introduction of compressible flow

COMPRESSIBLE FLOW

COMPRESSIBLE FLOW

Introduction

The compressibility of a fluid is, basically, a

measure of the change in density that will be

produced in the fluid by a specific change in

pressure and temperature.

In general, gases are highly compressible and

liquids have a very low compressibility.

Part one : Introduction of Compressible Flow 1

COMPRESSIBLE FLOW

Application ;

Aircraft design

Gas and steam turbines

Reciprocating engines

Natural gas transmission lines

Combustion chambers

Compressibility effect ;

Supersonic – the flow velocity is relatively

high compared to the speed of sound in the gas.

Subsonic


COMPRESSIBLE FLOW

Fundamental assumptions

1. The gas is continuous.

2. The gas is perfect (obeys the perfect gas law)

3. Gravitational effects on the flow field are

negligible.

4. Magnetic and electrical effects are negligible.

5. The effects of viscosity are negligible.

Applied principles

1. Conservation of mass (continuity equation)

2. Conservation of momentum (Newton’s law)

3. Conservation of energy (first law of

thermodynamics)

4. Equation of state


COMPRESSIBLE FLOW

Perfect gas law :

RTP=

ρ

P : Pressure ρ : Density R : Universal gas constant

)(04.287 KkgJ

airR ⋅=

T : Temperature


COMPRESSIBLE FLOW

Conservation laws :

Conservation of mass Rate mass enters

control volume Rate mass leaves

control volume = _

Rate of increase of mass of fluid in

control volume

Conservation of momentum :

Net force on gas in control volume

in direction considered

Rate momentum

leaves control volume in direction

considered

= +

Part one : Introduction of Compressible Flow

Rate of increase of momentum in

direction considered of

fluid in control l

Rate momentum leaves control

volume in direction

considered

_

5

COMPRESSIBLE FLOW

Conservation of energy :

Rate of increase in internal energy

and kinetic energy of gas in control volume

Rate enthalpy and kinetic

energy leave control volume

Rate enthalpy and kinetic

energy enter control volume

+ _

Rate heat is transferred into control volume

Rate work is done by gas in control

volume = _


COMPRESSIBLE FLOW

Definition :

A control volume is a volume in space (geometric

entity, independent of mass) through which fluid

may flow

Enthalpy H, is the sum of internal energy U and

the product of pressure P and volume V appears.

PVUH +=


COMPRESSIBLE FLOW

COMPRESSIBLE FLOW

Introduction

Many of the compressible flows that occur in

engineering practice can be adequately modeled as a

flow through a duct or streamtube whose cross-sectional

area is changing relatively slowly in the flow direction.

A duct is a solid walled channel, whereas a streamtube

is defined by considering a closed curve drawn in a

fluid flow.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow 8

COMPRESSIBLE FLOW

Quasi-one-dimensional flow is flows in which the flow

area is changing but in which the flow at any section

can be treated as one-dimensional.


COMPRESSIBLE FLOW

CONTINUITY EQUATION

The continuity equation is obtained by applying the

principle of conservation of mass to flow through a

control volume.

One-dimensional flow is being considered.

There is no mass transfer across the control volume.

The only mass transfer occurs through the ends of the

control volume.


COMPRESSIBLE FLOW

Mass enters through the left hand face of the control

volume be equal to the rate at which mass leaves

through the right hand face of the control volume.

21 mm && =

We know that VAm ρ=&

We considered ;

222111 AVAV ρρ =

For the differentially short control volume indicated,


COMPRESSIBLE FLOW

above equation gives ;

))()(( dAAdVVdVA +++= ρρρ

Neglecting higher order terms, we found ;

0=++ VdAAdVVAd ρρρ

ividing this equation by VAρ D then gives ;

0=+dVd

+A

dAVρ

ρ

This equation relates the fractional changes in density,

velocity and area over a short length of the control

volume.


COMPRESSIBLE FLOW

MOMENTUM EQUATION (Euler’s equation)

The flow is steady flow.

Gravitational forces are being neglected.

The only forces acting on the control volume are the

pressure forces and the frictional force exerted on the

surface of the control volume.


COMPRESSIBLE FLOW

The net force on the control volume in the x-direction

is ;

µdFAdAAdPPPdAAdPpPA −−++++++− ]))][(([())(( 21

Note :

dx is too small, dPdA have been neglected.

Mean pressure on the curved surface can be taken

as the average of the pressures acting on the two end

surfaces.

dFµ is the frictional force.

