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Problem 9.C1
The converging–diverging nozzle sketched in Fig. C9.1 is designed to have a Mach number of
2.00 at the exit plane (assuming the flow remains nearly isentropic). The flow travels from tank a
to tank b, where tank a is much larger than tank b. (a) Find the area at the exit Ae and the back
pressure pb that will allow the system to operate at design conditions. (b) As time goes on, the
back pressure will grow, since the second tank slowly fills up with more air. Since tank a is
huge, the flow in the nozzle will remain the same, however, until a normal shock wave appears at
the exit plane. At what back pressure will this occur? (c) If tank b is held at constant temperature,
T = 20°C, estimate how long it will take for the flow to go from design conditions to the
condition of part (b)—that is, with a shock wave at the exit plane.
Solution 9.C1
(a) Compute the isentropic pressure ratio and area ratio for Ma = 2.0:
2.0: Table B.1: 0.1278, 0.1278(1E6) (a)
e
ee
o
p
Ma p Ans.
p
= = = =
128,000Pa
Problem 9.C2
Two large air tanks, one at 400 K and 300 kPa and the other at 300 K and 100 kPa, are connected
by a straight tube 6 m long and 5 cm in diameter. The average friction factor is 0.0225.
Assuming adiabatic flow, estimate the mass flow through the tube.
Solution 9.C2
The higher-pressure tank denotes the inlet stagnation conditions, po = 300 kPa and To = 400 K.
The flow will be subsonic, but we have no idea whether it is choked. Assume that the tube exit
pressure equals the receiver pressure, 100 kPa. We must iterate—an ideal job for Excel! We do know
(f L/D):
Problem 9.C3*
Fig. C9.3 shows the exit of a converging-diverging nozzle, where an oblique shock pattern is
formed. In the exit plane, which has an area of 15 cm2, the air pressure is 16 kPa and the
temperature is 250 K. Just outside the exit shock, which makes an angle of 50° with the exit
plane, the temperature is 430 K. Estimate (a) the mass flow; (b) the throat area; (c) the turning
angle of the exit flow; and, in the tank supplying the air, (d) the pressure and (e) the
temperature.
Solution 9.C3
We know the temperature ratio and the shock wave angle, so we can muddle through oblique-
shock-wave theory to find the shock conditions:
,
o tank
Problem 9.C4
The properties of a dense gas (high pressure and low temperature) are often approximated by
van der Waals’ equation state [17, 18]:
Find an analytic expression for the speed of sound of a van der Waals gas. Assuming
k = 1.4, compute the speed of sound of air, in ft/s, at –100F and 20 atm, for (a) a perfect gas,
and (b) a van der Waals gas. What percentage higher density does the van der Waals relation
predict?
Solution 9.C4
For air, take R = 1716 ft-lbf/slugR. First evaluate the densities, T = 360R:
Problem 9.C5
2
1
1
11
22 4
12
3
1
1
where constants a and b can be found from the critical temperature and pressure
27 9 5 for air
64
and 0.65 for air
8
c
c
c
c
RT
pa
b
RT ft lb
aE
pslug
RT ft
bp slug
=−
−
−
==
==
Consider one-dimensional steady flow of a nonideal gas, steam, in a converging nozzle.
Stagnation conditions are po = 100 kPa and To = 200C. The nozzle exit diameter is 2 cm.
(a) If the nozzle exit pressure is 70 kPa, calculate the mass flow and the exit temperature for
real steam, from the steam tables. (As a first estimate, assume steam to be an ideal gas from
Table A.4.) Is the flow choked? (b) Find the nozzle exit pressure and mass flow for which
the steam flow is choked, using the steam tables.
Solution 9.C5
(a) For steam as an ideal gas, from Table A.4, k = 1.33 and R = 461 J/kgK. First use this
approximation to find the exit Mach number:
Problem 9.C6
Extend Prob. 9C.5 as follows. Let the nozzle be converging-diverging, with an exit diameter of
3 cm. Assume isentropic flow. (a) Find the exit Mach number, pressure, and temperature for an
ideal gas, from Table A.4. Does the mass flow agree with the value of 0.0452 kg/s in Prob. 9.C5?
Problem 9.C5
Consider one-dimensional steady flow of a nonideal gas, steam, in a converging nozzle.
Stagnation conditions are po = 100 kPa and To = 200C. The nozzle exit diameter is 2 cm.
(a) If the nozzle exit pressure is 70 kPa, calculate the mass flow and the exit temperature for
real steam, from the steam tables. (As a first estimate, assume steam to be an ideal gas from
Table A.4.) Is the flow choked? (b) Find the nozzle exit pressure and mass flow for which
the steam flow is choked, using the steam tables.
Solution 9.C6
(a) For steam as an ideal gas, from Table A.4, k = 1.33 and R = 461 J/kgK.
