life !

Aeropropulsion Unit


Repeating-Stage, Repeating-Row, Mean-Line Design

R-S, R-R, M-L

Repeating-row constraint : 1=2=3,1=2=3

2R 1 2v v r v

or

1 2v v r

3 2 3R 2 3 1v v ,v v

And also

Aeropropulsion Unit



Diffusion Factor

R-S, R-R, M-L

Diffusion factor : High velocity gradient ---> high boundary layer thicnkness ---> high losses

2R 1R 1R

1R 1R

V v v sD 1

V 2V c

csolidity

s

3 2 32R 1R 1R 2 2 12

1R 1R 2 2 1

V v vV v v cos tan tanD 1 1 1 cos ...

V 2V V 2V cos 2

2 1f ( D, , )

Aeropropulsion Unit



Stage Total Temperature/Pressure Ratio

R-S, R-R, M-L

2 2

t3 1 1s 2 2

t1 1 2

T ( 1)M cos1 1

T 1 ( 1) / 2 M cos

c

c

e /( 1 )

e /( 1 )t3 t3s s

t1 t1

P T

P T

s 1 1 2

s s c

f ( M , , , )

f ( , ,e )

From Euler Eq: p t2 t1 p t3 t1 2 1C (T T ) C (T T ) r( v v )

2 1r v v & Diffusion Factor ?

H

O

w

Aeropropulsion Unit



Degree of Reaction and Stage Efficiency

R-S, R-R, M-L

( 1 ) /

ss

s

1

1

s s s cf ( , , ,e )

2 2

2 3 p3 22 1 2 1c

3 1 3 1 3 1 t3 t1

V V / 2CT Th h T TR 1 1

h h T T T T T T

2 2

2 3 p

c 2 2

2 3 p

V V / 2C 1R 1

2V V / C

Euler Eq :

Stage Efficiency :

Aeropropulsion Unit



Stage Exit Mach Number

R-S, R-R, M-L

11 2

Vf ( , )

r

3 3 3 1

2 21 1 1 3 s 1 1

M V / RT T 11

M V / RT T 1 ( 1) / 2 M ( 1) / 2 M

1 1 1 1 1

1 2 1 1 2 1 1 2

V u / cos u / cos 1

r v v u (tan tan ) cos (tan tan )

Inlet velocity/Wheel speed ratio :

Aeropropulsion Unit



Stage Loading and Flow Coefficient

R-S, R-R, M-L

p t 2 1

2

2 1

C T tan tan

( r ) tan tan

1

1 2

u 1

r tan tan

Flow Coefficient : Axial velocity/rotor speed

Stage Loading : stage work/rotor speed squared

0.3 0.35

0.45 0.55

Aeropropulsion Unit



General Solution

R-S, R-R, M-L

c( D 0.5, 1,e 0.9 )

Aeropropulsion Unit



General Solution

R-S, R-R, M-L

c( D 0.5, 1,e 0.9 )

Aeropropulsion Unit


Annulus Area

Flow Path Dimensions

rm

Aeropropulsion Unit


Conclusion