Fundamentals of Fluid Mechanics

Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Chapter 11

Compressible Flow - all with Video Answers

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Chapter Questions

Problem 1

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

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

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)
(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?

Narayan Hari

Numerade Educator

Problem 3

Consider the flow process in Fig. $P 11.3 .$ Does the fluid flow from left to right or from right to left? Justify your answer. Assume as ideal gas with constant specific heats.

Narayan Hari

Numerade Educator

Problem 4

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

Problem 5

Describe briefly how a schlieren optical visualization system (Videos V11.1 and V11.5, also Fig. 1 1.4) works. How else might density changes in a fluid flow be made visible to the eye?

Chai Santi

Numerade Educator

Problem 6

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

Problem 7

Air expands adiabatically through a turbine from a pressure and temperature of 180 psia, $1600^{\circ} \mathrm{R}$ 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 ar and the actual enthalpy and entropy differences across the turbine.

Narayan Hari

Numerade Educator

Problem 8

An expression for the value of $c_{p}$ for carbon dioxide as 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 cnthalpy of carbon dioxide using
(a) $10^{\circ} \mathrm{R}$ the expression above, for $T_{2}-T_{1}$ equal to
(b) $1000^{\circ} \mathrm{R}$
(c) $3000^{\circ} \mathrm{R}$. Set $T_{1}=540^{\circ} \mathrm{R}$

Chai Santi

Numerade Educator

Problem 9

Air flows in a 15 -cm-diameter horizontal pipe. At section 1 $p=600 \mathrm{kPa}, T=70^{\circ} \mathrm{C},$ and $V=35 \mathrm{m} / \mathrm{s} .$ At section $2: T=42^{\circ} \mathrm{C}$
and $V=115 \mathrm{m} / \mathrm{s}$. Determine
(a) The pressure at section 2
(b) The total friction force on the pipe wall
(c) The heat transfer

Chai Santi

Numerade Educator

Problem 10

An air heater in a large coal-fired steam generator heats fresh air entering the steam generator by cooling flue gas leaving the steam generator. One million lbm/hr of air at $100^{\circ} \mathrm{F}$ and 1.1 million lbm/hr of flue gas at $720^{\circ} \mathrm{F}$ enters the air heater. The flue gas leaves at $310^{\circ} \mathrm{F}$. Flue gas has $c_{p}=0.26 \mathrm{Btu} / \mathrm{lbm}^{\circ} \mathrm{F}$ and $k=1.39 .$ Pressure changes are small and may be neglected. Calculate the temperature of the air leaving the air heater and the total entropy change for the process.

Narayan Hari

Numerade Educator

Problem 11

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 $40 \mathrm{mph}$
(c) a car moving at the rate of 65 mph,
(d) an airplane moving at the rate of $500 \mathrm{mph}$

Narayan Hari

Numerade Educator

Problem 12

The air velocity in the duct in Fig. $P 11.12$ is $750 \mathrm{ft} / \mathrm{s}$. The air static temperature is $100^{\circ} \mathrm{F}$. Use the mercury manometer measurement to calculate the static and stagnation pressures of the flowing air.

Narayan Hari

Numerade Educator

Problem 13

Air is flowing in a duct as shown in Fig. $\mathrm{P} 11.13 .$ A Pitotstatic tube and a thermocouple are inserted into the flow stream as shown. Calculate the air velocity and the air-mass flow rate.

Narayan Hari

Numerade Educator

Problem 14

Steam $\left(\mathrm{H}_{2} \mathrm{O} \text { vapor }\right)$ flows in a pipeline in a power station. The steam pressure is 150 psia, its temperature is $500^{\circ} \mathrm{F}$, and it flows with velocity $750 \mathrm{ft}$ /s. Calculate the stagnation pressure and stagnation temperature. If you are familiar with Steam Tables or steam property software, use these tools to make an "exact" calculation. If you are not familiar with these tools, model the steam as an ideal gas with molecular weight of 18 and $k=1.3$.

