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

Isentropic Processes of Ideal Gases

Several relations for the isentropic processes of ideal gases can be obtained

by setting the entropy-change relations developed previously equal to zero.

Again, this is done first for the case of constant specific heats and then for

the case of variable specific heats.

Constant Specific Heats (Approximate Analysis)

When the constant-specific-heat assumption is valid, the isentropic relations

for ideal gases are obtained by setting Eqs. 7–33 and 7–34 equal to zero.

From Eq. 7–33,

which can be rearranged as

(7–41)

or

(7–42)

since Rcpcv,kcp/cv, and thus R/cvk1.

Equation 7–42 is the first isentropic relationfor ideal gases under the

constant-specific-heat assumption. The second isentropic relationis obtained

in a similar manner from Eq. 7–34 with the following result:

(7–43)

The third isentropic relationis obtained by substituting Eq. 7–43 into Eq.

7–42 and simplifying:

(7–44)

Equations 7–42 through 7–44 can also be expressed in a compact form as

(7–45)

(7–46)

(7–47)

The specific heat ratio k, in general, varies with temperature, and thus an

average kvalue for the given temperature range should be used.

Note that the ideal-gas isentropic relations above, as the name implies, are

strictly valid for isentropic processes only when the constant-specific-heat

assumption is appropriate (Fig. 7–36).

Pvkconstant

TP^11 k2>kconstant¬¬ 1 ideal gas 2

Tvk^1 constant

a

P 2

P 1

b

sconst.

a

v 1

v 2

b

k

¬¬ 1 ideal gas 2

a

T 2

T 1

b

sconst.

a

P 2

P 1

b

1 k 1 2>k

¬¬ 1 ideal gas 2

a

T 2

T 1

b

sconst.

a

v 1

v 2

b

k 1

¬¬ 1 ideal gas 2

ln¬

T 2

T 1

ln¬a

v 1

v 2

b

R>cv

ln¬

T 2

T 1



R

cv¬¬

ln

v 2

v 1

358 | Thermodynamics

VALID FOR

*ideal gas

*isentropic process

*constant specific heats

T 2

(T 1 s( = const.

P 2

(P 1 (

(k –1)/k

=^1

( 2 (

k –1

= v

v

FIGURE 7–36

The isentropic relations of ideal gases
are valid for the isentropic processes
of ideal gases only.