Fundamentals of Fluid Mechanics

Bruce R. Munson, Theodore H. Okiishi, Wade W. Huebsch

Chapter 11 Compressible Flow - all with Video Answers

Educators

Chapter Questions

01:18

Problem 1

Distinguish between flow of an ideal gas and inviscid flow of a fluid.

Narayan Hari

Numerade Educator

01:01

Problem 2

Compare the density of standard air listed in Table 1.8 with the value of standard air calculated with the ideal gas equation of state, and comment on what you discover.

Dominador Tan

Numerade Educator

06:13

Five pounds mass of air are heated in a closed, rigid container from $80^{\circ} \mathrm{F}, 15$ psia to $500^{\circ} \mathrm{F}$. Estimate the final pressure of the air and the entropy rise involved.

Prabhat Tyagi

Numerade Educator

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Problem 4

Air flows steadily between two sections in a duct. At section $(1),$ the temperature and pressure are $T_{1}=80^{\circ} \mathrm{C}, p_{1}=301$ $\mathrm{kPa}(\mathrm{abs}),$ and at section $(2),$ the temperature and pressure are $T_{2}=180^{\circ} \mathrm{C}, p_{2}=181 \mathrm{kPa}(\mathrm{abs}) .$ Calculate the (a) change in internal energy between sections (1) and $(2),(\mathbf{b})$ change in enthalpy between sections (1) and $(2),(\mathrm{c})$ change in density between sections (1) and $(2),(\mathbf{d})$ change in entropy between sections (1) and
(2). How would you estimate the loss of available energy between the two sections of this flow?

Victor Salazar

Numerade Educator

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Problem 5

Determine, in SI units, nominal values of $c_{p}$ and $c_{v}$ for: $(\mathbf{a})$ air,
(b) carbon dioxide,
(c) helium. Use information provided in Table 1.8

Victor Salazar

Numerade Educator

01:17

Problem 6

As demonstrated in Video $V 11.1$ fluid density differences in a flow may be seen with the help of a schlieren optical system. Discuss what variables affect fluid density and the different ways in which a variable density flow can be achieved.

Narayan Hari

Numerade Educator

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Problem 7

Describe briefly how a schlieren optical visualization system (Videos $\mathrm{V} 11.1$ and $\mathrm{V} 11.5$, also Fig. 11.4 ) works. How else might density changes in a fluid flow be made visible to the eye?

Victor Salazar

Numerade Educator

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Problem 8

Three kilograms of hydrogen contained in a nondeforming sealed vessel are cooled from $400^{\circ} \mathrm{C}, 400 \mathrm{kPa}(\text { abs })$ until the hydrogen pressure is $100 \mathrm{kPa}(\text { abs }) .$ Calculate the change in internal energy, enthalpy, and entropy associated with this cooling process.

Victor Salazar

Numerade Educator

02:01

Problem 9

Helium is compressed isothermally from $121 \mathrm{kPa}(\text { abs })$ to $301 \mathrm{kPa}(\text { abs }) .$ Determine the entropy change associated with this process.

Anjali Tiwari

Numerade Educator

03:39

Problem 10

Air at 14.7 psia and $70^{\circ} \mathrm{F}$ is compressed adiabatically by a centrifugal compressor to a pressure of 100 psia. What is the minimum temperature rise possible? Explain.

Prabhat Tyagi

Numerade Educator

01:01

Problem 11

Methane is compressed adiabatically from 100 $\mathrm{kPa}(\mathrm{abs})$ and $25^{\circ} \mathrm{C}$ to $200 \mathrm{kPa}(\mathrm{abs}) .$ What is the minimum compressor exit temperature possible? Explain.

Narayan Hari

Numerade Educator

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Problem 12

Air expands adiabatically through a turbine from a pressure and temperature of 180 psia, 1600 R to a pressure of 14.7 psia. If the actual temperature change is $85 \%$ of the ideal temperature change, determine the actual temperature of the expanded air and the actual enthalpy and entropy differences across the turbine.

