SET 3 Flow in Ducts, (Nozzles and Diffusers) and Wind Tunnels

7/30/2019 SET 3 Flow in Ducts, (Nozzles and Diffusers) and Wind Tunnels

1/24

Flow in ducts, (Nozzles and diffusers) and wind tunnels

The flow can be assumed to be one-dimensional, that is, conditions across each sections are

uniform. The conditions at any two sections in a steady flow are related by the equation

222111AuAu

Using the sonic condition as reference

AuuA

When the flow is purely subsonic,

A is a fictitious area that does not occur in the flow. But, if sonic

and supersonic conditions are attained in the flow, then tAA area of the actual throat

Since au ,

u

a

u

a

A

A o

o

We have

1

12,

1o

2

1

2

1

12

M

u

a

1

1

2

2

11

Mo

The isentropic area-Mach number relation becomes

1

1

2

2

2

2

11

1

21

MMA

A

Area-pressure relation

1

1

2

1

2

1

12

1

1

1

2

2

1

1

oo p

p

pp

u

u

A

A

Mass flow rate per unit area

o

o

TT

T

RRT

puu

RT

pu

A

m 1

211

2o

pMM

RT


2/24

Defining a mass flow parameter as

Wp

T

A

m om

1

, where W is the molecular weight

211

2m M M

, = universal gas constant

In terms of stagnation quantities and Mach number

121

2

2

11

M

M

T

p

RA

m

o

o

Hence, for a given Mach number, the flow rate is proportional to the stagnation pressure and

inversely proportional to the square root of stagnation temperature.o

o

p

Tmis used as a non-

dimensional mass flow parameter for turbomachinery performances.

In can be seen that the mass flow rate attains a maxima when 1M . Hence,

o

o

T

p

RA

m

A

m 11

max1

2

Hence, for a given gas, the maximum flow per unit area depends ono

o

T

p. For fixed ando op T and

passage, the maximum flow that can pass is relatively large for gases of high molecular weight and

small for gases of low molecular weight.

The fact that the curve of mass flow rate per unit area has a maximum is connected with the

interesting and important effect called choking.

1

isentropic relations

chart orM

,

1

o

,

1

oT

T 1

A

A .

A is constant . Hence,

11

2

2

AA

A

A

AA

2

AA orchart

relationsisentropic

,2

M ,2

o

2

oTT

Since op and oT are constant, 2p & 2T can be obtained as


3/24

,

1

2

1

2

o

o

pp

pp

pp

1

2

1

2

o

o

TT

TT

T

T

Now, for a given area ratio1

2

AA , 2M can be computed for given 1M . The plotted results look like

(1) For a given initial Mach number1

M and a given area ratio1

2

AA

, there are either two solutions for

final2

M or none at all. When there are two solutions, one is subsonic and the other is supersonic.

Which one of the two occurs depends, the part, on whether a throat exists between sections (1) and(2), since in order to change the regime the flow must pass a throat at 1M .

For example if1

M is subsonic and the passage is converging, then2

M must be subsonic. But if the

passage is converging-diverging and has a throat between (1) and (2), the flow at section (2) may be

either subsonic (venturi) or supersonic (nozzle) depending on the pressures imposed at the inlet and

exit.

(ii) If there is no solution 2M for the chosen values of 1M and1

2

AA , that is, the solution is

imaginary in mathematical sense. This occurs only if2

A is smaller than1

A . Physically, this result

signifies that for a given flow at section 1, there is a maximum contraction which his possible: the

maximum contraction corresponds to sonic velocity at 2. If conditions at section (1) are specified, the

mass flow is fixed and there is then a minimum cross-sectional area required to pass this flow. This

phenomenon is called choking for a given area reduction, in subsonic flow there is a maximum

initial Mach number which can be maintained steady; and in supersonic flow a minimum initial Mach

A2 |A1 = 1

A2 |A1 = 1

M2

M1

No throat--- Throat

A2 |A1 = 2

A2 |A1 = 1.2

A2 |A1 = 0.8

A2 |A1 = 0.8


4/24

number which can be maintained steadily. At either of these limiting conditions, the flow at section (2)

is sonic and is said to be choked.

