TITLE.PM5

878 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-1.pm5

Throat

A=A 21

Fig. 16.7. Sonic flow (M = 1).

Large tank

Convergent

nozzle

2

V 2

p, ,T 222 r

p 1

r 1

T 1

V=0 1

1

V<V

p>p

>

T>T

21

21

21

21

rr

V>V

p<p

<

T<T

21

21

21

21

rr

V 1 V 2 V 1 V 2

( ) Divergent nozzle.a ( ) Convergent nozzle.b

Fig. 16.6. Supersonic flow (M > 1).

(iii)For sonic flow (M = 1) :

dA

A = 0 (straight flow passage

since dA must be zero)

and dp = (zero/zero) i.e., indeterminate, but

when evaluated, the change of pressure

p = 0, since dA = 0 and the flow is fric-

tionless.

16.8. Flow of Compressible Fluid Through a Convergent Nozzle

Fig. 16.8 shows a large tank/vessel fitted with a
short convergent nozzle and containing a compressible
fluid. Consider two points 1 and 2 inside the tank and
exit of the nozzle respectively.
Let p 1 = pressure of fluid at the point 1,
V 1 = velocity of fluid in the tank (= 0),
T 1 = temperature of fluid at point 1,
ρ 1 = density of fluid at point 1, and p 2 , V 2 , T 2 and
ρ 2 are corresponding values of pressure, velocity, tem-
perature and density at point 2.
Assuming the flow to take place adiabatically,
then by using Bernoulli’s equation (for adiabatic flow),
we have

γ

γρ−

F

HG

I

KJ

+

12

1

1

1

p^2

g

V

g + z^1 =

γ

γρ−

F

HG

I

KJ

+

12

2

2

2

p^2

g

V

g + z^2 [Eqn. (16.7)]

But z 1 = z 2 and V 1 = 0

∴

γ

γρ− 1

1

1

p

g =

γ

γρ−

F

HG

I

KJ

+

12

2

2

2

p^2

g

V

g

or,

γ

γ−

F

HG

I

1 KJ^

p

g

p

g

1

1

2

ρρ 2

−

L

N

M

O

Q

P =

V

g

22

2

or

γ

γ− 1

pp 1

1

2

ρρ 2

−

L

N

M

O

Q

P V^2

2

2

Fig. 16.8. Flow of fluid through a

convergent nozzle.