Modern Engineering Thermodynamics

43. 0.800 lbm/s of air passes through an insulated converging
nozzle that has an inlet to exit area ratio of 1.59 to 1. The
nozzle is choked, and the stagnation temperature is 80.0°F. The
exit pressure is 14.7 psia. Determine the force required to hold
the nozzle in place.
44.* Determine the horizontal and vertical forces on the stationary
turbine blade shown in Figure 16.27 when it is exposed to a
0.300 kg/s jet of air at 100. m/s.
0.300 kg/s
FIGURE 16.27
Problem 44.
100. m/s
F y = ?
60.0°
F x = ?
45.* Determine the horizontal and vertical restraining forces on the
air flow divider shown in Figure 16.28.
0.250 m diam
0.500 m diam
3.50 kg/s
200. m/s
1.00 MPa
1.00 m diam
FIGURE 16.28
Problem 45.
60.0 m/s
0.800 MPa
30.0°
60.0°
0.800 kg/s
150. m/s
0.600 MPa
F y = ?
F x = ?
46. What two conditions are required of an ideal gas to have
T osy = T osx across a normal shock wave?
47.* The flow conditions just downstream of a standing normal
shock wave in air in a wind tunnel are M y = 0.500, p y = 0.100
MPa, and T y = 450. K. Determine the flow conditions just
upstream of the shock (i.e., M x , p x , and T x ).
48.* A nuclear blast generates a normal shock wave that travels
through still air with a Mach number of 5.00. The pressure and
temperature in front of the shock (i.e., downstream) are 0.101
MPa and 20.0°C. Determine the air velocity relative to a
stationary observer (i.e., the wind velocity), the pressure, and the
temperature immediately after the shock wave has passed.
49.* The upstream and downstream temperatures across a normal
shock wave in air are measured and found to be 306.3 and
717.6 K, respectively. Determine the upstream and downstream
Mach numbers and the pressure ratio across the shock wave.
50.* The upstream and downstream static pressures across a normal
shock wave in air are measured and found to be 0.500 and
3.00 MPa, respectively. Determine the upstream and downstream
Mach numbers and the temperature ratio across the shock wave.
51.* The upstream and downstream isentropic stagnation
pressures across a normal shock wave in air are measured
and found to be 124.6801 kPa and 0.101325 MPa, respectively.
Determine the upstream and downstream Mach numbers and
static pressure and temperature ratios across the shock wave.
52. A converging-diverging nozzle has an exit to throat area ratio of
2.00. The inlet isentropic stagnation air pressure is 2.00 atm and
the exit static pressure is 1.00 atm. This flow is supersonic in a
portion of the nozzle, terminating in a normal shock inside the
nozzle. Determine the local area ratio A/A*at which the shock
occurs.
53.* Air with a velocity of 450. m/s and a static pressure and
temperature of 1.00 MPa and 200. K undergoes a normal shock.
Determine the velocity and static pressure and temperature after
the shock.
54. Use the conservation of mass condition across a shock wave
ð _m x = _m y Þ to show that
p x M x ðT x Þ −0:5 = p y M y ðT y Þ −0:5
55. Using Eqs. (16.34), (16.35), and (16.36), derive Eq. (16.37).
(Hint: Use Eqs. (16.35) and (16.36) to eliminate p x /p y and
pffiffiffiffiffiffiffiffiffiffiffi
T y /T x in Eq. (16.34). Then, square both sides of the resulting
equation and solve for M 2 y in terms of M2 x .)
56. Using Eqs. (16.13), (16.36), and (16.37), show that
P osy
P osx
=
"
ðk + 1ÞM 2 x /2
# k/ðk − 1Þ
1 + ðk − 1ÞM 2 x /2 × 2kM2 x
k + 1 − k − 1
k + 1
Problems 689
1/ð1 − kÞ
57. It may be shown algebraically that, across a normal shock,
V x
V y
c c = 1:0
p
where c =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kg c RT is the sonic velocity at the throat.
Consequently, it has become customary to use the rather
awkward notation M* = V/c* so that this relation can be written
as M x M y = 1:0: Table C.18 in Thermodynamic Tables to accompany
Modern Engineering Thermodynamics includes an M* column for
this purpose. Verify this relation for the normal shock data given
in Example 16.12 by
a. Calculating V x ,V y , c*and V x ,V y /(c*) 2 .
b. Using Table C.18 to calculate M x M y :
58. It can be shown, for a normal shock wave, that
ρ y
ρ x
= V x
V y
= ðk + 1ÞM2 x
ðk − 1ÞM 2 x + 2
Using this relation, determine the maximum density ratio
ðρ y /ρ x Þ max that can occur across a normal shock wave in air.
59. Use Eqs. (16.36) and (16.37) to develop the relation
p y
p x
= 1 +
2k
k + 1 M2 x − 1
60. The strength of a normal shock wave is defined as p y − p x
/px :
Using the results of Problem 55, show that this can be written as
p y − p x
p x
=
2k
k + 1