Chalmers University of Technology Elementary axial turbine theory –Velocity triangles –Degree of...

Chalmers University of Technology

• Elementary axial turbine theory– Velocity triangles– Degree of reaction– Blade loading coefficient, flow coefficient

• Problem 7.1

• Civil jet aircraft performance

Lecture 7 – Axial flow turbines


Axial flow turbines

• Working fluid is accelerated by the stator and decelerated by the rotor

– Expansion occurs in stator and in relative frame of rotor

• Boundary layer growth and separation does not limit stage loading as in axial compressor


Elementary theory• Energy equation for control

volumes (again):

0103

0103

21

1

23

3

00103

22TTchh

Vh

Vhwq p

gasPerfect

hh

• Adiabatic expansion process (work extracted from system - sign convention for added work = +w)– Rotor => -w = cp(T03-T02) <=>

w = cp(T02-T03)– Stator => 0 = cp(T02-T01)

=> T02= T01


How is the temperature drop related to the blade angles ?

• We study change of angular momentum at mid of blade (as approximation)


Governing equations and assumptions• Relative and absolute refererence

frames are related by:

23

velocityrelativefor direction

of change Assume

23

22332233

2233

radiusconstant at Flow

wwww

wwww

ww

CCUCCU

UCUCrCrCorkspecific wlTheoretica

torquespecificlTheoreticarCrC

momentumangularspecificofchangeofRate

UCV • We only study designs where:

– Ca2=Ca3

– C1=C3

• We repeat the derivation of theoretical work used for radial and axial compressors:


Principle of angular momentum

Stage work output w:

3322

32

tantan aa

ww

CCU

CCUw

Ca constant:

32

3322

tantan

tantan

a

aa

UC

CCUw


(1)

(2)

(1)+(2) =>

3232 tantantantan


Combine derived equations =>

32 tantan aUCw

stagep Tcw ,0

Exercise: derive the correct expression when 3 is small enough to allow 3 to be pointing in the direction of rotation.

(7.3) tantan 32,0 astagep UCTc

Energy equation

Energy equation:

We have a relation between temperature drop and blade angles!!! :


Dimensionless parametersBlade loading coefficient

(7.6) tantan2

7.3Equation

21 32

2

,0

U

C

U

Tcastagep

Degree of reaction31

32

TT

TT

Exercise: show that this expression is equal to =>when C3= C1 0301

32

TT

TT


can be related to the blade angles!

C3 = C1 =>

32,0 tantan UCTcTc stagepstagep

Relative to the rotor the flow does no work (in the relative frame the blade is fixed). Thus T0,relative is constant =>

22

3222

22

3 tantan2

1

2

1 arotorp CVVTc

Exercise: Verify this by using the definition of the relative total temperature: p

relative c

VTT

2

2

,0


can be related to the blade angles!

Plugging in results in definition of =>

(7.7) tantan2 23

31

32

U

C

TT

TT a

The parameter quantifies relative amount of ”expansion” in rotor. Thus, equation 7.7 relates blade angles to the relative amount of expansion. Aircraft turbine designs are typically 50% degree of reaction designs.


Dimensionless parameters Finally, the flow coefficient:

5.0

0.50.3

0.18.0

Current aircraft practice (according to C.R.S):

U

Ca

Aircraft practice => relatively high values on flow and stage loading coefficients limit efficiencies


Dimensionless parameters Using the flow coefficient in 7.6 and 7.7 we obtain:

(7.8) tantan2 32

The above equations and 7.1 can be used to obtain the gas and blade angles as a function of the dimensionless parameters

22

1

2

1 tan 2

22

1

2

1 tan 3

1

tan tan 22

1

tan tan 33

(7.9) tantan2

23


• Exercise: show that the velocity triangles become symmetric for = 0.5. Hint combine 7.1 and 7.9

• Exercise: use the “current aircraft practice” rules to derive bounds for what would be considered conventional aircraft turbine designs. What will be the range for 3? Assume = 0.5.

Two suggested exercises


Turbine loss coefficients:Nozzle (stator) loss coefficients:

202

0201

22

22

2

pp

ppY

cC

TT

N

p

N

3,03

,03,02

22

33

2

pp

ppY

cV

TT

rel

relrelR

p

R

Nozzle (rotor) loss coefficients:


Problem 7.1


Civil jet aircraft performance


Four forces of flight

L ift

W eight = m g

D rag

Thrust

V

x

y

Resulting force perpendicular

to the flight path

Net thrust from the engines

resulting force parallell to the flight path

α angle of attack

V velocity

sincos gmDFdt

dVm

Newton’s second law


Aerodynamic equations

• L=Lift = q·S·CL [N]

• D=Drag = q·S·CD [N]

• q = dynamic pressure [N/m²]

• S = reference wing area [m²]

• CL = coefficient of lift CL = f(α,Re,M)

• CD = coefficient of drag CD = f(α,Re,M)


Reference wing areaThe area is considered to extend without interruption

through the fuselage and is usually denoted S.


