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Fluid Mechanics

Frank M. White

Chapter 9

Compressible Flow - all with Video Answers

Educators

Chapter Questions

03:37

Problem 1

An ideal gas flows adiabatically through a duct. At section $1, p_{1}=140 \mathrm{kPa}, T_{1}=260^{\circ} \mathrm{C},$ and $V_{1}=75 \mathrm{m} / \mathrm{s},$ Farther downstream, $p_{2}=30 \mathrm{kPa}$ and $T_{2}=207^{\circ} \mathrm{C}$. Calculate $V_{2}$ in $\mathrm{m} / \mathrm{s}$ and $s_{2}-s_{1}$ in $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ if the gas is $(a)$ air, $k=1.4$ and $(b)$ argon, $k=1.67$

Chai Santi
Chai Santi
Numerade Educator
01:03

Problem 2

Solve Prob. P9.1 if the gas is steam. Use two approaches:
(a) an ideal gas from Table $A .4$ and $(b)$ real gas data from the steam tables [15]

Hast Aggarwal
Hast Aggarwal
Numerade Educator
07:41

Problem 3

If $8 \mathrm{kg}$ of oxygen in a closed tank at $200^{\circ} \mathrm{C}$ and $300 \mathrm{kPa}$ is heated until the pressure rises to $400 \mathrm{kPa}$, calculate
$(a)$ the new temperature, ( $b$ ) the total heat transfer, and $(c)$ the change in entropy.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
04:03

Problem 4

Consider steady adiabatic airflow in a duct. At section B, the pressure is 600 kPa and the temperature is $177^{\circ} \mathrm{C}$. At section $\mathrm{D}$, the density is $1.13 \mathrm{kg} / \mathrm{m}^{3}$ and the temperature is $156^{\circ} \mathrm{C} .
(a)$ Find the entropy change, if any.
(b) Which way is the air flowing?

Chai Santi
Chai Santi
Numerade Educator
06:55

Problem 5

Steam enters a nozzle at $377^{\circ} \mathrm{C}, 1.6 \mathrm{MPa}$, and a steady speed of $200 \mathrm{m} / \mathrm{s}$ and accelerates isentropically until it exits at saturation conditions. Estimate the exit velocity and temperature,

Dading Chen
Dading Chen
Numerade Educator
00:51

Problem 6

Methane, approximated as a perfect gas, is compressed adiabatically from $101 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $300 \mathrm{kPa}$. Estimate
$(a)$ the final temperature, and $(b)$ the final density.

Mayukh Banik
Mayukh Banik
Numerade Educator
04:15

Problem 7

Air flows through a variable-area duct. At section $1, A_{1}=$ $20 \mathrm{cm}^{2}, p_{1}=300 \mathrm{kP}_{\mathrm{a}}, \rho_{1}=1.75 \mathrm{kg} / \mathrm{m}^{3},$ and $V_{1}=122.5 \mathrm{m} / \mathrm{s}$ At section $2,$ the area is exactly the same, but the density is much lower: $\rho_{2}=0.266 \mathrm{kg} / \mathrm{m}^{3}$ and $T_{2}=281 \mathrm{K}$. There is no transfer of work or heat. Assume one-dimensional steady flow.
(a) How can you reconcile these differences?
(b) Find the mass flow at section 2. Calculate
(c) $V_{2},(d) p_{2},$ and
$(c) s_{2}-s_{1}$
[Hint. This problem requires the continuity equation.

Chai Santi
Chai Santi
Numerade Educator
06:34

Problem 8

Atmospheric air at $20^{\circ} \mathrm{C}$ enters and fills an insulated tank that is initially evacuated, Using a control volume analysis from Eq. $(3.67),$ compute the tank air temperature when it is full.

Gordon Ayadju
Gordon Ayadju
Numerade Educator
03:07

Problem 9

Liquid hydrogen and oxygen are burned in a combustion chamber and fed through a rocket nozzle that exhausts at $V_{\text {exit }}=1600 \mathrm{m} / \mathrm{s}$ to an ambient pressure of $54 \mathrm{kPa}$. The nozzle exit diameter is $45 \mathrm{cm},$ and the jet exit density is $0.15 \mathrm{kg} / \mathrm{m}^{3}$. If the exhaust gas has a molecular weight of 18 estimate ( $a$ ) the exit gas temperature, ( $b$ ) the mass flow, and (c) the thrust developed by the rocket.

Anand Jangid
Anand Jangid
Numerade Educator
01:49

Problem 10

A certain aircraft flies at $609 \mathrm{mi} / \mathrm{h}$ at standard sea level.
(a) What is its Mach number?
(b) If it flies at the same Mach number at $34,000 \mathrm{ft}$ altitude, how much slower for faster) is it flying, in $\mathrm{mi} / \mathrm{h} ?$

Narayan Hari
Narayan Hari
Numerade Educator
03:58

Problem 11

At $300^{\circ} \mathrm{C}$ and 1 atm, estimate the speed of sound of
$(a)$ nitrogen,
(b) hydrogen,
(c) helium,
$(d)$ steam, and
$(e)^{23 x} \mathrm{UF}_{6}\left(k^{\circ}-1,06\right)$

Vipender Yadav
Vipender Yadav
Numerade Educator
03:06

Problem 12

Assume that water follows Eq. (1.19) with $n \approx 7$ and $B$ ea $3000 .$ Compute the bulk modulus (in $\mathrm{kPa}$ ) and the speed of sound (in $\mathrm{m} / \mathrm{s}$ ) at $(a) 1$ atm and $(b) 1100$ atm (the deepest part of the ocean). (c) Compute the speed of sound at $20^{\circ} \mathrm{C}$ and 9000 atm and compare with the measured value of $2650 \mathrm{m} / \mathrm{s}(\mathrm{A} . \mathrm{H} . \mathrm{Smith} \text { and } \mathrm{A} . \mathrm{W} . \text { Lawson, } J .$ Chem. Phys. vol. $22,1954,$ p. 351 ).

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
00:49

Problem 13

Consider steam at $500 \mathrm{K}$ and $200 \mathrm{kPa}$. Estimate its speed of sound by two different methods: $(a)$ assuming an ideal gas from Table $\mathbf{B} .4,$ or $(b)$ using finite differences for isentropic densities between $210 \mathrm{kPa}$ and $190 \mathrm{kPa}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:46

Problem 14

Benzene, listed in Table A.3, has a measured density of $57.75 \mathrm{lbm} / \mathrm{ft}^{3}$ at a pressure of 700 bar. Use this data to estimate the speed of sound of benzene.

Emily Himsel
Emily Himsel
Numerade Educator
00:42

Problem 15

The pressure-density relation for ethanol is approximated by Eq- (1.19) with $B=1600$ and $n=7 .$ Use this relation to estimate the speed of sound of ethanol at 2000 atmospheres.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:21

Problem 16

A weak pressure pulse $\Delta p$ propagates through still air. Discuss the type of reflected pulse that occurs and the boundary conditions that must be satisfied when the wave strikes normal to, and is reflected from, (a) a solid wall and $(b)$ a free liquid surface.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
01:47

Problem 17

A submarine at a depth of $800 \mathrm{m}$ sends a sonar signal and receives the reflected wave back from a similar submerged object in 15 s. Using Prob. $\mathrm{P} 9.12$ as a guide, estimate the distance to the other object.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:27

Problem 18

Race cars at the Indianapolis Speedway average speeds of $185 \mathrm{mi} / \mathrm{h} .$ After determining the altitude of Indianapolis, find the Mach number of these cars and estimate whether compressibility might affect their aerodynamics.

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
View

Problem 19

In $1976,$ the $\mathrm{SR}-71 \mathrm{A},$ flying at $20 \mathrm{km}$ standard altitude, set a jet-powered aircraft speed record of $3326 \mathrm{km} / \mathrm{h}$. Estimate the temperature, in $^{\circ} \mathrm{C}$, at its front stagnation point. At what Mach number would it have a front stagnation-point temperature of $500^{\circ} \mathrm{C} ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:05

Problem 20

Air flows isentropically in a channel. Properties at section 1 are $V_{1}-250 \mathrm{m} / \mathrm{s}, T_{1}-330 \mathrm{K},$ and $p_{1}-80 \mathrm{kPa} . \mathrm{At}$ section 2 downstream, the temperature has dropped to $0^{\circ} \mathrm{C}$ Find $(a)$ the pressure, (b) velocity, and (c) Mach number at section 2.

Chai Santi
Chai Santi
Numerade Educator
01:43

Problem 21

$\mathrm{N}_{2} \mathrm{O}$ expands isentropically through a duct from $p_{1}=$ $200 \mathrm{kPa}$ and $T_{1}=250^{\circ} \mathrm{C}$ to a downstream section where $p_{2}=26 \mathrm{kPa}$ and $V_{2}=594 \mathrm{m} / \mathrm{s}$. Compute $(a) T_{2}:(b) \mathrm{Ma}_{2}$ $(c) T_{0} ;(d) p_{0} ;(e) V_{1} ;$ and $(f) \mathrm{Ma}_{1}$.

