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Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 12

COMPRESSIBLE FLOW - all with Video Answers

Educators

Chapter Questions

Problem 1

A high-speed aircraft is cruising in still air. How does the temperature of air at the nose of the aircraft differ from the temperature of air at some distance from the aircraft?

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Problem 2

How and why is the stagnation enthalpy $h_0$ defined? How does it differ from ordinary (static) enthalpy?

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Problem 3

What is dynamic temperature?

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Problem 4

In air-conditioning applications, the temperature of air is measured by inserting a probe into the flow stream. Thus, the probe actually measures the stagnation temperature. Does this cause any significant error?

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Problem 5

Determine the stagnation temperature and stagnation pressure of air that is flowing at $44 \mathrm{kPa}, 245.9 \mathrm{~K}$, and $470 \mathrm{~m} / \mathrm{s}$.

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Problem 6

Air at 300 K is flowing in a duct at a velocity of (a) 1 , (b) 10 , (c) 100 , and (d) $1000 \mathrm{~m} / \mathrm{s}$. Determine the temperature that a stationary probe inserted into the duct will read for each case.

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Problem 7

Calculate the stagnation temperature and pressure for the following substances flowing through a duct: (a) helium at $0.25 \mathrm{MPa}, 50^{\circ} \mathrm{C}$, and $240 \mathrm{~m} / \mathrm{s}$; (b) nitrogen at 0.15 MPa , $50^{\circ} \mathrm{C}$, and $300 \mathrm{~m} / \mathrm{s}$; and (c) steam at $0.1 \mathrm{MPa}, 350^{\circ} \mathrm{C}$, and $480 \mathrm{~m} / \mathrm{s}$.

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Problem 8

Air enters a compressor with a stagnation pressure of 100 kPa and a stagnation temperature of $27^{\circ} \mathrm{C}$, and it is compressed to a stagnation pressure of 900 kPa . Assuming the compression process to be isentropic, determine the power input to the compressor for a mass flow rate of $0.02 \mathrm{~kg} / \mathrm{s}$.

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Problem 9

Steam flows through a device with a stagnation pressure of 120 psia, a stagnation temperature of $700^{\circ} \mathrm{F}$, and a velocity of $900 \mathrm{ft} / \mathrm{s}$. Assuming ideal-gas behavior, determine the static pressure and temperature of the steam at this state.

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Problem 10

Products of combustion enter a gas turbine with a stagnation pressure of 1.0 MPa and a stagnation temperature of $750^{\circ} \mathrm{C}$, and they expand to a stagnation pressure of 100 kPa . Taking $k=1.33$ and $R=0.287 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$ for the products of combustion, and assuming the expansion process to be isentropic, determine the power output of the turbine per unit mass flow.

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Problem 11

Air flows through a device such that the stagnation pressure is 0.6 MPa , the stagnation temperature is $400^{\circ} \mathrm{C}$, and the velocity is $570 \mathrm{~m} / \mathrm{s}$. Determine the static pressure and temperature of the air at this state.

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Problem 12

What is sound? How is it generated? How does it travel? Can sound waves travel in a vacuum?

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Problem 13

Is it realistic to assume that the propagation of sound waves is an isentropic process? Explain.

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Problem 14

Is the sonic velocity in a specified medium a fixed quantity, or does it change as the properties of the medium change? Explain.

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Problem 15

In which medium does a sound wave travel faster: in cool air or in warm air?

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Problem 16

In which medium will sound travel fastest for a given temperature: air, helium, or argon?

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Problem 17

In which medium does a sound wave travel faster: in air at $20^{\circ} \mathrm{C}$ and 1 atm or in air at $20^{\circ} \mathrm{C}$ and 5 atm ?

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Problem 18

Does the Mach number of a gas flowing at a constant velocity remain constant? Explain.

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Problem 19

Determine the speed of sound in air at (a) 300 K and (b) 1000 K . Also determine the Mach number of an aircraft moving in air at a velocity of $240 \mathrm{~m} / \mathrm{s}$ for both cases.

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Problem 20

Carbon dioxide enters an adiabatic nozzle at 1200 K with a velocity of $50 \mathrm{~m} / \mathrm{s}$ and leaves at 400 K . Assuming constant specific heats at room temperature, determine the Mach number (a) at the inlet and (b) at the exit of the nozzle. Assess the accuracy of the constant specific heat assumption.

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Problem 21

Nitrogen enters a steady-flow heat exchanger at 150 $\mathrm{kPa}, 10^{\circ} \mathrm{C}$, and $100 \mathrm{~m} / \mathrm{s}$, and it receives heat in the amount of $120 \mathrm{~kJ} / \mathrm{kg}$ as it flows through it. Nitrogen leaves the heat exchanger at 100 kPa with a velocity of $200 \mathrm{~m} / \mathrm{s}$. Determine the Mach number of the nitrogen at the inlet and the exit of the heat exchanger.

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Problem 22

Assuming ideal gas behavior, determine the speed of sound in refrigerant-134a at 0.1 MPa and $60^{\circ} \mathrm{C}$.

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Problem 23

The Airbus A-340 passenger plane has a maximum takeoff weight of about $260,000 \mathrm{~kg}$, a length of 64 m , a wing span of 60 m , a maximum cruising speed of $945 \mathrm{~km} / \mathrm{h}$, a seating capacity of 271 passengers, maximum cruising altitude of $14,000 \mathrm{~m}$, and a maximum range of $12,000 \mathrm{~km}$. The air temperature at the crusing altitude is about $-60^{\circ} \mathrm{C}$. Determine the Mach number of this plane for the stated limiting conditions.

