978-0131206687 Chapter 2

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Problem 1. – Using the expansion wave and control volume depicted in Figs. 2.8 and 2.9

along with the continuity and momentum equations, rederive Eq. (2.4).

Continuity equation

moving wave

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18

Problem 2. – (a) Derive an expression for ks, for a perfect gas, substitute the result into

Eq. (2.10), and thereby demonstrate Eq. (2.7); (b) Derive an expression for kT, for a

perfect gas, substitute the result into Eq. (2.11), and thereby demonstrate Eq. (2.7) and

finally; (c) Derive an expression for βs, for a perfect gas, substitute your result into Eq.

(2.14), and thereby demonstrate Eq. (2.7).

1

⎞

⎛

ρ∂

k

s

ρ

(b)

T

Tp

1

k⎟

⎟

⎠

⎞

⎜

⎝

⎛

∂

ρ∂

ρ

=

p

=ρ

So,

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19

γ

ρ

Problem 3. – Use dimensional analysis to develop an expression for the speed of sound in

terms of the isentropic compressibility, the density and gc.

()

2

3

FT

L

F

T

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

1c2:T

1cb3a2:L

0ca:F

=−

=+−

=−−

Hence, 2

1

b

2

1

a

2

1

c−=−==

So,

s

c

k

g

aρ

=

Problem 4. – Using the data provided in Tables 2-1, 2-2 and 2-3, i.e., the density, and the

isentropic compressibility or the bulk modulus, calculate the velocity of sound at 20°C

and one atmosphere pressure in (a) helium, (b) turpentine, and (c) lead.

m

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20

10

1

a

9

=

=s/m 6.1027

=

300,11

ρ

Problem 5. – In Example Problem 2.3 the speed of sound of superheated steam was

determined by using a finite difference representation of the compressibility and steam

table data (Table 2-4). Using the same steam table data, determine the speed of sound of

superheated steam for the same pressure and temperature, i.e., at p = 500 kPa and T =

300˚C. However, use the following finite differences to obtain two estimates for the

speed of sound:

2

γ

T,ppv

p2

T,ppv

p

T

∆−

∆+

∆

∆−

∆+

⎥

⎦

⎢

⎣

∂

From Example 2.3

M

3

M

3

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=

21

2

s,ppv

p

s

∆−

∆+

⎦

⎣

∂

From Example 2.3

M

3

M

3

Problem 6. – Equation (2.16) provides a convenient expression for calculating the speed

of sound in air: a = 20.05 T, where T is the absolute temperature in degrees Kelvin.

Derive the following linear equation for the speed of sound in air:

+

=

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22

m

Problem 7. – Rather than measure the bulk modulus directly it may be easier to measure

the speed of sound as it propagates though a material and then use it to compute the bulk

modulus. For a Lucite plastic of density 1,200 kg/m3, the speed of sound is measured as

2,327 m/s. Determine the bulk modulus. What is the corresponding isentropic

compressibility?

kg

m

GPa

s

s=

β

Problem 8. – An object of diameter d (m) is rotated in air at a speed of N revolutions per

minute. Draw a plot of the rotational speed required for the velocity at the outer edge of

the object to just reach sonic velocity for a given diameter. Take the speed of sound of

the air to be 331m/s.

The highest speed will occur at R.

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5.4

5.6

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(a) ρa

p∆

V=∆ , Pa 100∆p=

air:

()

3

m

kg

1615.1

303

97.28

8314

000,101

ρ=

⎟

⎠

⎞

⎜

⎝

⎛

=

m/s 0.34930305.20a ==

Therefore,

()()

m/s 247.0

0.3491615.1

100

V==∆

(b) ρa∆V∆p=, m/s 1.0V =∆

hydrogen:

()

3

m

kg

0808.0

300

016.2

8314

000,100

RT

p

ρ=

⎟

⎠

⎞

⎜

⎝

⎛

==

() ()

m/s 8.1320300

016.1

8314

41.1a =

⎟

⎠

⎞

⎜

⎝

⎛

=

Therefore,

()()

(

)

Pa 68.101.08.13200808.0p

=

=∆

Problem 11. – (a) Helium at 35°C is flowing at a Mach number of 1.5. Find the velocity

and determine the local Mach angle. (b) Determine the velocity of air at 40°C to produce

a Mach angle of 38°

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25

⎟

⎠

⎞

⎜

⎝

⎛

=µ −

M

1

ins 1

a

V

ins

1

M=

µ

=

()

m/s 0.576

38ins

6.354

ins

a

V==

µ

=

Problem 12. – (a) A jet plane is traveling at Mach 1.8 at an altitude of 10 km where the

temperature is 223.3K. Determine the speed of the plane. (b) Air at 320 K flows in a

supersonic wind tunnel over a 2-D wedge. From a photograph the Mach angle is

measured to be 45°. Determine the flow velocity, the local speed of sound and the Mach

number of the tunnel.