Lift versus angle of attack


numberMachoffunctionasCD

DC

1M Mach


The ISA AtmosphereFrom lecture 5


Equations

222

2222

2

1

2

12

1

2

1

2

1

2

SLSL

a

SL

Sl

SLSLSL

aMT

TTRM

TRMaMVq

TRp

a

V

a

VM

TRap

p

T

T

SL

2

2

1Mp


Lift equation

LL

LLSL

LSLSLLSL

LL

CSMCSMconstL

CSMconstCSMp

CSMaCSaM

CSaMCSVL

242

22

constant

2222

222

100928,7

2

12

1

2

12

1

2

1


Drag equation

DD

DDSL

DSLSLDSL

DD

CSMCSMconstD

CSMconstCSMp

CSMaCSaM

CSaMCSVD

242

22

2222

222

100928,7

2

12

1

2

12

1

2

1


Drag polarL

D

R

R

C L

C D


High speed drag polar

LC

0.02 0.04 0.06 0.08 0.1 0.12 0.140.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Mach 0,63Mach 0,73Mach 0,80Mach 0,82Mach 0,83Mach 0,85

DC

LC


A flight consists of:

• Taxi

• Take off

• Climb

• Cruise

• Descent

• Approach and landing

• Diversion to alternate airport?

Sector Distance

Flight Time & Fuel

Block Time & Fuel

En route Climb

Descent

Approach &Landing

1500 ft

Initial Cruise

Step Cruise

Takeoff &Initial ClimbStart-up

&Taxi-out

Taxi-in


CruiseFor an airplane to be in level, unaccelerated flight, thrust and drag

must be equal and opposite, and the lift and weight must be equal andopposite according to the laws of motion, i.e.

Lift = Weight = mgThrust = Drag


Range

m

s

sN

kgnconsumptiofuelspecific

F

mSFC

s

kgflowfuelm

s

mvelocityV

mrangeR

f

f

a


Range

START

END

END

START

END

START

END

START

W

W

SLa

W

W

SL

W

W

W

W

a

a

af

a

ff

a

W

dW

D

LM

SFCg

aR

W

dW

D

LM

SFCg

a

W

dW

D

LM

SFCg

a

W

dW

D

LV

SFCgR

W

dW

D

LV

SFCgDFWL

WFSFCg

dWWVdR

FSFCg

dWV

FSFC

dmVdR

FSFC

dm

m

dm

V

dRdt

F

mSFC

dt

dmmV

dt

dR

1

1&


Breguet range equation

END

STARTSL

W

W

SLa W

W

D

LM

SFCg

a

W

dW

D

LM

SFCg

aR

START

END

ln

For a preliminary performance analysis is the range equation usually simplified. If we assume flight at constant altitude, M, SFC and L/D the range equation becomes

which is frequently called the Breguet range equation

END

STARTa W

W

D

LV

SFCgR ln

1


Breguet range equationThe Breuget range equation is written directly in terms of SFC. Clearly maximum range for a jetaircraft is not

dictated by maximum L/D, but rather the maximum value of the product M(L/D) or V(L/D).

ENDSTARTD

La

W

WD

L

D

La

D

La

D

L

D

L

L

LL

WWC

C

SSFCgR

W

dW

C

C

SSFCgW

dW

C

C

SSFCgdR

W

dW

C

C

S

W

SFCgW

dW

D

LV

SFCgdR

C

C

S

W

C

C

CS

W

D

LV

CS

WVCSVWL

END

START

22

2121

211

22

2

2

1 2


Breuget range equationFrom the simplified range equation, maximum range is obtained from

• Flight at maximum

• Low SFC

• High altitude, low ρ

• Carrying a lot of fuel

D

L

C

C


Range

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.82

4

6

8

10

12

14

16


LC

D

LM


Endurance

END

START

W

W

W

W

W

W

f

ff

W

W

D

L

SFCgt

simplifiedusuallyaboveequationtheisionapproximatfirstafor

W

dW

D

L

SFCgW

dW

D

L

SFCgW

dW

D

L

SFCgt

W

dW

D

L

SFCgDFWL

FSFCg

dW

FSFC

dm

m

dmdt

F

mSFC

dt

dmm

START

END

END

START

END

START

ln1

:

111

1&

Endurance is the amout of time that an aircraft can stay in the air on one given load of fuel


Endurance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.94

6

8

10

12

14

16

18

20


D

L

LC


Learning goals

• Understand how turbine blade angles relate to work output

• Be familiar with the three non-dimensional parameters: – Blade loading coefficient, degree of reaction,

flow coefficient

• Understand why civil aircraft are– operated at high altitude– at flight Mach numbers less than 1.0– How engine performance influences aircraft range

Chalmers University of Technology Elementary axial turbine theory –Velocity triangles –Degree of...

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