Consider subsonic flow at 1. If12

AA , all conditions at 2 will be identical to 1. A slight reduction in

2A will produce certain effects at 2 and will comprise an increase in

2M and decrease in

2p and

2T .

This slight reduction in 2A without a change in conditions at 1, must be accompanied by a reduction

in the back pressure2

p . Further reduction in2

A may be made in the same way until2

M reaches

unity. After this point is reached, there is no way of reducing the area further without simultaneous

change in the steady state conditions at section 1. If for example, the pressure and temperature at 1

are held constant a reduction in1

2

AA

beyond its limiting value will, after a transient period of wave

propagation, result in a reduced steady-state1

M , reducing the mass flow rate. The maximum

possible value of1

M (also, maximum flow rate) is obtained when 12

M . To obtain this limiting flow,

the back pressure2

p must of course be adjusted accordingly.

Converging Nozzles

A converging duct with a large entrance area at section 0 (reservoir) discharges into a region where

the back pressure Bp is controllable, by means of a valve. The values of oo Tp , will remain constant,

but Bp will vary. Ep denotes the pressure in the exit plane of the nozzle.

V0 = 0P0, T0Constant

To exhaust

pB varies

pB

pE

Valve

0

*

p

p 0p

p


5/24

The effects of back pressure variations on flow rate and exit pressure are as follows:

I. 1o

B

pp

, pressure is constant throughout and there is no flow.

II. Bp is slightly reduced There will be flow with a constantly decreasing pressure through the

nozzle. Since the exit flow is subsonic, ep must be Bp , except for minor secondary circulation

effects in the exhaust space. If ep is substantially larger than Bp , then the flow would expand

laterally after leaving the nozzle. But such an area increase at subsonic speeds causes the stream

pressure to rise further. Since, the back pressure is the pressure which the stream ultimately

achieves in the exhaust space it follows that ep can not be larger than Bp . Similarly, ep can not be

substantially less than Bp .

III. Further reduction in Bp - changes the pressure distribution, and increases the flow rate; but there is

no qualitative change in performance.

IV. 1

eoo

B Mp

pp

p, flow performances identical to II and III.

V. Reduction ino

B

pp

to this level cannot produce further change in conditions within the nozzle, for

the value ofo

e

pp

cannot be made less than the critical pressure ratio unless there is a throat

upstream of the exit section. Hence, at condition V, the pressure distribution within the nozzle, value

ofo

e

pp

, flow rate are identical with the corresponding quantities for condition (iv). The pressure-

distribution outside the nozzle can not be predicted using 1-D analysis.

0

0

pA

Tm

e

0p

pB

0

*p

p

0

*p

p

0p

pB

0

*p

p

0p

pe


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Regime I Regime II

oo

B

pp

pp

oo

B

pp

pp

o

B

o

e

p

p

p

p oo

e

p

p

p

p

01eM 01eM

oe

o

pA

Tmdepends on

o

B

pp

oe

o

pA

Tmindependent of

o

B

pp

Converging-Diverging Nozzles

In regime I the flow is entirely subsonic, and the duct behaves like a conventional venturi tube. The

flow rate is sensitive to changes in back pressure (conditions 1a, 1b).

V0 = 0P0, T0Constant

pB

pe

pB

pB

T

0

*p

p

0pp


7/24

At condition 2, which forms the dividing line between regimes I and II entered, a normal shock

appears downstream of the throat, and the process abt??? of the shock comprises subsonic

deceleration. As the back pressure is lowered, the shock moves down the nozzle at condition 4, it

appears in the exit plane of the nozzle. Both in regime I & II the exit pressure ep is virtually identical

with the back pressure Bp . But the flow rate in regime II, unlike regime I, is constant and is unaffected

by the back pressure (conditions at the throat are unaltered, sonic).

