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

and

(7–36)

Variable Specific Heats (Exact Analysis)

When the temperature change during a process is large and the specific

heats of the ideal gas vary nonlinearly within the temperature range, the

assumption of constant specific heats may lead to considerable errors in

entropy-change calculations. For those cases, the variation of specific heats

with temperature should be properly accounted for by utilizing accurate

relations for the specific heats as a function of temperature. The entropy

change during a process is then determined by substituting these cv(T) or

cp(T) relations into Eq. 7–31 or 7–32 and performing the integrations.

Instead of performing these laborious integrals each time we have a new

process, it is convenient to perform these integrals once and tabulate the

results. For this purpose, we choose absolute zero as the reference tempera-

ture and define a function s°as

(7–37)

Obviously,s° is a function of temperature alone, and its value is zero at

absolute zero temperature. The values of s° are calculated at various temper-

atures, and the results are tabulated in the appendix as a function of temper-

ature for air. Given this definition, the integral in Eq. 7–32 becomes

(7–38)

where s° 2 is the value of s° at T 2 and s° 1 is the value at T 1. Thus,

(7–39)

It can also be expressed on a unit-mole basis as

(7–40)

Note that unlike internal energy and enthalpy, the entropy of an ideal gas

varies with specific volume or pressure as well as the temperature. There-

fore, entropy cannot be tabulated as a function of temperature alone. The s°

values in the tables account for the temperature dependence of entropy (Fig.

7–33). The variation of entropy with pressure is accounted for by the last

term in Eq. 7–39. Another relation for entropy change can be developed

based on Eq. 7–31, but this would require the definition of another function

and tabulation of its values, which is not practical.

s 2 s 1 s° 2 s° 1 Ru ln¬

P 2

P 1

¬¬ 1 kJ>kmol#K 2

s 2 s 1 s° 2 s° 1 R ln¬

P 2

P 1

¬¬ 1 kJ>kg#K 2



2

1

cp 1 T 2 ¬

dT

T

s° 2 s° 1

s°

T

0

cp 1 T 2 ¬

dT

T

s 2 s 1 cp,avg ln¬

T 2

T 1

Ru ln¬

P 2

P 1

¬¬ 1 kJ>kmol#K 2

356 | Thermodynamics

(Table A-17)

T, K

.

.

.

300

310

320

.

.

.

s°, kJ/kg • K

.

.

.

1.70203

1.73498

1.76690

.

.

.

FIGURE 7–33

The entropy of an ideal gas depends
on both Tand P. The function s
represents only the temperature-
dependent part of entropy.