AERO ENGINE 2

7/30/2019 AERO ENGINE 2

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3. Diagram T-s or h-s

Enthalpy, Entropy: Stateparameters

2 parameters can describe state

of gas

By convention, p-v, T-s, h-s


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Diag T-s for compression

Equal pressure linep2>p1

1-2 comp s constant,

Dh=Cp(T2-T1) 1-2 irreversible comp,

under same p,

Dh=Cp(T2-T1) 31243 friction heatqin

1221 due to n


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Diag T-s for compression

Total loss 32243 Compressor efficiency

(neglect Cp variation)

Usually use total T

2 1

'

2 1

c

T T

T T


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Equal pressure linep2>p1

1-2 expan s constant,

Dh=Cp(T1-T2) 1-2 irreversible comp,under same p,

Dh=Cp(T1-T

2)

31243 friction heat qin

1221 due to n

Diag T-s for expansion


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Total loss 32243 Turbine efficiency

(neglect Cp variation)

TT

TTT

21

'

21

Diag T-s for expansion


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Diagram h-s

Using diagram h-s shows directly

relations of energy exchanges Compression efficiency C=A/C

Expansion efficiency T=C/A


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4. Cycle and cycle efficiency

Thermal machines transfer heatenergy to mechanical energy

SubstanceAir

ExpansionMechanical work

Thermal cycle


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Carnots cycle

DA

AB, heating

q1 BC

CD, release

heat q2unavoidably


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Carnots cycle

Efficiency

T1orT2

t

T

T

q

qqt

1

2

1

21 1


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Generic cycle

Efficiency

At the same T1

and T2, Carnots

cycle has thehighest efficiency.

ABA

ABA

t

S

S

431

21


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1.3 Fundamental equations of

aerodynamics

1. Continuity equationr1A1v1=r2A2v2=qm

(1-29)

Vis speed

For incompressible,r1=r2A1v1=A2v2 (1-30)


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2. Energy equationA volume of air from

1-2 to 1-2 in dt

Adding heat qdmOutput work Wdm

dm=vArdtismass

flow at any sectionKinetic energy

change

22d

2

1

2

2 vvm


aerodynamics


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2. Energy equation (Cont'd)

Internal energychange: dm(u2-u1)

Work to gas

p2v2A2dt-p1v1A1dt

Neglecting gravity,

thenW

vh

vhq

22

2

2

2

2

1

1


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2. Energy equation(Cont'd)

Ifq=W=0, then

or

22

2

2

2

2

1

1

vh

vh

constTv

cp

2

2


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2. Energy equation (Cont'd)

Means that when gasflowing in a tube (with

friction), enthalpy +

kinetic energy remains

unchanged if no work

gets into the act.

constTv

cp 2

2


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aerodynamics (Cont'd)

3. Bernoullis equation Differential for of above equation

First law of thermodynamics

dWdvdhdq 221

r

1pddudqdq in fin dWdq


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3. Bernoullis equation (Cont'd)

Enthalpy definition

Then

Integration

dppddhpddhdurrr111

02

2

fdWdW

vd

dp

r

02

1 21

2

2

2

1 fWWvv

dp

r


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First term depends on the process

If no work (W=0) and isentropic (Wf=0)pr-const

0

2

1 21

2

2

2

1

f

WWvvdp

r

02

11

1

1

2

2

1

1

21

vv

p

pRT


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First term depends on the process

Isentropic and incompressible, W=0:

0

2

1 21

2

2

2

1

f

WWvvdp

r

rrr12

2

1

ppdp

constvpvp 22

2

11

2

22 rr


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Isentropic

0212

122

2

1 fWWvvdpr

02

11

1

1

2

2

1

1

21

vv

p

pRT

Generic


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aerodynamics (Cont'd) 4. Sound speed and Mach number

Sound speed in fluid

In air, sound propagation is seen as adiabatic process, ie.

pr- const. Then

Ratio of specific heats

RGas constant

TAbsolute temperature

rd

dpc

Tc R


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4. Sound speed and M (Cont'd)

Mach Number

M>1Supersonic

M


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aerodynamics (Cont'd)

5. Stagnation parameters of flow andaerodynamic functions

From above equations, flow kineticenergy (speed), enthalpy and pressure

potential energy can be converted from

one to others.

hv and p


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5. Stagnation parameters andfunctions (Cont'd)

If flow stagnates (v=0) as isentropicprocess, the kinetic energy is

converted totally to enthalpy. It is

called stagnation enthalpy, or totalenthalpy

(1-38)2

2

* vTch p


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v: ordered movement

T: disordered movement

2

2* vTch p


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Corresponding stagnation temperatureor total temperature:

(1-39)

Since

1-40

Rcp1

pc

vTT

2

2*

Tc R

2

*

211 M

TT


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Stagnation process

Its isentropic, so

T

Tvv

pp

*

*

1*


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5. Stagnation parameters and

functions (Cont'd) Since stagnation is isentropic, so:

(1-41)

(1-42)

p*Total Pressure

r*Total Density

12

*

2

11

M

p

p

1

1

2

*

2

11

r

rM


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functions (Cont'd)According to the 3 equations above, for a

given gas flow, the ratios of the totalparameters and steady parameters are

function of Mach number. When air flows isentropically in a tube

without energy added, the totalparameters (Enthalpy, temperature,

pressure and density) remain unchanged.