Victor Salazar

Numerade Educator

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Problem 13

An expression for the value of $c_{p}$ for carbon dioxide as a function of temperature is $$c_{p}=286-\frac{1.15 \times 10^{5}}{T}+\frac{2.49 \times 10^{6}}{T^{2}}$$ where $c_{p}$ is in $(\mathrm{ft} \cdot \mathrm{lb}) /\left(\mathrm{lbm} \cdot^{\circ} \mathrm{R}\right)$ and $T$ is in $^{\circ} \mathrm{R}$. Compare the change in enthalpy of carbon dioxide using the constant value of $c_{p}$ (see Table 1.7 ) with the change in enthalpy of carbon dioxide using the
(b) $1000^{\circ} \mathrm{R},$ (c) expression above, for $T_{2}-T_{1}$ equal to
(a) $10^{\circ} \mathrm{R}$
$3000^{\circ} \mathrm{R} .$ Set $T_{1}=540^{\circ} \mathrm{R}$

Victor Salazar

Numerade Educator

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Problem 14

Confirm the speed of sound for air at $70^{\circ} \mathrm{F}$ listed in Table B.3.

Victor Salazar

Numerade Educator

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Problem 15

From Table $\mathrm{B} .1$ we can conclude that the speed of sound in water at $60^{\circ} \mathrm{F}$ is $4814 \mathrm{ft} / \mathrm{s}$. Is this value of $c$ consistent with the value of bulk modulus, $E_{v}$, listed in Table $1.5 ?$

Victor Salazar

Numerade Educator

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Problem 16

If the observed speed of sound in steel is $5300 \mathrm{m} / \mathrm{s}$ determine the bulk modulus of elasticity of steel in $\mathrm{N} / \mathrm{m}^{3}$. The density of steel is nominally $7790 \mathrm{kg} / \mathrm{m}^{3}$. How does your value of $E_{v}$ for steel compare with $E_{v}$ for water at $15.6^{\circ} \mathrm{C} ?$ Compare the speeds of sound in steel, water, and air at standard atmospheric pressure and $15^{\circ} \mathrm{C}$ and comment on what you observe.

Victor Salazar

Numerade Educator

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Problem 17

Using information provided in Table $\mathrm{C} .1,$ develop a table of speed of sound in $\mathrm{ft} / \mathrm{s}$ as a function of elevation for U.S. standard atmosphere.

Victor Salazar

Numerade Educator

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Problem 18

If a high-performance aircraft is able to cruise at a Mach number of 3.0 at an altitude $80,000 \mathrm{ft}$, how fast is this in
(a) mph,
(b) $\mathrm{ft} / \mathrm{s},(\mathrm{c}) \mathrm{m} / \mathrm{s} ?$

Victor Salazar

Numerade Educator

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Problem 19

Compare values of the speed of sound in $\mathrm{m} / \mathrm{s}$ at $20^{\circ} \mathrm{C}$ in the following gases:
(a) air, (b) carbon dioxide,
(c) helium.

Victor Salazar

Numerade Educator

01:07

Problem 20

Determine the Mach number of a car moving in standard air at a speed of
(a) $25 \mathrm{mph}$
(b) $55 \mathrm{mph},$ and
(c) $100 \mathrm{mph}$

Narayan Hari

Numerade Educator

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Problem 21

The flow of an ideal gas may be considered incompressible if the Mach number is less than $0.3 .$ Determine the velocity level in $\mathrm{ft} / \mathrm{s}$ and in $\mathrm{m} / \mathrm{s}$ for $\mathrm{Ma}=0.3$ in the following gases:
(a) standard air,
(b) hydrogen at $68^{\circ} \mathrm{F}$

Victor Salazar

Numerade Educator

01:36

Problem 22

A schlieren photo of a bullet moving through air (see Video $\mathrm{V} 11.10$ at 14.7 psia and $68^{\circ} \mathrm{F}$ indicates a Mach cone angle of $28^{\circ} .$ How fast was the bullet moving in:
(a) $\mathrm{m} / \mathrm{s}$
(b) $\mathrm{ft} / \mathrm{s}$
(c) mph?

Narayan Hari

Numerade Educator

15:41

Problem 23

where At a given instant of time, two pressure waves, each moving at the speed of sound, emitted by a point source moving with constant velocity in a fluid at rest are shown in Fig. $\mathrm{P} 11.23$ Determine the Mach number involved and indicate with a sketch the instantaneous location of the point source.

Paul A.

California State Polytechnic University, Pomona

01:18

Problem 24

At a given instant of time, a pressure wave moving at the speed of sound, which is emitted by a point source moving with constant velocity along the line $A-B$ in a fluid at rest, is shown in Fig. $\mathrm{P} 11.24 .$ If the point source Mach number is $0.8,$ sketch, for the given instant, the pressure wave emitted by the point source when it was at location $B$

Dominador Tan

Numerade Educator

15:41

Problem 25

At a given instant of time, two pressure waves, each moving at the speed of sound, emitted by a point source moving with constant velocity in a fluid at rest, are shown in Fig. P11.25. Determine the Mach number involved and indicate with a sketch the instantaneous location of the point source.

Paul A.

California State Polytechnic University, Pomona

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Problem 26

Sound waves are very small-amplitude pressure pulses that travel at the "speed of sound." Do very large-amplitude waves such as a blast wave caused by an explosion (see Video V11.8 travel less than, equal to, or greater than the speed of sound? Explain.

Victor Salazar

Numerade Educator

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Problem 27

How would you estimate the distance between you and an approaching storm front involving lightning and thunder?

Victor Salazar

Numerade Educator

01:49

Problem 28

if a new Boeing 787 Dreamliner cruises at a Mach number of 0.87 at an altitude of $30,000 \mathrm{ft}$, how fast is this in
(a) $\mathrm{mph}$
(b) $\mathrm{ft} / \mathrm{s}$
(c) $\mathrm{m} / \mathrm{s} ?$

Narayan Hari

Numerade Educator

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Problem 29

At the seashore, you observe a high-speed aircraft moving overhead at an elevation of $10,000 \mathrm{ft}$. You hear the plane 8 s after it passes directly overhead. Using a nominal air temperature of $40^{\circ} \mathrm{F}$, estimate the Mach number and speed of the aircraft.

Victor Salazar

Numerade Educator

01:00

Problem 30

Explain how you could vary the Mach number but not the Reynolds number in airflow past a sphere. For a constant Reynolds number of $300,000,$ estimate how much the drag coefficient will increase as the Mach number is increased from 0.3 to 1.0

Chai Santi

Numerade Educator

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Problem 31

Starting with the enthalpy form of the energy equation (Eq. 5.69)$,$ show that for isentropic flows, the stagnation temperature remains constant. Why is this important?

Victor Salazar

Numerade Educator

04:03

Problem 32

Starting with Eq. $11.52,$ prove that the stagnation enthalpy (and temperature) of an ideal gas remains constant during isentropic flow. Does this result fix the stagnation state for the flow? Explain. Using Eq. $5.69,$ comment on heat transfer and shaft work during any constant stagnation state process.

Chai Santi

Numerade Educator

01:45

Problem 33

Airflows steadily and isentropically from standard atmospheric conditions to a receiver pipe through a converging duct. The cross-sectional area of the throat of the converging duct is 0.05 $\mathrm{ft}^{2} .$ Determine the mass flowrate through the duct if the receiver pressure is (a) 10 psia, (b) 5 psia. Sketch temperature-entropy diagrams for situations (a) and (b). Verify results obtained with values from the appropriate graph in Appendix D with calculations involving ideal gas equations. Is condensation of water vapor a concern? Explain.

Dominador Tan

Numerade Educator

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Problem 34

Determine the critical pressure and temperature ratios for
(a) air, (b) carbon dioxide,
(c) helium.

Victor Salazar

Numerade Educator

04:45

Problem 35

Determine the static pressure to stagnation pressure ratio associated with the following motion in standard air: (a) a runner moving at the rate of $10 \mathrm{mph},$ (b) a cyclist moving at the rate of $40 \mathrm{mph},(\mathrm{c})$ a car moving at the rate of $65 \mathrm{mph}$
(d) an airplane moving at the rate of $500 \mathrm{mph}$

Narayan Hari

Numerade Educator

02:16

Problem 36

The static pressure to stagnation pressure ratio at a point in a gas flow field is measured with a Pitot-static probe as being equal to $0.6 .$ The stagnation temperature of the gas is $20^{\circ} \mathrm{C}$. Determine the flow speed in $\mathrm{m} / \mathrm{s}$ and the Mach number if the gas is air. What error would be associated with assuming that the flow is incompressible?

Narayan Hari

Numerade Educator

02:11

Problem 37

The stagnation pressure and temperature of air flowing past a probe are $120 \mathrm{kPa}(\mathrm{abs})$ and $100^{\circ} \mathrm{C},$ respectively. The air pressure is $80 \mathrm{kPa}(\text { abs }) .$ Determine the airspeed and the Mach number considering the flow to be
(a) incompressible,
(b) compressible.

Chai Santi

Numerade Educator

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Problem 38

The The stagnation pressure indicated by a Pitot tube mounted on an airplane in flight is $45 \mathrm{kPa}(\mathrm{abs}) .$ If the aircraft is cruising in standard atmosphere at an altitude of $10,000 \mathrm{m},$ determine the speed and Mach number involved.

Victor Salazar

Numerade Educator

13:01

Problem 39

Helium at $68^{\circ} \mathrm{F}$ and 14.7 psia in a large tank flows steadily and isentropically through a converging nozzle to a receiver pipe. The cross-sectional area of the throat of the converging passage is $0.05 \mathrm{ft}^{2} .$ Determine the mass flowrate through the duct if the receiver pressure is
(a) 10 psia,
(b) 5 psia. Sketch temperature-entropy diagrams for situations (a) and (b).

Susan Hallstrom

Numerade Educator

02:57

Problem 40

An ideal gas enters subsonically and flows isentropically through a choked converging-diverging duct having a circular cross-sectional area $A$ that varies with axial distance from the throat, $x,$ according to the formula $$A=0.1+x^{2}$$ where $A$ is in square feet and $x$ is in feet. For this flow situation, sketch the side view of the duct and graph the variation of Mach number, static temperature to stagnation temperature ratio, $T / T_{0}$ and static pressure to stagnation pressure ratio, $p / p_{0},$ through the duct from $x=-0.6 \mathrm{ft}$ to $x=+0.6 \mathrm{ft}$. Also show the possible fluid states at $x=-0.6 \mathrm{ft}, 0 \mathrm{ft},$ and $+0.6 \mathrm{ft}$ using temperature-entropy coordinates. Consider the gas as being helium (use $0.051 \leq \mathrm{Ma} \leq$ 5.193 ). Sketch on your pressure variation graph the nonisentropic paths that would occur with over- and underexpanded duct exit flows (see Video $V 11.7$ ) and explain when they will occur. When will isentropic supersonic duct exit flow occur?

Dominador Tan

Numerade Educator

01:36

Problem 41

An deal gas enters supersonically and flows isentropically through the choked converging-diverging duct described in Problem $11.40 .$ Graph the variation of $\mathrm{Ma}, T / T_{0},$ and $p / p_{0}$ from the entrance to the exit sections of the duct for helium (use 0.051 $\leq \mathrm{Ma} \leq 5.193$ ). Show the possible fluid states at $x=-0.6 \mathrm{ft}$
$0 \mathrm{ft},$ and $+0.6 \mathrm{ft}$ using temperature-entropy coordinates. Sketch on your pressure variation graph the nonisentropic paths that would occur with over- and underexpanded duct exit flows (see Video V11.7) and explain when they will occur. When will isentropic supersonic duct exit flow occur?

Dominador Tan

Numerade Educator

02:46

Problem 42

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of $59^{\circ} \mathrm{F}$ and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in ft/s if the gas is air.

Ajay Singhal

Numerade Educator

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Problem 43

An ideal gas flows isentropically through a convergingdiverging nozzle. At a section in the converging portion of the nozzle, $A_{1}=0.1 \mathrm{m}^{2}, p_{1}=600 \mathrm{kPa}(\text { abs }), T_{1}=20^{\circ} \mathrm{C},$ and $\mathrm{Ma}_{1}=0.6$ For section (2) in the diverging part of the nozzle, determine $A_{2}$ $p_{2},$ and $T_{2}$ if $\mathrm{Ma}_{2}=3.0$ and the gas is air.

Victor Salazar

Numerade Educator

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Problem 44

Upstream of the throat of an isentropic convergingdiverging nozzle at section $(1), V_{1}=150 \mathrm{m} / \mathrm{s}, p_{1}=100 \mathrm{kPa}(\mathrm{abs})$
and $T_{1}=20^{\circ} \mathrm{C} .$ If the discharge flow is supersonic and the throat area is $0.1 \mathrm{m}^{2},$ determine the mass flowrate in $\mathrm{kg} / \mathrm{s}$ for the flow of air.

Victor Salazar

Numerade Educator

01:26

Problem 45

The flow blockage associated with the use of an intrusive probe can be important. Determine the percentage increase in section velocity corresponding to a $0.5 \%$ reduction in flow area due to probe blockage for airflow if the section area is $1.0 \mathrm{m}^{2}, T_{0}$ $=20^{\circ} \mathrm{C},$ and the unblocked flow Mach numbers are
(a) $\mathrm{Ma}=0.2$
(b) $\mathrm{Ma}=0.8$
(c) $\mathrm{Ma}=1.5$
(d) $\mathrm{Ma}=30$

Chai Santi

Numerade Educator

01:46

Problem 46

(See Fluids in the News article titled "Rocket Nozzles," Section $11.4 .2 .)$ Comment on the practical limits of area ratio for the diverging portion of a converging-diverging nozzle designed to achieve supersonic exit flow.

Dominador Tan

Numerade Educator

02:00

Problem 47

An ideal gas enters [section (1)] an insulated, constant cross-sectional area duct with the following properties:
$$\begin{aligned}
T_{0} &=293 \mathrm{K} \\
p_{0} &=101 \mathrm{kPa}(\mathrm{abs}) \\
\mathrm{Ma}_{1} &=0.2
\end{aligned}$$
For Fanno flow, determine corresponding values of fluid temperature and entropy change for various levels of pressure and plot the Fanno line if the gas is helium.

Dominador Tan

Numerade Educator

14:33

Problem 48

For Fanno flow, prove that $$\frac{d V}{V}=\frac{f k\left(\mathrm{Ma}^{2} / 2\right)(d x / D)}{1-\mathrm{Ma}^{2}}$$ and in so doing show that when the flow is subsonic, friction accelerates the fluid, and when the flow is supersonic, friction decelerates the fluid.

David Morabito

Numerade Educator

01:17

Problem 49

Standard atmospheric air $\left(T_{0}=59^{\circ} \mathrm{F}, p_{0}=14.7 \mathrm{psia}\right)$ is drawn steadily through a frictionless and adiabatic converging nozzle into an adiabatic, constant cross-sectional area duct. The duct is $10 \mathrm{ft}$ long and has an inside diameter of $0.5 \mathrm{ft}$. The average friction factor for the duct may be estimated as being equal to $0.03 .$ What is the maximum mass flowrate in slugs/s through the duct? For this maximum flowrate, determine the values of static temperature, static pressure, stagnation temperature, stagnation pressure, and velocity at the inlet [section (1)] and exit [section (2)] of the constant area duct. Sketch a temperature-entropy diagram for this flow.

Dominador Tan

Numerade Educator

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Problem 50

The duct in Problem 11.49 is shortened by $50 \%$. The duct discharge pressure is maintained at the choked flow value determined in Problem $11.49 .$ Determine the change in mass flowrate through the duct associated with the $50 \%$ reduction in length. The average friction factor remains constant at a value of 0.03

Victor Salazar

Numerade Educator

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Problem 51

If the same mass flowrate of air obtained in Problem 11.49 is desired through the shortened duct of Problem $11.50,$ determine the back pressure, $p_{2},$ required. Assume $f$ remains constant at a value of 0.03

Victor Salazar

Numerade Educator

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Problem 52

If the average friction factor of the duct of Example 11.12 is changed to (a) 0.01 or (b) $0.03,$ determine the maximum mass flowrate of air through the duct associated with each new friction factor; compare with the maximum mass flowrate value of Example 11.12

Victor Salazar

Numerade Educator

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Problem 53

If the length of the constant area duct of Example 11.12 is changed to (a) $1 \mathrm{m}$ or (b) $3 \mathrm{m}$, and all other specifications in the problem statement remain the same, determine the maximum mass flowrate of air through the duct associated with each new length; compare with the maximum mass flowrate of Example 11.12

Victor Salazar

Numerade Educator

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Problem 54

The duct of Example 11.12 is lengthened by $50 \%$. If the duct discharge pressure is set at a value of $p_{d}=46.2 \mathrm{kPa}(\mathrm{abs}),$ determine the mass flowrate of air through the lengthened duct. The average friction factor for the duct remains constant at a value 0.02

Victor Salazar

Numerade Educator

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Problem 55

Air flows adiabatically between two sections in a constant area pipe. At upstream section (1), $p_{0,1}=100$ psia, $T_{0,1}=600$ $^{\circ} \mathrm{R},$ and $\mathrm{Ma}_{1}=0.5 .$ At downstream section $(2),$ the flow is choked. Estimate the magnitude of the force per unit cross-sectional area exerted by the inside wall of the pipe on the fluid between sections
(1) and (2)

Victor Salazar

Numerade Educator