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

Chapter 7 | 361

Alternative Solution The final temperature could also be determined from
Eq. 7–42 by assuming constant specific heats for air:

The specific heat ratio kalso varies with temperature, and we need to use
the value of kcorresponding to the average temperature. However, the final
temperature is not given, and so we cannot determine the average tempera-
ture in advance. For such cases, calculations can be started with a kvalue
at the initial or the anticipated average temperature. This value could be
refined later, if necessary, and the calculations can be repeated. We know
that the temperature of the air will rise considerably during this adiabatic
compression process, so we guessthe average temperature to be about 450 K.
The kvalue at this anticipated average temperature is determined from Table
A–2bto be 1.391. Then the final temperature of air becomes

This gives an average temperature value of 480.1 K, which is sufficiently
close to the assumed value of 450 K. Therefore, it is not necessary to repeat
the calculations by using the kvalue at this average temperature.
The result obtained by assuming constant specific heats for this case is in
error by about 0.4 percent, which is rather small. This is not surprising since
the temperature change of air is relatively small (only a few hundred
degrees) and the specific heats of air vary almost linearly with temperature
in this temperature range.

T 2  1 295 K 2182 1.391^1 665.2 K

a

T 2

T 1

b

sconst.

a

v 1

v 2

b

k 1

EXAMPLE 7–11 Isentropic Compression of an Ideal Gas

Helium gas is compressed by an adiabatic compressor from an initial state
of 14 psia and 50°F to a final temperature of 320°F in a reversible manner.
Determine the exit pressure of helium.

Solution Helium is compressed from a given state to a specified pressure
isentropically. The exit pressure of helium is to be determined.
Assumptions At specified conditions, helium can be treated as an ideal gas.
Therefore, the isentropic relations developed earlier for ideal gases are
applicable.
Analysis A sketch of the system and the T- s diagram for the process are
given in Fig. 7–40.
The specific heat ratio kof helium is 1.667 and is independent of tem-
perature in the region where it behaves as an ideal gas. Thus the final pres-
sure of helium can be determined from Eq. 7–43:

P 2 P 1 a

T 2

T 1

b

k>1k 12

 1 14 psia2a

780 R

510 R

b

1.667>0.667

40.5 psia

Process: isentropic

Given: 1 , T 1 , and 2

Find: T 2

T.........

r.........

T 2

T 1

read =

read

vv

v

v 2

vr (^2) v
1
vr 1
vr 1
FIGURE 7–39
The use of vrdata for calculating the
final temperature during an isentropic
process (Example 7–10).