978-0131206687 Chapter 1

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Problem 1. – Air is stored in a pressurized tank at a pressure of 120 kPa (gage) and a temperature

of 27°C. The tank volume is 1 m3. Atmospheric pressure is 101 kPa and the local acceleration

of gravity is 9.81 m/s2. (a) Determine the density and weight of the air in the tank, and (b)

determine the density and weight of the air if the tank was located on the Moon where the

acceleration of gravity is one sixth that on the Earth.

kpa 122101120PPP

atmgageabs

=+=+=

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32

m

kg

,

m

N

p

⎟

⎠

⎞

⎜

⎝

⎛≈ρ

⎟

⎠

⎞

⎜

⎝

⎛≈

c

g1000

1

factor

=

∴

Problem 3. – Air flows steadily through a circular jet ejector, refer to Figure 1.15. The primary

jet flows through a 10 cm diameter tube with a velocity of 20 m/s. The secondary flow is through

the annular region that surrounds the primary jet. The outer diameter of the annular duct is 30

cm and the velocity entering the annulus is 5 m/s. If the flows at both the inlet and exit are

uniform, determine the exit velocity. Assume the air speeds are small enough so that the flow

may be treated as an incompressible flow, i.e., one in which the density is constant.

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(

)

()

sp

2

o

2

p

s

2

o

s

2

p

2

op

2

p

e

sspp

eVV

D

V

D

VDDVD

A

VAVA

V−+=

−+

=

+

=

()

s/m6667.6520

30

10

52

2=−+=

Problem 4. – A slow leak develops in a storage bottle and oxygen slowly leaks out. The volume

of the bottle is 0.1 m3 and the diameter of the hole is 0.1 mm. The initial pressure is 10 MPa and

the temperature is 20˚C. The oxygen escapes through the hole according to the relation

ee A

T

p

04248.0m =

&

where p is the tank pressure and T is the tank temperature. The constant 0.04248 is based on the

gas constant and the ratio of specific heats of oxygen. The units are: pressure N/m2, temperature

K, area m2 and mass flow rate kg/s. Assuming that the temperature of the oxygen in the bottle

does not change with time, determine the time it takes to reduce the pressure to one half of its

initial value.

this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

ATR 80424.0

p

2⎟

⎞

⎛

hrs 5979.12sec4076.713,46

2

293)8219.259(

mm 0100

m

mm 1.0

4

)04248.0(

2

==

⎠

⎝

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛π

Problem 5. – A normal shock wave occurs in a nozzle in which air is steadily flowing. Because

the shock has a very small thickness, changes in flow variables across the shock may be assumed

to occur without change of cross-sectional area. The velocity just upstream of the shock is 500

m/s, the static pressure is 50 kPa and the static temperature is 250 K. On the downstream side of

the shock the pressure is 137 kPa and the temperature is 343.3 K. Determine the velocity of the

air just downstream of the shock.

s/m 050V

1= ?V2

=

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Problem 6. – A gas flows steadily in a 2.0 cm diameter circular tube with a uniform velocity of

1.0 cm/s and a density ρo. At a cross section farther down the tube, the velocity distribution is

given by V = Uo[1-( r/R)2], with r in centimeters. Find Uo, assuming the gas density to be

ρo[1+( r/R)2].

⎤

⎡⎟

⎞

⎛

2

r

o

22

1o

o111

o11 RRVrdr2VdAVm ρπ=πρ=πρ=ρ= ∫∫

()

oo

22

oo

1

o

522

oo

2

R

UR

3

2

6

1

2

1

RU2

R

r

wheredR2U

rdr2

r

1U

r

1dAVm

ρπ=

⎟

⎠

⎞

⎜

⎝

⎛−πρ=

=ξξξ−ξπρ=

π

⎟

⎞

⎜

⎛−

⎟

⎞

⎜

⎛+ρ=ρ=

∫

&

2

2UR

2

π

2

Problem 7. – For the rocket shown in Figure 1.6, determine the thrust. Assume that exit plane

pressure is equal to ambient pressure.

(

)

2

oH

mm

+

⎞

⎛

+

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Problem 8. – Determine the force F required to push the flat plate of Figure Pl.8 against the

round air jet with a velocity of 10 cm/s. The air jet velocity is 100 cm/s, with a jet diameter of 5.0

cm. Air density is 1.2 kg/m3.

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Problem 10. – A high-pressure oxygen cylinder, typically found in most welding shops,

accidentally is knocked over and the valve on top of the cylinder breaks off. This creates a hole

with a cross-sectional area of 6.5 x 10-4 m

2. Prior to the accident, the internal pressure of the

oxygen is 14 MPa and the temperature is 27˚C. Based on critical flow calculations, the velocity

of the oxygen exiting the cylinder is estimated to be 300 m/s, the exit pressure 7.4 MPa and the

exit temperature 250 K. How much thrust does the oxygen being expelled from the cylinder

generate? What percentage is due to the pressure difference? What percentage due to the exiting

momentum? Atmospheric pressure is 101 kPa. Also note that 0.2248 lbf = 1 N.

Figure P1.10

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8

The thrust due to the pressure is 41% of total and that due to momentum 59%.