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Problem 24

Steam flows through a device with a pressure of 120 psia , a temperature of $700^{\circ} \mathrm{F}$, and a velocity of $900 \mathrm{ft} / \mathrm{s}$. Determine the Mach number of the steam at this state by assuming ideal-gas behavior with $k=1.3$. Answer: 0.441

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Problem 25

Reconsider Prob. 12-24E. Using EES (or other) software, compare the Mach number of steam flow over the temperature range 350 to $700^{\circ} \mathrm{F}$. Plot the Mach number as a function of temperature.

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Problem 26

The isentropic process for an ideal gas is expressed as $P V^k=$ constant. Using this process equation and the definition of the speed of sound (Eq. 12-9), obtain the expression for the speed of sound for an ideal gas (Eq. 12-11).

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Problem 27

Air expands isentropically from 1.5 MPa and $60^{\circ} \mathrm{C}$ to 0.4 MPa . Calculate the ratio of the initial to final speed of sound.

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Problem 28

Repeat Prob. 12-27 for helium gas.

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Problem 29

Air expands isentropically from 170 psia and $200^{\circ} \mathrm{F}$ to 60 psia. Calculate the ratio of the initial to final speed of sound.

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Problem 30

Consider a converging nozzle with sonic speed at the exit plane. Now the nozzle exit area is reduced while the nozzle inlet conditions are maintained constant. What will happen to (a) the exit velocity and (b) the mass flow rate through the nozzle?

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Problem 31

A gas initially at a supersonic velocity enters an adiabatic converging duct. Discuss how this affects (a) the velocity, (b) the temperature, (c) the pressure, and (d) the density of the fluid.

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Problem 32

A gas initially at a supersonic velocity enters an adiabatic diverging duct. Discuss how this affects (a) the velocity, (b) the temperature, (c) the pressure, and (d) the density of the fluid.

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Problem 33

A gas initially at a subsonic velocity enters an adiabatic converging duct. Discuss how this affects (a) the velocity, (b) the temperature, (c) the pressure, and (d) the density of the fluid.

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Problem 34

A gas initially at a subsonic velocity enters an adiabatic diverging duct. Discuss how this affects (a) the velocity, (b) the temperature, (c) the pressure, and (d) the density of the fluid.

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Problem 35

A gas at a specified stagnation temperature and pressure is accelerated to $\mathrm{Ma}=2$ in a converging-diverging nozzle and to $\mathrm{Ma}=3$ in another nozzle. What can you say about the pressures at the throats of these two nozzles?

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Problem 36

Is it possible to accelerate a gas to a supersonic velocity in a converging nozzle?

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Problem 37

Air enters a converging-diverging nozzle at a pressure of 1.2 MPa with negligible velocity. What is the lowest pressure that can be obtained at the throat of the nozzle?

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Problem 38

Helium enters a converging-diverging nozzle at $0.7 \mathrm{MPa}, 800 \mathrm{~K}$, and $100 \mathrm{~m} / \mathrm{s}$. What are the lowest temperature and pressure that can be obtained at the throat of the nozzle?

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Problem 39

Calculate the critical temperature, pressure, and density of (a) air at $200 \mathrm{kPa}, 100^{\circ} \mathrm{C}$, and $250 \mathrm{~m} / \mathrm{s}$, and (b) helium at $200 \mathrm{kPa}, 40^{\circ} \mathrm{C}$, and $300 \mathrm{~m} / \mathrm{s}$.

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Problem 40

Quiescent carbon dioxide at 800 kPa and 400 K is accelerated isentropically to a Mach number of 0.6 . Determine the temperature and pressure of the carbon dioxide after acceleration.

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Problem 41

Air at $200 \mathrm{kPa}, 100^{\circ} \mathrm{C}$, and Mach number $\mathrm{Ma}=0.8$ flows through a duct. Calculate the velocity and the stagnation pressure, temperature, and density of the air.

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Problem 42

Reconsider Prob. 12-41. Using EES (or other) software, study the effect of Mach numbers in the range 0.1 to 2 on the velocity, stagnation pressure, temperature, and density of air. Plot each parameter as a function of the Mach number.

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Problem 43

Air at $30 \mathrm{psia}, 212^{\circ} \mathrm{F}$, and Mach number $\mathrm{Ma}=0.8$ flows through a duct. Calculate the velocity and the stagnation pressure, temperature, and density of air. Answers: $1016 \mathrm{ft} / \mathrm{s}, 758 \mathrm{R}, 45.7 \mathrm{psia}, 0.163 \mathrm{lbm} / \mathrm{ft}^3$

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Problem 44

An aircraft is designed to cruise at Mach number Ma $=1.4$ at 8000 m where the atmospheric temperature is 236.15 K. Determine the stagnation temperature on the leading edge of the wing.

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Problem 45

Consider subsonic flow in a converging nozzle with fixed inlet conditions. What is the effect of dropping the back pressure to the critical pressure on (a) the exit velocity, (b) the exit pressure, and (c) the mass flow rate through the nozzle?

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Problem 46

Consider subsonic flow in a converging nozzle with specified conditions at the nozzle inlet and critical pressure at the nozzle exit. What is the effect of dropping the back pressure well below the critical pressure on (a) the exit
velocity, (b) the exit pressure, and (c) the mass flow rate through the nozzle?

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Problem 47

Consider a converging nozzle and a convergingdiverging nozzle having the same throat areas. For the same inlet conditions, how would you compare the mass flow rates through these two nozzles?

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Problem 48

Consider gas flow through a converging nozzle with specified inlet conditions. We know that the highest velocity the fluid can have at the nozzle exit is the sonic velocity, at which point the mass flow rate through the nozzle is a maximum. If it were possible to achieve hypersonic velocities at the nozzle exit, how would it affect the mass flow rate through the nozzle?

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Problem 49

How does the parameter Ma* differ from the Mach number Ma ?

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Problem 50

What would happen if we attempted to decelerate a supersonic fluid with a diverging diffuser?

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Problem 51

What would happen if we tried to further accelerate a supersonic fluid with a diverging diffuser?