Chai Santi
Chai Santi
Numerade Educator
02:11

Problem 22

Given the pitot stagnation temperature and pressure and the static pressure measurements in Fig. $\mathrm{P} 9.22,$ estimate the air velocity $V,$ assuming $(a)$ incompressible flow and $(b)$ compressible flow.

Chai Santi
Chai Santi
Numerade Educator
04:44

Problem 23

A gas, assumed ideal, flows isentropically from point 1 where the velocity is negligible, the pressure is $200 \mathrm{kPa}$ and the temperature is $300^{\circ} \mathrm{C},$ to point $2,$ where the pressure is $40 \mathrm{kPa}$. What is the Mach number $\mathrm{Ma}_{2}$ if the gas is $(a)$ air, (b) argon, or $(c) \mathrm{CH}_{4} ?(d)$ Can you tell, without calculating, which gas will be the coldest at point $2 ?$

Shalini Tyagi
Shalini Tyagi
Numerade Educator
02:16

Problem 24

For low-speed (nearly incompressible) gas flow, the stagnation pressure can be computed from Bernoulli's equation:
\[
p_{0}=p+\frac{1}{2} \rho V^{2}
\]
(a) For higher subsonic speeds, show that the isentropic relation $(9.28 a)$ can be expanded in a power series as follows:
\[
p_{0}^{\prime}=p+\frac{1}{2} \rho V^{2}\left(1+\frac{1}{4} M a^{2}+\frac{2-k}{24} M a^{4}+\dots\right)
\]
(b) Suppose that a pitot-static tube in air measures the pressure difference $p_{0}-p$ and uses the Bernoulli relation, with stagnation density, to estimate the gas velocity. At what Mach number will the error be 4 percent?

Narayan Hari
Narayan Hari
Numerade Educator
02:34

Problem 25

If it is known that the air velocity in the duct is $750 \mathrm{ft} / \mathrm{s}$, use the mercury manometer measurement in Fig. P9.25 to estimate the static pressure in the duct in 1 bf $/$ in $^{2}$ absolute.

Narayan Hari
Narayan Hari
Numerade Educator
32:10

Problem 26

Show that for isentropic flow of a perfect gas if a pitotstatic probe measures $p_{0}, p,$ and $T_{0},$ the gas velocity can be calculated from
\[
V^{2}=2 c_{p} T_{0}\left[1-\left(\frac{p}{p_{0}}\right)^{(2-1) N}\right]
\]
What would be a source of error if a shock wave were formed in front of the probe?

Shalini Tyagi
Shalini Tyagi
Numerade Educator
View

Problem 27

A pitot tube, mounted on an airplane flying at $8000 \mathrm{m}$ standard altitude, reads a stagnation pressure of 57 kPa. Estimate the plane's (a) velocity and (b) Mach number.

Victor Salazar
Victor Salazar
Numerade Educator
03:10

Problem 28

Air flows isentropically through a duct. At section $1,$ the pressure and temperature are $250 \mathrm{kPa}$ and $125^{\circ} \mathrm{C},$ and the velocity is $200 \mathrm{m} / \mathrm{s}$. At section 2 , the area is $0.25 \mathrm{m}^{2}$ and the Mach number is $2.0 .$ Determine
$(a) \mathrm{Ma}_{1} ;(b) T_{2} ;(c) V_{2}$ and $(d)$ the mass flow.

Narayan Hari
Narayan Hari
Numerade Educator
07:11

Problem 29

Steam from a large tank, where $T=400^{\circ} \mathrm{C}$ and $p=1 \mathrm{MPa}$ expands isentropically through a nozzle until, at a section of $2-\mathrm{cm}$ diameter, the pressure is $500 \mathrm{kPa}$. Using the steam tables $[15],$ estimate
$(a)$ the temperature, $(b)$ the velocity, and $(c)$ the mass flow at this section. Is the flow subsonic?

Dading Chen
Dading Chen
Numerade Educator
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Problem 30

When does the incompressible-flow assumption begin to fail for pressures? Construct a graph of $p$ b for incompressible flow of a perfect gas as compared to Eq. $(9.28 a)$ Plot both versus Mach number for $0 \leq \mathrm{Ma} \leq 0.6$ and decide for yourself where the deviation is too great.

Victor Salazar
Victor Salazar
Numerade Educator
01:31

Problem 31

Air flows adiabatically through a duct, At one section $V_{1}=$ $400 \mathrm{ft} / \mathrm{s}, T_{1}=200^{\circ} \mathrm{F},$ and $p_{1}=35 \mathrm{lbf} / \mathrm{in}^{2}$ absolute, while
farther downstream $V_{2}=1100 \mathrm{ft} / \mathrm{s}$ and $p_{2}=18 \mathrm{lbf} / \mathrm{in}^{2}$ absolute. Compute $(a) \mathrm{Ma}_{2},(b) U_{\max },$ and $(c) p_{u} / p_{01}$.

Dominador Tan
Dominador Tan
Numerade Educator
01:42

Problem 32

The large compressed-air tank in Fig. $\mathrm{P} 9.32$ exhausts from a nozzle at an exit velocity of $235 \mathrm{m} / \mathrm{s}$. The mercury manometer reads $h=30 \mathrm{cm}$. Assuming isentropic flow, compute the pressure $(a)$ in the tank and $(b)$ in the atmosphere. (c) What is the exit Mach number?

Chai Santi
Chai Santi
Numerade Educator
02:05

Problem 33

Air flows isentropically from a reservoir, where $p=$ $300 \mathrm{kPa}$ and $T=500 \mathrm{K},$ to section 1 in a duct, where $A_{1}=$ $0.2 \mathrm{m}^{2}$ and $V_{1}=550 \mathrm{m} / \mathrm{s}$. Compute $(a) \mathrm{Ma}_{1},(b) T_{1},(c) p_{1}$ $(d)$ in and $(e) A^{*}$. Is the flow choked?

Chai Santi
Chai Santi
Numerade Educator
07:23

Problem 34

Air in a large tank, at $300^{\circ} \mathrm{C}$ and $400 \mathrm{kPa}$, flows through a converging-diverging nozzle with throat diameter $2 \mathrm{cm} .$ It exits smoothly at a Mach number of 2.8 . According to onedimensional isentropic theory, what is $(a)$ the exit diameter, and ( $b$ ) the mass flow?

Dading Chen
Dading Chen
Numerade Educator
02:33

Problem 35

Helium, at $T_{0}=400 \mathrm{K}$, enters a nozzle isentropically. At section $1,$ where $A_{1}=0.1 \mathrm{m}^{2},$ a pitot-static arrangement (see Fig. $\mathrm{P} 9.25$ ) measures stagnation pressure of $150 \mathrm{kPa}$ and static pressure of 123 kPa. Estimate
$(a) \mathrm{Ma}_{1},(b)$ mass flow $\dot{m}$ $(c) T_{1},$ and $(d) A^{*}$

Anand Jangid
Anand Jangid
Numerade Educator
02:25

Problem 36

An air tank of volume $1.5 \mathrm{m}^{3}$ is initially at $800 \mathrm{kPa}$ and $20^{\circ} \mathrm{C} .$ At $t=0,$ it begins exhausting through a converging nozzle to sea-level conditions. The throat area is $0.75 \mathrm{cm}^{2}$ Estimate
(a) the initial mass flow in kg/s, (b) the time required to blow down to $500 \mathrm{kPa}$, and (c) the time at which the nozzle ceases being choked.

Chai Santi
Chai Santi
Numerade Educator
01:09

Problem 37

Make an exact control volume analysis of the blowdown process in Fig $P 9.37,$ assuming an insulated tank with negligible kinctic and potential energy within. Assume critical flow at the exit, and show that both $p_{0}$ and $T_{0}$ decrease during blowdown. Set up first-order differential equations for $p_{d}(t)$ and $T_{0}(t),$ and reduce and solve as far as you can.

Manik Pulyani
Manik Pulyani
Numerade Educator
11:29

Problem 38

Prob. $\mathrm{P} 9.37$ makes an ideal senior project or combined laboratory and computer problem, as described in Ref. $27,$ Sec. $8.6 .$ In Bober and Kenyon's lab experiment, the tank had a volume of $0.0352 \mathrm{ft}^{3}$ and was initially filled with air at $50 \mathrm{lb} / \mathrm{in}^{2}$ gage and $72^{\circ} \mathrm{F}$. Atmospheric pressure was 14.5 Ib/in absolute, and the nozzle exit diameter was 0.05 in. After 2 s of blowdown, the measured tank pressure was $20 \mathrm{lb} / \mathrm{in}^{2}$ gage and the tank temperature was $-5^{\circ} \mathrm{F}$. Compare these values with the theoretical analysis of Prob. P9.37.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
12:56

Problem 39

Consider isentropic flow in a channel of varying area, from section 1 to section $2 .$ We know that $\mathrm{Ma}_{1}=2.0$ and desire that the velocity ratio $V_{2} / V_{1}$ be $1.2 .$ Estimate
$(a) \mathrm{Ma}_{2}$ and (b) $A_{2} / A_{1}$. (c) Sketch what this channel looks like. For example, does it converge or diverge? Is there a throat?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:20

Problem 40

Steam, in a tank at $300 \mathrm{kPa}$ and $600 \mathrm{K}$, discharges isentropically to a low-pressure atmosphere through a converging nozzle with exit area $5 \mathrm{cm}^{2}$
(a) Using an ideal gas approximation from Table $B .4,$ estimate the mass flow.
(b) Without actual calculations, indicate how you would use real properties of steam to find the mass flow.