Problem 11. – Air enters a hand held hair dryer with a velocity of 3 m/s at a temperature of 20°C

and a pressure of 101 kPa. Internal resistance heaters warm the air and it exits through an area of

20 cm2 with a velocity of 10 m/s at a temperature of 80°C. Assume that internal obstructions do

not appreciably affect the pressure between inlet and exit and that heat transfer to the

surroundings are negligible. Determine the power in kW needed to operate the hair dryer at

steady state.

i

e

this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

⎟

⎞

⎜

⎛+−

⎟

⎞

⎜

⎛+=− 2

V

hm

V

hmWQ

2

1

2

2&&

&

() ()( )

W2051.203,1

kg

kJ

203205.1

0455.03.60019939.0

2

hhmW

12

==

+=

⎥

⎦

⎢

⎣

⎟

⎠

⎜

⎝

+−=− &

Problem 12. – Air is expanded isentropically in a horizontal nozzle from an initial pressure of

1.0 MPa, of a temperature of 800 K, to an exhaust pressure of 101 kPa. If the air enters the

nozzle with a velocity of 100 m/s, determine the air exhaust velocity. Assume the air behaves as

a perfect gas, with R = 0.287 kJ/kg · K and γ = 1.4. Repeat for a vertical nozzle with exhaust

plane 2.0 m above the intake plane.

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Problem 13. – Nitrogen is expanded isentropically in a nozzle from a pressure of 2000 kPa, at a

temperature of 1000 K, to a pressure of 101 kPa. If the velocity of the nitrogen entering the

nozzle is negligible, determine the exit nozzle area required for a nitrogen flow of 0.5 kg/s.

Assume the nitrogen to behave as a perfect gas with constant specific heats, mean molecular

mass of 28.0, and γ = 1.4.

p1 = 2000kPa

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()( )

22

2.1092798.0

V

ρ

Problem 14. – Air enters a compressor with a pressure of 100 kPa and a temperature 20°C; the

mass flow rate is 0.25 kg/s. Compressed air is discharged from the compressor at 800 kPa and

50°C. Inlet and exit pipe diameters are 4.0 cm. Determine the exit velocity of the air at the

compressor outlet and the compressor power required. Assume an adiabatic, steady, flow and

that the air behaves as a perfect gas with constant specific heats; cp = 1.005 kJ/kg · K and

R = 0.287kJ/kg·K.

kJ

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s

Problem 15. – Hot gases enter a jet engine turbine with a velocity of 50 m/s, a temperature of

1200 K, and a pressure of 600 kPa. The gases exit the turbine at a pressure of 250 kPa and a

velocity of 75 m/s. Assume isentropic steady flow and that the hot gases behave as a perfect gas

with constant specific heats (mean molecular mass 25, γ = 1.37). Find the turbine power output

in kJ/(kg of mass flowing through the turbine).

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13

⎟

⎞

⎜

⎛+−

⎟

⎞

⎜

⎛+=− 2

V

hm

V

hmWQ

2

1

11

2

22 &&

&

kg

2000

⎠

⎝

Problem 16. – Hydrogen is stored in a tank at 1000 kPa and 30°C. A valve is opened, which

vents the hydrogen and allows the pressure in the tank to fall to 200 kPa. Assuming that the

hydrogen that remains in the tank has undergone an isentropic process, determine the amount of

hydrogen left in the tank. Assume hydrogen is a perfect gas with constant specific heats; the ratio

of specific heats is 1.4, and the gas constant is 4.124 kJ/kg · K. The tank volume is 2.0m3.

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Problem 17. – Methane enters a constant-diameter, 3 cm duct at a pressure of 200 kPa, a

temperature of 250 K, and a velocity of 20 m/s. At the duct exit, the velocity reaches 25 m/s. For

isothermal steady flow in the duct, determine the exit pressure, mass flow rate, and rate at which

heat is added to the methane. Assume methane behaves as a perfect gas; the ratio of specific

heats is 1.32 (constant) and the mean molecular mass is 16.0.

d = 3cm

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Problem 19. – Two streams of air mix in a constant-area mixing tube of a jet ejector. The

primary jet enters the tube with a speed of 600 m/s, a pressure of 200 kPa and a temperature of

400˚C. The secondary stream enters with a velocity of 30 m/s, a pressure of 200 kPa and a

temperature of 100˚C. The ratio of the area of the secondary flow to the primary jet is 5:1. The

air behaves as a perfect gas with constant specific heats, cp = 1.0045 kJ/kg· K. Using the iterative

numerical procedure described in Example 1.9 determine the velocity, pressure and temperature

of the air leaving the mixing tube.

γ 1.4

n Ve (m/s) Pe (Pa) Te (K)

1 0.0000 101,000.0 293.1500

2 125.1620 244,722.8 695.5757

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16

Problem 20. – The flow exiting a jet ejector was determined by utilizing an iterative numerical

procedure. A more direct approach is possible however. Eliminate pressure Pe between Eqs.

(1.53) and (1.54). Solve for the temperature Te in the resulting expression, and equate it to Eq.

(1.55). This produces a quadratic equation for the velocity Ve. Solve the quadratic to determine

Vm for the same set of conditions given in Example 1.9.

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