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Problem 52

Consider the isentropic flow of a fluid through a converging-diverging nozzle with a subsonic velocity at the throat. How does the diverging section affect (a) the velocity, (b) the pressure, and (c) the mass flow rate of the fluid?

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Problem 53

Is it possible to accelerate a fluid to supersonic velocities with a velocity other than the sonic velocity at the throat? Explain

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Problem 54

Explain why the maximum flow rate per unit area for a given ideal gas depends only on $P_0 / \sqrt{T_0}$. For an ideal gas with $k=1.4$ and $R=0.287 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, find the constant $a$ such that $\dot{m} / A^*=a P_0 / \sqrt{T_0}$.

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Problem 55

For an ideal gas obtain an expression for the ratio of the speed of sound where $\mathrm{Ma}=1$ to the speed of sound based on the stagnation temperature, $c^* / c_0$.

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Problem 56

An ideal gas flows through a passage that first converges and then diverges during an adiabatic, reversible, steady-flow process. For subsonic flow at the inlet, sketch the variation of pressure, velocity, and Mach number along the length of the nozzle when the Mach number at the minimum flow area is equal to unity.

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Problem 57

Repeat Prob. 12-56 for supersonic flow at the inlet.

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Problem 58

Air enters a nozzle at $0.2 \mathrm{MPa}, 350 \mathrm{~K}$, and a velocity of $150 \mathrm{~m} / \mathrm{s}$. Assuming isentropic flow, determine the pressure and temperature of air at a location where the air velocity equals the speed of sound. What is the ratio of the area at this location to the entrance area?

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Problem 59

Repeat Prob. 12-58 assuming the entrance velocity is negligible.

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Problem 60

Air enters a nozzle at $30 \mathrm{psia}, 630 \mathrm{R}$, and a velocity of $450 \mathrm{ft} / \mathrm{s}$. Assuming isentropic flow, determine the pressure and temperature of air at a location where the air velocity equals the speed of sound. What is the ratio of the area at this location to the entrance area?

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Problem 61

Air enters a converging-diverging nozzle at 0.8 MPa with a negligible velocity. Assuming the flow to be isentropic, determine the back pressure that will result in an exit Mach number of 1.8 .

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Problem 62

Nitrogen enters a converging-diverging nozzle at 700 kPa and 400 K with a negligible velocity. Determine the critical velocity, pressure, temperature, and density in the nozzle.

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Problem 63

An ideal gas with $k=1.4$ is flowing through a nozzle such that the Mach number is 2.4 where the flow area is $25 \mathrm{~cm}^2$. Assuming the flow to be isentropic, determine the flow area at the location where the Mach number is 1.2 .

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Problem 64

Repeat Prob. 12-63 for an ideal gas with $k=1.33$.

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Problem 65

Air at 900 kPa and 400 K enters a converging nozzle with a negligible velocity. The throat area of the nozzle is $10 \mathrm{~cm}^2$. Assuming isentropic flow, calculate and plot the exit pressure, the exit velocity, and the mass flow rate versus the back pressure $P_b$ for $0.9 \geq P_b \geq 0.1$ MPa .

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Problem 66

Reconsider Prob. 12-65. Using EES (or other) software, solve the problem for the inlet conditions of 1 MPa and 1000 K .

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Problem 67

Air enters a converging-diverging nozzle of a supersonic wind tunnel at 150 psia and $100^{\circ} \mathrm{F}$ with a low velocity. The flow area of the test section is equal to the exit area of the nozzle, which is $5 \mathrm{ft}^2$. Calculate the pressure, temperature, velocity, and mass flow rate in the test section for a Mach number $\mathrm{Ma}=2$. Explain why the air must be very dry for this application.

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Problem 68

Can a shock wave develop in the converging section of a converging-diverging nozzle? Explain.

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Problem 69

What do the states on the Fanno line and the Rayleigh line represent? What do the intersection points of these two curves represent?

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Problem 70

Can the Mach number of a fluid be greater than 1 after a normal shock wave? Explain.

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Problem 71

How does the normal shock affect (a) the fluid velocity, (b) the static temperature, (c) the stagnation temperature, (d) the static pressure, and (e) the stagnation pressure?

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Problem 72

How do oblique shocks occur? How do oblique shocks differ from normal shocks?

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Problem 73

For an oblique shock to occur, does the upstream flow have to be supersonic? Does the flow downstream of an oblique shock have to be subsonic?

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Problem 74

It is claimed that an oblique shock can be analyzed like a normal shock provided that the normal component of velocity (normal to the shock surface) is used in the analysis. Do you agree with this claim?

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Problem 75

Consider supersonic airflow approaching the nose of a two-dimensional wedge and experiencing an oblique shock. Under what conditions does an oblique shock detach from the nose of the wedge and form a bow wave? What is the numerical value of the shock angle of the detached shock at the nose?

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Problem 76

Consider supersonic flow impinging on the rounded nose of an aircraft. Will the oblique shock that forms in front of the nose be an attached or detached shock? Explain.

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Problem 77

Are the isentropic relations of ideal gases applicable for flows across (a) normal shock waves, (b) oblique shock waves, and (c) Prandtl-Meyer expansion waves?

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Problem 78

For an ideal gas flowing through a normal shock, develop a relation for $V_2 / V_1$ in terms of $k, \mathrm{Ma}_1$, and $\mathrm{Ma}_2$.

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Problem 79

Air enters a converging-diverging nozzle of a supersonic wind tunnel at 1 MPa and 300 K with a low velocity. If a normal shock wave occurs at the exit plane of the nozzle at $\mathrm{Ma}=2$, determine the pressure, temperature, Mach number, velocity, and stagnation pressure after the shock wave. Answers: $0.575 \mathrm{MPa}, 281 \mathrm{~K}, 0.577,194 \mathrm{~m} / \mathrm{s}, 0.721 \mathrm{MPa}$

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Problem 80

Air enters a converging-diverging nozzle with low velocity at 2.0 MPa and $100^{\circ} \mathrm{C}$. If the exit area of the nozzle is 3.5 times the throat area, what must the back pressure be to produce a normal shock at the exit plane of the nozzle?

