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Problem 9.60
When a pitot tube such as Fig. (6.30) is placed in a supersonic flow, a normal shock will stand in
front of the probe. Suppose the probe reads po = 190 kPa and p = 150 kPa. If the stagnation
temperature is 400 K, estimate the (supersonic) Mach number and velocity upstream of the shock.
Solution 9.60
We can immediately find Ma inside the shock:
( )
3.5
2
o2 2 2 2
190
p /p 1.267 1 0.2Ma , solve Ma 0.591
150
= = = +
Problem 9.61
Air flows from a large tank, where T = 376 K and p = 360 kPa, to a design condition where the
pressure is 9800 Pa. The mass flow is 0.9 kg/s. However, there is a normal shock in the exit
plane just after this condition is reached. Estimate (a) the throat area; and, just downstream of
the shock, (b) the Mach number, (c) the temperature, and (d) the pressure.
Solution 9.61
The low design pressure definitely indicates a supersonic condition:
Problem 9.62
An atomic explosion propagates into still air at 14.7 psia and 520R. The pressure just inside the
shock is 5000 lbf/in2. Assuming k = 1.4, what are the speed C of the shock and the velocity V
just inside the shock?
Solution 9.62
The pressure ratio tells us the Mach number of the shock motion:
.
o
p. ( . Ma ) , solve for Ma .
p
−
= = = + =
2 3 5
11
9800 0 0272 1 0 2 3 00
360000
Problem 9.63
Sea-level standard air is sucked into a vacuum tank through a nozzle, as in Fig. P9.63. A normal
shock stands where the nozzle area is 2 cm2, as shown. Estimate (a) the pressure in the tank; and
(b) the mass flow.
Solution 9.63
The flow at the exit section (“3”) is subsonic (after a shock) therefore must equal the tank
pressure. Work our way to 1 and 2 at the shock and thence to 3 in the exit:
Problem 9.64
Air, from a reservoir at 350 K and 500 kPa, flows through a converging-diverging nozzle. The
throat area is 3 cm2. A normal shock appears, for which the downstream Mach number is
0.6405. (a) What is the area where the shock appears? Calculate (b) the pressure and (c) the
temperature downstream of the shock.
Solution 9.64
This Ma2 = 0.6405 occurs right in Table B.2, and we read Ma1 = 1.70. (a) Hence, for isentropic
flow up to that point, from Eq. (9.45),
1
Problem 9.65
Air flows through a converging-diverging nozzle between two large reservoirs, as in Fig. P9.65.
A mercury manometer between the throat and the downstream reservoir reads h = 15 cm.
Estimate the downstream reservoir pressure. Is there a shock wave in the flow? If so, does it
stand in the exit plane or farther upstream?
Solution 9.65
The manometer reads the pres-sure drop between throat and exit tank:
Problem 9.66
In Prob. 9.65 what would be the mercury manometer reading if the nozzle were operating exactly
at supersonic “design” conditions?
Problem 9.65
Air flows through a converging-diverging nozzle between two large reservoirs, as in Fig. P9.65.
A mercury manometer between the throat and the downstream reservoir reads h = 15 cm.
Estimate the downstream reservoir pressure. Is there a shock wave in the flow? If so, does it
stand in the exit plane or farther upstream?
Solution 9.66
We worked out this idealized isentropic-flow condition in Prob. 9.65:
Problem 9.67
A supply tank at 500 kPa and 400 K feeds air to a converging diverging nozzle whose throat area
is 9 cm2. The exit area is 46 cm2. State the conditions in the nozzle if the pressure outside the
exit plane is (a) 400 kPa; (b) 120 kPa; and (c) 9 kPa. (d) In each of these cases, find the mass
flow.
Solution 9.67
For reference purposes, find the design (isentropic supersonic) condition for this nozzle:
Problem 9.68
Air in a tank at 120 kPa and 300 K exhausts to the atmosphere through a 5-cm2-throat converging
nozzle at a rate of 0.12 kgs. What is the atmospheric pressure? What is the maximum mass flow
possible at low atmospheric pressure?
