Modern Engineering Thermodynamics

16.4 The Mach Number 655
Note that p os is the same as p to three significant figures. At 20.0°C and 0.101 MPa flow stream conditions, the flow stream
density is
ρ =
Then, Eq. (16.4) gives the isentropic stagnation density as
p
RT =
101 kPa
= 1:21 kg/m3
ð0:286 kJ/kg.KÞð20:0 + 273:15 KÞ
ρ os = ρ 1 + V 2
1
k−1
2g c c p T
0
1
ð25 m/sÞ 2 1 kJ/kg
= ð1:21 kg/m 3 1000 m
Þ 1 +
2 /s 2
B
C
@ 2ð1Þð1:004 kJ/kg.KÞð20:0 + 273:15 KÞA
1
1:40−1
= 1:1213 kg/m 3
Note that ρ os is the same as ρ to three significant figures. The following example shows that the use of an isentropic stagnation
state is not limited to ideal gases.
EXAMPLE 16.3
A new steam turbine design contains a flow nozzle that produces a flow of steam at 14.7 psia and 1000.°F at a velocity of
1612 ft/s. Determine the isentropic stagnation temperature, pressure, and density of this flow.
Solution
Equations (16.2, 16.3), and (16.4) are only for ideal gases. However, we can use the steam tables to solve this problem
as follows. Let station 1 be the exit flow stream from the nozzle and let station os be the same flow stream isentropically
decelerated to a zero velocity (stagnation) state. From Eq. (16.1), we have h os = h 1 + V1 2/2g c = h 1 + ð1612 ft/sÞ 2 /
½2ð32:174 lbm.ft/lbf .s 2 Þð778:16 ft.lbf/BtuÞŠ = h 1 + 51:90 Btu/lbm. Then, the station data become
Station 1
p 1 ¼ 14:7 psia
T 1 ¼ 1000°F
h 1 ¼ 1534:4 Btu=lbm
Station os
s os ¼ s 1 ¼ 2:1332 Btu=lbm.R
h os ¼ h 1 þ V 2 1 =2g c ¼ 1534:4 þ 51:90 Btu=lbm ¼ 1586 Btu=lbm
p os ¼ ? psia
s 1 ¼ 2:1332 Btu=lbm.R T os ¼ ? °F
ρ os ¼ ? lbm=ft 3
and using these values of s os and h os and the steam table, Table C.3a, in Thermodynamic Tables to accompany Modern Engineering
Thermodynamics a Mollier diagram for steam, or computerized steam tables, we find that p os ≈ 20. psia, T os ≈ 1100°F, and ρ os =
1/v os ≈ 1/(46.4 ft 3 /lbm) = 0.022 lbm/ft 3 .
Exercises
4. Determine the isentropic stagnation pressure in Example 16.2 if the air stream is moving at 500. km/h instead of
90.0 km/h. Assume all the other variables remain unchanged. Answer: p os = 0.112 MPa.
5. Use the steam tables or a Mollier diagram for steam to estimate the isentropic stagnation pressure of steam at 1100°F
and 600. psia traveling at 1600. ft/s. Answer: p os ≈ 800 psia.
6. If the isentropic stagnation temperature and pressure of the steam in Example 16.3 are 60.0 psia and 1500.°F, determine
the corresponding velocity of the steam at station 1, assuming all the other variables remain unchanged. Answer:
V ≈ 3670 ft/s.
16.4 THE MACH NUMBER
The Mach number M was introduced in 1929. It was named in honor of Ernst Mach (1838–1916), an Austrian
physicist and philosopher who studied high-speed compressible flow in the1870s. It is the dimensionless ratio
of the local fluid velocity V to the velocity of sound, c:
M = V/c (16.5)