Modern Engineering Thermodynamics

690 CHAPTER 16: Compressible Fluid Flow
61. Use Eqs. (16.35) and (16.37) to develop the relation
T y
= ðk − 1ÞM2
x + 2
2kM
2
x − ðk − 1Þ
T x ½ðk + 1ÞM x Š 2
62. In a supersonic wind tunnel utilizing air similar to that shown
in Figure 16.12, the converging-diverging nozzle section has
an inlet isentropic stagnation pressure of 3.20 atm. The test
section has a Mach number of 2.70, and the convergingdiverging
diffuser section has an exit pressure of 1.00 atm.
Determine the efficiency of the converging-diverging diffuser
section.
63. Measurements on a prototype nozzle using air produce inlet and
exit temperatures of 70.0°F and 60.0°F, respectively, while the
exit velocity is 325 ft=s. Determine the nozzle’s efficiency and
velocity coefficient.
64.* An air diffuser has inlet and exit isentropic stagnation pressures of
3.50 and 3.10 MPa, respectively. The inlet velocity is 300. m/s and
the inlet static temperature is 27.0°C. Determine the diffuser
efficiency, pressure recovery coefficient, and exit static temperature,
if the air leaves the diffuser with a negligible velocity.
65.* 8.00 kg/s of air flows through a diffuser with an inlet diameter
of 0.0350 m and a static pressure and temperature of 0.500 MPa
and 22.0°C. The air exits through a diameter of 0.900 m at static
conditions of 0.540 MPa and 25.0°C. Determine the diffuser’s
efficiency and pressure recovery coefficient.
66. A diffuser decelerates 15.0 kg/s of carbon dioxide from 200. m/s
at 20.0°C and 0.800 MPa to 1.00 m/s at 30.0°C and 1.00 MPa.
Determine the diffuser efficiency, pressure recovery coefficient,
and the inlet and exit areas.
67.* Experimental measurements on a new methane fuel nozzle for a
furnace produce an exit velocity, pressure, and temperature for
methane of 335 m/s, 0.100 MPa, and 0.00°C. The upstream
stagnation pressure and temperature are 0.150 MPa and 22.0°C.
Determine
a. The nozzle efficiency.
b. The nozzle’s velocity coefficient.
c. The nozzle’s discharge coefficient.
68. A diffuser having an efficiency of 92.0% is to be used to reduce
the velocity of an air stream initially at 450. ft/s, 65.0°F, and
50.0 psia down to a Mach number of 0.100. Calculate
a. The exit to inlet area ratio ðA exit /A inlet Þ required.
b. The pressure recovery factor for the diffuser.
69. A sonic converging nozzle with a negligible inlet velocity and
inlet and throat areas of 2.00 in 2 and 0.500 in 2 , respectively, has
a velocity coefficient of 0.820 when the upstream stagnation
pressure and temperature are 100. psia and 70.0°F. Determine
the thrust produced by the nozzle in atmospheric air.
Design Problems
The following are open-ended design problems. The objective is to carry
out a preliminary design as indicated. A detailed design with working
drawings is not expected unless otherwise specified. These problems do
not have specific answers, so each student’s designisunique.
70. Design a converging-diverging nozzle system that can be used to
demonstrate supersonic flow in the classroom. Choose a
convenient gas, inlet conditions, and exit Mach number.
Determine the necessary area ratios, pressures, and temperatures
throughout the system.
71.* Design an attitude control nozzle for a spacecraft that produces
50.0 N of thrust (see Problem 42) using compressed helium gas
stored at 50.0 MPa and 0.00°C. Assume the nozzle discharges
into a total vacuum. Specify the nozzle inlet, throat, and exit
areas as well as the exit Mach number.
72.* Design a system that has no moving mechanical parts to cool
machine shop compressed air at 0.500 MPa, 25.0°C to 0.00°C at
0.101 MPa. The outlet velocity must be at least 10.0 m/s. If
possible, fabricate and test your design.
73. Design a small demonstration wind tunnel to be driven from a
standard compressed air supply line at 100. psia and 70.0°F.
Assume that the maximum volumetric air flow rate available
from this supply is 10.0 ft 3 /min at 14.7 psia and 70.0 o F. The
wind tunnel test section must be at least 1.00 inch in diameter
and must reach a Mach number of at least 2.25. The air may be
exhausted to the atmosphere, but it first must be decelerated to
subsonic velocity to minimize noise generation. If possible,
fabricate and test your design.
74.* Design a converging-diverging nozzle for a spacecraft thruster
that has an exit Mach number of 5.00 when using compressed
helium at 50.0 MPa, and 0.00°C. Assume the nozzle exhausts
into a total vacuum. Plot the nozzle diameter vs. length along
the nozzle, keeping the angle of the diverging wall to less than
10 o with the horizontal to prevent flow separation. Show the
positions along the nozzle where M = 1, 2, 3, 4, and 5.
Determine the mass flow rate through your nozzle.
75. Design a system that produces a constant mass flow rate of
1.00 × 10 –2 lbm/s of oxygen from one or more 3.00 ft 3 highpressure
storage bottles initially at 2000. psia and 400. R. The
oxygen must be delivered at 50.0 ft=s at 60.0°F and 175 psia.
The system must operate continuously for six months and must
have a fail-safe backup.
Computer Problems
The following open-ended computer problems are designed to be
done on a personal computer using a spreadsheet or problem
solver.
76. Develop a computer program that returns values for
T /T os , p /p os , and ρ /ρ os , when k and the remaining variables
are input in response to a screen prompt.
77. Develop an interactive computer program that returns values for
p/p os , T/T os , ρ/ρ os , and A/A , when k and M are input from the
keyboard in response to a screen prompt.
78. Develop an interactive computer program that returns values for
M y , p y /p x , T y /T x , ρ y /ρ x , _S / _m , and p osy /p osx , when k and M x are
input from the keyboard in response to a screen prompt.
79. Using Eqs. (16.35), (16.36), (16.37), and (16.38), plot _Sp/ð _mRÞ
vs. M x for 1 ≤ M x ≤ 50 for (a) air, (b) carbon dioxide,
(c) methane, and (d) water vapor (use k = 1:33 here).
80.* Fanno line. An analysis of the adiabatic aergonic flow of a
viscous ideal gas with constant specific heats traveling through a
constant area duct can be carried out by combining the
continuity equation, the energy rate balance (Eq. (16.1)), and
the entropy rate balance (using Eq. (7.36)) to obtain the
following relation:
_S p
ð _mc v Þ = s out − s in
c v
" # ðk−1Þ/2
T os − T out
T os − T in
= ln T out
T in
≥ 0