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Problem 81

What must the back pressure be in Prob. 12-80 for a normal shock to occur at a location where the cross-sectional area is twice the throat area?

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Problem 82

Air flowing steadily in a nozzle experiences a normal shock at a Mach number of $\mathrm{Ma}=2.5$. If the pressure and temperature of air are 61.64 kPa and 262.15 K , respectively, upstream of the shock, calculate the pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock. Compare these results to those for helium undergoing a normal shock under the same conditions.

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Problem 83

Calculate the entropy change of air across the normal shock wave in Prob. 12-82.

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Problem 84

Air flowing steadily in a nozzle experiences a $=25$. If thermal shock at a Mach number of Ma $=2.5$. If the pressure and temperature of air are 10.0 psia and 440.5 R , respectively, upstream of the shock, calculate the pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock. Compare these results to those for helium undergoing a normal shock under the same conditions.

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Problem 85

Reconsider Prob. 12-84E. Using EES (or other) software, study the effects of both air and helium flowing steadily in a nozzle when there is a normal shock at a Mach number in the range $2<\mathrm{Ma}_1<3.5$. In addition to the required information, calculate the entropy change of the air and helium across the normal shock. Tabulate the results in a parametric table.

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Problem 86

Air enters a normal shock at $22.6 \mathrm{kPa}, 217 \mathrm{~K}$, and $680 \mathrm{~m} / \mathrm{s}$. Calculate the stagnation pressure and Mach number upstream of the shock, as well as pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock.

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Problem 87

Calculate the entropy change of air across the normal shock wave in Problem 12-86.

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Problem 88

Calculate and plot the entropy change of air across the normal shock for upstream Mach numbers between 0.5 and 1.5 in increments of 0.1 . Explain why normal shock waves can occur only for upstream Mach numbers greater than $\mathrm{Ma}=1$.

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Problem 89

Consider supersonic airflow approaching the nose of a two-dimensional wedge at a Mach number of 5 . Using Fig. 12-41, determine the minimum shock angle and the maximum deflection angle a straight oblique shock can have.

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Problem 90

Air flowing at $60 \mathrm{kPa}, 240 \mathrm{~K}$, and a Mach number of 3.4 impinges on a two-dimensional wedge of half-angle $12^{\circ}$. Determine the two possible oblique shock angles, $\beta_{\text {weak }}$ and $\beta_{\text {strong }}$, that could be formed by this wedge. For each case, calculate the pressure, temperature, and Mach number downstream of the oblique shock.

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Problem 91

Consider the supersonic flow of air at upstream conditions of 70 kPa and 260 K and a Mach number of 2.4 over a two-dimensional wedge of half-angle $10^{\circ}$. If the axis of the wedge is tilted $25^{\circ}$ with respect to the upstream air flow, determine the downstream Mach number, pressure, and temperature above the wedge. Answers: $3.105,23.8 \mathrm{kPa}, 191 \mathrm{~K}$

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Problem 92

Reconsider Prob. 12-91. Determine the downstream Mach number, pressure, and temperature below the wedge for a strong oblique shock for an upstream Mach number of 5.

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Problem 93

Air at $8 \mathrm{psia}, 20^{\circ} \mathrm{F}$, and a Mach number of 2.0 is forced to turn upward by a ramp that makes an $8^{\circ}$ angle off the flow direction. As a result, a weak oblique shock forms. Determine the wave angle, Mach number, pressure, and temperature after the shock.

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Problem 94

Air flowing at $P_1=40 \mathrm{kPa}, T_1=280 \mathrm{~K}$, and $\mathrm{Ma}_1$ $=3.6$ is forced to undergo an expansion turn of $15^{\circ}$. Determine the Mach number, pressure, and temperature of air after the expansion. Answers: $4.81,8.31 \mathrm{kPa}, 179 \mathrm{~K}$

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Problem 95

Air flowing at $P_1=6 \mathrm{psia}, T_1=480 \mathrm{R}$, and $\mathrm{Ma}_1$ $=2.0$ is forced to undergo a compression turn of $15^{\circ}$. Determine the Mach number, pressure, and temperature of air after the compression.

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Problem 96

What is the characteristic aspect of Rayleigh flow? What are the main assumptions associated with Rayleigh flow?

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Problem 97

On a T-s diagram of Rayleigh flow, what do the points on the Rayleigh line represent?

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Problem 98

What is the effect of heat gain and heat loss on the entropy of the fluid during Rayleigh flow?

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Problem 99

Consider subsonic Rayleigh flow of air with a Mach number of 0.92 . Heat is now transferred to the fluid and the Mach number increases to 0.95 . Will the temperature $T$ of the fluid increase, decrease, or remain constant during this process? How about the stagnation temperature $T_0$ ?

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Problem 100

What is the effect of heating the fluid on the flow velocity in subsonic Rayleigh flow? Answer the same questions for supersonic Rayleigh flow.

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Problem 101

Consider subsonic Rayleigh flow that is accelerated to sonic velocity $(\mathrm{Ma}=1)$ at the duct exit by heating. If the fluid continues to be heated, will the flow at duct exit be supersonic, subsonic, or remain sonic?