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
04:08

Problem 41

Air, with a stagnation pressure of $100 \mathrm{kPa}$, flows through the nozzle in Fig. $P 9.41$, which is $2 \mathrm{m}$ long and has an area variation approximated by
\[
A \approx 20-20 x+10 x^{2}
\]
with $A$ in $\mathrm{cm}^{2}$ and $x$ in $\mathrm{m}$. It is desired to plot the complete family of isentropic pressures $p(x)$ in this nozzle, for the range of inlet pressures $1<p(0)<100 \mathrm{kPa}$. Indicate which inlet pressures are not physically possible and discuss briefly. If your computer has an online graphics routine, plot at least 15 pressure profiles; otherwise just hit the highlights and explain.

Chai Santi
Chai Santi
Numerade Educator
06:31

Problem 42

A bicycle tire is filled with air at an absolute pressure of $169.12 \mathrm{kPa}$, and the temperature inside is $30.0^{\circ} \mathrm{C}$. Suppose the valve breaks, and air starts to exhaust out of the tire into the atmosphere $\left(p_{e}=100 \mathrm{kPa} \text { absolute and } T_{a}=\right.$ $20.0^{\circ} \mathrm{C}$ ). The valve exit is $2.00 \mathrm{mm}$ in diameter and is the smallest cross-sectional area of the entire system. Frictional losses can be ignored here; one-dimensional isentropic flow is a reasonable assumption.
(a) Find the Mach number, velocity, and temperature at the exit plane of the valve (initially).
(b) Find the initial mass flow rate out of the tire.
(c) Estimate the velocity at the exit plane using the incompressible Bernoulli equation. How well does this estimate agree with the "exact" answer of part (a)? Explain.

Chai Santi
Chai Santi
Numerade Educator
03:10

Problem 43

Air flows isentropically through a variable-area duct. At section $1, A_{1}=20 \mathrm{cm}^{2}, p_{1}=300 \mathrm{kPa}, \rho_{1}=1.75 \mathrm{kg} / \mathrm{m}^{3},$ and
$\mathrm{Ma}_{1}=0.25,$ At section $2,$ the area is exactly the same, but the flow is much faster, Compute (b) $(b) \operatorname{Ma}_{2},(c) T_{2}$ $(a) V_{2}$ and $(d)$ the mass flow. (e) Is there a sonic throat between sections 1 and $2 ?$ If $s 0,$ find its area.

Narayan Hari
Narayan Hari
Numerade Educator
03:17

Problem 44

In Prob, $P 3.34$ we knew nothing about compressible flow at the time, so we merely assumed exit conditions $p_{2}$ and $T_{2}$ and computed $V_{2}$ as an application of the continuity equation. Suppose that the throat diameter is 3 in. For the given stagnation conditions in the rocket chamber in Fig. $\mathrm{P} 3.34$ and assuming $k=1.4$ and a molecular weight of $26,$ compute the actual exit velocity, pressure, and temperature according to one-dimensional theory. If $p_{a}=14.7 \mathrm{lbf} / \mathrm{in}^{2}$ absolute, compute the thrust from the analysis of Prob. P3.68. This thrust is entirely independent of the stagnation temperature (check this by changing $T_{0}$ to $2000^{\circ} \mathrm{R}$ if you like). Why?

James Kiss
James Kiss
Numerade Educator
12:21

Problem 45

It is desired to have an isentropic airflow achieve a velocity of $550 \mathrm{m} / \mathrm{s}$ at a $6-\mathrm{cm}$ -diameter section where the pressure is $87 \mathrm{kPa}$ and the density $1.3 \mathrm{kg} / \mathrm{m}^{3} .(a)$ Is a sonic throat needed? $(b)$ If $s 0,$ estimate its diameter, and compute $(c)$ the stagnation temperature and $(d)$ the mass flow.

Rachel B.
Rachel B.
Numerade Educator
04:03

Problem 46

A one-dimensional isentropic airflow has the following properties at one section where the area is $53 \mathrm{cm}^{2}$ : $p=$ $12 \mathrm{kPa}, \rho=0.182 \mathrm{kg} / \mathrm{m}^{3},$ and $V=760 \mathrm{m} / \mathrm{s}$. Determine (a) the throat area, ( $b$ ) the stagnation temperature, and
$(c)$ the mass flow.

Chai Santi
Chai Santi
Numerade Educator
03:34

Problem 47

In wind tunnel testing near Mach $1,$ a small area decrease caused by model blockage can be important. Suppose the test section area is $1 \mathrm{m}^{2}$, with unblocked test conditions $\mathrm{Ma}=$ 1.10 and $T=20^{\circ} \mathrm{C}$. What model area will first cause the test section to choke? If the model cross section is $0.004 \mathrm{m}^{2}$ $(0.4 \text { percent blockage }),$ what percentage change in test section velocity results?

Chai Santi
Chai Santi
Numerade Educator
00:53

Problem 48

A force $F=1100 \mathrm{N}$ pushes a piston of diameter $12 \mathrm{cm}$ through an insulated cylinder containing air at $20^{\circ} \mathrm{C},$ as in Fig. $\mathrm{P} 9.48 .$ The exit diameter is $3 \mathrm{mm},$ and $p_{a}=1 \mathrm{atm}$ Estimate $(a) V_{e},(b) V_{p-}$ and $(c) \dot{m}_{e}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
09:26

Problem 49

Consider the venturi nozzle of Fig. $6.40 c_{-}$ with $D=5 \mathrm{cm}$ and $d=3 \mathrm{cm} .$ Stagnation temperature is $300 \mathrm{K},$ and the upstream velocity $V_{1}=72 \mathrm{m} / \mathrm{s}$. If the throat pressure is $124 \mathrm{kPa},$ estimate, with isentropic flow theory,
$(a) p_{1}$ $(b) \mathrm{Ma}_{2},$ and $(c)$ the mass flow.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:42

Problem 50

Methane is stored in a tank at $120 \mathrm{kPa}$ and $330 \mathrm{K}$. It discharges to a second tank through a converging nozzle whose exit area is $5 \mathrm{cm}^{2}$. What is the initial mass flow rate if the second tank has a pressure of
(a) $70 \mathrm{kPa}$ or
$(b) 40 \mathrm{kPa} ?$

Chai Santi
Chai Santi
Numerade Educator
02:20

Problem 51

The scramjet engine is supersonic throughout. A sketch is shown in Fig. $\mathrm{C} 9.8 .$ Test the following design. The flow enters at $\mathrm{Ma}=7$ and air properties for $10,000 \mathrm{m}$ altitude. Inlet area is $1 \mathrm{m}^{2}$, the minimum area is $0.1 \mathrm{m}^{2}$ and the exit area is $0.8 \mathrm{m}^{2}$. If there is no combustion,
(a) will the flow still be supersonic in the throat? Also, determine ( $b$ ) the exit Mach number,
$(c)$ exit velocity and $(d)$ exit pressure.

Chai Santi
Chai Santi
Numerade Educator
04:47

Problem 52

A converging-diverging nozzle exits smoothly to sealevel standard atmosphere. It is supplied by a $40-\mathrm{m}^{3}$ tank initially at $800 \mathrm{kPa}$ and $100^{\circ} \mathrm{C}$. Assuming isentropic flow in the nozzle, estimate
(a) the throat area and
(b) the tank pressure after 10 s of operation. The exit area is $10 \mathrm{cm}^{2}$

Hubert Agamasu
Hubert Agamasu
Numerade Educator
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Problem 53

Air flows steadily from a reservoir at $20^{\circ} \mathrm{C}$ through a nozzle of exit area $20 \mathrm{cm}^{2}$ and strikes a vertical plate as in Fig. $\mathrm{P} 9.53 .$ The flow is subsonic throughout. A force of 135 $\mathrm{N}$ is required to hold the plate stationary. Compute $(a) V_{c}$
$(b) M a_{c,}$ and $(c) p_{0}$ if $p_{a}=101 \mathrm{kPa}$.

Victor Salazar
Victor Salazar
Numerade Educator
02:26

Problem 54

The airflow in Prob. P9.46 undergoes a normal shock just past the section where data was given. Determine the
(a) Mach number, (b) pressure, and (c) velocity just downstream of the shock.

Chai Santi
Chai Santi
Numerade Educator
03:15

Problem 55

Air, supplied by a reservoir at $450 \mathrm{kPa}$, flows through a converging-diverging nozzle whose throat area is $12 \mathrm{cm}^{2}$ A normal shock stands where $A_{1}=20 \mathrm{cm}^{2}$
(a) Compute the pressure just downstream of this shock. Still farther downstream, at $A_{3}=30 \mathrm{cm}^{2}$, estimate ( $b$ ) $p_{3}$, (c) $A_{3}^{*}$, and $(d) \mathbf{M a}_{3}$.

