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

Variable Specific Heats (Exact Analysis)

When the constant-specific-heat assumption is not appropriate, the isen-
tropic relations developed previously yields results that are not quite accu-
rate. For such cases, we should use an isentropic relation obtained from Eq.
7–39 that accounts for the variation of specific heats with temperature. Set-
ting this equation equal to zero gives

or

(7–48)

where s° 2 is the s° value at the end of the isentropic process.

Relative Pressure and Relative Specific Volume

Equation 7–48 provides an accurate way of evaluating property changes of
ideal gases during isentropic processes since it accounts for the variation of
specific heats with temperature. However, it involves tedious iterations
when the volume ratio is given instead of the pressure ratio. This is quite an
inconvenience in optimization studies, which usually require numerous
repetitive calculations. To remedy this deficiency, we define two new
dimensionless quantities associated with isentropic processes.
The definition of the first is based on Eq. 7–48, which can be
rearranged as

or

The quantity exp(s°/R) is defined as the relative pressurePr. With this def-
inition, the last relation becomes

(7–49)

Note that the relative pressure Pris a dimensionlessquantity that is a func-
tion of temperature only since s° depends on temperature alone. Therefore,
values of Prcan be tabulated against temperature. This is done for air in
Table A–17. The use of Prdata is illustrated in Fig. 7–37.
Sometimes specific volume ratios are given instead of pressure ratios.
This is particularly the case when automotive engines are analyzed. In such
cases, one needs to work with volume ratios. Therefore, we define another
quantity related to specific volume ratios for isentropic processes. This is
done by utilizing the ideal-gas relation and Eq. 7–49:

P 1 v 1

T 1



P 2 v 2

T 2

S

v 2

v 1



T 2

T 1

¬

P 1

P 2



T 2

T 1

¬

Pr 1

Pr 2



T 2 >Pr 2

T 1 >Pr 1

a

P 2

P 1

b

sconst.



Pr 2

Pr 1

P 2

P 1



exp 1 s° 2 >R 2

exp 1 s° 1 >R 2

P 2

P 1

exp¬

s° 2 s° 1

R

s° 2 s° 1 
R ln¬

P 2

P 1

0 s° 2 s° 1 R ln¬

P 2

P 1

Chapter 7 | 359

Process: isentropic

Given: PP 1 , T

P

P

P

P

P

1 , and 2

Find: T 2

T.........

Pr

.........

T 2

T 1

r 2 = r 1

r 1

2

1

read

read

FIGURE 7–37

The use of Prdata for calculating the

final temperature during an isentropic

process.