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AREA RATIO AS FUNCTION OF MACH NUMBER

For Isentropic flow, look at effect of area change on M by comparing A at any

point to area at sonic pointA*

Use mass conservation to find relation

c

c

A

A

cAAc

**

*

***

21

2

2

2

1

2

11

*

1**

M

M

Mc

a

c

c

21

2

1

1

1

21*

MMc

c

We know that

(2)

(1)

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Cont..

11

1

1

1

2*

(3)

Substitute 2 and 3 in 1

)1(2

1

2

1

1

1

21

*

M

MA

A

In many compressible flow calculations the function A/A* ( p/p0 )occurs frequently.

This can also be obtained as a function of Mach number from the above andbelow equation.

1

2

0

M2

11

p

p

(4)

(5)

21

2

121

0M

2

11

1

2

M

1

P

P

A

A

Using 4 & 5

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Impulse Function (F)

The thrust experienced by the duct in the direction shown is a

result of changes in pressure and M between the cross

sections 1 and 2.

The terms pA and AC2 occur frequently in compressible flow

equations. Since both these terms have same unit, they are

defined by a parameter F, Impulse function or the wall forcefunction

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Cont..

By momentum equation the Thrust is given by

)()(2

11111

2

22222 cAApcAAp (1)

For a perfect gas

2

2

22 pMRTcpc

RTpc

(2)

The impulse function is 2AcpAF (3)

Equations (3) and (2) give )M1(pAF 2 (4)

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Substituting Equation (4) in (1) yields

)M1(Ap)M1(ApFF

2

111

2

222

12

The thrust due to the fluid flow between two sections of aduct can be obtained by the change of the impulse function

between these sections.

At M=1, F=F* and eqn. (4) reduces to

)1(ApF

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Cont..

Since the M at exit is less than 1, the given nozzle isconvergent type; the throat and exit are same.

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Cont..