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

as heat transfer through the duct wall at the same rate and disregarding any

changes in chemical composition. This simplified problem is still too com-

plicated for an elementary treatment of the topic since the flow may involve

friction, variations in duct area, and multidimensional effects. In this sec-

tion, we limit our consideration to one-dimensional flow in a duct of con-

stant cross-sectional area with negligible frictional effects.

Consider steady one-dimensional flow of an ideal gas with constant spe-

cific heats through a constant-area duct with heat transfer, but with negligible

friction. Such flows are referred to as Rayleigh flowsafter Lord Rayleigh

(1842–1919). The conservation of mass, momentum, and energy equations

for the control volume shown in Fig. 17–51 can be written as follows:

Mass equationNoting that the duct cross-sectional area Ais constant, the

relation m

.

1 
m

.

2 or r 1 A 1 V 1 
r 2 A 2 V 2 reduces to

(17–50)

x-Momentum equationNoting that the frictional effects are negligible

and thus there are no shear forces, and assuming there are no external

and body forces, the momentum equation

in the flow (or x-) direction becomes a balance between static pressure

forces and momentum transfer. Noting that the flows are high speed

and turbulent, the momentum flux correction factor is approximately 1

(b
1) and thus can be neglected. Then,

or

(17–51)

Energy equationThe control volume involves no shear, shaft, or other

forms of work, and the potential energy change is negligible. If the rate

of heat transfer is Q

.

and the heat transfer per unit mass of fluid is q


Q

.

/m

.

, the steady-flow energy balance E

.

in
E

.

outbecomes

(17–52)

For an ideal gas with constant specific heats, h
cp T, and thus

(17–53)

or

(17–54)

Therefore, the stagnation enthalpy h 0 and stagnation temperature T 0

change during Rayleigh flow (both increase when heat is transferred to

the fluid and thus qis positive, and both decrease when heat is trans-

ferred from the fluid and thus qis negative).

Entropy changeIn the absence of any irreversibilities such as friction,

the entropy of a system changes by heat transfer only: it increases with

heat gain, and decreases with heat loss. Entropy is a property and thus

q
h 02 h 01 
cp 1 T 02 T 012

q
cp 1 T 2 T 12 

V^22 V^21

2

Q

#

m#ah 1 

V^21

2

b
m#ah 2 

V^22

2

b S qh 1 

V 12

2

h 2 

V^22

2

P 1 r 1 V^21 
P 2 r 2 V^22

P 1 A 1 P 2 A 2 
m

#

V 2 m

#

V 1 S P 1 P 2 
 1 r 2 V 22 V 2  1 r 1 V 12 V 1

aF

!


a

out

bm

#

V

!

a

in

bm

#

V

!

r 1 V 1 
r 2 V 2

Chapter 17 | 861

P 1 , T 1 , r 1 P 2 , T 2 , r 2

V 1

Control

volume

Q

.

V 2

FIGURE 17–51

Control volume for flow in a constant-

area duct with heat transfer and

negligible friction.

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