In regime III, as for condition 5, the flow within the entire nozzle is supersonic, and the pressure in the

exit plane is lower than the back pressure over expanded nozzle. The subsequent compression

occurs outside the nozzle by oblique shocks and their reflections

Condition 6, is the design condition for the nozzle under supersonic conditions. The exit pressure is

identical with the back pressure. A reduction in the back pressure below that corresponding to

0

*p

p

0p

pe

0p

pB

0p

pB

0

*p

p

0p

pB

0p

pB

0

*p

p

0

0

pA

Tm

t


8/24

condition 6 has no effects whatsoever on the flow pattern within the nozzle. In regime IV the

expansion from ep to Bp occurs outside the nozzle in the form of expansion waves.

In both regimes III and IV the flow within the nozzle is independent ofBp and corresponds to the flow

pattern for the design condition. Adjustment to the back pressure is made in the jet.

For subsonic flow there are an infinite number of possible pressure-distance curve. For the

supersonic region of flow, the pressure-distance curve is unique. Since, in subsonic flow the pressure

ratio does not depend solely on area ratio, but in supersonic flow the pressure ratio depends solely on

area ratio.

Laval nozzle as a supersonic wind tunnel

As the exit pressure is described the shock moves downstream, finally reaching the exit, pressure

there then reaches a value of4

p . If pressure at the exit decreases further, the flow in the nozzle is not

affected; the pressure adjustment being made through system of oblique shock waves. For exit

pressure lower than4

p , flow up to the exit is completely supersonic. Thus a Laval nozzle may be

used as supersonic wind tunnel provided

4ppe

This is the principle of the open-circuit type of supersonic wind tunnel, operating from a high-pressure

reservoir or into a vacuum receiver or both. Continuous flow may be obtained if enough power is

available; otherwise, it is used as an intermittent or blow-down wind tunnel.

If the nozzle discharges directly into the receiver, the minimum pressure ratio for full supersonic flow

in the test section is

4p

pp

p oe

o

But if a diffuser is attached to the exit, operation at a lower pressure ratio is possible, since the

subsonic flow downstream of the shock may be decelerated isentropically to the stagnation pressure

op in principle.

Testsection

M1P1P0

M2P2 = p4P0


9/24

The pressure ratio required then is the ratio of stagnation pressures across a normal shock at the test

section Mach number1

M , i.e.,

1

2

1

2

1

11

2

11

211

1

21

M

MM

p

p

o

o

s

Practically, this type of diffuser does not give the expected recovery, the interaction of shock wave

and boundary layer develops a flow different from this model.

However, a long constant area duct ahead of the subsonic diffuser nearly realizes the normal shock

recovery. Such a duct, provided it is long enough, gives nearly the same recompression as a normal

shock. The compression occurs through a system of shocks interacting with the thickened boundary

layer. Through this recovery through a dissipative system is not the most efficient, but it is often most

practical. It is quite stable with respect to variations of inlet conditions.

The normal shock pressure recovery is an ideal, convenient reference or standard for comparison of

performance.

If the supersonic flow at the test section could be isentropically compressed to sonic conditions at a

second throat, if could then be decelerated sub-sonically in the diffuser. Ideally, then it is possible to

operate at even a lower pressure ratio than the one for normal shock recovery. In the idealized case,

no shock complete recovery, oo pp .

12

AA

No pressure difference No power

But necessary to create a pressure difference to start.

Initial normal shock at the test section layer second throat

Minimum starting area forth second throat is

so

o

o

o

pp

p

p

A

A

2

1

1

2

Diffuser contraction ratio, 2

1

AA

1

1

1

2

1

1

1

2

1

max

1Mf

A

A

A

A

A

A

A

A

s

With minimum area for the second throat and with larger ones, the shock wave jump from the test

section to the downstream side of the diffuser throat swallowing the shock test section is

supersonic, but also the second throat and part of the diffuser. The second throat area can be


10/24

reduced after the flow has started to move the shock toward the start.

12AA will now make it ideal,

but not possible to achieve the reduction, but some contraction is possible.

Flow in constant Area ducts with friction

Stationary power plants, aircraft propulsion, high-vacuum technology, fluid transport in chemical

process plants, natural gas transport in long pipe.