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Problem 102

Consider a 12 -cm-diameter tubular combustion chamber. Air enters the tube at $500 \mathrm{~K}, 400 \mathrm{kPa}$, and $70 \mathrm{~m} / \mathrm{s}$. Fuel with a heating value of $39,000 \mathrm{~kJ} / \mathrm{kg}$ is burned by spraying it into the air. If the exit Mach number is 0.8 , determine the rate at which the fuel is burned and the exit temperature. Assume complete combustion and disregard the increase in the mass flow rate due to the fuel mass.
Figure Can't Copy

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Problem 103

Air enters a rectangular duct at $T_1=300 \mathrm{~K}, P_1$ $=420 \mathrm{kPa}$, and $\mathrm{Ma}_1=2$. Heat is transferred to the air in the amount of $55 \mathrm{~kJ} / \mathrm{kg}$ as it flows through the duct. Disregarding frictional losses, determine the temperature and Mach number at the duct exit.
Figure Can't Copy

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Problem 104

Repeat Prob. 12-103 assuming air is cooled in the amount of $55 \mathrm{~kJ} / \mathrm{kg}$.

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Problem 105

Air is heated as it flows subsonically through a duct. When the amount of heat transfer reaches $60 \mathrm{~kJ} / \mathrm{kg}$, the flow is observed to be choked, and the velocity and the static pressure are measured to be $620 \mathrm{~m} / \mathrm{s}$ and 270 kPa . Disregarding frictional losses, determine the velocity, static temperature, and static pressure at the duct inlet.

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Problem 106

Air flows with negligible friction through a 4-indiameter duct at a rate of $5 \mathrm{lbm} / \mathrm{s}$. The temperature and pressure at the inlet are $T_1=800 \mathrm{R}$ and $P_1=30 \mathrm{psia}$, and the Mach number at the exit is $\mathrm{Ma}_2=1$. Determine the rate of heat transfer and the pressure drop for this section of the duct.

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Problem 107

Air enters a frictionless duct with $V_1=70 \mathrm{~m} / \mathrm{s}$, $T_1=600 \mathrm{~K}$, and $P_1=350 \mathrm{kPa}$. Letting the exit temperature $T_2$ vary from 600 to 5000 K , evaluate the entropy change at intervals of 200 K , and plot the Rayleigh line on a $T-s$ diagram.

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Problem 108

Air is heated as it flows through a 4 in $\times 4$ in square duct with negligible friction. At the inlet, air is at $T_1$ $=700 \mathrm{R}, P_1=80 \mathrm{psia}$, and $V_1=260 \mathrm{ft} / \mathrm{s}$. Determine the rate at which heat must be transferred to the air to choke the flow at the duct exit, and the entropy change of air during this process.

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Problem 109

Compressed air from the compressor of a gas turbine enters the combustion chamber at $T_1=550 \mathrm{~K}, P_1$ $=600 \mathrm{kPa}$, and $\mathrm{Ma}_1=0.2$ at a rate of $0.3 \mathrm{~kg} / \mathrm{s}$. Via combustion, heat is transferred to the air at a rate of $200 \mathrm{~kJ} / \mathrm{s}$ as it flows through the duct with negligible friction. Determine the Mach number at the duct exit, and the drop in stagnation pressure $P_{01}-P_{02}$ during this process.

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Problem 110

Repeat Prob. 12-109 for a heat transfer rate of $300 \mathrm{~kJ} / \mathrm{s}$.

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Problem 111

Argon gas enters a constant cross-sectional area duct at $\mathrm{Ma}_1=0.2, P_1=320 \mathrm{kPa}$, and $T_1=400 \mathrm{~K}$ at a rate of $0.8 \mathrm{~kg} / \mathrm{s}$. Disregarding frictional losses, determine the highest rate of heat transfer to the argon without reducing the mass flow rate.

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Problem 112

Consider supersonic flow of air through a $6-\mathrm{cm}-$ diameter duct with negligible friction. Air enters the duct at $\mathrm{Ma}_1=1.8, P_{01}=210 \mathrm{kPa}$, and $T_{01}=600 \mathrm{~K}$, and it is decelerated by heating. Determine the highest temperature that air can be heated by heat addition while the mass flow rate remains constant.

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Problem 113

What is the characteristic aspect of Fanno flow? What are the main assumptions associated with Fanno flow?

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Problem 114

On a $T-s$ diagram of Fanno flow, what do the points on the Fanno line represent?

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Problem 115

What is the effect of friction on the entropy of the fluid during Fanno flow?

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Problem 116

Consider subsonic Fanno flow of air with an inlet Mach number of 0.70 . If the Mach number increases to 0.90 at the duct exit as a result of friction, will the (a) stagnation temperature $T_0$, (b) stagnation pressure $P_0$, and (c) entropy $s$ of the fluid increase, decrease, or remain constant during this process?

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Problem 117

Consider supersonic Fanno flow of air with an inlet Mach number of 1.8 . If the Mach number decreases to 1.2 at the duct exit as a result of friction, will the $(a)$ stagnation temperature $T_0$, (b) stagnation pressure $P_0$, and (c) entropy $s$ of the fluid increase, decrease, or remain constant during this process?

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Problem 118

What is the effect of friction on flow velocity in subsonic Fanno flow? Answer the same question for supersonic Fanno flow.

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Problem 119

Consider subsonic Fanno flow accelerated to sonic velocity $(\mathrm{Ma}=1)$ at the duct exit as a result of frictional effects. If the duct length is increased further, will the flow at the duct exit be supersonic, subsonic, or remain sonic? Will the mass flow rate of the fluid increase, decrease, or remain constant as a result of increasing the duct length?

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Problem 120

Consider supersonic Fanno flow that is decelerated to sonic velocity $(\mathrm{Ma}=1)$ at the duct exit as a result of frictional effects. If the duct length is increased further, will the flow at the duct exit be supersonic, subsonic, or remain sonic? Will the mass flow rate of the fluid increase, decrease, or remain constant as a result of increasing the duct length?

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Problem 121

Air enters a 5 -cm-diameter adiabatic duct at $\mathrm{Ma}_1$ $=0.2, T_1=400 \mathrm{~K}$, and $P_1=200 \mathrm{kPa}$. The average friction factor for the duct is estimated to be 0.016 . If the Mach number at the duct exit is 0.8 , determine the duct length, temperature, pressure, and velocity at the duct exit.

