882 ENGINEERING THERMODYNAMICS

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\M-therm\Th16-2.pm5

Fig. 16.9. depicts the variation of actual and

theoretical mass flow rate versus
p
p

2

1

. Following
points are worthnoting :
(i) The flow rate increases with a decrease in

the pressure ratio
p
p

2

1

and attains the maximum

value of the critical pressure ratio
p
p

2

1

= 0.528 for

air.

(ii) With further decrease in exit pressure

below the critical value, the theoretical mass flow

rate decreases. This is contrary to the actual re-

sults where the mass flow rate remains constant

after attaining the maximum value. This may be

explained as follows :

At critical pressure ratio, the velocity V 2 at

the throat is equal to the sonic speed (derived below).

For an accelerating flow of a compressible fluid in a convergent nozzle the velocity of flow within

the nozzle is subsonic with a maximum velocity equal to the sonic velocity at the throat : Thus

once the velocity V 2 at the throat has attained the sonic speed at the critical pressure ratio, it

remains at the same value for all the values of

p

p

2

1

F

HG

I

KJ less than critical pressure ratio, since the flow

in the nozzle is being continuously accelerated with the reduction in the throat pressure below the

critical values and hence the velocity cannot reduce. Thus, the mass flow rate for all values of

p

p

2

1

F

HG

I
KJ less than critical pressure ratio remains constant at the maximum value (indicated by the
solid horizontal line in Fig. 16.9). This fact has been verified experimentally too.
Velocity at outlet of nozzle for maximum flow rate :
The velocity at outlet of nozzle for maximum flow rate is given by,

V 2 =

2

1

1

1

γ

γρ+

F

HG

I

KJ

p

...[Eqn. (16.31)]

Now pressure ratio,

p

p

2

1

=

2

1

1

γ

γ

γ

+

F

HG

I

KJ

−

∴ p 1 =

p

(^2) p
1
2
1
2
1
2
1
γ
γγ γ
γ
γ

• 
  

  F
  HG
  I
  KJ

  
  


• 
  

  F
  HG
  I
  − KJ
  −
  For adiabatic flow : p^1
  ρ 1
  γ =
  p 2
  ρ 2
  γ or
  p
  p
  1
  2

  
  

  ρ
  ρ
  γ
  1
  2
  F
  HG
  I
  KJ
  or
  ρ
  ρ
  1
  2
  = p
  p
  1
  2
  1
  F
  HG
  I
  KJ
  γ
  = p
  p
  2
  1
  1
  F
  HG
  I
  KJ
  −γ
  Actual
  Choked
  flow Subsonicflow
  m
  A
  max
  2
  Theoretical
  0.528
  Pressure ratio
  p
  p
  2
  1
  m/A
  2
  Fig. 16.9. Mass flow rate through
  a convergent nozzle.