Chai Santi
Chai Santi
Numerade Educator
03:15

Problem 56

Air from a reservoir at $20^{\circ} \mathrm{C}$ and $500 \mathrm{kPa}$ flows through a duct and forms a normal shock downstream of a throat of area $10 \mathrm{cm}^{2} .$ By an odd coincidence it is found that the stagnation pressure downstream of this shock exactly equals the throat pressure. What is the area where the shock wave stands?

Chai Santi
Chai Santi
Numerade Educator
01:06

Problem 57

Air flows from a tank through a nozzle into the standard atmosphere, as in Fig. P9.57. A normal shock stands in the exit of the nozzle, as shown. Estimate (a) the pressure in the $\tan \mathrm{k}$ and $(b)$ the mass flow.

Penny Riley
Penny Riley
Numerade Educator
06:01

Problem 58

Downstream of a normal shock wave, in airflow, the conditions are $T_{2}=603 \mathrm{K}, V_{2}=222 \mathrm{m} / \mathrm{s},$ and $p_{2}=900 \mathrm{kPa}$
Estimate the following conditions just upstream of the shock:
$(a) \mathrm{Ma}_{1} ;(b) T_{1} ;(c) p_{1} ;(d) p_{\mathrm{ol}} ;$ and $(e) T_{\mathrm{ol}}$.

Vipender Yadav
Vipender Yadav
Numerade Educator
02:52

Problem 59

Air, at stagnation conditions of $450 \mathrm{K}$ and $250 \mathrm{kPa}$, flows through a nozzle. At section $1,$ where the area is $15 \mathrm{cm}^{2}$ there is a normal shock wave. If the mass flow is $0.4 \mathrm{kg} / \mathrm{s}$ estimate ( $a$ ) the Mach number and $(b)$ the stagnation pressure just downstream of the shock.

Chai Santi
Chai Santi
Numerade Educator
02:58

Problem 60

When a pitot tube such as in Fig. 6.30 is placed in a supersonic flow, a normal shock will stand in front of the probe. Suppose the probe reads $p_{0}=190 \mathrm{kPa}$ and $p=150 \mathrm{kPa} .$ If the stagnation temperature is $400 \mathrm{K}$ estimate the (supersonic) Mach number and velocity upstream of the shock.

Chai Santi
Chai Santi
Numerade Educator
02:29

Problem 61

Air flows from a large tank, where $T=376 \mathrm{K}$ and $p=$ $360 \mathrm{kPa}$, to a design condition where the pressure is $9800 \mathrm{Pa}$ The mass flow is $0.9 \mathrm{kg} / \mathrm{s}$. However, there is a normal shock in the exit plane just after this condition is reached. Estimate
(a) the throat area and, just downstream of the shock, ( $b$ ) the Mach number, (c) the temperature, and ( $d$ ) the pressure.

Chai Santi
Chai Santi
Numerade Educator
03:14

Problem 62

An atomic explosion propagates into still air at $14.7 \mathrm{lbf} / \mathrm{in}^{2}$ absolute and $520^{\circ} \mathrm{R}$. The pressure just inside the shock is 5000 lbf/in $^{2}$ absolute. Assuming $k=1.4,$ what are the speed $C$ of the shock and the velocity $V$ just inside the shock?

Chai Santi
Chai Santi
Numerade Educator
01:06

Problem 63

Sea-level standard air is sucked into a vacuum tank through a nozzle, as in Fig. $\mathrm{P} 9.63 .$ A normal shock stands where the nozzle area is $2 \mathrm{cm}^{2},$ as shown. Estimate ( $a$ ) the pressure in the tank and ( $b$ ) the mass flow.

Penny Riley
Penny Riley
Numerade Educator
03:15

Problem 64

Air, from a reservoir at $350 \mathrm{K}$ and $500 \mathrm{kPa}$, flows through a converging-diverging nozzle. The throat area is $3 \mathrm{cm}^{2}$. A normal shock appears, for which the downstream Mach number is 0.6405
(a) What is the area where the shock appears? Calculate
(b) the pressure and ( $c$ ) the temperature downstream of the shock.

Chai Santi
Chai Santi
Numerade Educator
01:36

Problem 65

Air flows through a converging-diverging nozzle between two large reservoirs, as shown in Fig. P9.65. A mercury manometer between the throat and the downstream reservoir reads $h=15 \mathrm{cm} .$ Estimate the downstream reservoir pressure. Is there a normal shock in the flow? If so, does it stand in the exit plane or farther upstream?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:33

Problem 66

In Prob. P9.65 what would be the mercury manometer reading $h$ if the nozzle were operating exactly at supersonic design conditions?

Averell Hause
Averell Hause
Carnegie Mellon University
04:47

Problem 67

A supply tank at $500 \mathrm{kPa}$ and $400 \mathrm{K}$ feeds air to a convergingdiverging nozzle whose throat area is $9 \mathrm{cm}^{2}$. The exit area is $46 \mathrm{cm}^{2}$. State the conditions in the nozzle if the pressure outside the exit plane is $(a) 400 \mathrm{kPa},(b) 120 \mathrm{kPa},$ and $(c) 9$ kPa. $(d)$ In each of these cases, find the mass flow.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
01:49

Problem 68

Air in a tank at $120 \mathrm{kPa}$ and $300 \mathrm{K}$ exhausts to the atmosphere through a $5-\mathrm{cm}^{2}$ -throat converging nozzle at a rate of $0.12 \mathrm{kg} / \mathrm{s}$. What is the atmospheric pressure? What is the maximum mass flow possible at low atmospheric pressure?

Dading Chen
Dading Chen
Numerade Educator
02:38

Problem 69

With reference to Prob. $\mathrm{P} 3.68,$ show that the thrust of a rocket engine exhausting into a vacuum is given by
\[
F=\frac{p_{0} A_{e}\left(1+k \mathrm{Ma}_{e}^{2}\right)}{\left(1+\frac{k-1}{2} \mathrm{Ma}_{e}^{2}\right)^{N(k-1)}}
\]
\[
\begin{array}{l}
\text { where } A_{e}=\text { exit area } \\
\mathrm{Ma}_{e}=\text { exit Mach number }
\end{array}
\]
$p_{0}=$ stagnation pressure in combustion chamber
Note that stagnation temperature does not enter into the thrust.

Vipender Yadav
Vipender Yadav
Numerade Educator
04:47

Problem 70

Air, with $p_{\mathrm{o}}=500 \mathrm{kPa}$ and $T_{\mathrm{o}}=600 \mathrm{K},$ flows through a converging-diverging nozzle. The exit area is $51.2 \mathrm{cm}^{2}$ and mass flow is $0.825 \mathrm{kg} / \mathrm{s}$. What is the highest possible back pressure that will still maintain supersonic flow inside the diverging section?

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:15

Problem 71

A converging-diverging nozzle has a throat area of $10 \mathrm{cm}^{2}$ and an exit area of $28.96 \mathrm{cm}^{2} .$ A normal shock stands in the exit when the back pressure is sea-level standard. If the upstream tank temperature is $400 \mathrm{K}$, estimate ( $a$ ) the tank pressure and $(b)$ the mass flow.

Chai Santi
Chai Santi
Numerade Educator
01:15

Problem 72

A large tank at $500 \mathrm{K}$ and 165 kPa feeds air to a converging nozzle. The back pressure outside the nozzle exit is sealevel standard. What is the appropriate exit diameter if the desired mass flow is $72 \mathrm{kg} / \mathrm{h} ?$

James Kiss
James Kiss
Numerade Educator
01:54

Problem 73

Air flows isentropically in a converging-diverging nozzle with a throat area of $3 \mathrm{cm}^{2} .$ At section 1 , the pressure is $101 \mathrm{kPa}$, the temperature is $300 \mathrm{K},$ and the velocity is $868 \mathrm{m} / \mathrm{s}$
$(a)$ Is the nozzle choked? Determine
(b) $A_{1}$ and $(c)$ the mass flow. Suppose, without changing stagnation conditions or $A_{1},$ the (flexible) throat is reduced to $2 \mathrm{cm}^{2}$. Assuming shock-free flow, will there be any change in the gas properties at section $1 ?$ If $s o$ compute new $p_{1}, V_{1},$ and $T_{1}$ and explain.

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 74

Use your strategic ideas, from part ( $b$ ) of Prob. P9.40, to actually carry out the calculations for mass flow of steam, with $p_{\mathrm{o}}=300 \mathrm{kPa}$ and $T_{\mathrm{o}}=600 \mathrm{K},$ discharging through a converging nozzle of choked exit area $5 \mathrm{cm}^{2}$.

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:13

Problem 75

A double-tank system in Fig. $\mathrm{P} 9.75$ has two identical converging nozzles of $1-$ in $^{2}$ throat area. Tank 1 is very large, and tank 2 is small enough to be in steady-flow equilibrium with the jet from tank $1 .$ Nozzle flow is isentropic, but entropy changes between 1 and 3 due to jet dissipation in tank $2 .$ Compute the mass flow. (If you give up, Ref. $9,$ pp. $288-290,$ has a good discussion.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
07:00

Problem 76

A large reservoir at $20^{\circ} \mathrm{C}$ and $800 \mathrm{kPa}$ is used to fill a small insulated tank through a converging-diverging nozzle with $1-\mathrm{cm}^{2}$ throat area and $1.66-\mathrm{cm}^{2}$ exit area. The small tank has a volume of $1 \mathrm{m}^{3}$ and is initially at $20^{\circ} \mathrm{C}$ and $100 \mathrm{kPa}$. Estimate the elapsed time when $(a)$ shock waves begin to appear inside the nozzle and ( $b$ ) the mass flow begins to drop below its maximum value.

