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

Explicit relations for differential changes in entropy are obtained by solv-
ing for dsin Eqs. 7–23 and 7–24:

(7–25)

and

(7–26)

The entropy change during a process can be determined by integrating
either of these equations between the initial and the final states. To perform
these integrations, however, we must know the relationship between duor
dhand the temperature (such as ducvdTand dhcpdTfor ideal gases)
as well as the equation of state for the substance (such as the ideal-gas
equation of state PvRT). For substances for which such relations exist,
the integration of Eq. 7–25 or 7–26 is straightforward. For other substances,
we have to rely on tabulated data.
The T dsrelations for nonsimple systems, that is, systems that involve
more than one mode of quasi-equilibrium work, can be obtained in a similar
manner by including all the relevant quasi-equilibrium work modes.

7–8 ■ ENTROPY CHANGE OF LIQUIDS AND SOLIDS

Recall that liquids and solids can be approximated as incompressible sub-
stancessince their specific volumes remain nearly constant during a process.
Thus,dv0 for liquids and solids, and Eq. 7–25 for this case reduces to

(7–27)

since cpcvcand duc dTfor incompressible substances. Then the
entropy change during a process is determined by integration to be

Liquids, solids: (7–28)

where cavgis the averagespecific heat of the substance over the given tem-
perature interval. Note that the entropy change of a truly incompressible
substance depends on temperature only and is independent of pressure.
Equation 7–28 can be used to determine the entropy changes of solids and
liquids with reasonable accuracy. However, for liquids that expand consider-
ably with temperature, it may be necessary to consider the effects of volume
change in calculations. This is especially the case when the temperature
change is large.
A relation for isentropic processes of liquids and solids is obtained by set-
ting the entropy change relation above equal to zero. It gives

Isentropic: (7–29)

That is, the temperature of a truly incompressible substance remains con-
stant during an isentropic process. Therefore, the isentropic process of an
incompressible substance is also isothermal. This behavior is closely
approximated by liquids and solids.

s 2 s 1 cavg ln¬

T 2

T 1

 0 ¬S¬T 2 T 1

s 2 s 1 

2

1

c 1 T 2 ¬

dT

T

cavg ln¬

T 2

T 1

¬¬ 1 kJ>kg#K 2

ds

du

T



c¬dT

T

ds

dh

T



v¬dP

T

ds

du

T

P¬dv

T

Chapter 7 | 351

SEE TUTORIAL CH. 7, SEC. 8 ON THE DVD.

INTERACTIVE

TUTORIAL