880 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-1.pm5

m = A 2

2

1 11

2

1

2

2

1

1

γ

γ

ρ

γ γγ

−

F

HG

I

KJ

−

F

HG

I

KJ

L

N

M

M

M

O

Q

P

P

P

+

p p

p

p

p

/

...(16.28)

The mass rate of flow (m) depends on the value of p

p

2

1

(for the given values of p 1 and ρ 1 at

point 1).

Value of

p

p

2

1

for maximum value of mass rate of flow :

For maximum value of m, we have d

d

p

p

m

2

1

F

HG

I

KJ

() = 0

As other quantities except the ratio p

p

2

1

are constant

∴

d

d

p

p

m

2

1

F

HG

I

KJ

() =

p

p

p

p

2

1

2

2

1

1

F

HG

I

KJ

−

F

HG

I

KJ

L

N

M

M

M

O

Q

P

P

P

/γ γ+

γ

= 0

or,

(^22)
1
(^21)
γ
p γ
p
F
HG
I
KJ
−

• γ
  γ

γ

F + γ

HG

I

KJ

F

HG

I

KJ

+ −

(^12)
1
(^11)
p
p
= 0
or, p
p
2
1
(^21)
F
HG
I
KJ
γ−
= γ
F + γ
HG
I
KJ
F
HG
I
KJ
1
2
2
1
1
p
p
or p
p
2
1
2
F
HG
I
KJ
−γ
γ
= γ

• F γ
  HG
  I
  KJ
  1
  2
  2
  1
  1
  p
  p
  or,
  p
  p
  2
  1
  F^2
  HG
  I
  KJ
  −γ
  = γ


• 
  

  γ
  F
  HG
  I
  KJ
  F
  HG
  I
  KJ
  1
  2
  2
  1
  p
  p
  or,
  p
  p
  2
  1
  F^21
  HG
  I
  KJ
  −−γ

  
  

  Fγ+ γ
  HG
  I
  KJ
  1
  2
  or
  p
  p
  2
  1
  1
  1
  2
  F
  HG
  I
  KJ
  =F +
  HG
  I
  KJ
  −γ γ
  γ
  or,
  p
  p
  2
  1
  1
  F
  HG
  I
  KJ
  −γ

  
  

  2
  γ 1
  γ

  
  


• 
  

  F
  HG
  I
  KJ
  or,
  p
  p
  2
  1
  F
  HG
  I
  KJ
  =^2
  1
  1
  γ
  γ
  γ

  
  


• 
  

  F
  HG
  I
  KJ
  −
  ...(16.29)
  Eqn. (16.29) is the condition for maximum mass flow rate through the nozzle.
  It may be pointed out that a convergent nozzle is employed when the exit pressure is equal
  to or more than the critical pressure, and a convergent-divergent nozzle is used when the dis-
  charge pressure is less than the critical pressure.
  For air with γ = 1.4, the critical pressure ratio,
  p
  p
  2
  1

  
  

  2
  41
  4
  41

  
  

1. 
   


2. 
   


3. 
   

• F
  HG
  I
  KJ
  −
  = 0.528 ...(16.30)