978-0131206687 Chapter 3 Part 1

32

C

Ch

ha

ap

pt

te

er

r

T

Th

hr

re

ee

e

I

IS

SE

EN

NT

TR

RO

OP

PI

IC

C

F

FL

LO

OW

W

O

OF

F

A

P

PE

ER

RF

FE

EC

CT

T

G

GA

AS

S

Problem 1. – Air flows at Mach 0.25 through a circular duct with a diameter of 60 cm.

The stagnation pressure of the flow is 500 kPa; the stagnation temperature is 175°C.

Calculate the mass flow rate through the channel, assuming γ = 1.4 and that the air

behaves as a perfect gas with constant specific heats.

p

⎞

⎛

Problem 2. – Helium flows at Mach 0.50 in a channel with cross-sectional area of 0.16

m2. The stagnation pressure of the flow is 1 MPa, and stagnation temperature is 1000 K.

Calculate the mass flow rate through the channel, with γ = 5/3.

p

⎞

⎛

T

o

⎟

⎠

⎜

⎝

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

33

Problem 3. – In Problem 2, the cross-sectional area is reduced to 0.12 m2. Calculate the

Mach number and flow velocity at the reduced area. What percent of further reduction in

area would be required to reach Mach 1 in the channel?

12.0

A

1

2

2=

⎞

⎛

Problem 4. – (a) For small Mach numbers, determine an expression for the density ratio

ρ /ρo. (b) Using Eqs. (3.15) and (3.17), prove that

oo

o

a

T

p

⎠

⎜

⎝

=

⎟

⎠

⎜

⎝

=

⎟

⎠

⎜

⎝

ρ

⎟

⎠

⎜

⎝

(a)

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

34

⎟

⎠

⎜

⎝

−γ

⎠

⎝

⎠

⎝

ρ

−

2

1

2

1

o

2

o

2

oo

a

Ra

T

⎟

⎠

⎞

⎜

⎝

⎛

=

γ

==

Problem 5. – An airflow at Mach 0.6 passes through a channel with a cross-sectional area

of 50 cm2. The static pressure in the airstream is 50 kPa; static temperature is 298 K.

(a) Calculate the mass flow rate through the channel.

(b) What percent of reduction in area would be necessary to increase the flow

Mach number to 0.8? to 1.0?

(c) What would happen if the area were reduced more than necessary to reach

Mach 1?

(a)

()

3

kg/m 5846.0

K 298Km/kgkN 287.0

kPa50

RT

p=

⋅⋅

==ρ

(b) For 0382.1*A/A,8.0M ==

1882.1

(c) Flow would be reduced.

Problem 6. – A converging nozzle with an exit area of 1.0 cm2 is supplied from an

oxygen reservoir in which the pressure is 500 kPa and the temperature is 1200 K.

Calculate the mass flow rate of oxygen for back pressures of 0, 100, 200, 300, and 400

kPa. Assume that γ = 1.3.

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

For γ = 1.3, the critical pressure ratio is: 5457.0

p

*p

o

=. So, the back pressure is

p

o

b==

⎟

⎠

⎜

⎝

Thus, the nozzle is choked for back-pressures below 272.85 kPa, i.e., for 0, 100, and 200

kPa. For these back pressures, pe = 272.8 kPa and

For pb = pe =300 kPa; thus, ,6.0

500

300

p

o

e== from which we find 9133.0Me

=

For pb = pe = 400 kPa, 5935.0M ,8.0

p

e

o

e==

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

400

⎞

⎛

Problem 7. – Compressed air is discharged through the converging nozzle as shown in

Figure P3.7. The tank pressure is 500 kPa, and local atmospheric pressure is 101 kPa.

The inlet area of the nozzle is 100 cm2; the exit area is 34 cm2. Find the force of the air on

the nozzle, assuming the air to behave as a perfect gas with constant γ = 1.4. Take the

temperature in the tank to be 300 K.

