Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

17–2 ■ SPEED OF SOUND AND MACH NUMBER

An important parameter in the study of compressible flow is the speed of

sound(or the sonic speed), which is the speed at which an infinitesimally

small pressure wave travels through a medium. The pressure wave may be

caused by a small disturbance, which creates a slight rise in local pressure.

To obtain a relation for the speed of sound in a medium, consider a pipe

that is filled with a fluid at rest, as shown in Fig. 17–7. A piston fitted in the

pipe is now moved to the right with a constant incremental velocity dV, cre-

ating a sonic wave. The wave front moves to the right through the fluid at

the speed of sound cand separates the moving fluid adjacent to the piston

from the fluid still at rest. The fluid to the left of the wave front experiences

an incremental change in its thermodynamic properties, while the fluid on

the right of the wave front maintains its original thermodynamic properties,

as shown in Fig. 17–7.

To simplify the analysis, consider a control volume that encloses the wave

front and moves with it, as shown in Fig. 17–8. To an observer traveling

with the wave front, the fluid to the right will appear to be moving toward

the wave front with a speed of cand the fluid to the left to be moving away

from the wave front with a speed of cdV. Of course, the observer will

think the control volume that encloses the wave front (and herself or him-

self) is stationary, and the observer will be witnessing a steady-flow process.

The mass balance for this single-stream, steady-flow process can be

expressed as

or

By canceling the cross-sectional (or flow) area Aand neglecting the higher-

order terms, this equation reduces to

(a)

No heat or work crosses the boundaries of the control volume during this

steady-flow process, and the potential energy change, if any, can be

neglected. Then the steady-flow energy balance ein
eoutbecomes

h

c^2

2

hdh

1 cdV 22

2

c drr dV
 0

rAc
 1 rdr 2 A 1 cdV 2

m#right
m#left

Chapter 17 | 827

Disregarding potential energy changes and heat transfer, the compressor

work per unit mass of air is determined from Eq. 17–8:

Thus the work supplied to the compressor is 233.9 kJ/kg.

Discussion Notice that using stagnation properties automatically accounts

for any changes in the kinetic energy of a fluid stream.

233.9 kJ/kg

1 1.005 kJ>kg#K 21 519.5 K286.8 K 2

win
cp 1 T 02 T 012

x

dV

rrr + d

Moving

Piston wave front

Stationary

P + dP fluid

h + dh

P

h

dV

V

(^0) x
P + dP
P
P
c
FIGURE 17–7
Propagation of a small pressure wave
along a duct.
dV
rrr + d
Control volume
traveling with
the wave front
P + dP
h + dh
P
c – c h
FIGURE 17–8
Control volume moving with the small
pressure wave along a duct.
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