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864 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-1.pm5

∴ C =

dp

dρ ...(16.10)

16.3.2. Sonic velocity in terms of bulk modulus

The bulk modulus of elasticity of fluid (K) is defined as

K =

dp

dv

v

F

HG

I

KJ

...(i)

where, dv = decrease in volume, and v = original volume
(– ve sign indicates that volume decreases with increase in pressure)

Also, volume v ∝

1

ρ, or vρ = constant

Differentiating both sides, we get

vdρ + ρdv = 0or–

dv

v

=dρ

ρ

Substituting the value of – dv

v

dp

K

FHG= IKJ from eqn. (i), we get

dp

K =

dρ

ρ

or

dp

dρ

=

K

ρ

Substituting this value of

dp

dρ in eqn. (16.10), we get

C =

K

ρ ...(16.11)

Eqn. (16.11) is applicable for liquids and gases.

16.3.3. Sonic velocity for isothermal process

For isothermal process :

p

ρ = constant

Differentiating both sides, we get

ρρ

ρ

..dp−p d

2 = 0 or

dp p d

ρ

ρ

ρ

−. 2 = 0

or,
dp
ρ
= pd. ρ
ρ^2

or

dp

d

p

ρρ

= = RT ...(16.12)

p RT

ρ

=

F

HG

I

KJ

...equation of state

Substituting the value of

dp

dρ

in eqn. (16.10), we get

C =

p

ρ = RT ...(16.13)