Modern Engineering Thermodynamics

Summary 685
Since for a diffuser, M exit ≪ M inlet , we can assume (p exit ) actual = p e ≈ p ose = 309.3 kPa. The inlet pressure can then be
calculated as
and Eq. (16.43) gives
p inlet = p i =
p osi
1 + k − 1
2 M2 inlet
=
k
k−1
314:7kPa
1 + 1:40 − 1 1:40
0:890 2
2
1:40 − 1
= 188 kPa
C p = ðp exitÞ actual − p inlet
p osi − p inlet
=
249:3kPa− 188 kPa
314:7kPa− 188 kPa = 0:484
Exercises
41. If the diffuser shape is modified in Example 16.15 so that the exit stagnation pressure is increased from 249.3 kPa to
261.5 kPa, determine the new values for the diffuser’s efficiency and pressure recovery coefficient, assuming all other
variables remain unchanged. Answer: η D = 0.623 = 62.3% and C p = 0.580.
42. A supersonic diffuser is also tested with the same inlet and exit conditions as the subsonic diffuser in Example 16.15,
except that the inlet Mach number is 2.45 instead of 0.890. Determine the efficiency and pressure recovery coefficient of
the supersonic diffuser. Answer: η D = 0.882 = 88.2% and C p = 0.778.
43. What value does the diffuser efficiency η D approach as the exit isentropic stagnation pressure approaches the inlet
isentropic stagnation pressure (i.e., η D → ?asp ose → p osi )? Answer: η D → 1.00 = 100.%.
SUMMARY
In this chapter, we investigate the basic phenomena that occur in high-speed compressible flows of gases and
vapors. New concepts, such as the stagnation state, Mach number, choked flow, and shock waves, are introduced
to fully explain the basic characteristics of these flows. We focus our attention on converging-diverging nozzle
and diffuser flow geometries because of their industrial value and their ability to generate supersonic flows and
shock waves. Finally, we consider the overall performance of nozzles and diffusers in terms of their actual operating
efficiencies.
We also introduce the Reynolds transport theorem in this chapter. This allows us to generalize our open system
balance concept and subsequently to easily develop a linear momentum rate balance for open systems.
Some of the more important equations introduced in this chapter follow. Do not attempt to use them blindly
without understanding their limitations. Please refer to the text material where they were introduced to gain an
understanding of their use and limitations.
1. Stagnation state specific enthalpy h 0 of a fluid with specific enthalpy h and velocity V:
h o = h + V 2 /ð2g c Þ
2. Stagnation state temperature T o of an ideal gas with a velocity V at a temperature T and a constant pressure
specific heat c p :
T o = T 1 + V2
2g c c p T
3. Isentropic stagnation state properties denoted by an os subscript:
T os
T = p ðk−1Þ/k
os
= v 1 − k
os ρ = os
p
v ρ
where
k
p os
p = 1 + V2 k−1
2g c c p T
and
k−1
ρ 1
os
ρ = 1 + V2 k−1
2g c c p T
4. The Mach number M is the ratio of the fluid velocity V to the velocity of sound c in the fluid:
M = V/c