Aeronautical Engineering Chapter 9 Homework Now Need The Mach Number This Section

Solution 9.37

For a CV around the tank, write the mass and the energy equations:

Problem 9.38

Prob. 9.37 makes an ideal senior project or combined laboratory and computer problem, as

described in Ref. 27, sec. 8.6. In Bober and Kenyon’s lab experiment, the tank had a volume of

0.0352 ft3 and was initially filled with air at 50 lb/in2 gage and 72F. Atmospheric pressure was

14.5 lb/in2 absolute, and the nozzle exit diameter was 0.05 in. After 2 s of blowdown, the

measured tank pressure was 20 lb/in2 gage and the tank temperature was –5F. Compare these

values with the theoretical analysis of Prob. 9.37.

Problem 9.37

Make an exact control volume analysis of the blowdown process in Fig. P9.37, assuming an insulated

tank with negligible kinetic and potential energy within. Assume critical flow at the exit and

show that both po and To decrease during blowdown. Set up first-order differential equations for

po(t) and To(t) and reduce and solve as far as you can.

Solution 9.38

Use the formulas derived in Prob. 9.37 above, with the given data:

o

T (0) 72 460 532 R,

= + = 

Problem 9.39

Consider isentropic flow in a channel of varying area, from sections 1 to section 2. We know that

Ma1 = 2.0, and desire that the velocity ratio V2/V1 equal 1.2. Estimate (a) Ma2 and (b) A2/A1.

(c) Sketch what this channel looks like, for example, does it converge or diverge? Is there a

throat?

Solution 9.39

This is a problem in iteration, ideally suited for Excel. Algebraically,

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 9.40

The code words “low–pressure atmosphere” mean that the flow is choked at the exit. For steam,

from Table B.4, assume k = 1.33 and R = 461 m2/s2-K. Then we are at maximum mass flow:

Problem 9.41

Air, with a stagnation pressure of 100 kPa, flows through the nozzle in Fig. P9.41, which is 2 m

long and has an area variation approximated by

2

20 20 10A x x − +

with A in cm2 and x in m. It is desired to plot the complete family of isentropic pressures p(x) in

this nozzle, for the range of inlet pressures 1  p(0)  100 kPa. Indicate which inlet pressures are

not physically possible and discuss briefly. If your computer has an online graphics routine, plot

at least 15 pressure profiles; otherwise just hit the highlights and explain.

Solution 9.41

There is a subsonic entrance region of high pressure and a supersonic entrance region of low

pressure, both of which are bounded by a sonic (critical) throat, and both of which have a ratio

x0

A /A* 2.0.

==

From Table B.1 or Eq. (9.44), we find these two conditions to be bounded by

Problem 9.42

A bicycle tire is filled with air at an absolute pressure of 169.12 kPa (abs) and the temperature

inside is 30C. Suppose the valve breaks, and air starts to exhausts out of the tire into the

atmosphere (pa = 100 kPa absolute and Ta = 20C). The valve exit is 2.00 mm in diameter and is

the smallest cross-sectional area in the entire system. Frictional losses can be ignored here; one-

dimensional isentropic flow is a reasonable assumption. (a) Find the Mach number, velocity, and

temperature at the exit plane of the valve (initially). (b) Find the initial mass flow rate out of the

tire. (c) Estimate the velocity at the exit plane using the incompressible Bernoulli equation. How

well does this estimate agree with the “exact” answer of part (a)? Explain.

Solution 9.42

(a) Flow is not choked, because the pressure ratio is less than 1.89:

Problem 9.43

Air flows isentropically through a variable-area duct. At section 1, A1 = 20 cm2, p1 = 300 kPa,



1 = 1.75 kg/m3, and Ma1 = 0.25. At section 2, the area is exactly the same, but the flow is much

faster. Compute (a) V2; (b) Ma2; and (c) T2, and (d) the mass flow. (e) Is there a sonic throat

between sections 1 and 2? If so, find its area.

Solution 9.43

If the areas are the same but the velocities different, there must be a sonic throat in between.

Problem 9.44

In Prob. 3.34 we knew nothing about compressible flow at the time so merely assumed exit conditions

p2 and T2 and computed V2 as an application of the continuity equation. Suppose that the throat

diameter is 3 in. For the given stagnation conditions in the rocket chamber in Fig. P3.34 and

assuming k = 1.4 and a molecular weight of 26, compute the actual exit velocity, pressure, and

temperature according to one-dimensional theory.

If pa = 14.7 lbf/in2 absolute, compute the thrust from the analysis of Prob. 3.68. This thrust is

entirely independent of the stagnation temperature (check this by changing To to 2000°R if you

like). Why?

Problem 3.34

A rocket motor is operating steadily, as shown in Fig. P3.34. The products of combustion

s

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 9.44

If M = 26, then Rgas = 49720/26 = 1912 ftlbf/slugR. Assuming choked flow in the throat (to

Problem 9.45

It is desired to have an isentropic airflow achieve a velocity of 550 m/s at a 6-cm-diameter

section where the pressure is 87 kPa and the density 1.3 kg/m3. (a) Is a sonic throat needed?

(b) If so, estimate its diameter, and compute (c) the stagnation temperature and (d) the mass

flow.

Solution 9.45

We need the Mach number at this section and can readily find the temperature:

Problem 9.46

A one-dimensional isentropic airflow has the following properties at one section where the area

is 53 cm2: p = 12 kPa,



= 0.182 kg/m3, and V = 760 m/s. Determine (a) the throat area; (b) the

stagnation temperature; and (c) the mass flow.

