fluid_mechanics

582 Chapter 11 ■ Compressible Flow
Differentiating Eq. 11.10 leads to
or
dȟ dǔ R dT
dȟ
dT dǔ
dT R
From Eqs. 11.3, 11.7, and 11.11 we conclude that
c p c v R
(11.11)
(11.12)
Equation 11.12 indicates that the difference between c p and c v is constant for each ideal gas regardless
of temperature. Also c p 7 c v . If the specific heat ratio, k, is defined as
then combining Eqs. 11.12 and 11.13 leads to
c p
k c p
c v
Rk
k 1
(11.13)
(11.14)
and
c v
R
k 1
(11.15)
The gas constant is
related to the specific
heat values.
Actually, c p , c v , and k are all somewhat temperature dependent for any ideal gas. We will assume
constant values for these variables in this book. Values of k and R for some commonly used gases at
nominal temperatures are listed in Tables 1.7 and 1.8. These tabulated values can be used with Eqs.
11.13 and 11.14 to determine the values of c p and c v . Example 11.1 demonstrates how internal energy
and enthalpy changes can be calculated for a flowing ideal gas having constant and c v .
c p
E XAMPLE 11.1
Internal Energy, Enthalpy, and Density for an Ideal Gas
GIVEN Air flows steadily between two sections in a long
straight portion of 4-in.-diameter pipe as is indicated in Fig. E11.1.
The uniformly distributed temperature and pressure at each section
are T 1 540 °R, p 1 100 psia, and T 2 453 °R, p 2 18.4 psia.
Flow
Pipe Control volume
Section (1) Section (2)
D 1 = D 2 = 4 in.
D
SOLUTION
(a) Assuming air behaves as an ideal gas, we can use Eq. 11.5 to
evaluate the change in internal energy between sections 112 and 122.
Thus
ǔ 2 ǔ 1 c v 1T 2 T 1 2
(1)
From Eq. 11.15 we have
c v
R
k 1
and from Table 1.7, R 1716 1ft # lb21slug # °R2 and k 1.4.
Throughout this book, we use the nominal values of k for
common gases listed in Tables 1.7 and 1.8 and consider these values
as being representative. Since English Engineering System
(2)
units are used more often than British Gravitational System units
in compressible flow discussions, we use
to get
F I G U R E E11.1
FIND Calculate the 1a2 change in internal energy between sections
112 and 122, 1b2 change in enthalpy between sections 112 and
122, and 1c2 change in density between sections 112 and 122.
1 slug 32 .174 lbm
R 1716 1ft # lb21slug # °R2
32.174 1lbmslug2
53.3 1ft # lb21lbm # °R2