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)