Rearranging above equation, we found the net force on

the control volume in the x-direction is ;

µdFAdP −−


COMPRESSIBLE FLOW

Since the rate at which momentum crosses any section

of the duct is equal to , we found that ; Vm&VAdVVdVVVA ρρ =−+ ])[(

The above equation can be written as ;

VAdVdFAd ρρ µ =−−

Frictional force is assumed to be negligible. The Euler’s

equation for steady flow through a duct becomes;

VdVdP=−

ρ


COMPRESSIBLE FLOW

Integrating Euler’s equation ;

CdPV=+ ∫ ρ2

2

(For compressible)

And if density can be assumed constant, Euler’s

equation become ;

CPV=+

ρ2

2

(For incompressible)


COMPRESSIBLE FLOW

STEADY FLOW ENERGY EQUATION

For flow through the type of control volume

considered as before, we found ;

wqVhVh −++=+22

21

1

22

2

h = enthalpy per mass

V = velocity

q = heat transferred into the control volume per unit mass of fluid

flowing through it w = work done by the fluid per unit mass


COMPRESSIBLE FLOW

Assumption ;

No work is done, w=0

Perfect gases is considered, Tch p=

Steady flow energy equation ;

qVTcVTc pp ++=+22

21

1

22

2

Applying this equation to the flow through the

differentially short control volume gives ;

2)()(

2

22 dVVdTTcdqVTc pp+

++=++


COMPRESSIBLE FLOW

Neglecting higher order terms gives ;

dqVdVdTcp =+

This equation indicates that in compressible

flows, changes in velocity will, in general, induce

changes in temperature and that heat addition

can cause velocity changes as well as

temperature changes.

If the flow is adiabatic i.e., if there is no heat

transfer to of from the flow, it gives ;

22

21

1

22

2VTcVTc pp +=+


COMPRESSIBLE FLOW

Steady flow energy equation for adiabatic flow

becomes ;

0=+VdVdTcp

This equation shows that in adiabatic flow, an increase

in velocity is always accompanied by a decrease in

temperature.


COMPRESSIBLE FLOW

EQUATION OF STATE

When applied between any two points in the flow ;

22

2

11

1

TP

TP

ρρ=

When applied between the inlet and the exit of a

differentially short control volume, this equation

becomes ;

))(( dTTddPP

TP

+++

=ρρρ


COMPRESSIBLE FLOW

Higher order terms are neglected and it gives ;

( )( )( )TdTd

PdP

TP

TP

−−+= 111 ρρ

ρρ

0=−−TdTd

PdP

ρρ

This equation shows how the changes in pressure,

density and temperature are interrelated in compressible

flow.


COMPRESSIBLE FLOW

ENTROPY CONSIDERATIONS

In studying compressible flows, another variable, the

entropy, s, has to be introduced. The entropy basically

places limitations on which flow processes are

physically possible and which are physically excluded.

The entropy change between any two points in the flow

is given by ;

[ ] [ ]1

2

1

2 lnln12 PP

TT

p Rcss −=− (1)

Since , this equation can be written; vp ccR −=

( )( ) ⎥⎦

⎤⎢⎣

⎡=

− −−

γγ 1

121

2

1

2ln PP

TT

pcss

If there is no change in entropy, i.e., if the flow is


COMPRESSIBLE FLOW

isentropic, this equation requires that :

γγ 1

1

2

1

2

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

PP

TT

hence, since the perfect gas law gives ;

2

1

1

2

1

2

ρρ

PP

TT

=

it follows that in isentropic flow : γ

ρρ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2

1

2

PP

in isentropic flows, then γρP is a constant.

If equation (1) is applied between the inlet and the exit


COMPRESSIBLE FLOW

of a differentially short control volume, it gives ;

[ ] [ ]PdPP

TdTT

p Rcsdss ++ −=−+ lnln)(

neglecting small value, the above equation gives;

PdP

TdT

p Rcds −= (2)

which can be written as ;

PdP

TdT

cds

p⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

γγ 1

lastly, it is noted that in an isentropic flow, equation (2)

gives;

dPP

RTdTcp =

using the perfect gas law ;


COMPRESSIBLE FLOW

ρdPdTcp =

(3)

but the energy equation for isentropic flow, i.e., for flow

with no heat transfer, it gives ;

0=+VdVdTcp

which using equation (3) gives ;

0=+VdVdPρ


introduction of compressible flow

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