Modern Engineering Thermodynamics

688 CHAPTER 16: Compressible Fluid Flow
27.* Air enters a supersonic isentropic diffuser with a Mach number of
3.00, a temperature of 0.00°C, and a pressure of 1.00 × 10 –2 MPa.
Assuming the air exits with negligible velocity, determine the exit
temperature, pressure, and mass flow rate per unit inlet area.
28.* Propane at 100. kPa 40.0°C is expanded isentropically through a
converging-diverging nozzle that has an exit to throat diameter
ratio of 2.00. The propane enters with a negligible velocity but
reaches sonic velocity at the throat. Determine the exit
temperature, pressure, and Mach number, if (a) the exit pressure
is high enough so that the exit velocity is subsonic and (b) the
exit pressure is low enough that the exit velocity is supersonic.
29. An isentropic converging-diverging nozzle that reaches a Mach
number of 4.00 at the exit, when the exit pressure is
atmospheric (14.7 psia) and the inlet isentropic stagnation
temperature is 70.0°F, is to be built using air as the working
fluid. Determine the required inlet isentropic stagnation
pressure, the exit static temperature, and the exit to throat area
ratio.
30. Acetylene at 50.0 psia, 65.0°F is accelerated through a
converging-diverging nozzle isentropically until it reaches an exit
pressure of 14.7 psia. Assuming the flow enters the nozzle with
a negligibly small velocity, determine the exit Mach number,
temperature, and exit to throat area ratio.
31.* Carbon dioxide gas at 13.8 MPa, 20.0°C is expanded
isentropically through a converging-diverging nozzle until its
exit temperature reaches –100.°C. Assuming the flow enters with
a negligible inlet velocity, determine the exit Mach number,
pressure, and exit to throat area ratio.
32. Steam at 800. psia, 600.°F expands through an uninsulated
nozzle to a saturated vapor at 600. psia at a rate of 100. lbm/h.
The surface temperature of the nozzle is measured and found to
be 450.°F. The entropy production rate of the nozzle is equal to
10.0% of the magnitude of the heat transport of entropy for the
nozzle. The potential energies and the inlet velocity can be
neglected, but the exit velocity cannot be neglected. Determine
a. The nozzle’s heat transfer rate.
b. The nozzle’s entropy production rate.
c. The nozzle’s exit velocity.
d. The exit area of the nozzle.
33. Determine the maximum possible mass flow rate of air through
a nozzle with a 1.00 × 10 –3 m diameter throat and inlet
stagnation conditions of 5.00 MPa and 30.0°C.
34.* Determine the maximum flow rate of helium that passes
through a nozzle with a 1.00 × 10 –2 m diameter throat from an
upstream stagnation state of 35.0 MPa at 27.0°C.
35. Determine the minimum throat diameter required for a nozzle
to pass 0.250 lbm/s of air from a stagnation state of 100. psia at
70.0°F.
36. Atmospheric air (14.7 psia, 70.0°F) leaks into an initially
evacuated 2.00 ft 3 tank through a tiny hole whose area is
1.00 × 10 –6 ft 2 . Determine the time required for the pressure in
the tank to rise to 0.528 times the atmospheric pressure, if the
air inside the tank is maintained at 70.0°F.
37.* A tiny leak in a 1.00 m 3 vacuum chamber causes the internal
pressure to rise from 1.00 Pa to 10.0 Pa in 3.77 h when the
vacuum pump is not operating. Air leaks into the chamber from
the atmosphere at 101.3 kPa, 20.0°C, and the air inside the
chamber is maintained at 20.0°C by heat transfer with the walls.
Determine the diameter of the leak hole.
38. An initially evacuated 1.50 ft 3 tank is to be isothermally filled
with air to 50.0 psia and 70.0°F. It is to be filled from a very
large constant pressure source at 100. psia and 70.0°F. The tank
is connected to the source by a single 0.125-inch inside
diameter tube. How long will it take to fill the tank?
39.* An initially evacuated 0.500 m 3 tank is to be isothermally filled
with air to 1.00 MPa and 20.0°C. It is to be filled from a very
large constant pressure source at 2.50 MPa and 20.0°F. The tank
is connected to the source by a single 1.00 × 10 –3 m inside
diameter tube. How long will it take to fill the tank?
40. Use the Reynolds transport equation (Eq. (16.31)) to develop a
formula for a one-dimensional angular momentum rate balance
(AMRB), and show that, for steady flow with a single inlet and a
single outlet, the AMRB reduces to
∑T ext = _m ðV × rÞ out
− ðV × rÞ in /gc
where T ext = F ext × r is the torque vector due to external forces,
V is the average velocity vector, and r is the radius vector. Hint:
Start with X = m(V × r) and utilize the conservation of angular
momentum principle.
41. Use the linear momentum rate balance (LMRB) to show that
the thrust force F of a rocket engine nozzle (Figure 16.26) is
given by
p a
F = _m ðV exit /g c
Þ+ ðp exit − p a ÞA exit
and show that the absolute maximum thrust produced as
M exit → ∞ is given by
(Hint: See Problem 3.)
FIGURE 16.26
Problem 41.
F
F max = 2c p ðρ exit A exit T os
Rocket
Þ+ ðp exit − p a
ÞA exit
p ex
V ex
Area A ex
42. The thrust F produced by the supersonic flow in the convergingdiverging
nozzle of a rocket engine is
F = _m ðV exit /g c Þ+ ðp exit − p a ÞA exit , where p exit and A exit are the
pressure and area at the nozzle exit and p a is the local
atmospheric pressure.
a. Suppose the stagnation temperature is increased by 100%
while maintaining the stagnation and exit pressures and
nozzle geometry constant. What is the percent increase in
thrust?
b. Suppose the stagnation pressure is increased by 100% while
maintaining the stagnation and exit temperatures and nozzle
geometry constant. What is the percent increase in thrust?
Hint: Assume the nozzle is choked in each case, and use air
as the exhaust gas.