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
03:05

Problem 77

A perfect gas (not air) expands isentropically through a supersonic nozzle with an exit area 5 times its throat area. The exit Mach number is $3.8 .$ What is the specific-heat ratio of the gas? What might this gas be? If $p_{0}=300 \mathrm{kPa}$ what is the exit pressure of the gas?

Chai Santi
Chai Santi
Numerade Educator
04:49

Problem 78

The orientation of a hole can make a difference. Consider holes $A$ and $B$ in Fig. $\mathrm{P} 9.78,$ which are identical but reversed. For the given air properties on either side, compute the mass flow through each hole and explain why they are different.

Sophie S
Sophie S
Numerade Educator
04:29

Problem 79

A large tank, at $400 \mathrm{kPa}$ and $450 \mathrm{K}$, supplies air to a converging-diverging nozzle of throat area $4 \mathrm{cm}^{2}$ and exit area $5 \mathrm{cm}^{2}$. For what range of back pressures will the flow
(a) be entirely subsonic;
(b) have a shock wave inside the nozzle; $(c)$ have oblique shocks outside the exit; and
(d) have supersonic expansion waves outside the exit?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:02

Problem 80

A sea-level automobile tire is initially at 32 lbf/in $^{2}$ gage pressure and $75^{\circ} \mathrm{F}$. When it is punctured with a hole that resembles a converging nozzle, its pressure drops to $15 \mathrm{lbf} / \mathrm{in}^{2}$ gage in 12 min. Estimate the size of the hole, in thousandths of an inch. The tire volume is $2.5 \mathrm{ft}^{2}$.

Amit Srivastava
Amit Srivastava
Numerade Educator
View

Problem 81

Air, at $p_{\mathrm{o}}=160 \mathrm{lbf} / \mathrm{in}^{2}$ and $T_{\mathrm{o}}=300^{\circ} \mathrm{F}$, flows isentropically through a converging-diverging nozzle. At section 1 where $A_{1}=288$ in $^{2},$ the velocity is $V_{1}=2068 \mathrm{ft} / \mathrm{s}$. Calculate
$(a) \mathrm{Ma}_{1},(b) A^{*},(c) p_{1},$ and $(d)$ the mass flow, in slug/s.

Victor Salazar
Victor Salazar
Numerade Educator
06:09

Problem 82

Air at $500 \mathrm{K}$ flows through a converging-diverging nozzle with throat area of $1 \mathrm{cm}^{2}$ and exit area of $2.7 \mathrm{cm}^{2} .$ When the mass flow is $182.2 \mathrm{kg} / \mathrm{h}$, a pitot-static probe placed in the exit plane reads $p_{0}=250.6 \mathrm{kPa}$ and $p=240.1 \mathrm{kPa}$. Estimate the exit velocity. Is there a normal shock wave in the duct? If so, compute the Mach number just downstream of this shock.

Dading Chen
Dading Chen
Numerade Educator
06:09

Problem 83

Air at $500 \mathrm{K}$ flows through a converging-diverging nozzle with throat area of $1 \mathrm{cm}^{2}$ and exit area of $2.7 \mathrm{cm}^{2} .$ When the mass flow is $182.2 \mathrm{kg} / \mathrm{h}$, a pitot-static probe placed in the exit plane reads $p_{0}=250.6 \mathrm{kPa}$ and $p=240.1 \mathrm{kPa}$. Estimate the exit velocity. Is there a normal shock wave in the duct? If so, compute the Mach number just downstream of this shock.

Dading Chen
Dading Chen
Numerade Educator
01:54

Problem 84

Air flows through a duct as in Fig. P9.84, where $A_{1}=$ $24 \mathrm{cm}^{2}, A_{2}=18 \mathrm{cm}^{2},$ and $A_{3}=32 \mathrm{cm}^{2} .$ A normal shock stands at section 2. Compute (a) the mass flow, (b) the Mach number, and (c) the stagnation pressure at section 3.

Chai Santi
Chai Santi
Numerade Educator
02:25

Problem 85

A typical carbon dioxide tank for a paintball gun holds about 12 oz of liquid $\mathrm{CO}_{2}$. The tank is filled no more than one-third with liquid, which, at room temperature, maintains the gaseous phase at about 850 psia. $(a)$ If a valve is opened that simulates a converging nozzle with an exit diameter of 0.050 in, what mass flow and exit velocity results? (b) Repeat the calculations for helium.

Chai Santi
Chai Santi
Numerade Educator
02:18

Problem 86

Air enters a 3 -cm-diameter pipe $15 \mathrm{m}$ long at $V_{1}=73 \mathrm{m} / \mathrm{s}$ $p_{1}=550 \mathrm{kPa},$ and $T_{1}=60^{\circ} \mathrm{C} .$ The friction factor is 0.018 Compute $V_{2}, p_{2}, T_{2},$ and $p_{02}$ at the end of the pipe. How much additional pipe length would cause the exit flow to be sonic?

Chai Santi
Chai Santi
Numerade Educator
03:28

Problem 87

Problem C6.9 gives data for a proposed Alaska-to-Canada natural gas (assume $\mathrm{CH}_{4}$ ) pipeline. If the design flow rate is $890 \mathrm{kg} / \mathrm{s}$ and the entrance conditions are $2500 \mathrm{lbf} / \mathrm{in}^{2}$ and $140^{\circ} \mathrm{F},$ determine the maximum length of adiabatic pipe before choking occurs.

Chai Santi
Chai Santi
Numerade Educator
01:31

Problem 88

Air flows adiabatically, with $\bar{f}=0.024,$ down a long $6-\mathrm{cm}-$ diameter pipe. At section $1,$ conditions are $T_{1}=$ $300 \mathrm{K}, p_{1}=400 \mathrm{kPa},$ and $V_{1}=104 \mathrm{m} / \mathrm{s} .$ At section 2 $V_{2}=233 \mathrm{m} / \mathrm{s} .$ (a) How far downstream is section 2? Estimate $(b) \mathrm{Ma}_{2},(c) p_{2},$ and $(d) T_{2}$.

Dominador Tan
Dominador Tan
Numerade Educator
03:02

Problem 89

Carbon dioxide flows through an insulated pipe $25 \mathrm{m}$ long and $8 \mathrm{cm}$ in diameter. The friction factor is $0.025 .$ At the entrance, $p=300 \mathrm{kPa}$ and $T=400 \mathrm{K}$. The mass flow is $1.5 \mathrm{kg} / \mathrm{s} .$ Estimate the pressure drop by
(a) compressible and $(b)$ incompressible (Sec. 6.6 ) flow theory. (c) For what pipe length will the exit flow be choked?

Mayukh Banik
Mayukh Banik
Numerade Educator
17:07

Problem 90

Air flows through a rough pipe $120 \mathrm{ft}$ long and 3 in in diameter. Entrance conditions are $p=90 \mathrm{lbf} / \mathrm{in}^{2}, T=68^{\circ} \mathrm{F}$ and $V=225 \mathrm{ft} / \mathrm{s}$. The flow chokes at the end of the pipe.
(a) What is the average friction factor?
(b) What is the pressure at the end of the pipe?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:30

Problem 91

Air flows steadily from a tank through the pipe in Fig. $\mathrm{P} 9.91 .$ There is a converging nozzle on the end. If the mass flow is $3 \mathrm{kg} / \mathrm{s}$ and the nozzle is choked, estimate
(a) the Mach number at section 1 and ( $b$ ) the pressure inside the tank.

Naman Kumar
Naman Kumar
Numerade Educator
02:18

Problem 92

Air enters a 5 -cm-diameter pipe at $380 \mathrm{kPa}, 3.3 \mathrm{kg} / \mathrm{m}^{3}$ and $120 \mathrm{m} / \mathrm{s}$. The friction factor is 0.017 . Find the pipe length for which the velocity
( $a$ ) doubles,
$(b)$ triples, and
$(c)$ quadruples.

Chai Santi
Chai Santi
Numerade Educator
01:31

Problem 93

Air flows adiabatically in a 3 -cm-diameter duct, with $\bar{f}=0.018 .$ At the entrance, $T_{1}=323 \mathrm{K}, p_{1}=200 \mathrm{kPa},$ and $V_{1}=72 \mathrm{m} / \mathrm{s}$. (a) What is the mass flow? ( $b$ ) For what tube length will the flow choke? $(c)$ If the tube length is increased to $112 \mathrm{m}$, with the same inlet pressure and temperature, what will be the new mass flow?