Solution 9.68
Let us answer the second question first, to see where 0.12 kgs stands:
Problem 9.69
With reference to Prob. 3.68, show that the thrust of a rocket engine exhausting into a vacuum is
given by
( )
2
0
/( 1)
2
1 Ma
1
1 Ma
2
ee
kk
e
p A k
F
k−
+
=−
+
where Ae = exit area
Mae = exit Mach number
p0 = stagnation pressure in combustion chamber
Note that stagnation temperature does not enter into the thrust.
Problem 3.68
The rocket in Fig. P3.68 has a super-sonic exhaust, and the exit pressure pe is not necessarily
equal to pa. Show that the force F required to hold this rocket on the test stand is
F =
eAeVe2 + Ae(pe − pa).
Is this force F what we term the thrust of the rocket?
Solution 9.69
In a vacuum, patm = 0, the solution to Prob. 3.68 is
Problem 9.70
Air, with po = 500 kPa and To = 600 K, flows through a converging-diverging nozzle. The exit
area is 51.2 cm2, and the mass flow is 0.825 kg/s. What is the highest possible back pressure that
will still maintain supersonic flow inside the diverging section?
Solution 9.70
Naturally assume isentropic flow with k = 1.40. If the flow is to be supersonic, there must be a
choked throat. Find its area:
Problem 9.71
A converging-diverging nozzle has a throat area of 10 cm2 and an exit area of 28.96 cm2. A
normal shock stands in the exit when the back pressure is sea-level standard. If the upstream
tank temperature is 400 K, estimate (a) the tank pressure; and (b) the mass flow.
Solution 9.71
The throat must be choked. Just before the shock, the Mach number is determined by the area
Problem 9.72
A large tank at 500 K and 165 kPa feeds air to a converging nozzle. The back pressure outside
the nozzle exit is sea-level standard. What is the appropriate exit diameter if the desired mass
flow is 72 kg/h?
Solution 9.72
Given To = 500 K and po = 165 kPa. The pressure ratio across the nozzle is
Problem 9.73
Air flows isentropically in a converging-diverging nozzle with a throat area of 3 cm2. At section 1,
the pressure is 101 kPa, the temperature is 300 K, and the velocity is 868 ms. (a) Is the nozzle
choked? Determine (b) A1; and (c) the mass flow. Suppose, without changing stagnation
conditions of A1, the (flexible) throat is reduced to 2 cm2. Assuming shock-free flow, will there
be any changes in the gas properties at section 1? If so, calculate the new p1, V1, and T1 and
explain.
Solution 9.73
Check the Mach number. If choked, calculate the mass flow:
Problem 9.74
Use your strategic ideas, from part (b) of Prob. P9.40, to actually carry out the calculations for
mass flow of steam, with po = 300 kPa and To = 600 K, discharging through a converging nozzle
of choked exit area 5 cm2. Note: The online site www.spiraxsarco has the speed of sound of
steam.
Problem 9.40
Steam, in a tank at 300 kPa and 600 K, discharges isentropically to a low-pressure atmosphere
through a converging nozzle with exit area 5 cm2. (a) Using an ideal gas approximation from
Table B.4, estimate the mass flow. (b) Without actual calculations, indicate how you would use
properties of steam to find the mass flow.
Solution 9.74
Using spiraxsarco for real steam, we don’t have these nice power-law formulas, but we can use
the energy equation and the continuity equation and the fact that the entropy is constant:
Problem 9.75*
A double-tank system in Fig. P9.75 has two identical converging nozzles of 1-in2 throat area.
Tank 1 is very large, and tank 2 is small enough to be in steady-flow equilibrium with the jet
from tank 1. Nozzle flow is isentropic, but entropy changes between 1 and 3 due to jet
dissipation in tank 2. Compute the mass flow. (If you give up, Ref. 9, pp. 288–290, has a good
discussion.)