sin

a

µ

Problem 13. – A supersonic aircraft, flying horizontally a distance H above the earth,

passes overhead. ∆t later the sound wave from the aircraft is heard. In this time

increment, the plane has traveled a distance L. Show that the Mach number of the

aircraft can be computed from:

tV

L

22

⎞

⎛∆

⎞

⎛

H

L

µ

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26

Problem 14. – Given speeds and temperatures, determine the corresponding Mach

numbers of the following (note: 1 mile = 5,280 ft = 1,609.3 m; 1 mi/hr = 1.6093 km/hr =

0.447 m/s):

(a) A cheetah running at top speed of 60 mi/hr; the local temperature is 40°C

(b) A Peregrine falcon in a dive at 217 mi/hr; local temperature of 25°C

(c) In June 1999 in Athens Greece, Maurice Greene became the world’s fastest human

by running 100 m in 9.79 s; the temperature was 21°C

(d) In June 1999, Alexander Popov became the world’s fastest swimmer by swimming

50 m in 21.64s; the temperature of the water was 20°C

(a) m/s 7.35431305.20a ==

() ()

()

076.0

m/s 7.354

mi/hr

m/s

447.0

hr

mi

60

a

V

M===

(b) m/s 1.34629805.20a ==

()( )

()

28.0

1.346

447.0217

a

V

M===

(c) m/s 8.34329405.20a == m/s 21.10

97.9

100

V== ⎟

⎠

⎞

⎜

⎝

⎛== hr

mi

9.22

447.0

21.10

03.0

8.343

21.10

M==

(d) 2)–2 Table (from m/s 481,1a = m/s 31.2

21.64

50

V== ⎟

⎠

⎞

⎜

⎝

⎛== hr

mi

17.5

447.0

31.2

00156.0

481,1

31.2

M==

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Problem 15. – Given speeds and Mach numbers, assuming air is a perfect gas, determine

the corresponding local temperature (note: 1 mi/hr = 0.447 m/s) for the following:

(a) A Boeing 747-400 at a cruise speed of 910 km/hr; M = 0.85.

(b) Concorde at a cruise speed of 1,320 mi/hr; M = 2.0

(c) The fastest airplane, the Lockheed SR-71 Blackbird, flying at 2,200 mi/hr; M =

3.3

(d) The fastest boat, the Spirit of Australia, that averaged 317.6 mi/hr; M = 0.41

(e) The fastest car, the ThrustSSC, averaged 760.035 mi/hr; M = 0.97

m

m000,910

m

8.252

V

05.20

⎠

⎝

⎠

⎝

m

142

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Problem 16. – A baseball, which has a mass of 145 grams and a diameter of 3.66 cm,

when dropped from a very tall building reaches high speeds. If the building is tall

enough the speed will be controlled by the drag, as the baseball will reach terminal speed.

At this state

D

FW

=

Where W (weight) = mg, g (acceleration of gravity) = 9.81 m/s2, FD (drag force) =

CDρairAV2/2, CD (drag coefficient) = 0.5 and A (projected area of sphere) = πR2. Find the

terminal speed of the baseball and determine the corresponding Mach number if the

ambient air temperature is 23°C and the ambient air pressure is 101 kPa..

The density of the air is first determined:

Problem 17. – Derive the following equation for the speed of sound of a real gas from

Berthelot’s equation of state:

2

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29

Since T is treated as a constant, we may simply use information from Section 2.6 where

()

⎥

⎦

⎢

⎣

βρ−

1

Problem 18. – Using the speed of sound expression from the previous problem and the

following constants for nitrogen

determine the speed of sound for the two cases described in Example 2.4.

Case (1) p 0.3 MPa and T = 300K

Case (2): p 30.0 MPa and T = 300K

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Iteration v f(v) df /dv v-f/(df/dv) ρ a

1 0.002968 -8.2594E-09 3.0708E-06 0.005658 336.9016 604.3973

2 0.005658 5.2436E-08 4.9296E-05 0.004594 176.7430 426.1798

3 0.004594 1.3084E-08 2.5826E-05 0.004088 217.6647 457.9898

4 0.004088 2.2920E-09 1.7035E-05 0.003953 244.6426 483.2795

5 0.003953 1.4088E-10 1.4959E-05 0.003944 252.9695 491.8702

6 0.003944 6.6552E-13 1.4817E-05 0.003944 253.5736 492.5088

7 0.003944 1.5099E-17 1.4817E-05 0.003944 253.5765 492.5118

The result differs from the experimental value 483.18 m/s by 1.9%.

Problem 19. –Employ the finite difference method of Example 2.5 to determine the

speed of sound in nitrogen using the Redlich-Kwong equation of state

()

T1

1

βρ+

βρ−

The Redlich-Kwong equation of state is:

()

Tvv

a

v

RT

po

β+

−

β−

=. Rearrange to obtain:

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31

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