Dominador Tan
Dominador Tan
Numerade Educator
10:13

Problem 94

Compressible pipe flow with friction, Sec. $9.7,$ assumes constant stagnation enthalpy and mass flow but variable momentum. Such a flow is often called Fanno flow, and a line representing all possible property changes on a temperature-entropy chart is called a Fanno line. Assuming a perfect gas with $k=1.4$ and the data of Prob. $\mathrm{P} 9.86$ draw a Fanno curve of the flow for a range of velocities from very low $(\mathrm{Ma}<1)$ to very high $(\mathrm{Ma} \gg 1) .$ Comment on the meaning of the maximum-entropy point on this curve.

Vipender Yadav
Vipender Yadav
Numerade Educator
03:47

Problem 95

Helium (Table A.4) enters a 5 -cm-diameter pipe at $p_{1}=$ $550 \mathrm{kPa}, V_{1}=312 \mathrm{m} / \mathrm{s},$ and $T_{1}=40^{\circ} \mathrm{C} .$ The friction factor is $0.025 .$ If the flow is choked, determine
$(a)$ the length of the duct and ( $b$ ) the exit pressure.

Chai Santi
Chai Santi
Numerade Educator
02:33

Problem 96

Methane $\left(\mathrm{CH}_{4}\right)$ flows through an insulated 15 -cm-diameter pipe with $f=0.023 .$ Entrance conditions are $600 \mathrm{kPa}$ $100^{\circ} \mathrm{C},$ and a mass flow of $5 \mathrm{kg} / \mathrm{s} .$ What lengths of pipe will
$(a)$ choke the flow,
$(b)$ raise the velocity by 50 percent, or
$(c)$ decrease the pressure by 50 percent?

Naman Kumar
Naman Kumar
Numerade Educator
02:42

Problem 97

By making a few algebraic substitutions, show that Eq. (9.74) may be written in the density form
$$\rho_{1}^{2}=\rho_{2}^{2}+\rho^{* 2}\left(\frac{2 k}{k+1} \frac{f L}{D}+2 \ln \frac{\rho_{1}}{\rho_{2}}\right)$$
Why is this formula awkward if one is trying to solve for the mass flow when the pressures are given at sections 1 and $2 ?$

Narayan Hari
Narayan Hari
Numerade Educator
03:35

Problem 98

Compressible laminar flow, $f \approx 64 / \mathrm{Re},$ may occur in capillary tubes. Consider air, at stagnation conditions of $100^{\circ} \mathrm{C}$ and $200 \mathrm{kPa}$, entering a tube $3 \mathrm{cm}$ long and $0.1 \mathrm{mm}$ in diameter. If the receiver pressure is near vacuum, estimate
$(a)$ the average Reynolds number,
(b) the Mach number at the entrance, and $(c)$ the mass flow in $\mathrm{kg} / \mathrm{h}$

Chai Santi
Chai Santi
Numerade Educator
08:22

Problem 99

A compressor forces air through a smooth pipe $20 \mathrm{m}$ long and $4 \mathrm{cm}$ in diameter, as in Fig. $\mathrm{P} 9.99$. The air leaves at $101 \mathrm{kPa}$ and $200^{\circ} \mathrm{C} .$ The compressor data for pressure rise versus mass flow are shown in the figure. Using the Moody chart to estimate $\bar{f},$ compute the resulting mass flow.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:28

Problem 100

Natural gas, approximated as $\mathrm{CH}_{4}$, flows through a Schedule 40 six-inch pipe from Providence to Narragansett, $\mathrm{RI}$, a distance of 31 miles. Gas companies use the barg as a pressure unit, meaning a bar of pressure gage, above ambient pressure. Assuming isothermal flow at $68^{\circ} \mathrm{F}$, with $f \approx$ 0.019 , estimate the mass flow if the pressure is 5 bargs in Providence and 1 barg in Narragansett.

Chai Santi
Chai Santi
Numerade Educator
03:35

Problem 101

How do the compressible pipe flow formulas behave for small pressure drops? Let air at $20^{\circ} \mathrm{C}$ enter a tube of diameter $1 \mathrm{cm}$ and length $3 \mathrm{m}$. If $\bar{f}=0.028$ with $p_{1}=102 \mathrm{kPa}$ and $p_{2}=100 \mathrm{kPa}$, estimate the mass flow in kg/h for
$(a)$ isothermal flow,
($b$ ) adiabatic flow, and
$(c)$ incompressible flow (Chap. 6 ) at the entrance density.

Chai Santi
Chai Santi
Numerade Educator
08:22

Problem 102

Air at $550 \mathrm{kPa}$ and $100^{\circ} \mathrm{C}$ enters a smooth 1 -m-long pipe and then passes through a second smooth pipe to a $30-\mathrm{kPa}$ reservoir, as in Fig. $\mathrm{P} 9.102 .$ Using the Moody chart to compute $\bar{f}$, estimate the mass flow through this system. Is the flow choked?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:28

Problem 103

Natural gas, with $k \approx 1.3$ and a molecular weight of $16,$ is to be pumped through $100 \mathrm{km}$ of 81 -cm-diameter pipeline. The downstream pressure is 150 kPa. If the gas enters at $60^{\circ} \mathrm{C},$ the mass flow is $20 \mathrm{kg} / \mathrm{s},$ and $\bar{f}=0.024,$ estimate the required entrance pressure for $(a)$ isothermal flow and $(b)$ adiabatic flow.

Chai Santi
Chai Santi
Numerade Educator
01:35

Problem 104

A tank of oxygen (Table A.4) at $20^{\circ} \mathrm{C}$ is to supply an astronaut through an umbilical tube $12 \mathrm{m}$ long and $1.5 \mathrm{cm}$ in diameter. The exit pressure in the tube is $40 \mathrm{kPa}$. If the desired mass flow is $90 \mathrm{kg} / \mathrm{h}$ and $\bar{f}=0.025,$ what should be the pressure in the tank?

Chai Santi
Chai Santi
Numerade Educator
07:17

Problem 105

Modify Prob. P9.87 as follows: The pipeline will not be allowed to choke. It will have pumping stations about every 200 miles. ( $a$ ) Find the length of pipe for which the pressure has dropped to $2000 \mathrm{lbf} / \mathrm{in}^{2}$ (b) What is the temperature at that point?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:04

Problem 106

Air, from a 3 cubic meter tank initially at $300 \mathrm{kPa}$ and $200^{\circ} \mathrm{C},$ blows down adiabatically through a smooth pipe $1 \mathrm{cm}$ in diameter and $2.5 \mathrm{m}$ long. Estimate the time required to reduce the tank pressure to $200 \mathrm{kPa}$. For simplicity, assume constant tank temperature and $f \approx 0.020$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
07:14

Problem 107

A fuel-air mixture, assumed equivalent to air, enters a duct combustion chamber at $V_{1}=104 \mathrm{m} / \mathrm{s}$ and $T_{1}=300 \mathrm{K}$ What amount of heat addition in $\mathrm{kJ} / \mathrm{kg}$ will cause the exit flow to be choked? What will be the exit Mach number and
temperature if $504 \mathrm{kJ} / \mathrm{kg}$ are added during combustion?

Dading Chen
Dading Chen
Numerade Educator
01:53

Problem 108

What happens to the inlet flow of Prob. P9.107 if the combustion yields $1500 \mathrm{kJ} / \mathrm{kg}$ heat addition and $p_{01}$ and $T_{01}$ remain the same? How much is the mass flow reduced?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:46

Problem 109

A jet engine at 7000 -m altitude takes in $45 \mathrm{kg} / \mathrm{s}$ of air and adds $550 \mathrm{kJ} / \mathrm{kg}$ in the combustion chamber. The chamber cross section is $0.5 \mathrm{m}^{2}$, and the air enters the chamber at $80 \mathrm{kPa}$ and $5^{\circ} \mathrm{C}$. After combustion the air expands through an isentropic converging nozzle to exit at atmospheric pressure. Estimate (a) the nozzle throat diameter, (b) the nozzle exit velocity, and ( $c$ ) the thrust produced by the engine.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 110

Compressible pipe flow with heat addition, Sec. 9.8 , assumes constant momentum $\left(p+\rho V^{2}\right)$ and constant mass flow but variable stagnation enthalpy. Such a flow is often called Rayleigh flow, and a line representing all possible property changes on a temperature-entropy chart is called a Rayleigh line. Assuming air passing through the flow state $p_{1}=$ $548 \mathrm{kPa}, T_{1}=588 \mathrm{K}, V_{1}=266 \mathrm{m} / \mathrm{s},$ and $A=1 \mathrm{m}^{2},$ draw a
Rayleigh curve of the flow for a range of velocities from very low (Ma $\ll 1$ ) to very high (Ma $\geqslant 1$ ). Comment on the meaning of the maximum-entropy point on this curve.

Victor Salazar
Victor Salazar
Numerade Educator
02:33

Problem 111

Add to your Rayleigh line of Prob. P9.110 a Fanno line (see Prob. $\mathrm{P} 9.94$ ) for stagnation enthalpy equal to the value associated with state 1 in Prob. P9.110. The two curves will intersect at state $1,$ which is subsonic, and at a certain state $2,$ which is supersonic. Interpret these two states vis-ã-vis Table B.2.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:19

Problem 112

Air enters a duct at $V_{1}=144 \mathrm{m} / \mathrm{s}, p_{1}=200 \mathrm{kPa},$ and $T_{1}=$
$323 \mathrm{K}$. Assuming frictionless heat addition, estimate
(a) the heat addition needed to raise the velocity to $372 \mathrm{m} / \mathrm{s}$ and
$(b)$ the pressure at this new section 2.