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Problem 122

Air enters a $15-\mathrm{m}$-long, $4-\mathrm{cm}$-diameter adiabatic duct at $V_1=70 \mathrm{~m} / \mathrm{s}, T_1=500 \mathrm{~K}$, and $P_1=300 \mathrm{kPa}$. The average friction factor for the duct is estimated to be 0.023 . Determine the Mach number at the duct exit, the exit velocity, and the mass flow rate of air.

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Problem 123

Air in a room at $T_0=290 \mathrm{~K}$ and $P_0=95 \mathrm{kPa}$ is drawn steadily by a vacuum pump through a $1-\mathrm{cm}$-diameter, $50-\mathrm{cm}$-long adiabatic tube equipped with a converging nozzle at the inlet. The flow in the nozzle section can be assumed to be isentropic, and the average friction factor for the duct can be taken to be 0.018 . Determine the maximum mass flow rate of air that can be sucked through this tube and the Mach number at the tube inlet.

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Problem 124

Repeat Prob. 12-123 for a friction factor of 0.025 and a tube length of 1 m .

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Problem 125

Air enters a 5-cm-diameter, 4 -m-long adiabatic duct with inlet conditions of $\mathrm{Ma}_1=2.8, T_1=380 \mathrm{~K}$, and $P_1=80 \mathrm{kPa}$. It is observed that a normal shock occurs at a location 3 m from the inlet. Taking the average friction factor to be 0.007 , determine the velocity, temperature, and pressure at the duct exit.

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Problem 126

Helium gas with $k=1.667$ enters a $6-\mathrm{in}$-diameter duct at $\mathrm{Ma}_1=0.2, P_1=60 \mathrm{psia}$, and $T_1=600 \mathrm{R}$. For an average friction factor of 0.025 , determine the maximum duct length that will not cause the mass flow rate of helium to be reduced.

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Problem 127

Air enters a 20 -cm-diameter adiabatic duct with inlet conditions of $V_1=150 \mathrm{~m} / \mathrm{s}, T_1=500 \mathrm{~K}$, and $P_1=200$
kPa . For an average friction factor of 0.014 , determine the duct length from the inlet where the inlet velocity doubles. Also determine the pressure drop along that section of the duct.

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Problem 128

Air flows through a 6 -in-diameter, 50 - ft -long adiabatic duct with inlet conditions of $V_1=500 \mathrm{ft} / \mathrm{s}, T_{01}=650$ R , and $P_1=50$ psia. For an average friction factor of 0.02 , determine the velocity, temperature, and pressure at the exit of the duct.

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Problem 129

Consider subsonic airflow through a $10-\mathrm{cm}$ diameter adiabatic duct with inlet conditions of $T_1=330 \mathrm{~K}, P_1=180 \mathrm{kPa}$, and $\mathrm{Ma}_1=0.1$. Taking the average friction factor to be 0.02 , determine the duct length required to accelerate the flow to a Mach number of unity. Also, calculate the duct length at Mach number intervals of 0.1 , and plot the duct length against the Mach number for 0.1 $\leq \mathrm{Ma} \leq 1$. Discuss the results.

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Problem 130

Argon gas with $k=1.667, c_p=0.5203 \mathrm{~kJ} / \mathrm{kg}$ . K, and $R=0.2081 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$ enters an 8 cm -diameter adiabatic duct with $V_1=70 \mathrm{~m} / \mathrm{s}, T_1=520 \mathrm{~K}$, and $P_1=350 \mathrm{kPa}$. Taking the average friction factor to be 0.005 and letting the exit temperature $T_2$ vary from 540 K to 400 K , evaluate the entropy change at intervals of 10 K , and plot the Fanno line on a T-s diagram.

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Problem 131

Air in an automobile tire is maintained at a pressure of 220 kPa (gage) in an environment where the atmospheric pressure is 94 kPa . The air in the tire is at the ambient temperature of $25^{\circ} \mathrm{C}$. Now a 4 -mm-diameter leak develops in the tire as a result of an accident. Assuming isentropic flow, determine the initial mass flow rate of air through the leak.

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Problem 132

The thrust developed by the engine of a Boeing 777 is about 380 kN . Assuming choked flow in the nozzles, determine the mass flow rate of air through the nozzle. Take the ambient conditions to be 295 K and 95 kPa .

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Problem 133

A stationary temperature probe inserted into a duct where air is flowing at $250 \mathrm{~m} / \mathrm{s}$ reads $85^{\circ} \mathrm{C}$. What is the actual temperature of air?

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Problem 134

Nitrogen enters a steady-flow heat exchanger at $150 \mathrm{kPa}, 10^{\circ} \mathrm{C}$, and $100 \mathrm{~m} / \mathrm{s}$, and it receives heat in the amount of $150 \mathrm{~kJ} / \mathrm{kg}$ as it flows through it. The nitrogen leaves the heat exchanger at 100 kPa with a velocity of $200 \mathrm{~m} / \mathrm{s}$. Determine the stagnation pressure and temperature of the nitrogen at the inlet and exit states.

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Problem 135

Derive an expression for the speed of sound based on van der Waals' equation of state $P=R T(v-b)-a / v^2$. Using this relation, determine the speed of sound in carbon dioxide at $50^{\circ} \mathrm{C}$ and 200 kPa , and compare your result to that obtained by assuming ideal-gas behavior. The van der Waals constants for carbon dioxide are $a=364.3 \mathrm{kPa} \cdot \mathrm{m}^6 / \mathrm{kmol}^2$ and $b=0.0427 \mathrm{~m}^3 / \mathrm{kmol}$.

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Problem 136

Obtain Eq. 12-10 by starting with Eq. 12-9 and using the cyclic rule and the thermodynamic property relations $\frac{c_p}{T}=\left(\frac{\partial s}{\partial T}\right)_P$ and $\frac{c_V}{T}=\left(\frac{\partial s}{\partial T}\right)_V$.

