TITLE.PM5

886 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-2.pm5

Example 16.13. (a) In case of isentropic flow of a compressible fluid through a variable

duct, show that

A

Ac =

1

M

1 1

2

(1)M

1

2

(1)

2

1

+− 2( 1)

+

L

N

M

M

M

O

Q

P

P

P

+

γ −

γ

γ

γ

where γ is the ratio of specific heats, M is the Mach number at a section whose area is A and Ac is
the critical area of flow.
(b) A supersonic nozzle is to be designed for air flow with Mach number 3 at the exit
section which is 200 mm in diameter. The pressure and temperature of air at the nozzle exit are
to be 7.85 kN/m^2 and 200 K respectively. Determine the reservoir pressure, temperature and the
throat area. Take : γ = 1.4. (U.P.S.C. Exam.)
Sol. (a) Please Ref. to Art. 16.9.
(b) Mach number, M = 3
Area at the exit section, A = π/4 × 0.2^2 = 0.0314 m^2
Pressure of air at the nozzle, (p)nozzle = 7.85 kN/m^2
Temperature of air at the nozzle, (T)nozzle = 200 K
Reservoir pressure, (p)res. :

From eqn. (16.17), (p)res. = (p)nozzle 1 1

2

+F −^21

HG

I

KJ

L

N

M

O

Q

P

−

F

HG

I

γ KJ

γ

γ

M

or, (p)res. = 7.85 1 14 1

2

32

1

11

+F −

HG

I

KJ

L ×

N

M

O

Q

P

−

F

HG

I

. KJ

.4

.4

= 288.35 kN/m^2 (Ans.)

Reservoir temperature, (T)res. :

From eqn. (16.22), (T)res. = (T)nozzle 1

1

2

+F −^2

HG

I

KJ

L

NM

O

QP

γ M

or, (T)res. = 200 1 14 1

2

+F − 32

HG

I

KJ

L ×

N

M

O

Q

P

. = 560 K (Ans.)

Throat area (critical), Ac :

From eqn. (16.41),

A

Ac =

12 1

1

2

1

21

M

+−M

+

L

N

M

M

O

Q

P

P

+

()γ ()−

γ

γ

γ

or,

0 0314.

Ac =

1

3

21413

14 1

2

11

+−^211

+

L

N

M

M

O

Q

P

P

+

(. ) −

.4

(.4 )

or 0 0314^1

3

Ac

= (2.333)^3 = 4.23

or, Ac =
0 0314
423
= 0.00742 m^2 (Ans.)

16.10.Flow Through Laval Nozzle (Convergent-divergent Nozzle)

Laval nozzle is a convergent-divergent nozzle (named after de Laval, the swedish scientist

who invented it) in which subsonic flow prevails in the converging section, critical or transonic

conditions in the throat and supersonic flow in the diverging section.