Modern Engineering Thermodynamics

16.4 The Mach Number 657
Table 16.1 lists the jargon terms developed to describe the different flow regimes. Transonic flow is a flow in
which the Mach number fluctuates around 1.0 by a small amount (±ε).
We can deduce a relation for the dependence of the isentropic speed of sound (or sonic velocity) c on local thermodynamic
properties from a mass and energy balance analysis of a moving acoustical wave. Figure 16.5 shows
an open system attached to an isentropic sound wave moving at velocity c through a stationary fluid in a duct
that has a constant cross-sectional area A. Our coordinate system is attached to the moving wave, so the fluid
appears to be approaching the wave with velocity c. For an adiabatic, aergonic, reversible, steady state, steady
flow, single-inlet, single-outlet open system, the mass rate balance (MRB) reduces to
or
_m in = _m out
ρAc = ðρ + ∂ρ s ÞðAÞðc −∂V s Þ
= Aðρc − ρ ∂V s + c ∂ρ s −∂ρ s ∂V s Þ
Where the subscript s indicates that entropy is held constant during the differentiation process. Neglecting
second-order differential terms (i.e., setting ∂ρ s ∂V s = 0), this equation can be rearranged to give
∂V
= c/ρ (16.6)
∂ρ
s
Similarly, the energy rate balance for this system becomes
h + c 2 /2g c = ðh + ∂h s Þ + ðc −∂V s Þ 2 /ð2g c Þ
Again neglecting second-order differential terms, this can be expanded and rearranged to give
∂h
= c/g c (16.7)
∂V s
Finally, the Gibbs Eq. (7.21) for an isentropic process yields
T∂s s = ∂h s − v∂p s = 0
Table 16.1 Compressible Flow Mach Number Regimes (ε Is a Small Fluctuation)
Flow Regime
Name
M < 1
Subsonic
M = 1
Sonic
1 − ε ≤ M ≤ 1 + ε Transonic
M >1
Supersonic
M ≥ 5
Hypersonic
System boundary attached to a moving sound wave
Upstream
properties
p
ρ
h
and s
Downstream
properties
c c − ∂ V s p + ∂p s
ρ + ∂ρ s
h + ∂ p s
and s
FIGURE 16.5
An isentropic sound wave moving through a stationary fluid in a horizontal duct of constant cross-sectional area A. The coordinate
system is fixed to the moving wave so that the surrounding fluid appears to be moving. The s subscript is used to indicate isentropic
changes in the properties.