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Problem 137

For ideal gases undergoing isentropic flows, obtain expressions for $P / P^*, T / T^*$, and $\rho / \rho^*$ as functions of $k$ and Ma.

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Problem 138

Using Eqs. 12-4, 12-13, and 12-14, verify that for the steady flow of ideal gases $d T_0 / T=d A / A$ $+\left(1-\mathrm{Ma}^2\right) d V / V$. Explain the effect of heating and area changes on the velocity of an ideal gas in steady flow for (a) subsonic flow and (b) supersonic flow.

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Problem 139

A subsonic airplane is flying at a $3000-\mathrm{m}$ altitude where the atmospheric conditions are 70.109 kPa and 268.65 K . A Pitot static probe measures the difference between the static and stagnation pressures to be 22 kPa . Calculate the speed of the airplane and the flight Mach number.

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Problem 140

Plot the mass flow parameter $\dot{m} \sqrt{R T_0} /\left(A P_0\right)$ versus the Mach number for $k=1.2,1.4$, and 1.6 in the range of $0 \leq \mathrm{Ma} \leq 1$.

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Problem 141

Helium enters a nozzle at $0.8 \mathrm{MPa}, 500 \mathrm{~K}$, and a velocity of $120 \mathrm{~m} / \mathrm{s}$. Assuming isentropic flow, determine the pressure and temperature of helium at a location where the velocity equals the speed of sound. What is the ratio of the area at this location to the entrance area?

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Problem 142

Repeat Problem 12-141 assuming the entrance velocity is negligible.

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Problem 143

Air at 0.9 MPa and 400 K enters a converging nozzle with a velocity of $180 \mathrm{~m} / \mathrm{s}$. The throat area is $10 \mathrm{~cm}^2$. Assuming isentropic flow, calculate and plot the mass flow rate through the nozzle, the exit velocity, the exit Mach number, and the exit pressure-stagnation pressure ratio versus the back pressure-stagnation pressure ratio for a back pressure range of $0.9 \geq P_b \geq 0.1 \mathrm{MPa}$.

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Problem 144

Steam at 6.0 MPa and 700 K enters a converging nozzle with a negligible velocity. The nozzle throat area is $8 \mathrm{~cm}^2$. Assuming isentropic flow, plot the exit pressure, the exit velocity, and the mass flow rate through the nozzle versus the back pressure $P_b$ for $6.0 \geq P_b$ $\geq 3.0 \mathrm{MPa}$. Treat the steam as an ideal gas with $k=1.3, c_p$ $=1.872 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, and $R=0.462 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$.

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Problem 145

Find the expression for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of $k$ and the Mach number upstream of the shock wave $\mathrm{Ma}_1$.

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Problem 146

Nitrogen enters a converging-diverging nozzle at 700 kPa and 300 K with a negligible velocity, and it experiences a normal shock at a location where the Mach number is $\mathrm{Ma}=3.0$. Calculate the pressure, temperature, velocity,

Mach number, and stagnation pressure downstream of the shock. Compare these results to those of air undergoing a normal shock at the same conditions.

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Problem 147

An aircraft flies with a Mach number $\mathrm{Ma}_1=0.8$ at an altitude of 7000 m where the pressure is 41.1 kPa and the temperature is 242.7 K . The diffuser at the engine inlet has an exit Mach number of $\mathrm{Ma}_2=0.3$. For a mass flow rate of $65 \mathrm{~kg} / \mathrm{s}$, determine the static pressure rise across the diffuser and the exit area.

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Problem 148

Helium expands in a nozzle from $1 \mathrm{MPa}, 500 \mathrm{~K}$, and negligible velocity to 0.1 MPa . Calculate the throat and exit areas for a mass flow rate of $0.25 \mathrm{~kg} / \mathrm{s}$, assuming the nozzle is isentropic. Why must this nozzle be convergingdiverging?

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Problem 149

Helium expands in a nozzle from $150 \mathrm{psia}, 900 \mathrm{R}$, and negligible velocity to 15 psia. Calculate the throat and exit areas for a mass flow rate of $0.2 \mathrm{lbm} / \mathrm{s}$, assuming the nozzle is isentropic. Why must this nozzle be convergingdiverging?

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Problem 150

Using the EES software and the relations in Table A-13, calculate the one-dimensional compressible flow functions for an ideal gas with $k=1.667$, and present your results by duplicating Table A-13.

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Problem 151

Using the EES software and the relations in Table A-14, calculate the one-dimensional normal shock functions for an ideal gas with $k=1.667$, and present your results by duplicating Table A-14.

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Problem 152

Consider an equimolar mixture of oxygen and nitrogen. Determine the critical temperature, pressure, and density for stagnation temperature and pressure of 800 K and 500 kPa .

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Problem 153

Using EES (or other) software, determine the shape of a converging-diverging nozzle for air for a mass flow rate of $3 \mathrm{~kg} / \mathrm{s}$ and inlet stagnation conditions of 1400 kPa and $200^{\circ} \mathrm{C}$. Assume the flow is isentropic. Repeat the calculations for $50-\mathrm{kPa}$ increments of pressure drops to an exit pressure of 100 kPa . Plot the nozzle to scale. Also, calculate and plot the Mach number along the nozzle.

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Problem 154

Using EES (or other) software and the relations given in Table A-13, calculate the onedimensional isentropic compressible-flow functions by varying the upstream Mach number from 1 to 10 in increments of 0.5 for air with $k=1.4$.

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Problem 155

Repeat Prob. 12-154 for methane with

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Problem 156

Using EES (or other) software and the relations given in Table A-14, generate the one-dimensional normal shock functions by varying the upstream Mach number from 1 to 10 in increments of 0.5 for air with $k=1.4$.

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Problem 157

Repeat Prob. 12-156 for methane with $k=1.3$.