Solution 9.75
We know that
1V1 =
2V2 from continuity. Since patm is so low, we may assume that the
second nozzle is choked, but the first nozzle is probably not choked. We may guess values of p2
and compare the computed values of flow through each nozzle:
Problem 9.76
A large reservoir at 20C and 800 kPa is used to fill a small insulated tank through a converging-
diverging nozzle with 1-cm2 throat area and 1.66-cm2 exit area. The small tank has a volume of
1 m3 and is initially at 20C and 100 kPa. Estimate the elapsed time when (a) shock waves begin
to appear inside the nozzle; and (b) the mass flow begins to drop below its maximum value.
Solution 9.76
During this entire time the nozzle is choked, so let’s compute the mass flow:
Problem 9.77
A perfect gas (not air) expands isentropically through a supersonic nozzle with an exit area 5 times
its throat area. The exit Mach number is 3.8. What is the specific heat ratio of the gas? What might
this gas be? If po = 300 kPa, what is the exit pressure of the gas?
Solution 9.77
We must iterate the area-ratio formula, Eq. (9.44), for k:
Problem 9.78
The orientation of a hole can make a difference. Consider holes A and B in Fig. P9.78, which are
identical but reversed. For the given air properties on either side, compute the mass flow through
each hole and explain why they are different.
Solution 9.78
Case B is a converging nozzle
Problem 9.79
A large tank, at 400 kPa and 450 K, supplies air to a converging-diverging nozzle of throat area
4 cm2 and exit area 5 cm2. For what range of back pressures will the flow be (a) be entirely
subsonic; (b) have a shock wave inside the nozzle; (c) have oblique shocks outside the exit; and
(d) have supersonic expansion waves outside the exit.
Solution 9.79
We are given po = 400 kPa and To = 450 K. Assume steady one-dimensional flow. All of these
various conditions are illustrated in Fig. 9.12. The area ratio is (5cm2)/ (4cm2) = 1.25.
Problem 9.80
A sea-level automobile tire is initially at 32 lbfin2 gage pressure and 75F. When it is punctured
with a hole which resembles a converging nozzle, its pressure drops to 15 lbfin2 gage in
12 min. Estimate the size of the hole, in thousandths of an inch. The tire volume is 2.5ft2.
Solution 9.80
The volume of the tire is 2.5 ft2. With patm 14.7 psi, the absolute pressure drops from 46.7 psia
Problem 9.81
Air, at po = 160 lbf/in2 and To = 300F, flows isentropically through a converging- diverging
nozzle. At section 1, where A1 = 288 in2, the velocity is V1 = 2068 ft/s. Calculate (a) Ma1;
(b) A*; (c) p1; and (d) the mass flow, in slug/s.
Solution 9.81
That is a high velocity, 2068 ft/s, even higher than the stagnation speed of sound,
Problem 9.82
Air at 500 K flows through a converging-diverging nozzle with throat area of 1 cm2 and exit
area of 2.7 cm2. When the mass flow is 182.2 kgh, a pitot-static probe placed in the exit plane
reads po = 250.6 kPa and p = 240.1 kPa. Estimate the exit velocity. Is there a normal shock wave in
the duct? If so, compute the Mach number just downstream of this shock.
Solution 9.82
These numbers just don’t add up to a purely isentropic flow. For example, pop = 250.6240.1
yields Ma 0.248, whereas AA* = 2.7 gives Ma 0.221. If the mass flow is maximum, we can
estimate the upstream stagnation pressure:
Problem 9.83
When operating at design conditions (smooth exit to sea-level pressure), a rocket engine has
a thrust of 1 million lbf. The chamber pressure and temperature are 600 lbfin2 absolute and
4000R, respectively. The exhaust gases approximate k = 1.38 with a molecular weight of 26.
Estimate (a) the exit Mach number and (b) the throat diameter.
Solution 9.83
“Design conditions” mean isentropic expansion to pe = 14.7 psia = 2116 lbfft2:
Problem 9.84