Figure P3.7

Assume the nozzle is choked. Accordingly, pe = 0.5283 (500 kPa) = 264.15 kPa. Since

this pressure exceeds the back pressure, the assumption is valid.

At the nozzle inlet, 2038.0M findwewhich, from9412.2

34

100

*A

A

i

i===

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

()( )

s/kg9673.39321.3160034.0

15.264

me==

&

Problem 8. – A converging nozzle has an exit area of 56 cm. Nitrogen stored in a

reservoir is to be discharged through the nozzle to an ambient pressure of 100 kPa.

Determine the flow rate through the nozzle for reservoir pressures of 120 kPa, 140 kPa,

200 kPa, and 1 MPa. Assume isentropic nozzle flow. In each case, determine the increase

in mass flow to be gained by reducing the back pressure from 100 to 0 kPa. Reservoir

temperature is 298 K.

For N2, γ = 1.40. The nozzle is choked for

()

5283.0

100

p*p

p

o

b

Case 1. po = 120 kPa and pb = 100 kPa

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

eee =×=ρ= −

Case 2. po = 140 kPa and pb = 100 kPa

100

p

e=====

Case 3. po = 200 kPa and pb = 100 kPa

Since po is above the critical reservoir pressure the nozzle is choked, therefore Me = 1.0

Case 4. po = 1 MPa = 1000 kPa and pb = 100 kPa

1000

⎞

⎛

Case 5. po = 120 kPa and pb = 0 kPa

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

Case 6. po = 140 kPa and pb = 0 kPa

120

Case 7.

120

Case 8.

120

Problem 9. – Pressurized liquid water flows from a large reservoir through a converging

nozzle. Assuming isentropic nozzle flow with a negligible inlet velocity and a back

pressure of 101 kPa, calculate the reservoir pressure necessary to choke the nozzle.

Assume that the isothermal compressibility of water is constant at 5 × 10-7 (kPa)-l and

equal to the isentropic compressibility. Exit density of the water is 1000 kg/m3.

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

1

Problem 10. – Calculate the stagnation temperature in an airstream traveling at Mach 5

with a static temperature of 273 K (see Figure P3.10). An insulated flat plate is inserted

into this flow, aligned parallel with the flow direction, with a boundary layer building up

along the plate. Since the absolute velocity at the plate surface is zero, would you expect

the plate temperature to reach the free stream stagnation temperature? Explain.

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

Problem 11. – A gas stored in a large reservoir is discharged through a converging

nozzle. For a constant back pressure, sketch a plot of mass flow rate versus reservoir

pressure. Repeat for a converging-diverging nozzle.

Problem 12. – A converging-diverging nozzle is designed to operate isentropically with

air at an exit Mach number of 1.75. For a constant chamber pressure and temperature of 5

MPa and 200°C, respectively, calculate the following:

(a) Maximum back pressure to choke nozzle

(b) Flow rate in kilograms per second for a back pressure of 101 kPa

(c) Flow rate for a back pressure of 1 MPa Nozzle exit area is 0.12 m2.

A=5

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

Problem 13. – A supersonic flow is allowed to expand indefinitely in a diverging

channel. Does the flow velocity approach a finite limit, or does it continue to increase

indefinitely? Assume a perfect gas with constant specific heats.

V

2

Problem 14. – A converging-diverging frictionless nozzle is used to accelerate an

airstream emanating from a large chamber. The nozzle has an exit area of 30 cm2 and a

throat area of 15 cm2. If the ambient pressure surrounding the nozzle is 101 kPa and the

chamber temperature is 500 K, calculate the following:

(a) Minimum chamber pressure to choke the nozzle

(b) Mass flow rate for a chamber pressure of 400 kPa

(c) Mass flow rate for a chamber pressure of 200 kPa

A

exit =

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

400

⎠

⎝

Problem 15. – Sketch p versus x for the case shown in Figure P3.15.

Figure P3.15

x

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