Analysis of A Disturbance in A Gas Flow

P M V SubbaraoAssociate Professor

Mechanical Engineering DepartmentI I T Delhi

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Analysis of Plane Disturbance

• A control volume for this analysis is shown, and the gas flows from left to right.

• The conditions to the right of the disturbance are uniform, but different from the left side and vice versa.

•The thickness of disturbance is very small.•No chemical reactions.

•There is no friction or heat loss at the disturbance.

Conservation of Mass Applied to 1 D Steady Flow

0.

Vt

Conservation of Mass:

Conservation of Mass for 1DSF:

0. V

Integrate from inlet to exit :

onstant. CVdVV

x

x

x

x

u

T

u

y

y

y

y

u

T

u

Gauss Divergence Theorem

Constantˆ.. AV

dAnVVdV

If the velocity is normal to the area :

ConstantA

udA

yyyxxx AuAu

The area of the disturbance is constant.

yyxx uu

Conservation of mass:

Conservation of momentum: The momentum is the quantity that remains constant because there are no external forces.

x

x

x

x

u

T

u

y

y

y

y

u

T

u

Conservation of Momentum Applied to 1 D Steady Flow

pVV

.

VdpVdVVVV

.

Using gauss divergence theorem:

AA

dAnpVdAnV ˆ.ˆ.

x

x

x

x

u

T

u

y

y

y

y

u

T

u

If the velocity is normal to the area :

AA

pdAdAu2

Steady, Inviscid 1-D Flow, Body Forces negligible

xxxyyyyyxx AuAuApAp 22

22xxyyyx uupp

The area of the disturbance is constant.

Conservation of Energy Applied to 1 D Steady Flow

wqVet

e

.

Steady flow with negligible Body Forces and no heat transfer is an adiabatic flow

wVe

.

For a blissful fluid the rate of work transfer is only due to pressure.

VdAnpVe

.ˆ.

For a total change from inlet to exit :

AV

VdAnpVdVe

.ˆ.

Using gauss divergence theorem:

AA

VdAnpdAVe

.ˆ

One dimensional flow normal to the area of cross section

0 xxxxyyyyxxxyyy AueAueuApuAp

0 xyxxxyyy ememuApuAp

Using conservation of mass

With negligible body forces:

2

2uie

022

22

x

xy

yxxxyyy

uim

uimuApuAp

022

22

x

xy

yxxx

xxx

yyy

yyy ui

ui

uA

uAp

uA

uApm

022

22

x

xy

yx

x

y

y ui

ui

ppm

022

22

x

x

xx

y

y

yy

upi

upim

022

22

x

xy

y

uh

uhm

The process is adiabatic, or nearly adiabatic, and therefore the energy equation can be written as:

22

22y

yx

x

uh

uh

For calorically perfect gas:

0

22

22TC

uTC

uTC p

yyp

xxp

The equation of state for perfect gas reads

yyyxxx RTpRTp :

Solution of Simultaneous Equations• If the conditions upstream are known, then there are four

unknown conditions downstream. • A system of four unknowns and four equations is solvable.• There exist multiple solutions because of the quadratic

form of equations. • Out of these multiple solutions, some are physically

possible and some are not.• These Physically possible solutions refer to the universal

law of direction of happening.• Different Physically possible solutions will lead to

development of different products or processes. • The only tool that brings us to the right direction of

happening is the second law of thermodynamics.• This law dictates the direction of happening : Across the

disturbance the entropy can increase or remain constant.

•In mathematical terms, it can be written as follows:

0 xyxy ssss

For an ideal gas :

0lnln

x

y

x

yp p

pR

T

TC

0ln1

ln

x

y

x

y

p

p

T

T

• We will not use isentropic conditions.• Use more algebra to reduce the number of variables.

Summary of Equations

yyxx uu Conservation of mass:

22xxyyyx uupp Conservation of momentum:

Conservation of Energy:

22

22y

yx

x

uh

uh

The equation of state for perfect gas

yyyxxx RTpRTp :

0ln1

ln

x

y

x

y

p

p

T

T

Constraint:

Change in Mach Number between points x & y

yyxx uu Conservation of mass:

Dividing this equation by cx

x

yy

x

xx c

u

c

u x

y

y

yy

x

xx c

c

c

u

c

u

x

yyyxx c

cMM

y

x

y

x

x

y

c

c

M

M

22xxyyyx uupp Conservation of momentum:

2

2 cp

pc

22yyyxxx upup

22

22

yyyy

xxxx u

cu

c

22

22

yyyy

xxxx u

cu

c

Dividing this through by cx2/

2

2

2

2

x

yyy

x

yyxx c

cM

c

cM

2

2

2

1

1

y

x

y

x

x

y

c

c

M

M

2

2

2

1

1

x

y

y

x

x

y

c

c

M

M

y

x

y

x

x

y

c

c

M

M

&

2

2

2

1

1

y

x

y

x

y

x

y

x

c

c

M

M

c

c

M

M

x

y

y

x

x

y

M

M

M

M

c

c2

2

1

1

Momentum Equation : Continuity Equation :