Wall friction is the chief factor, with the assumption that no special attempt is made to transfer heat to

or from the stream. When the ducts are reasonably short, the flow is approximately adiabatic, but for

extremely long ducts; there is sufficient area for heat transfer to make the flow non-adiabatic and

approximately isothermal.

Assumptions: - One dimensional steady flow,

Neither external heat exchange nor external shaft work

ohuh 2

2

1

Gm

u 2

, hu,, are measured at he same section

2

2

2

Ghh o , a relation between &h for a particular flow ( Gho , are constant)

All possible states of the fluid for a given adiabatic, constant area flow lie on one of these lines.

1

h0


11/24

Since for a pure substance ,s s h , the curves may be transferred to the enthalpy-entropy

diagram. The curves are called Fanno curves or Fanno lines.

For all substances, the Fanno curves have the general shape

The upper branch of each Fanno curve corresponds to the subsonic flow, and the lower branch

corresponds to supersonic flow, and the Mach number is unity at the point of maximum entropy on

each Fanno curve.

Since the flow is adiabatic, the second law of thermodynamics states that entropy can not decrease;

thus the path of states along any Fanno curves must be towards the right. Consequently, if the flow at

a point in the duct is subsonic (a), the effects of friction will be to increase the velocity and Machnumber and to decrease the pressure and enthalpy. If the flow is initially supersonic (b), the friction

will decrease the velocity and Mach number and will increase the enthalpy and pressure. A subsonic

flow, therefore, will never become supersonic and a supersonic flow will not become subsonic, unless

a discontinuity is present.

The limiting pressure, beyond which the entropy would suffer a decrease, occurs at Mach number

unity and is denoted by

p .

p denotes the state where 1M for the adiabatic flow at constant area.

Referring to a state '' a the value of

ap will be different for an isentropic flow as compared with the

value for an adiabatic constant area flow.

The isentropic stagnation pressure is reduced as a result of friction, irrespective to whether the flow is

subsonic or supersonic.

Choking due to friction Consider the stagnation enthalpy, flow per unit area and length of duct are

such that Mach number unity is reached at the end of duct. If the duct length is increased, it is evident

from the foregoing considerations that some sort of adjustment in the flow is necessary. When the

p0a=p0b p0a*=p0b*

pa*=pb*

hopa

Fanno curves

Small Gpb

b

Large G

s

h

a


12/24

flow is subsonic, this adjustment is in the form of a reduction in the flow rate, that is, the flow is

chocked. When the flow is supersonic, the adjustment at first involves the appearance of shock

waves, and for sufficiently large increase in duct length, involves ultimately a choking of the flow.

Adiabatic, Constant-Area Flow of a Perfect Gas

T

dTd

p

dpRTp

--- (1)

T

dT

u

du

M

dM

RT

uMRTu

2

2

2

22

22,

--- (2)

Energy:- 02

2

uddTCp

Divide by TCp , and use definition of Mach number

02

12

2

2

u

duM

T

dT --- (3)

Mass conservation

uA

mG

02

12

2

u

dud

--- (4)

Momentum Conservation:

AududAAdp ww

+ d, M + dM

dx

w


13/24

A: Cross sectional area

dAw: wetted wall area over which w acts.

The drag coefficient, usually called the coefficient of friction for duct flow, is defined as

2

2

1u

fw

The hydraulic diameter is defined as 4 times the ratio for c.s. area to wetted perimeter,

dxdA

A

dxdA

AD

ww

44

Using fD, and continuity equation into the momentum equation to give

u

duudu

A

m

D

dxufdp

2

2

24

Dividing by p and using22 pMu

02

42

2

222

u

duM

D

dxf

M

p

dp --- (5)

From the definition of isentropic stagnation pressure

12

2

11

Mppo

2

2

2

2

2

11

2

M

dM

M

M

p

dp

p

dp

o

o

--- (6)

Impulse function is defined as

22 1 MpAAupAF

2

2

2

2

1 M

dM

M

M

p

dp

F

dF

--- (7)


14/24

We have now seven simultaneous linear algebraic relations involving eight differential variables

FdF

pdp

kdk

MdM

TdTd

pdp

o

o,,,,, 22

2

2

1

and

Ddx or

Ddxf4 . The physical

phenomenon causing changes in state is viscous friction. Hence, the variableD

dxf4 is the physically

independent variable.

