Modern Engineering Thermodynamics

658 CHAPTER 16: Compressible Fluid Flow
or
∂h
= v = 1 ∂p
s
ρ
(16.8)
Multiplying Eq. (16.6) by Eq. (16.7) and dividing by Eq. (16.8) gives a relation for the sonic velocity c in terms
of the measurable properties p and ρ:
∂V ∂h ∂p
= ∂p
= ρðc/g c Þðc/ρÞ = c 2 /g c
∂ρ
s
∂V s ∂h
s
∂ρ
s
or
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∂p
c = g c
∂ρ
s
(16.9)
Equation (16.9) is a valid equation for the isentropic sonic velocity in any compressible substance. In particular,
in the case of an ideal gas, Eq. (7.39) relates pressure and density for an isentropic process by
pv k = pρ –k = constant (7.39)
Solving for p = constant × ρ k and taking the partial differential of p with respect to ρ gives
∂p
= kp/ρ = kRT
∂ρ
s
where we use the ideal gas law, p = ρRT. Then, from Eq. (16.9), the speed of sound (sonic velocity) in an ideal
gas is
p
= ffiffiffiffiffiffiffiffiffiffiffiffi
kg c RT
(16.10)
and from Eq. (16.5),
c ideal
gas
M ideal
gas
V
= p ffiffiffiffiffiffiffiffiffiffiffiffi
(16.11)
kg c RT
EXAMPLE 16.4
Methane (CH 4 ) gas at 35.0°C flows through a pipe with a velocity of 300. m/s. Determine the Mach number of the
methane.
Solution
The velocity of sound in methane at 35.0°C is given by Eq. (16.10). Using Table C.13b in Thermodynamic Tables to accompany
Modern Engineering Thermodynamics for the values of the specific heat ratio and the gas constant for methane, we get
p
c methane =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðk methane Þðg c ÞðR methane ÞðTÞ
p
= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1:30Þð1Þð518 J/kgCKÞð35:0 + 273:15 KÞ
= 456 m/s
then, Eq. (16.5) (or (16.11)) gives the Mach number as
M methane =
300: m/s
456 m/s = 0:658
Exercises
7. Determine the Mach number of the methane in Example 16.4 if its velocity is reduced from 300. m/s to 3.00 m/s and
all the other variables remained unchanged. Answer: M methane = 6.58 × 10 –3 .
8. If the methane in the pipe in Example 16.4 is cooled to 0.00°C and all the other variables remain unchanged, determine
the new Mach number of the methane. Answer: M methane = 0.700.
9. If the gas in Example 16.4 is changed from methane to argon, determine the Mach number of the argon assuming all
the other variables remain unchanged. Answer: M argon = 0.917.