2121

22y

y

yx

x

xupup

Energy equation in terms for pressure and velocity for a perfect gas

2

22

2

1

2

1

2y

yx

x uc

uc

Dividing this by

1

2 2

xc

1

2

11

2

1 2

2

2y

x

yx M

c

cM

22

22y

ypx

xp

uTc

uTc

12

1

12

1

2

22

y

x

x

y

M

M

c

c

x

y

y

x

x

y

M

M

M

M

c

c2

2

1

1

22

2

2

2

2

1

1

12

1

12

1

x

y

y

x

y

x

M

M

M

M

M

M

Energy Equation :

Combined Mass, Momentum and Energy Conservation :

Combined Mass & Momentum Equation :

Combined Mass, Momentum and Energy Conservation

x

x

x

x

u

T

u

y

y

y

y

u

T

u

22

2

2

2

2

1

1

12

1

12

1

x

y

y

x

y

x

M

M

M

M

M

M

22222222 112

11

2

11 yxyxy MMMMMM

x

0221 222222 xyxyxy MMMMMM

0221 22222244 xyxyxyxy MMMMMMMM

Nothing Happening :

2222 0 xyxy MMMM

12

2)1(2

22

x

xy

M

MM

With a disturbance between x & y,

This equation relates the downstream Mach number to the upstream.

It can be used to derive pressure ratio, the temperature ratio, and density ratio across the disturbance.

0221 2222 xyxy MMMM

2)1(21 222 xxy MMM

If there is something happening between x & y

x

y

y

x

x

y

T

T

M

M

c

c

12

1

12

1

2

22

12

2)1(2

22

x

xy

M

MM

Substitute value of My

112

2)1(

21

12

1

2

2

2

x

x

x

x

y

M

M

M

T

T

1222)1(1

122122

22

xx

xx

x

y

MM

MM

T

T

22

22

1

1221

x

xx

x

y

M

MM

T

T

1

12 2

x

xx

yy

x

y M

T

T

p

p

21

12

2

x

x

x

y

M

M

y

x

y

x

x

y

c

c

M

M

Change in Entropy Across the disturbance

11

ln

x

y

x

yxy

p

p

T

T

R

ss

x

y

x

ypxy p

pR

T

TCss lnln

x

y

x

ypxy

p

p

T

T

R

C

R

sslnln

x

y

x

yxy

p

p

T

T

R

sslnln

1

121

22

22

1

12

1

1221ln

x

x

xxxy M

M

MM

R

ss

R

ss xy

Mx

Infeasible

Physically possible solution 2

Solution - 1

The Nature of Irreversible Phenomenon

Mx

My constant=1.4

This Strong Irreversibility is called as Normal Shock.

Nature of Normal Shock

• The flow across the shock is adiabatic and the stagnation temperature is constant across a shock.

• The effect of increase in entropy across a shock will result in change of supersonic to subsonic flow.

• The severity of a shock is proportional to upstream Mach Number.

• Normal Shock is A severe irreversible Diffuser.• No capital investment.• Can we promote it ?

Normal Shock Past F-18

Reentry Interface Gas Dynamics

Occurrence of Normal Shock Reentry Vehicles

Weakening of Normal Shock

Brayton Cycle for Jet Propulsion

A BC

D

1'-11" 10" 1'-2" 1 2 3 4Compressor

Turbine +Nozzle

Turbofan

Burner

Gas Dynamic Analysis of Turbojet Engine

Jet Engine Inlet Duct

• All jet engines have an inlet to bring free stream air into the engine.

• The inlet sits upstream of the compressor and, while the inlet does no work on the flow.

• Inlet performance has a strong influence on engine net thrust.

• Inlets come in a variety of shapes and sizes with the specifics usually dictated by the speed of the aircraft.

• The inlet duct has two engine functions and one aircraft function .

• First : it must be able recover as much of the total pressure of the free air stream as possible and deliver this pressure to the front of the engine compressor .

• Second : the duct must deliver air to the compressor under all flight conditions with a little turbulence .

• Third : the aircraft is concerned , the duct must hold to a minimum of the drag.

• The duct also usually has a diffusion section just ahead of the compressor to change the ram air velocity into higher static pressure at the face of the engine .

• This is called ram recovery .

• SUBSONIC INLETS• A simple, straight, short inlet works quite well. • On a typical subsonic inlet, the surface of the inlet

from outside to inside is a continuous smooth curve with some thickness from inside to outside.

• The most upstream portion of the inlet is called the highlight, or the inlet lip.

• A subsonic aircraft has an inlet with a relatively thick lip.

Subsonic Inlet Duct

Axi-Symmetric Inlets

Rectangular Inlets

High Supersonic Flight

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