Narayan Hari

Numerade Educator

Problem 15

An ideal gas flows with velocity $V$, pressure $p$, temperature $T,$ and density $\rho .$ Determine a set of equations for stagnation properties, including entropy, if the stagnation process is defined to be isothermal $(T=\text { constant ) rather than isentropic }(s=\text { constant })$.

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 16

If the observed speed of sound in steel is $5300 \mathrm{m} / \mathrm{s}$, deter mine 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 $E_{\mathrm{v}}$ for steel compare with $E_{p}$ 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.

Narayan Hari

Numerade Educator

Problem 17

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

Chai Santi

Numerade Educator

Problem 18

An airplane is flying at a flight (or local) Mach number of 0.70 at $10,000 \mathrm{m}$ in the Standard Atmosphere. Find the ground speed (a) if the air is not moving relative to the ground and (b) if the air is moving at $30 \mathrm{km} / \mathrm{hr}$ in the opposite direction from the airplane.

Narayan Hari

Numerade Educator

Problem 19

The stagnation pressure in a Mach 2 wind tunnel operating with air is 900 kPa. A 1.0 -cm-diameter sphere positioned in the wind tunnel has a drag coefficient of $0.95 .$ Calculate the drag force on the sphare.

Narayan Hari

Numerade Educator

Problem 20

Calculate the speed of sound in air, helium, and hydrogen. The temperature is $70^{\circ} \mathrm{F}$.

Narayan Hari

Numerade Educator

Problem 21

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

Problem 22

A jet-propelled airliner flies at an altitude of $15,000 \mathrm{m}$ and a velocity of $265 \mathrm{m} / \mathrm{s}$. What is the stagnation pressure and stagnation temperature of the air entering the engine? If the airliner is stationary on the ground, the engines are running with the same Mach number flow into the engines, and the air flow to each engine is $40 \mathrm{kg} / \mathrm{s},$ what is the airflow to each engine at the above flight conditions?

Narayan Hari

Numerade Educator

Problem 23

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) ft/s, (c) mph?

Narayan Hari

Numerade Educator

Problem 24

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. $P 11.24$ Determine the Mach number involved and indicate with a sketch the instantaneous location of the point source.

Paul A.

California State Polytechnic University, Pomona

Problem 25

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 $\vee 11.8$ ) travel less than, equal to, or greater than the speed of sound? Explain.

Chai Santi

Numerade Educator

Problem 26

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) $\mathrm{ft} / \mathrm{s},(\mathrm{c}) \mathrm{m} / \mathrm{s} ?$

Narayan Hari

Numerade Educator

Problem 27

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

Problem 28

Air flows in a constant-area, insulated duct. The air enters the duct at $520^{\circ} \mathrm{R}, 50$ psia, and $\mathrm{Ma}=0.45 .$ At a downstream location, the Mach number is one. Find:
(a) The pressure and temperature at the downstream location
(b) The change in specific entropy
(c) The frictional force if the duct is circular and $1 \mathrm{ft}$ in diameter.

Chai Santi

Numerade Educator

Problem 29

A normal shock occurs in a stream of oxygen. The oxygen flows at $\mathrm{Ma}=1.8$ and the upstream pressure and temperature are 15 psia and $85^{\circ} \mathrm{F}$
(a) Calculate the following on the downstream side of the shock:
static pressure, stagnation pressure, static temperature, stagnation temperature, static density, and velocity.
(b) If the Mach number is doubled to $3.6,$ what will be the resulting values of the parameters listed in part (b)?

Chai Santi

Numerade Educator

Problem 30

The Pitot tube on a supersonic aircraft (see Video $\mathbf{V} 3.8$ ) cruising at an altitude of 30,000 ft senses a stagnatior pressure of 12 psia. If the atmosphere is considered standard, determine the airspeed and Mach number of the aircraft. A shock wave is present just upstream of the probe impact hole.

Narayan Hari

Numerade Educator

Problem 31

An aircraft cruises at a Mach number of 2.0 at an altitude of $15 \mathrm{km} .$ Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is $1.2 .$ For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.