Solution 9.46

We already have what we need to compute the mass flow:

4

Problem 9.47

In wind-tunnel testing near Mach 1, a small area decrease caused by model blockage can be

important. Suppose the test section area is 1 m2 with unblocked test conditions Ma = 1.10 and

T = 20°C. What model area will first cause the test section to choke? If the model cross-section

is 0.004 m2, (0.4 percent blockage), what percentage change in test-section velocity results?

Solution 9.47

First evaluate the unblocked test conditions:

Problem 9.48

A force F = 1100 N pushes a piston of diameter 12 cm through an insulated cylinder containing

air at 20°C, as in Fig. P9.48. The exit diameter is 3 mm, and pa = 1 atm. Estimate (a) Ve, (b) Vp,

and (c)

.

e

m

Solution 9.48

First find the pressure inside the large cylinder:

Problem 9.49

Consider the venturi nozzle of Fig. 6.40c, with D = 5 cm and d = 3 cm. Stagnation temperature is

300 K, and the upstream velocity V1 = 72 m/s. If the throat pressure is 124 kPa, estimate, with

isentropic flow theory, (a) p1; (b) Ma2; and (c) the mass flow.

Solution 9.49

Given one-dimensional isentropic flow of air. The problem looks sticky— sparse, scattered

information, implying laborious iteration. But the energy equation yields V1 and Ma1:

Problem 9.50

Methane is stored in a tank at 120 kPa and 330 K. It discharges to a second tank through a

converging nozzle whose exit area is 5 cm2. What is the initial mass flow rate if the second tank

has a pressure of (a) 70 kPa, or (b) 40 kPa?

Solution 9.50

For methane, CH4, from Table A.4, R = 518 m2/(s2·K) and k = 1.32. First find p* to see how it

compares to these two exit pressures:

Problem 9.51

The scramjet engine of is supersonic throughout. A sketch is shown in Fig. C9.8. Test the

following design. The flow enters at Ma = 7 and air properties for 10,000 m altitude. Inlet area

is 1 m2, the minimum area is 0.1 m2, and the exit area is 0.8 m2. If there is no combustion,

(a) will the flow still be supersonic in the throat? Also, determine (b) the exit Mach number,

(c) exit velocity, and (d) exit pressure.

Solution 9.51

From Table B.6 at 10,000 m, read p1 = 26416 Pa, T1 = 223.16 K, and



1 = 0.4125 kg/m3.

Establish area ratio and stagnation conditions at the inlet, section 1:

Problem 9.52

A converging-diverging nozzle exits smoothly to sea–level standard atmosphere. It is supplied

by a 40-m3 tank initially at 800 kPa and 100C. Assuming isentropic flow, estimate (a) the

throat area; and (b) the tank pressure after 10 s of operation. The exit area is 10 cm2.

Solution 9.52

The phrase “exits smoothly” means that exit pressure = atmospheric pressure, which is 101 kPa.

Then the pressure ratio specifies the exit Mach number:

Problem 9.53

Air flows steadily from a reservoir at 20C through a nozzle of exit area 20 cm2 and strikes a

vertical plate as in Fig. P9.53. The flow is subsonic throughout. A force of 135 N is required to hold

the plate stationary. Compute (a) Ve, (b) Mae, and (c) p0 if pa = 101 kPa.

Solution 9.53

Assume pe = 1 atm. For a control volume surrounding the plate, we deduce that

Problem 9.54

The airflow in Prob. P9.46 undergoes a normal shock just past the section where data was given.

Determine the (a) Mach number, (b) pressure, and (c) velocity just downstream of the shock.

Problem 9.46

A one-dimensional isentropic airflow has the following properties at one section where the area

is 53 cm2: p = 12 kPa,



= 0.182 kg/m3, and V = 760 m/s. Determine (a) the throat area; (b) the

stagnation temperature; and (c) the mass flow.

Solution 9.54

In Prob. P9.46 we found that the local Mach number at the section was 2.5.

Problem 9.55

Air, supplied by a reservoir at 450 kPa, flows through a converging-diverging nozzle whose

throat area is 12 cm2. A normal shock stands where A1 = 20 cm2. (a) Compute the pressure just

downstream of this shock. Still farther downstream, where A3 = 30 cm2, estimate (b) p3; (c) A3*;

and (d) Ma3.

Solution 9.55

If a shock forms, the throat must be choked (sonic). Use the area ratio at (1):

Problem 9.56

Air from a reservoir at 20C and 500 kPa flows through a duct and forms a normal shock

downstream of a throat of area 10 cm2. By an odd coincidence it is found that the stagnation pressure

downstream of this shock exactly equals the throat pressure. What is the area where the shock wave

stands?

Solution 9.56

If a shock forms, the throat is sonic, A* = 10 cm2. Now

Problem 9.57

Air flows from a tank through a nozzle into the standard atmosphere, as in Fig. P9.57. A normal

shock stands in the exit of the nozzle, as shown. Estimate (a) the tank pressure; and (b) the mass

flow.

Solution 9.57

The throat must be sonic, and the area ratio at the shock gives the Mach number:

Problem 9.58

Downstream of a normal shock wave, in airflow, the conditions are T2 = 603 K, V2 = 222 m/s,

and p2 = 900 kPa. Estimate the following conditions just upstream of the shock:

(a) Ma1; (b) T1; (c) p1; (d) po1; and (e) To1.

Solution 9.58

We have enough information to compute the downstream Mach number:

2

222 222 / 0.451

492 /

1.4(287)(603)

Vms

Ma ms

kRT

= = = =

Problem 9.59

Air, at stagnation conditions of 450 K and 250 kPa, flows through a nozzle. At section 1, where

the area = 15 cm2, there is a normal shock wave. If the mass flow is 0.4 kgs, estimate (a) the

Mach number; and (b) the stagnation pressure just downstream of the shock.

Solution 9.59

If there is a shock wave, then the mass flow is maximum: