COMPRESSIBLE FLOW 877

dharm

\M-therm\Th16-1.pm5

Substituting the value of

dρ

ρ in eqn. (16.24), we get

dV

V +

dA

A –

VdV

C^2 = 0

or,
dA
A

VdV

C^2

• dV
  V
  =
  dV
  V

V

C

2

2 −^1

F

HG

I

KJ

∴

dA

A

=

dV

V

(M^2 – 1) Q M

V

C

FHG = IKJ ...(16.25)

This important equation is due to Hugoniot.

Eqns. (16.23) and (16.25) give variation of

dA

A

F

HG

I

KJ for the flow of incompressible and com-

pressible fluids respectively. The ratios

dA

A

F

HG

I

KJ and

dV

V

F

HG

I

KJ are respectively fractional variations in

the values of area and flow velocity in the flow passage.

Further, in order to study the variation of pressure with the change in flow area, an expres-

sion similar to eqn. (16.25), as given below, can be obtained.

dp = ρV^2

1

1 −^2

F

HG

I

M KJ

dA
A ...(16.26)
From eqns. (16.25) and (16.26), it is possible to formulate the following conclusions of prac-
tical significance.
(i)For subsonic flow (M < 1) :
dV
V > 0 ;

dA

A < 0 ; dp < 0 (convergent nozzle)

dV

V

< 0 ;

dA

A

> 0 ; dp > 0 (divergent diffuser)

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

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

Fig. 16.5. Subsonic flow (M < 1).

(ii)For supersonic flow (M > 1) :

dV

V > 0 ;

dA

A > 0 ; dp < 0 (divergent nozzle)

dV

V < 0 ;

dA

A < 0 ; dp > 0 (convergent diffuser)