Chai Santi

Numerade Educator

Problem 32

At some point for air flow in a duct, $p=20$ psia, $T=500^{\circ} \mathrm{R}$ and $V=500 \mathrm{ft} / \mathrm{s}$. Can a normal shock occur at this point?

Narayan Hari

Numerade Educator

Problem 33

A normal shock propagates at $2000 \mathrm{ft} / \mathrm{s}$ into the still air in a tube. The temperature and pressure of the air are $80^{\circ} \mathrm{F}$ and 14.7 psia before "hit" by the shock. Calculate the air temperature, pressure, and velocity after the shock, the stagnation temperature and pressure relative to the shock ahead of and behind the shock, and the stagnation temperature and pressure relative to the tube ahead of and behind the shock.

Anand Jangid

Numerade Educator

Problem 34

Air at $V_{1}=800 \mathrm{m} / \mathrm{s}, p_{1}=100 \mathrm{kP} \epsilon$, and $T_{1}=300 \mathrm{K}$ passes
through a normal shock. Calculate the velocity $V_{2}$, temperature $T_{2}$ and pressure $p_{2}$ after the shock. What would be the values of $T_{2}$ and $p_{2}$ if the same velocity change were accomplished isentropically?

Chai Santi

Numerade Educator

Problem 35

A normal shock occurs in a perfect gas. Sketch a Temperature - Entropy $(T-s)$ diagram of the process and show the following:
Static and stagnation pressure and temperature before and after the shock and gas velocity before and after the shock.

Chai Santi

Numerade Educator

Problem 36

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

Problem 37

Air flows isentropically through a duct as shown in Fig. P11.37. For the conditions shown, find the Mach number at both stations 1 and 2 and the flow rate. The diameter is not necessarily constant.

Chai Santi

Numerade Educator

Problem 38

A simplified schematic diagram of a carburetor of a gasoline engine is shown in Fig. $\mathrm{P} 11.38 .$ The throat area is $0.5 \mathrm{in}^{2}$. The engine draws air downward through the carburetor Venturi and maintains a throat pressure of 14.3 psia. This low throat pressure draws fuel from the float chamber and into the air stream. The energy losses in the 0.06 -in. I.D. fuel line and valve are given by
\[h_{L}=\frac{\mathrm{K} V^{2}}{2 g}\]
with $\mathrm{K}=6.0 .$ The fuel specific gravity is $0.75 .$ Assume an atmospheric pressure of 14.7 psia. Find the air-fuel ratio (that is, the ratio of the air mass flow rate to the fuel mass flow rate) based on an (a) ideal (constant density and inviscid) air flow and
(b) an isentropic air flow.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 39

An engineering student wants to satisfy her curiosity about the compressibility of air in motion. She has set up a converging nozzle in which air discharges into the atmosphere. Figure $\mathrm{Pl} 1.39$ shows the nozzle with the necessary information. For these conditions, find the velocity at the exit by using the incompressible Bernoulli's equation, a compressible isothermal process, and a compressible isentropic process. Comment on your results.

Ajay Singhal

Numerade Educator

Problem 40

A nozzle for a supersonic wind tunnel is designed to achieve a Mach number of $3.0,$ with a velocity of $2000 \mathrm{m} / \mathrm{s},$ and a density of $1.0 \mathrm{kg} / \mathrm{m}^{3}$ in the test section. Find the temperature and pressure in the test section and the upstream stagnation conditions. The fluid is helium.

Narayan Hari

Numerade Educator

Problem 41

Air flows isentropically through a duct to a section where $p_{1}=25 \mathrm{kPa}, T_{1}=300 \mathrm{K},$ and $V_{1}=900 \mathrm{m} / \mathrm{s} .$ For these conditions:
(a) Determine the stagnation conditions for the flow.
(b) What is the Mach number at station $1 ?$ Show a $T-s$ diagram displaying stagnation and static conditions.
(c) Is the flow choked? Is the throat behind or ahead of section $1 ?$ Label this state on the $T-s$ diagram.

Chai Santi

Numerade Educator

Problem 42

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

Problem 43

Air flows 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

Problem 44

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 temperatureentropy diagrams for situations (a) and (b).

Chai Santi

Numerade Educator

Problem 45

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 $\mathrm{ft} / \mathrm{s}$ if the gas is air.

Ajay Singhal

Numerade Educator

Problem 46

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) $\mathrm{Ma}=1.5,(\mathrm{d}) \mathrm{Ma}=30$

Chai Santi

Numerade Educator

Problem 47

At a certain point in a pipe, air flows steadily with a velocity of $150 \mathrm{m} / \mathrm{s}$ and has a static pressure of $70 \mathrm{kPa}$ and a static temperature of $4^{\circ} \mathrm{C}$. The flow is adiabatic and frictionless.

Dominador Tan

Numerade Educator

Problem 48

A tank of oxygen has a hole of area $0.5 \mathrm{cm}^{2}$ in its wall. The temperature of the oxygen in the tank is $25^{\circ} \mathrm{C}$. Calculate the rate $(\mathrm{kg} / \mathrm{s})$ at which oxygen leaks to the atmosphere for tank pressures of 135 kPa and 375 kPa. Assume frictionless flow.

Chai Santi

Numerade Educator

Problem 49

The gas entering a rocket nozzle has a stagnation pressure of $1500 \mathrm{kPa}$ and a stagnation temperature of $3000^{\circ} \mathrm{C}$. The rocket is traveling in the still Standard Atmosphere at $30,000 \mathrm{m}$. Find the throat and exit area for a flow rate of $10 \mathrm{kg} / \mathrm{s}$. Assume $k=1.35, R=$ $287.0 \mathrm{N} \cdot \mathrm{m} / \mathrm{kg} \cdot \mathrm{K} .$ The gas is perfectly, expanded to the ambient pressure.

Chai Santi

Numerade Educator

Problem 50

Air flows in the channel in Fig. $\mathrm{P} 11.50$. Determine the Mach number, static pressure, and stagnation pressures at station 3. Assume isentropic flow except for the normal shock wave.

Chai Santi

Numerade Educator

Problem 51

A jet engine is to be designed for an altitude of $12,000 \mathrm{m}$ where the atmospheric pressure is $19.3 \mathrm{kPa}$. The jet nozzle has a supersonic exit Mach number and is perfectly expanded. The stagnation pressure and temperature of the gas are 100 kpa and $600^{\circ} \mathrm{C}$ The flow rate of gas is $45 \mathrm{kg} / \mathrm{s}$. Calculate the throat area, exit area, and exit velocity. Use $k=1.4$ and $R=260 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ for the gas.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 52

Air is flowing in the converging-diverging nozzle shown in Fig. $\mathrm{P} 11.52 .$ Determine the three critical pressure ratios and the Mach numbers immediately upstream and immediately downstream from the shock.

Chai Santi

Numerade Educator

Problem 53

A convergent-divergent nozzle has an exit thrcat area ratio of $3.0 .$ It is to be supplied with air. Find:
(a) The first, second, and third critical pressure ratios;
(b) The exit plane Mach number in each case:
(c) The throat Mach number in each case.

Chai Santi

Numerade Educator

Problem 54

An ideal gas flows isentropically through a convergingdiverging nozzle. At a section in the converging portion of the nozzle. $A_{1}=0.1 \mathrm{m}^{2}, p_{1}=600 \mathrm{kPa}(\mathrm{abs}), T_{1}=20^{\circ} \mathrm{C},$ and $\mathrm{M} \varepsilon_{1}=0.6 .$ 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.

Chai Santi

Numerade Educator

Problem 55

Air is supplied to a convergent-divergent nozzle from a reservoir where the pressure is $100 \mathrm{kPa}$. The air is then discharged through a short pipe into another reservoir where the pressure can be varied. The cross-sectional area of the pipe is twice the area of the throat of the nozzle. Friction and heat transfer may be neglected throughout the flow. If the discharge pipe anstant cross-sectional area, determine the range of static pressure in the pipe for which a normal shock will stand in the divergent section of the nozzle. If the discharge pipe tapers so that its cross-sectional area is reduced by $25 \%$, show that a normal shock cannot be drawn to the end of the divergent section of the nozzle. Find the maximum strength of shock (as expressed by the upstream Mach number) that can be formed.

Chai Santi

Numerade Educator

Problem 56

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 valies 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

Problem 57

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 anit cross-sectional area exerted by the inside wall of the pipe on the fluid between sections (1) and (2).

Narayan Hari

Numerade Educator

Problem 58

Supersonic airflow enters an adiabatic, constant area (inside diameter $=1 \mathrm{ft}$ ) 30 -ft-long pipe with $\mathrm{Ma}_{1}=3.0 .$ The pipe friction factor is estimated to be $0.02 .$ What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at
(a) $x=5$ ft, or $(\mathbf{b}) x=10 \mathrm{ft},$ where $x$ is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.

Victor Salazar

Numerade Educator

Problem 59

Supersonic airflow enters an adiabatic, constant area pipe (inside diameter $=0.1 \mathrm{m}$ ) with $\mathrm{Ma}_{1}=2.0 .$ The pipe friction factor is $0.02 .$ If a standing normal shock is localed right at the pipe exit. and the Mach number just upstream of the shock is $1.2,$ determine the length of the pipe.

Narayan Hari

Numerade Educator

Problem 60

Consider the flow of air through the piping system shown in Fig. $\mathrm{P} 11.60 .$ If the system is choked, determine the location where the Mach number is $1 .$ Calculate the pressure ratio $p_{B} / p_{0},$ that causes choking in this system.

Chai Santi

Numerade Educator

Problem 61

Estimate the maximum mass flow rate of air that can be passed by the duct shown in Fig. P1 1.61 for $\mu=4 \times 10^{-7} 1 \mathrm{b}$ $\mathrm{s} / \mathrm{ft}^{2}$.

Chai Santi

Numerade Educator

Problem 62

Waste gas $\left(\mathrm{CO}_{2}\right)$ is vented to outer space from a spacecraft through a circular pipe $0.2 \mathrm{m}$ long. The pressure and temperature in the spacecraft are $35 \mathrm{kPa}$ and $25^{\circ} \mathrm{C}$. The gas must be vented at the rate of $0.01 \mathrm{kg} / \mathrm{s}$. The friction factor for the flow in the pipe is given by
\[\begin{array}{c}f=64 / \mathrm{Re}, \quad \mathrm{Re}<5000 \\f=0.013, \quad \mathrm{Re}>5000, \quad \mathrm{Re}=\frac{ \dot{m}}{\pi \mu D}\end{array}\]
The viscosity $(\mu)$ is $4 \times 10^{-4} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$. Determine the required pipe diameter.

Chai Santi

Numerade Educator

Problem 63

Air enters a 4 -cm-square galvanized steel duct with $p_{0}=$ $150 \mathrm{kPa}, T_{0}=400 \mathrm{K},$ and $V_{1}=120 \mathrm{m} / \mathrm{s},$ (Note: $\mu=2.2 \times 10^{-5}$ $\left.\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$
(a) Compute the maximum possible duct length for these conditions.
(b) If the actual duct length is 0.75 times the maximum value, calculate the mass flow rate, the exit pressure, and the stagnation pressure.
(c) If the actual duct length is 1.3 times the maximum value for the stated conditions, compute the new mass flow rate and inlet velocity. Assume a low back pressure and use the same value of $f$ as used in the maximum possible duct length case.

James Kiss

Numerade Educator

Problem 64

Figure $P 11.64$ shows an insulated pipe attached to $\varepsilon$ tank of air. Estimate the maximum mass flow rate that the pipe could exhaust from the tank.

Chai Santi

Numerade Educator

Problem 65

Standard atmospheric air $\left[T_{0}=288 \mathrm{K}, p_{0}=101 \mathrm{kPa}(\mathrm{abs})\right]$ is drawn steadily through an isentropic converging nozzle into a frictionless diabatic $(q=500 \mathrm{kJ} / \mathrm{kg})$ constant area duct. For maximum flow, determine the values of static temperature, static pressure, stagnation temperature, stagnation pressure, and flow veloc$\text { ity at the inlet [section }(1)] \text { and exit [section }(2)]$ of the constant area duct. Sketch a temperature-entropy diagram for this flow.