Narayan Hari
Narayan Hari
Numerade Educator
04:22

Problem 113

Air enters a constant-area duct at $p_{1}=90 \mathrm{kPa}, V_{1}=$ $520 \mathrm{m} / \mathrm{s},$ and $T_{1}=558^{\circ} \mathrm{C} .$ It is then cooled with negligible friction until it exits at $p_{2}=160 \mathrm{kPa}$. Estimate
$(a) V_{2}$
$(b) T_{2},$ and
$(c)$ the total amount of cooling in $\mathrm{kJ} / \mathrm{kg}$

Jincy M Saji
Jincy M Saji
Numerade Educator
03:36

Problem 114

The scramjet of Fig. $\mathrm{C} 9.8$ operates with supersonic flow throughout. Assume that the heat addition of $500 \mathrm{kJ} / \mathrm{kg}$ between sections 2 and $3,$ is frictionless and at constant area of $0.2 \mathrm{m}^{2} .$ Given $\mathrm{Ma}_{2}=4.0, p_{2}=260 \mathrm{kPa},$ and $T_{2}=$ $420 \mathrm{K} .$ Assume airflow at $k=1.40 .$ At the combustion section exit, find $(a) \mathrm{Ma}_{3},(b) p_{3},$ and $(c) T_{3}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:18

Problem 115

Air enters a 5 -cm-diameter pipe at $380 \mathrm{kPa}, 3.3 \mathrm{kg} / \mathrm{m}^{3},$ and $120 \mathrm{m} / \mathrm{s}$. Assume frictionless flow with heat addition. Find the amount of heat addition for which the velocity
$(a)$ doubles,
(b) triples, and
$(c)$ quadruples

Chai Santi
Chai Santi
Numerade Educator
05:03

Problem 116

An observer at sea level does not hear an aircraft flying at 12,000 -ft standard altitude until it is 5 (statute) $\mathrm{mi}$ past her. Estimate the aircraft speed in $\mathrm{ft} / \mathrm{s}$.

Vipender Yadav
Vipender Yadav
Numerade Educator
06:09

Problem 117

A tiny scratch in the side of a supersonic wind tunnel creates a very weak wave of angle $17^{\circ},$ as shown in Fig. P9.1 $17,$ after which a normal shock occurs. The air temperature in region (1) is $250 \mathrm{K}$. Estimate the temperature in region ( 2 ).

Dading Chen
Dading Chen
Numerade Educator
01:30

Problem 118

A particle moving at uniform velocity in sea-level standard air creates the two disturbance spheres shown in Fig. $\mathrm{P} 9.118 .$ Compute the particle velocity and Mach number.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
00:44

Problem 119

The particle in Fig. $\mathrm{P} 9.119$ is moving supersonically in sea-level standard air. From the two given disturbance spheres, compute the particle Mach number, velocity, and Mach angle.

Nidhi Singhi
Nidhi Singhi
Numerade Educator
02:00

Problem 120

The particle in Fig. $\mathrm{P} 9.120$ is moving in sea-level standard air. From the two disturbance spheres shown, estimate (a) the position of the particle at this instant and $(b)$ the temperature in $^{\circ} \mathrm{C}$ at the front stagnation point of the particle.

Surendra Kumar
Surendra Kumar
Numerade Educator
02:32

Problem 121

A thermistor probe, in the shape of a needle parallel to the flow, reads a static temperature of $-25^{\circ} \mathrm{C}$ when inserted into a supersonic airstream. A conical disturbance cone of half-angle $17^{\circ}$ is created. Estimate
( $a$ ) the Mach number,
(b) the velocity, and
$(c)$ the stagnation temperature of the stream.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
04:35

Problem 122

Supersonic air takes a $5^{\circ}$ compression turn, as in Fig. $\mathrm{P} 9.122 .$ Compute the downstream pressure and Mach number and the wave angle, and compare with smalldisturbance theory.

Chai Santi
Chai Santi
Numerade Educator
02:53

Problem 123

The $10^{\circ}$ deflection in Example 9.17 caused a final Mach number of 1.641 and a pressure ratio of $1.707 .$ Compare this with the case of the flow passing through two $5^{\circ}$ deflections. Comment on the results and why they might be higher or lower in the second case.

Chai Santi
Chai Santi
Numerade Educator
04:57

Problem 124

When a sea-level flow approaches a ramp of angle $20^{\circ},$ an oblique shock wave forms as in Figure $\mathrm{P} 9.124$. Calculate
$(a) \mathrm{Ma}_{1},(b) p_{2},(c) T_{2},$ and $(d) V_{2}$

Anand Jangid
Anand Jangid
Numerade Educator
05:57

Problem 125

We saw in the text that, for $k=1.40$, the maximum possible deflection caused by an oblique shock wave occurs at infinite approach Mach number and is $\theta_{\max }=45.58^{\circ}$ Assuming an ideal gas, what is $\theta_{\max }$ for $(a)$ argon and $(b)$ carbon dioxide?

Bruce Edelman
Bruce Edelman
Numerade Educator
01:49

Problem 126

Airflow at $\mathrm{Ma}=2.8, p=80 \mathrm{kPa},$ and $T=280 \mathrm{K}$ undergoes a $15^{\circ}$ compression turn. Find the downstream values of $(a)$ Mach number (b) pressure, and ( $c$ ) temperature.

James Kiss
James Kiss
Numerade Educator
00:23

Problem 127

Do the Mach waves upstream of an oblique shock wave intersect with the shock? Assuming supersonic downstream flow, do the downstream Mach waves intersect the shock? Show that for small deflections the shock wave angle $\beta$ lies halfway between $\mu_{1}$ and $\mu_{2}+\theta$ for any Mach number.

Dading Chen
Dading Chen
Numerade Educator
01:48

Problem 128

Air flows past a two-dimensional wedge-nosed body as in Fig. $\mathrm{P} 9.128 .$ Determine the wedge half-angle $\delta$ for which the horizontal component of the total pressure force on the nose is $35 \mathrm{kN} / \mathrm{m}$ of depth into the paper.

James Kiss
James Kiss
Numerade Educator
01:03

Problem 129

Air flows at supersonic speed toward a compression ramp, as in Fig. $\mathrm{P} 9.129 .$ A scratch on the wall at point $a$ creates a wave of $30^{\circ}$ angle, while the oblique shock created has a $50^{\circ}$ angle. What is $(a)$ the ramp angle $\theta$ and $(b)$ the wave angle $\phi$ caused by a scratch at $b ?$

Narayan Hari
Narayan Hari
Numerade Educator
00:00

Problem 130

A supersonic airflow, at a temperature of $300 \mathrm{K},$ strikes a wedge and is deflected $12^{\circ} .$ If the resulting shock wave is attached, and the temperature after the shock is $450 \mathrm{K}$
(a) estimate the approach Mach number and wave angle.
(b) Why are there two solutions?

Jincy M Saji
Jincy M Saji
Numerade Educator
01:31

Problem 131

The following formula has been suggested as an alternate to Eq. (9.86) to relate upstream Mach number to the oblique shock wave angle $\beta$ and turning angle $\theta$
\[
\sin ^{2} \beta=\frac{1}{\mathrm{Ma}_{1}^{2}}+\frac{(k+1) \sin \beta \sin \theta}{2 \cos (\beta-\theta)}
\]
Can you prove or disprove this relation? If not, try a few numerical values and compare with the results from Eq. (9.86)

Chai Santi
Chai Santi
Numerade Educator
01:13

Problem 132

Air flows at $\mathrm{Ma}=3$ and $p=10 \mathrm{lbf} / \mathrm{in}^{2}$ absolute toward a wedge of $16^{\circ}$ angle at zero incidence in Fig. $\mathrm{P} 9.132 .$ If the pointed edge is forward, what will be the pressure at point $A ?$ If the blunt edge is forward, what will be the pressure at point $B ?$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:22

Problem 133

Air flows supersonically toward the double-wedge system in Fig. $\mathrm{P} 9.133 .$ The $(x, y)$ coordinates of the tips are given. The shock wave of the forward wedge strikes the tip of the aft wedge. Both wedges have $15^{\circ}$ deflection angles. What is the free-stream Mach number?

Dading Chen
Dading Chen
Numerade Educator
01:10

Problem 134

When an oblique shock strikes a solid wall, it reflects as a shock of sufficient strength to cause the exit flow $\mathrm{Ma}_{3}$ to be parallel to the wall, as in Fig. P9.134. For airflow with Ma $_{1}=$ 2.5 and $p_{1}=100 \mathrm{kPa},$ compute $\mathrm{Ma}_{3}, p_{3},$ and the angle $\phi$.

Chai Santi
Chai Santi
Numerade Educator
06:09

Problem 135

A bend in the bottom of a supersonic duct flow induces a shock wave that reflects from the upper wall, as in Fig. $\mathrm{P} 9.135 .$ Compute the Mach number and pressure in region 3.

