fluid_mechanics

584 Chapter 11 ■ Compressible Flow
If and are assumed to be constant for a given gas, Eqs. 11.19 and 11.20 can be integrated to get
c p
c v
Changes in entropy
are related to
changes in temperature,
pressure, and
density.
and
s 2 s 1 c v ln T 2
T 1
R ln r 1
r 2
(11.21)
s 2 s 1 c p ln T 2
T 1
R ln p 2
p 1
(11.22)
Equations 11.21 and 11.22 allow us to calculate the change of entropy of an ideal gas flowing from
one section to another with constant specific heat values 1 and 2.
c p
c v
E XAMPLE 11.2
Entropy for an Ideal Gas
GIVEN Consider the air flow of Example 11.1.
FIND Calculate the change in entropy, s 2 s 1 , between sections
112 and 122.
SOLUTION
Assuming that the flowing air in Fig. E11.1 behaves as an ideal
gas, we can calculate the entropy change between sections by using
either Eq. 11.21 or Eq. 11.22. We use both to demonstrate that
the same result is obtained either way.
From Eq. 11.21,
To evaluate s 2 s 1 from Eq. 1 we need the density ratio, r 1r 2 ,
which can be obtained from the ideal gas equation of state
1Eq. 11.12 as
and thus from Eqs. 1 and 2,
s 2 s 1 c v ln T 2
T 1
R ln r 1
r 2
By substituting values already identified in the Example 11.1
problem statement and solution into Eq. 3 with
a p 1
T 1
b a T 2
p 2
b a
r 1
r 2
a p 1
T 1
b a T 2
p 2
b
s 2 s 1 c v ln T 2
T 1
R ln ca p 1
T 1
b a T 2
p 2
bd
100 psia
540 °R
b a
453 °R
18.4 psia b 4.56
(1)
(2)
(3)
we get
or
or
s 2 s 1 3133 1ft # lb21lbm #
453 °R
°R24 ln a
540 °R b
353.3 1ft # lb21lbm # °R24 ln 4.56
From Eq. 11.22,
s 2 s 1 57.5 1ft # lb21lbm # °R2
s 2 s 1 c p ln T 2
R ln p 2
T 1 p 1
By substituting known values into Eq. 4 we obtain
s 2 s 1 3186 1ft # lb21lbm #
453 °R
°R24 ln a
540 °R b
18.4 psia
353.3 1ft # lb21lbm # °R24 ln a
100 psia b
s 2 s 1 57.5 1ft # lb21lbm # °R2
(Ans)
(4)
(Ans)
COMMENT As anticipated, both Eqs. 11.21 and 11.22 yield
the same result for the entropy change, s 2 s 1 .
Note that since the ideal gas equation of state was used in the
derivation of the entropy difference equations, both the pressures
and temperatures used must be absolute.
If internal energy, enthalpy, and entropy changes for ideal gas flow with variable specific heats
are desired, Eqs. 11.4, 11.8, and 11.19 or 11.20 must be used as explained in Ref. 2. Detailed tables
1see, for example, Ref. 32 are available for variable specific heat calculations.
The second law of thermodynamics requires that the adiabatic and frictionless flow of any fluid
results in ds 0 or s 2 s 1 0. Constant entropy flow is called isentropic flow. For the isentropic
flow of an ideal gas with constant and c v , we get from Eqs. 11.21 and 11.22
c p
c v ln T 2
T 1
R ln r 1
r 2
c p ln T 2
T 1
R ln p 2
p 1
0
(11.23)