Dominador Tan

Numerade Educator

Problem 66

Air enters a 0.5 -ft inside diameter duct with $p_{1}=20$ psia. $T_{1}=80^{\circ} \mathrm{F},$ and $V_{1}=200 \mathrm{ft} / \mathrm{s}$. What frictionless heat addition rate in $\mathrm{Btu} / \mathrm{s}$ is necessary for an exit gas temperature $T_{2}=1500^{\circ} \mathrm{F} ?$ Determine $p_{2}, V_{2},$ and $\mathrm{Ma}_{2}$ also.

Narayan Hari

Numerade Educator

Problem 67

Air enters a length of constant area pipe with $p_{1}=200$ $\mathrm{kPa}(\mathrm{abs}), T_{1}=500 \mathrm{K},$ and $V_{1}=400 \mathrm{m} / \mathrm{s} .$ If $500 \mathrm{kJ} / \mathrm{kg}$ of energy is removed from the air by frictionless heat transfer between sections (1) and $(2),$ determine $p_{2}, T_{2},$ and $V_{2} .$ Sketch a temperature-ertropy diagram for the flow between sections (1) and (2).

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 68

Air enters a frictionless, constant area duct with $\mathrm{Ma}_{1}=2.0$ $T_{0,1}=59^{\circ} \mathrm{F},$ and $p_{0,1}=14.7$ psia. The air is decelerated by heating until a normal shock wave occurs where the local Mach number is 1.5. Downstream of the normal shock, the subscnic flow is accelerated with heating until it chokes at the duct exit. Determine the static temperature and pressure, the stagnation temperature and pressure, and the fluid velocity at the duct entrance, just upstream and downstream of the normal shock, and at the duct exit. Sketch the temperature-entropy diagram for this flow.

Dominador Tan

Numerade Educator

Problem 69

Air enters a frictionless, constant area duct with $\mathrm{Ma}=2.5$ $T_{0}=20^{\circ} \mathrm{C},$ and $p_{0}=101 \mathrm{kPa}(\mathrm{abs}) .$ The gas is decelerated by heating until a normal shock occurs where the local Mach number is $1.3 .$ Downstream of the shock, the subsonic fow is accelerated with heating until it exits with a Mach number of $0.9 .$ Determine the static temperature and pressure, the stagnation temperature and pressure, and the fluid velocity at the duct entrance, just upstream and downstream of the normal shock, and at the duct exit. Sketch the temperature-entropy diagram for this flow.

Dominador Tan

Numerade Educator

Problem 70

Air enters a 15 -cm pipe with velocity $120 \mathrm{m} / \mathrm{s}, 1$ atmosphere pressure, and $T=100^{\circ} \mathrm{C}$. How much heat must be added to bring the air to the maximum static temperature? What are the values of temperature and pressure at this point?

Saurabh Kumar Gupta

Saurabh Kumar Gupta

Numerade Educator

Problem 71

Show that for Rayleigh flow, the maximum amount of heat that may be added to the gas is given by:
\[\frac{q_{\max }}{c_{p} T_{1}}=\frac{\left(\mathrm{Ma}_{1}^{2}-1\right)^{2}}{2(k+1) \mathrm{Ma}_{1}^{2}}\]

Mukesh Devi

Numerade Educator

Problem 72

Air is stored in a tank where the pressure is 40 psia and the temperature is $500^{\circ} \mathrm{R}$. A converging-diverging nozzle with an exitto-throat area ratio of 2.5 attaches the tank to a duct where heat is exchanged with the air. The exit pressure is 15 psia and a normal shock stands at the exit of the nozzle. Determine the magnitude and direction of the heat exchange.

James Kiss

Numerade Educator

Problem 73

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is $1 / \sqrt{k}$.

Hunza Gilgit

Numerade Educator