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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of Aerospace Engineering

2

Flow in ducts with heating or cooling factors tending to produce continuous changes in the state of a

flowing stream are (i) changes in cross-sectional area, (ii) wall friction and (iii) energy effects such as

external heat exchange, combustion, or moisture condensation. Simple oT change is difficult to

achieve in practice. If oT is changed through external heat exchange, the connection between the

mechanisms of friction and of heat transfer assure that frictional effects will be present. Combustion

change in mass rate, chemical composition Simple oT change is an ideal case.

With constant area and no friction, the momentum equation is ==+A

Fup2

constant

Continuity ==A

mu

constant G=

CombiningA

FGp =+

2

For fixed mass flow rate per unit area and constant impulse function per unit area, the above equation

defines a unique relation between p and called the Rayleigh line. Since both enthalpy and

entropy are functions of p and , the above equation can be used for representing the Rayleigh line

on the sh diagram. All fluids have Rayleigh curves of the general form.

isentrope

Fanno

p1

p01

p*

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The relation above, in the differential form, becomes

===

ddp

uuG

d

dp 22

2

ddp

represents the local velocity of sound only for a special circumstances, namely, when the

infinitesimal variation of pressure with density is such that there is no change of entropy. This

condition is fulfilled at the point of maximum entropy on the Rayleigh line. This point represents the

state of Mach number of unity for the process of simple oT - change.

Beginning with state 1, Mach number unity might be reached in several way (isentropically,

adiabatically at constant area, etc), and it is only for simple heating the * point will correspond to

Mach number unity. The branch of the Rayleigh curve about the point of maximum entropy generally

corresponds to subsonic flow. Since the process of simple heating is thermodynamically reversible,

heat addition must corresponds to an entropy increase and heat rejection must corresponds to an

entropy decrease. Thus at subsonic speeds the Mach number is increased by heating and decreased

by cooling. The reverse happens in case of supersonic flow. Hence, heat addition, like friction, always

tends to make the Mach number approach unity. Cooling causes the Mach number to change always

in the direction away from unity.

For heat addition at either subsonic or supersonic speeds, the amount of heat input can not be

greater than that for which the leaving Mach number is unity. If the heat addition is too great, the flow

will be choked, the initial Mach number will be reduced to a magnitude that is consistent with the

amount of heatthermal choking.

Mass Conservation2

1

1

2

u

u=

p1, T1

M1, T01

p2, T2

M2, T02

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Momentum Equation ( )1221 uuA

mpp =

Using uA

m=

and pMu =2 the momentum equation can be arranged to give

2

2

2

1

1

2

1

1

M

M

p

p

+

+=

Equation of state: 2 2 2

1 1 1

p T

p T

= or 2 2 1 2 2

1 1 2 1 1

T p p u

T p p u

= =

Definition of Mach number: 2 2 1 2 1

1 2 1 1 2

M u a u T

M a u u T

= =

Impulse function( )( )

11

12

11

2

22

1

2 =+

+=

Mp

Mp

FF

Definition of isentropic pressure

12

2

2 2

1 1 12

1

11

2

11

2

o

o

Mp p

p pM

+

=

+

Change in entropy

2

2 1 1

12

1

lnp

Ts s T

cp

p

=

When the process involves heat exchange, the change in stagnation temperature is a direct measure

of the amount of heat transfer. Form the energy equation

( ) ( )122

1

2

212

2 oopp TTC

uuTTCQ =

+=

When the process involves combustion or evaporation, it is usually possible to devise an

approximately equivalent process of simple oT change. In such cases the initial and final stagnation

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temperatures would be made respectively identical for the real process and for the equivalent

process. For a Rayleigh process, the change in stream properties are due primarily to changes in

stagnation temperature, u , the rate of change of stream properties along the Rayleigh line is a

function of the rate of change of stagnation temperature.

Now2

2

11 M

T

To +=

2

2

1

2

1

2

2

11

2

11

M

M

T

T

T

T

o

o

+

+

=

Substituting momentum equation and continuity into the equation of state

1

2

2

2

2

1

1

2

1

1

u

u

M

M

T

T

+

+=

Using1

2

u

ufrom the definition of Mach number

( )( )222

22

1

2

1

2

2

1

2

1

1

M

M

M

M

T

T

+

+=

Substituting this into the stagnation temperature ratio

( )( ) 2

1

2

2

22

2

22

1

2

1

2

2

1

2

2

11

2

11

1

1

M

M

M

M

M

M

T

T

o

o

+

+

+

+=

Similar expression for1

2

,

1

2

pp

,1

2

uu

may be found in terms of 1M and 2M . It is convenient to

normalize the equation by setting the Mach number equal to unity at one of the sections, say at 1.

( )

( )2222

1

1

M

M

T

T

+

+=

( )( ) 22

2

1

1

1

1

Mp

p

M

M

u

u

+

+=

+

+==

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12

2 1

2

112

1

1

+

+

+

+=

M

Mp

p

o

o

1

2

2

1ln

1p

s sM

c M

+ +

= +

The ratio of properties at two sections where the Mach numbers are 1M and 2M are found using

these normalized expressions

2

1

1

2

MT

T

MT

T

T

T

o

o

o

o

o

o

=

and so on...

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