0.5( 1)/( 1)
2
2
2
1 0.5( 1)
( /4)(0.03) 1
2.25 for 1.33
* 0.5( 1)
( /4)(0.02)
kk
e
exit
e
k Ma
Ak
A Ma k
+−
+−
= = = =
+
Problem 9.C7
Professor Gordon Holloway and his student, Jason Bettle, of the University of New
Brunswick, obtained the following tabulated data for blow-down air flow through a
converging-diverging nozzle similar in shape to Fig. P3.22. The supply tank pressure and
temperature were 29 psig and 74F, respectively. Atmospheric pressure was 14.7 psia. Wall
pressures and centerline stagnation pressures were measured in the expansion section,
which was a frustrum of a cone. The nozzle throat is at x = 0.
x (cm):
0
1.5
3
4.5
6
7.5
9
Diameter (cm):
1.00
1.098
1.195
1.293
1.390
1.488
1.585
pwall (psig):
7.7
–2.6
–4.9
–7.3
–6.5
–10.4
–7.4
pstagnation
(psig):
29
26.5
22.5
18
16.5
14
10
Use the stagnation pressure data to estimate the local Mach number. Compare the measured
Mach numbers and wall pressures with the predictions of one-dimensional theory. For x 9 cm,
the stagnation pressure data was not thought by Holloway and Bettle to be a valid measure of
Mach number. What is the probable reason?
Solution 9.C7
From the cone’s diameters we can determine A/A* and compute theoretical Mach numbers and
pressures from Table B.1. From the measured stagnation pressures we can compute measured
(supersonic) Mach numbers, because a normal shock forms in front of the probe. The ratio
po2/po1 from Eq. (9.58) or Table B.2 is used to estimate the Mach number.
x (cm):
0
1.5
3
4.5
6
7.5
9
Problem 9.C8
Engineers call the supersonic combustion, in the scramjet engine almost miraculous, “like lighting
a match in a hurricane”. Figure C9.8 is a crude idealization of the engine. Air enters, burns fuel
in the narrow section, then exits, all at supersonic speeds. There are no shock waves. Assume
areas of 1 m2 at sections 1 and 4 and 0.2 m2 at sections 2 and 3. Let the entrance conditions be
Ma1 = 6, at 10,000 m standard altitude. Assume isentropic flow from 1 to 2, frictionless heat
transfer from 2 to 3 with Q = 500 kJ/kg, and isentropic flow from 3 to 4. Calculate the exit
conditions and the thrust produced.
Solution 9.C8
From Table B.6 at 10,000 m, p1 = 26416 Pa, T1 = 223.16 K,
1 = 0.4125 kg/m3, and a1 = 299.5
m/s. Calculate the inlet velocity and mass flow:
Problem 9.W1
Notice from Table 9.1 that (a) water and mercury and (b) aluminum and steel have nearly the
same speeds of sound, yet the second of each pair of materials is much denser. Can you account
for this oddity? Can molecular theory explain it?
Solution 9.W1
Problem 9.W2
When an object approaches you at Ma = 0.8, you can hear it, according to Fig. 9.18a. But would
there be a Doppler shift? For example, would a musical tone seem to you to have a higher or a
lower pitch?
Solution 9.W2
Problem 9.W3
The subject of this chapter is commonly called gas dynamics. But can liquids not perform in this
manner? Using water as an example, make a rule-of-thumb estimate of the pressure level needed
to drive a water flow at velocities comparable to the sound speed.
Solution 9.W3
Problem 9.W4
Suppose a gas is driven at compressible subsonic speeds by a large pressure drop, p1 to p2.
Describe its behavior on an appropriately labeled Mollier chart for (a) frictionless flow in a
converging nozzle and (b) flow with friction in a long duct.
Solution 9.W4
Problem 9.W5
Describe physically what the “speed of sound” represents. What kind of pressure changes occur
in air sound waves during ordinary conversation?
Solution 9.W5
Problem 9.W6
Give a physical description of the phenomenon of choking in a converging-nozzle gas flow.
Could choking happen even if wall friction were not negligible?
Solution 9.W6
The solution to this word problem is not provided.
Problem 9.W7
Shock waves are treated as discontinuities here, but they actually have a very small finite
thickness. After giving it some thought, sketch your idea of the distribution of gas velocity,
pressure, temperature, and entropy through the inside of a shock wave.
Solution 9.W7
The solution to this word problem is not provided.
Problem 9.W8
Describe how an observer, running along a normal shock wave at finite speed V, will see what
appears to be an oblique shock wave. Is there any limit to the running speed?
Solution 9.W8
Problem 9.1
An ideal gas flows adiabatically through a duct. At section 1, p1 = 140 kPa, T1 = 260C, and
V1 = 75 m/s. Farther downstream, p2 = 30 kPa and T2 = 207C. Calculate V2 in m/s and s2 − s1 in
J/(kg K) if the gas is (a) air, k = 1.4, and (b) argon, k = 1.67.