From (1) and (3)

2

2

2

2

1

k

dkM

d

p

dp

Introducing (4) into it

2

22

2

11

u

duM

p

dp

With (5) this gives,

Ddx

fM

MM

p

dp4

12

112

22

--- (8)

Similarly,

D

dxf

M

MM

M

dM4

1

2

11

2

22

2

2

--- (9)

Ddx

fM

M

v

dv4

122

2

--- (10)

D

dxf

M

M

a

da

T

dT4

12

1

2

12

4

--- (11)

Ddx

fM

Md4

12 2

2

--- (12)

D

dxf

M

p

dpo 42

2 --- (13)

Ddx

fM

M

F

dF4

122

2

--- (14)


15/24

Also,

2

1

1

2

1

1ln

o

o o

p p oo

o

TT dps ds

c c pp

p

Since oT is constant in an adiabatic flow

D

dxfM

c

ds

p

42

1 2

--- (15)

Since entropy can not decrease in an adiabatic process, (15) tells that '' f must be positive, thus the

shearing stress must always act on the stream in a direction opposite to the direction of flow, as

assumed.

The isentropic stagnation pressure and impulse function must decrease if friction is present both in

the subsonic or supersonic flow. Wall friction reduces the effectiveness of all types of flow machinery

and also reduces the thrust obtainable from jet propulsion devices.

Summary Subsonic Supersonic

p decreases increases

M increases decreases

V increases decreases

T decreases increases

decreases increases

Fpo , decreases decreases

The Mach number always tends toward unity. Continuous transition from one regime to the other is

impossible. For given conditions at an initial section of the duct, the maximum possible duct lengthwhich can be employed without attiring the given initial conditions and without introducing

discontinuities is that length for which the exit Mach number is exactly unity.

M is chosen as independent variable to integrate the relations above.

21

22

2

2

max

2

11

14 dM

MM

M

D

dxf

M

L

o


16/24

Defining a mean friction coefficient with respect to length as

dxfL

fL

omax

max

1; we have

2

2

2

2

max

2

112

1

2

114

M

Mlu

M

M

D

Lf

max value ofD

Lf4 for given M

Hence, the length of duct L required for the flow to pass from1

M to2

M can be found from

21

maxmax 444MM

D

Lf

D

Lf

D

Lf

Combining (8) and (9)

222

2

2

112

11dM

MM

M

p

dp

2

2

112

11

MMp

p

Similarly,

;

2

112

1

2

M

Mv

v

2

2

2

2

112

1

Ma

a

T

T

1

2

112

1

2

M

Mu

u

1

1

1

2

112

1

2

M

Mp

p

o

o


17/24

2

2

2

1112

1

MM

M

FF

1

22

2

2

112

1

MMMlc

ssu

p

Hence,

1

2

1

2

Mp

p

Mp

p

pp

and so on.

Isothermal flow in long ducts

Isothermal flow with friction is of interest in connection with pipe lines for transporting gas over long

distances. Mach number is usually quite low, but substantial pressure changes because of friction

over great lengths.

Energy equation: - opp dTcv

ddTcdQ

2

2

--- (1)

oT is not a constant now, it is the local stagnation temperature

2

2

2

2

2

2

112

1

2

11

M

dM

M

M

T

dTMTT

o

oo

--- (2)

Equation of state of a perfect gas is isothermal flow

d

pdp --- (3)

Alsou

du

M

dM2

2

2

--- (4)

The continuity, momentum and definition of stagnation pressure (local) remain unaltered (e.g. 4, 5, 6

earlier)


18/24

Solving the algebraic equations

Ddx

fM

M

M

dM

u

dud

p

dp4

122

12

2

2

2

Ddxf

MM

MM

p

dp

o

o 4

2

1112

2

11

22

22

Ddx

f

MM

M

T

dT

o

o4

2

1112

1

22

2

In this case, the direction of change depends not on M alone, but or2

M . Since

D

dxf4 is always

+ve, the direction of change of the parameters is as follows

1M (subsonic) Sub or Supersonic

1M

p decreases increases

decreases increases

u increases decreases

M increases decreases

oT increases decreases

op decreases increases

The Mach number always tends towards

1 . When M is less than

1, heat is added to the

stream, when M is greater than

1heat is rejected from the stream.