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Problem 158

Air in a room at $T_0=300 \mathrm{~K}$ and $P_0=100 \mathrm{kPa}$ is to be drawn by a vacuum pump through a $3-\mathrm{cm}$-diameter, 2 m -long adiabatic tube equipped with a converging nozzle at the inlet. The flow in the nozzle section can be assumed to be isentropic. The static pressure is measured to be 97 kPa at the tube inlet and 55 kPa at the tube exit. Determine the mass flow rate of air through the duct, the air velocity at the duct exit, and the average friction factor for the duct.

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Problem 159

Air enters a 4-cm-diameter adiabatic duct with inlet conditions of $\mathrm{Ma}_1=2.2, T_1=250 \mathrm{~K}$, and $P_1=80 \mathrm{kPa}$, and exits at a Mach number of $\mathrm{Ma}_2=1.8$. Taking the average friction factor to be 0.03 , determine the velocity, temperature, and pressure at the exit.

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Problem 160

Air is cooled as it flows through a 20 -cm-diameter duct. The inlet conditions are $\mathrm{Ma}_1=1.2, T_{01}=350 \mathrm{~K}$, and $P_{01}$ $=240 \mathrm{kPa}$ and the exit Mach number is $\mathrm{Ma}_2=2.0$. Disregarding frictional effects, determine the rate of cooling of air.

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Problem 161

Air is heated as it flows subsonically through a $10 \mathrm{~cm} \times 10 \mathrm{~cm}$ square duct. The properties of air at the inlet are maintained at $\mathrm{Ma}_1=0.4, P_1=400 \mathrm{kPa}$, and $T_1=360 \mathrm{~K}$ at all times. Disregarding frictional losses, determine the highest rate of heat transfer to the air in the duct without affecting the inlet conditions. Answer: 1958 kW

Figure Can't Copy

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Problem 162

Repeat Prob. 12-161 for helium.

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Problem 163

Air is accelerated as it is heated in a duct with negligible friction. Air enters at $V_1=100 \mathrm{~m} / \mathrm{s}, T_1=400 \mathrm{~K}$, and $P_1=35 \mathrm{kPa}$ and the exits at a Mach number of $\mathrm{Ma}_2=0.8$. Determine the heat transfer to the air, in $\mathrm{kJ} / \mathrm{kg}$. Also determine the maximum amount of heat transfer without reducing the mass flow rate of air.

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Problem 164

Air at sonic conditions and at static temperature and pressure of 500 K and 420 kPa , respectively, is to be accelerated to a Mach number of 1.6 by cooling it as it flows through a channel with constant cross-sectional area. Disregarding frictional effects, determine the required heat transfer from the air, in kJ/kg. Answer: $69.8 \mathrm{~kJ} / \mathrm{kg}$

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Problem 165

Combustion gases with an average specific heat ratio of $k=1.33$ and a gas constant of $R=0.280 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$
enter a $10-\mathrm{cm}$-diameter adiabatic duct with inlet conditions of $\mathrm{Ma}_1=2, T_1=510 \mathrm{~K}$, and $P_1=180 \mathrm{kPa}$. If a normal shock occurs at a location 2 m from the inlet, determine the velocity, temperature, and pressure at the duct exit. Take the average friction factor of the duct to be 0.010 .

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Problem 166

Ces Consider supersonic airflow through a $18-\mathrm{cm}-$ diameter adiabatic duct with inlet conditions of $T_1=530 \mathrm{~K}, P_1=80 \mathrm{kPa}$, and $\mathrm{Ma}_1=3$. Taking the average friction factor to be 0.03 , determine the duct length required to decelerate the flow to a Mach number of unity. Also, calculate the duct length at Mach number intervals of 0.25 , and plot the duct length against the Mach number for 1 $\leq \mathrm{Ma} \leq 3$. Discuss the results.

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Problem 167

Air is flowing through a 6-cm-diameter adiabatic duct with inlet conditions of $V_1$ $=120 \mathrm{~m} / \mathrm{s}, T_1=400 \mathrm{~K}$, and $P_1=100 \mathrm{kPa}$ and an exit Mach number of $\mathrm{Ma}_2=1$. To study the effect of duct length on the mass flow rate and the inlet velocity, the duct is now extended until its length is doubled while $P_1$ and $T_1$ are held constant. Taking the average friction factor to be 0.02 , calculate the mass flow rate, and the inlet velocity, for various extension lengths, and plot them against the extension length. Discuss the results.

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Problem 168

In compressible flow, velocity measurements with a Pitot probe can be grossly in error if relations developed for incompressible flow are used. Therefore, it is essential that compressible flow relations be used when evaluating flow velocity from Pitot probe measurements. Consider supersonic flow of air through a channel. A probe inserted into the flow causes a shock wave to occur upstream of the probe, and it measures the stagnation pressure and temperature to be 620 kPa and 340 K , respectively. If the static pressure upstream is 110 kPa , determine the flow velocity.

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Problem 169

Find out if there is a supersonic wind tunnel on your campus. If there is, obtain the dimensions of the wind tunnel and the temperatures and pressures as well as the Mach number at several locations during operation. For what typical experiments is the wind tunnel used?

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Problem 170

Assuming you have a thermometer and a device to measure the speed of sound in a gas, explain how you can determine the mole fraction of helium in a mixture of helium gas and air.

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Problem 171

Design a 1 -m-long cylindrical wind tunnel whose diameter is 25 cm operating at a Mach number of 1.8. Atmospheric air enters the wind tunnel through a convergingdiverging nozzle where it is accelerated to supersonic velocities. Air leaves the tunnel through a converging-diverging diffuser where it is decelerated to a very low velocity before entering the fan section. Disregard any irreversibilities. Spec-
ify the temperatures and pressures at several locations as well as the mass flow rate of air at steady-flow conditions. Why is it often necessary to dehumidify the air before it enters the wind tunnel?

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