Solution 9.1
(a) For air, take k = 1.40, R = 287 J/kg K, and cp = 1005 J/kg K. The adiabatic steady-flow energy
equation (9.23) is used to compute the downstream velocity:
Problem 9.2
Solve Prob. 9.1 if the gas is steam. Use two approaches: (a) an ideal gas from Table A.4; and (b) real
steam from the steam tables [15].
Problem 9.1
An ideal gas flows adiabatically through a duct. At section 1, p1 = 140 kPa, T1 = 260C, and
V1 = 75 m/s. Farther downstream, p2 = 30 kPa and T2 = 207C. Calculate V2 in m/s and s2 − s1 in
J/(kg K) if the gas is (a) air, k = 1.4, and (b) argon, k = 1.67.
Solution 9.2
For steam, take k = 1.33, R = 461 J/kg K, and cp = 1858 J/kg K. Then
Problem 9.3
If 8 kg of oxygen in a closed tank at 200C and 300 kPa is heated until the pressure rises to
400 kPa, calculate (a) the new temperature; (b) the total heat transfer; and (c) the change in
entropy.
Solution 9.3
For oxygen, take k = 1.40, R = 260 J/kg K, and cv = 650 J/kg K. Then
Problem 9.4
Consider steady adiabatic airflow in a duct. At section B, the pressure is 600 kPa and the
temperature is 177ºC. At section D, the density is 1.13 kg/m3 and the temperature is 156C.
(a) Find the entropy change, if any. (b) Which way is the air flowing?
Solution 9.4
Convert the temperatures to TB = 177+273 = 450 K and TD = 156+273 = 429 K. For the entropy
change, we need either two densities or two pressures:
Problem 9.5
Steam enters a nozzle at 377C, 1.6 MPa, and a steady speed of 200 m/s and accelerates
isentropically until it exits at saturation conditions. Estimate the exit velocity and temperature.
Solution 9.5
At saturation, steam is not ideal. Use spiraxsarco or the Steam Tables:
Problem 9.6
Methane, approximated as a perfect gas, is compressed adiabatically from 101 kPa and 20ºC to
300 kPa. Estimate (a) the final temperature, and (b) the final density.
Solution 9.6
For methane, CH4, from Table A.4, R = 518 m2/(s2·K) and k = 1.32. The initial density is
Problem 9.7
Air flows through a variable-area duct. At section 1, A1 = 20 cm2, p1 = 300 kPa,
1 = 1.75 kg/m3,
and V1 = 122.5 m/s. At section 2, the area is exactly the same, but the density is much lower:
2 = 0.266 kg/m3, and T2 = 281 K. There is no transfer of work or heat. Assume one-
dimensional steady flow. (a) How can you reconcile these differences? (b) Find the mass flow
at section 2. Calculate (c) V2, (d) p2, and (e) s2 – s1. [Hint: This problem requires the continuity
equation.]
Solution 9.7
Part (a) is too confusing, let’s try (b, c, d, e) first. (b) The mass flow must be constant:
Problem 9.8
Atmospheric air at 20C enters and fills an insulated tank that is initially evacuated. Using a
control volume analysis from Eq. (3.67), compute the tank air temperature when it is full.
Solution 9.8
The energy equation during filling of the adiabatic tank is
Problem 9.9
Liquid hydrogen and oxygen are burned in a combustion chamber and fed through a rocket
nozzle that exhausts at 1600 m/s to an ambient pressure of 54 kPa. The nozzle exit diameter is
45 cm, and the jet exit density is 0.15 kg/m3. If the exhaust gas has a molecular weight of 18,
estimate (a) the exit gas temperature; (b) the mass flow; and (c) the thrust generated by the
rocket.
Solution 9.9
(a) From Eq. (9.3), estimate Rgas and hence the gas exit temperature:
Problem 9.10
A certain aircraft flies at 609 mi/hr at standard sea-level. (a) What is its Mach number? (b) If it
flies at the same Mach number at 34,000 ft altitude, how much slower (or faster) is it flying, in
mi/h?
Solution 9.10
At sea-level, from Table A.6, the speed of sound, a, equals 340.3 m/s = 761.2 mi/h. (a) Thus
h
Problem 9.11
At 300C and 1 atm, estimate the speed of sound of (a) nitrogen; (b) hydrogen; (c) helium;
(d) steam; and (e) uranium hexafluoride 238UF6 (k 1.06).
Solution 9.11
The gas constants are listed in Appendix Table A.4 for all but uranium gas (e):
Problem 9.12
Assume that water follows Eq. (1.19) with n 7 and B 3000. Compute the bulk modulus (in
kPa) and the speed of sound (in m/s) at (a) 1 atm; and (b) 1100 atm (the deepest part of the