5 St ti t d


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functions (Cont'd)

Critical sound speed ccr(in tunnel)

v increases along the tunnel (ex. Laval nozzle)

When v increases, Tdecreases. Sound speed

cis function ofT, itgoes down.

5 St ti t d


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functions (Cont'd)


When v=c, ie M=1, this chas special meaning,

called critical sound speed ccr. This section is

called critical section and it is the smallest

section in the tunnel, also called throat.

5 St ti t d


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functions (Cont'd)


ccr is a parameter of the isentropic flow of

the tunnel. ccr is constant in this kind of flow.

5 St ti t d


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functions (Cont'd)

Definition of speed coefficient in asection

(1-43)

From (1-40) , we obtaincr

c

v

*

1

2

TTcr

2*

2

11 M

T

T

5 St ti t d


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functions (Cont'd)

And*

1

2TTcr

*2

1

2RTccr

*

1

2RTccr

C d C


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Cand Ccr

C Ccr

Apply to local tunnel

Depending on T T*

Speed ratio M

Relation see followingM

5 Stagnation parameters and


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functions (Cont'd)

Using

in , we obtain

and

2

222

c

cM cr

*2

12 RTccr

RTc 2

2*

2

11 M

T

T

2

2

2

2

11

2

1

M

M

2

2

2

1

11

1

2

M

5 St ti t d


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functions (Cont'd)

Change (1-40), (1-41) & (1-42) to

(1-46)

(1-47)

(1-48)

t, p and e three aerodynamic func

2

* 1

11)(

tT

T

12

* 1

11)(

p

p

p

1

1

2

*

1

11)(

r

re

5 St ti t d


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functions (Cont'd)

Flow density function

kg/svAqm r

1

1

2*

111

rr

*

1

2RTcv cr

*1

1

2*

1

2

1

11 RTv

rr



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functions (Cont'd)

Flow density function

(1-49) q() presents relative flow density in

section A to the critical section even

though the critical section does not exist. Ratio of the sections

1

1

21

1

1

11

2

1

)()(

rr

A

A

v

vq cr

cr



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functions (Cont'd)

Using flow density function and totalparameters, mass flow can be expressed:

1-50

where

Air 0.04042, gas 0.03968

*

* )(

T

AqpKqm

1

1

1

2

RK J

Kkg.



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functions (Cont'd)

At the critical section, q()=1.

*

*

T

Ap

Kqcr

m

1 3 F d t l ti f


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aerodynamics

6. Equation of momentum

Based on second Newtons law

Momentum change of an object at a

period of time is equal to the applied

force

In aircraft engines

)( 12 vvqF m

1 3 F d t l ti f


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aerodynamics

7. Equation of moment of momentum Similar with above equation, but

rotational movement

vmrdt

dFr


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1.3 Fundamental equations ofaerodynamics

8. Shock waves and expansion waves

Ex. The tail trace when a boat goes with

a high speed.

M>1M=1

8 S


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8. Shock waves and expansion

waves (Contd)

Or bridge pier when water flows.

Accumulation of disturbances

M>1M=1

8 Sh k d i


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waves (Contd)

Intakes: Fig (a) normal shock wave, due to

intakes form; Fig (b) oblique shock wave

The anglebdepends on Mach number of the

flow and geometrical angle of the cone .

8 Sh k d i


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waves (Contd)

When Mreduces orincreases,will increase until the wave

becomes a normal shock wave.

8 Sh k d i


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waves (Contd)

When supersonic flow passes

through the shock wave, sharply

speed decreases, pressure and

temperature increase.

After normal wave, the flow is

certainly subsonic. But after obliqueshock wave, it is still supersonic.

8 Sh k d i


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waves (Contd)

Strength of the shock wave is

described by pressure ratio of after

and before. It is only function ofM

for normal shock wave, the greater

M, the stronger the wave.

For oblique shock wave, the greaterMand , the stronger the wave.

8 Sh k d i


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waves (Contd) Supersonic flow passing through the shock

wave is NOT isentropic process. Partial

mechanical energy Irreversibly changes to

heat, and total pressure decreases. This is shock wave loss, and usually total

pressure recoverys is used to present the

loss. It is function of the wave strength, the

stronger the wave, the greater the loss.


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8. Expansion waves (Contd)

When a supersonicair flows to a lowerpressure zone, thereare expansion wavesdue to air continuousexpansion.

In Fig, turbine

cascade passage. Inthe throat AA,critical section, flowbecomes supersonic.


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In downstream, it islow pressure zone.The flow accelerates,it passes through aseries of expansionwaves, and speedincreases,

temperature andpressure decrease.


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The flow changesalso the direction. Thebigger the turnedangle, the moreexpansion and flowparameters changemore.


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The turned angledepends on exit

pressure. The lower

the pressure, thebigger the angle.

If pressure increases,

expansion waves maydisappear and the

flow may be subsonic.

Summary


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Summary

1.1 First law of thermodynamics Gas, state parameters, gas constants, processes

and parameters

Enthalpy and first law

1.2 Second law of thermodynamics Entropy and second law

Cycle and efficiency

1.3 Aerodynamics fundamental equations

Fundamental equations Sound speed and Mach number

Stagnation parameters and aerodynamic functions

Shockwaves and expansion waves

AERO ENGINE 2

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