Dading Chen
Dading Chen
Numerade Educator
01:28

Problem 136

Figure $\mathrm{P} 9.136$ is a special application of Prob. $\mathrm{P} 9.135 .$ With careful design, one can orient the bend on the lower wall so that the reflected wave is exactly canceled by the return bend, as shown. This is a method of reducing the Mach number in a channel (a supersonic diffuser). If the bend angle is $\phi=10^{\circ},$ find $(a)$ the downstream width $h$ and $(b)$ the downstream Mach number. Assume a weak shock wave.

Chai Santi
Chai Santi
Numerade Educator
01:57

Problem 137

$A 6^{\circ}$ half-angle wedge creates the reflected shock system in Fig. $P 9.137 .$ If $\mathrm{Ma}_{3}=2.5,$ find $(a) \mathrm{Ma}_{1}$ and $(b)$ the angle $\alpha$.

Chai Santi
Chai Santi
Numerade Educator
02:12

Problem 138

The supersonic nozzle of Fig. $\mathrm{P} 9.138$ is overexpanded (case $G$ of Fig. $9.12 b$ ) with $A_{e} / A_{t}=3.0$ and a stagnation pressure of $350 \mathrm{kPa}$. If the jet edge makes a $4^{\circ}$ angle with the nozzle centerline, what is the back pressure $p_{r}$ in $\mathrm{kPa} ?$

Chai Santi
Chai Santi
Numerade Educator
03:16

Problem 139

Airflow at $\mathrm{Ma}=2.2$ takes a compression turn of $12^{\circ}$ and then another turn of angle $\theta$ in Fig. $\mathrm{P} 9.139 .$ What is the maximum value of $\theta$ for the second shock to be attached? Will the two shocks intersect for any $\theta$ less than $\theta_{\max } ?$

Keshav Singh
Keshav Singh
Numerade Educator
03:00

Problem 140

The solution to Prob. $\mathrm{P} 9.122$ is $\mathrm{Ma}_{2}=2.750$ and $p_{2}=$ $145.5 \mathrm{kPa} .$ Compare these results with an isentropic compression turn of $5^{\circ},$ using Prandtl-Meyer theory.

Averell Hause
Averell Hause
Carnegie Mellon University
06:52

Problem 141

Supersonic airflow takes a $5^{\circ}$ expansion turn, as in Fig. $\mathrm{P} 9.141 .$ Compute the downstream Mach number and pressure, and compare with small-disturbance theory.

Dading Chen
Dading Chen
Numerade Educator
View

Problem 142

A supersonic airflow at $\mathrm{Ma}_{1}=3.2$ and $p_{1}=50 \mathrm{kPa}$ under goes a compression shock followed by an isentropic expansion turn. The flow deflection is $30^{\circ}$ for each turn. Compute $\mathrm{Ma}_{2}$ and $p_{2}$ if $(a)$ the shock is followed by the expansion and $(b)$ the expansion is followed by the shock.

Victor Salazar
Victor Salazar
Numerade Educator
03:46

Problem 143

Airflow at $\mathrm{Ma}=3.4$ and $300 \mathrm{K}$ encounters a $28^{\circ}$ oblique shock turn. What subsequent isentropic expansion turn will bring the temperature back to $300 \mathrm{K} ?$

Jordan Vanevery
Jordan Vanevery
Numerade Educator
02:22

Problem 144

The $10^{\circ}$ deflection in Example 9.17 caused the Mach number to drop to $1.64 .(a)$ What turn angle will create a Prandtl-Meyer fan and bring the Mach number back up to $2.0 ?$ (b) What will be the final pressure?

Chai Santi
Chai Santi
Numerade Educator
01:44

Problem 145

Air at $\mathrm{Ma}_{1}=2.0$ and $p_{1}=100 \mathrm{kPa}$ undergoes an isentropic expansion to a downstream pressure of $50 \mathrm{kPa}$. What is the desired turn angle in degrees?

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:33

Problem 146

Air flows supersonically over a surface that changes direction twice, as in Fig. P9.146. Calculate
$(a) \mathrm{Ma}_{2}$ and $(b) p_{3}$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:17

Problem 147

A converging-diverging nozzle with a 4: 1 exit-area ratio and $p_{0}=500 \mathrm{kPa}$ operates in an underexpanded condition (case $I$ of Fig. $9.12 b$ ) as in Fig. $\mathrm{P} 9.147 .$ The receiver pressure is $p_{a}=10 \mathrm{kPa}$, which is less than the exit pressure, so that expansion waves form outside the exit. For the given conditions, what will the Mach number $\mathrm{Ma}_{2}$ and the angle
$\phi$ of the edge of the jet be? Assume $k=1.4$ as usual.

James Kiss
James Kiss
Numerade Educator
01:13

Problem 148

Air flows supersonically over a circular-arc surface as in Fig. $\mathrm{P} 9.148 .$ Estimate $(a)$ the Mach number $\mathrm{Ma}_{2}$ and $(b)$ the pressure $p_{2}$ as the flow leaves the circular surface.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:28

Problem 149

Air flows at $\mathrm{Ma}_{\infty}=3.0$ past a doubly symmetric diamond airfoil whose front and rear included angles are both $24^{\circ}$.For zero angle of attack, compute the drag coefficient obtained using shock-expansion theory and compare with Ackeret theory.

Chai Santi
Chai Santi
Numerade Educator
03:37

Problem 150

A flat-plate airfoil with $C=1.2 \mathrm{m}$ is to have a lift of $30 \mathrm{kN} / \mathrm{m}$ when flying at 5000 -m standard altitude with $U_{x}=641 \mathrm{m} / \mathrm{s} .$ Using Ackeret theory, estimate $(a)$ the angle of attack and $(b)$ the drag force in $\mathrm{N} / \mathrm{m}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
03:42

Problem 151

Air flows at $\mathrm{Ma}=2.5$ past a half-wedge airfoil whose angles are $4^{\circ},$ as in Fig. $\mathrm{P} 9.151 .$ Compute the lift and drag coefficient at $\alpha$ equal to $(a) 0^{\circ}$ and $(b) 6^{\circ}$.

Chai Santi
Chai Santi
Numerade Educator
01:52

Problem 152

The $X-43$ model A scramjet aircraft in Fig. $C 9.8$ is small $\mathrm{W}=3000 \mathrm{lbf}$, and unmanned, only $12.33 \mathrm{ft}$ long and $5.5 \mathrm{ft}$ wide. The aerodynamics of a slender arrowhead-shaped hypersonic vehicle is beyond our scope. Instead, let us assume it is a flat plate airfoil of area $2.0 \mathrm{m}^{2}$. Let $\mathrm{Ma}=7$ at $12,000 \mathrm{m}$ standard altitude. Estimate the drag, by shockexpansion theory. Hint: Use Ackeret theory to estimate the angle of attack.

Narayan Hari
Narayan Hari
Numerade Educator
05:51

Problem 153

A supersonic transport has a mass of $65 \mathrm{Mg}$ and cruises at 1 1-km standard altitude at a Mach number of 2.25. If the angle of attack is $2^{\circ}$ and its wings can be approximated by flat plates, estimate
$(a)$ the required wing area in $\mathrm{m}^{2}$ and (b) the thrust required in $\mathrm{N}$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:22

Problem 154

The $F-22$ supersonic fighter cruises at $11,000 \mathrm{m}$ altitude, with a weight of 50,000 lbf and thrust of 10,000 lbf. Its wing area is $840 \mathrm{ft}^{2}$. Assume the wing is a 6 -percent-thick diamond shape and provides all lift and thrust. Use Ackeret theory to estimate the resulting Mach number.

Ernest Castorena
Ernest Castorena
Numerade Educator
01:30

Problem 155

The $F-35$ airplane in Fig. 9.30 has a wingspan of $10 \mathrm{m}$ and a wing area of $41.8 \mathrm{m}^{2}$. It cruises at about $10 \mathrm{km}$ altitude with a gross weight of about $200 \mathrm{kN}$. At that altitude, the engine develops a thrust of about $50 \mathrm{kN}$. Assume the wing has a symmetric diamond airfoil with a thickness of 8 percent, and accounts for all lift and drag. Estimate the cruise Mach number of the airplane. For extra credit, explain why there are rwo solutions.

Dominador Tan
Dominador Tan
Numerade Educator
01:28

Problem 156

Consider a flat-plate airfoil at an angle of attack of $6^{\circ} .$ The Mach number is $\mathrm{Ma}_{\mathrm{x}}=3.2$ and the stream pressure $p_{\infty}$ is unspecified. Calculate the predicted lift and drag coefficients by
(a) shock-expansion theory and
(b) Ackeret theory.

Chai Santi
Chai Santi
Numerade Educator
01:14

Problem 157

The Ackeret airfoil theory of Eq. (9.104) is meant for moderate supersonic speeds, $1.2<\mathrm{Ma}<4 .$ How does it fare for hypersonic speeds? To illustrate, calculate $(a) C_{L}$ and $(b) C_{D}$ for a flat-plate airfoil at $a=5^{\circ}$ and $\mathrm{Ma}_{\mathrm{a}_{\infty}}=8.0,$ using shock-expansion theory, and compare with Ackeret theory. Comment.

Hunza Gilgit
Hunza Gilgit
Numerade Educator