Again


19/24

21

4

2

2

max 14 M

M

M

D

dxf

M

L

o

2

2

2

max 14 MlM

M

D

Lf u

Denoting properties at

12 M by symbols tptV , , it can be written as 2

2

21

tVVM

Since TRT

uM ,2

2

being a constant here

MV

ut

Also *1

t

t

u

u M

Hence,*t from the perfect gas relationships

Mp

ptt

1

From the formula for isentropic stagnation pressure

1

12

1

2

11

2

11

M

p

ptt

o

o

M

M1

2

1 2

11

1321

From stagnation temperature

2

2

2

11

13

2

1

2

11

2

11

M

M

T

T

T

Ttt

o

o


20/24

In long commercial pipelines, the Mach numbers employed are so low that the loss in stagnation

pressure is virtually identical with the loss in static pressure.

2

2

2

1

2

2

2

2

2

1

2

1

2

max

1

max11

444M

Ml

M

M

M

M

D

Lf

D

Lf

D

Lf n

Since2

112

1

2

2

1

ppMM

MM

pp

2

2

1

2

1

2

1

21

4

p

pl

M

pp

D

Lf n

Since2

M cannot exceed

1 , it follows form the pressure ratio relation that2

1

2

1

2 Mp

p

For given values of1

M andD

Lf4 , there are two solutions for1

2

pp

. However, one of these is

not acceptable as it involves a violation of the second law of thermodynamics.

For a small pressure drop (in percentage), employing power series of the fractional pressure

drop 1 2

1

p p

p

1

21

2

12

1

1

21

2

12

11

24

p

ppMM

p

pp

MD

Lf

The conventional pressure-drop formula for incompressible flow is similar to above except the square

bracket on the rhs is unity. Solving the quadratic equation

D

Lf

M

M

M

M

M

M

p

pp4

11

1

1

12

1

2

1

2

2

1

2

1

2

1

2

1

1

21

It is convenient to use

1

1

2

2

1P

RT

A

mM

For a given value of1

M , there is a maximum length for continuous isothermal flow, hence, it follows

that choking effects may occur in similar fashion to those for adiabatic flow.


21/24

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.

The relation above, in the differential form, becomes

ddp

uuG

d

dp 22

2

is

entrope

Fanno

p1

p01

p*


22/24

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 heat thermal choking.

Mass Conservation2

1

1

2

u

u

Momentum Equation 1221 uuAmpp

Using, uA

m

, and pMu 2 momentum equation can be arranged to give

2

2

2

1

1

2

1

1

M

M

p

p

p1, T1

M1, T01

p2, T2

M2, T02


23/24

Equation of state:21

22

1

2

T

T

p

p

or

1

2

1

2

2

2

1

2

1

2

u

u

p

p

p

p

T

T

Definition of Mach number:2

1

1

2

1

1

1

2

1

2

T

T

u

u

u

a

a

u

M

M

Impulse function 1

1

12

11

222

1

2

Mp

MpF

F

Definition of isentropic pressure

12

1

12

1

1

2

1

2

2

11

2

11

M

M

p

p

p

p

o

o

Change in entropy

2

1 1

12

1

lno

p

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

12

2

1

2

2

12 2 ooppTTC

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

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


24/24

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

22

2

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 of1

M and2

M . 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

12

21

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 usingthese normalized expressions

2

1

1

2

MT

T

MT

T

T

T

o

o

o

o

o

o

and so on...

xxx

Documents

SET 3 Flow in Ducts, (Nozzles and Diffusers) and Wind Tunnels