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TABLE 12.4 Ideal Gas Expressions for ∆h, ∆u, and ∆s
Variable Specific Heats Constant Specific Heats b
(
(
(
(
(
hT 2 ) – hT 1 ) = ∫ T 2 c p T() T (1) hT 2 ) – hT 1 ) = c p T 2 – T 1 ) (1′)
d
T 1
c p T() p 2 a p 2
T 2
(
(
(
(
sT 2 , p 2 ) – sT 1 , p 1 ) = ∫ ------------- T –d R ln ---- (2) sT 2 , p 2 ) – sT 1 , p 1 ) = c p ln T 2 R ln ---- (2′)
----- –
T 1 T p 1 T 1 p 1
(
(
(
(
(
d
uT 2 ) – uT 1 ) = ∫ T 2 c v T() T (3) uT 2 ) – uT 1 ) = c v T 2 – T 1 ) (3′)
T 1
c v T() v 2 T 2 v 2
T 2
(
(
(
(
sT 2 , v 2 ) – sT 1 , v 1 ) = ∫ ------------ T +d R ln ---- (4) sT 2 , v 2 ) – sT 1 , v 1 ) = c v ln ----- + R ln ---- (4′)
T 1 T v 1 T 1 v 1
s 2 = s 1 s 2 = s 1
p r T 2 ) ( k−1) /k
(
--------------- = p 2 (5) T 2 (5′)
p 2
----
----
----- =
(
p r T 1 ) p 1 T 1
p 1
(
v r T 2 ) k−1
-------------- = v 2 (6) T 2 (6′)
v 2
----
----
----- =
v r T 1 ) v 1 T 1
(
v 1
(
----
a ( sT 1 ,p 1 ) = ( ( p 2
Alternatively, sT 2 ,p 2 ) – s° T 2 ) – s° T 1 ) – R ln .
p 1
b
c p and c v are average values over the temperature interval from T 1 to T 2 .
It can be shown that (∂u/∂v ) T vanishes identically for a gas whose equation of state is exactly given
by Eq. (12.21), and thus the specific internal energy depends only on temperature. This conclusion is
supported by experimental observations beginning with the work of Joule, who showed that the internal
energy of air at low density depends primarily on temperature.
The above considerations allow for an ideal gas model of each real gas: (1) the equation of state is given
by Eq. (12.21) and (2) the internal energy, enthalpy, and specific heats (Table 12.2) are functions of
temperature alone. The real gas approaches the model in the limit of low reduced pressure. At other
states the actual behavior may depart substantially from the predictions of the model. Accordingly, caution
should be exercised when invoking the ideal gas model lest error is introduced.
Specific heat data for gases can be obtained by direct measurement. When extrapolated to zero pressure,
ideal gas-specific heats result. Ideal gas-specific heats also can be calculated using molecular models of
matter together with data from spectroscopic measurements. The following ideal gas-specific heat rela-
tions are frequently useful:
c p T() = c v T() + R (12.22a)
kR
R
c p = -----------, c v = ----------- (12.22b)
k 1 k 1
–
–
where k = c p /c v .
For processes of an ideal gas between states 1 and 2, Table 12.4 gives expressions for evaluating the
changes in specific enthalpy, ∆h, specific entropy, ∆s, and specific internal energy, ∆u. Relations also are
provided for processes of an ideal gas between states having the same specific entropy: s 2 = s 1 . Property
relations and data required by the expressions of Table 12.4: h, u, c p , c v , p r , v r , and s° are obtainable from
the literature—see, for example, Moran and Shapiro (2000).
12.4 Vapor and Gas Power Cycles
Vapor and gas power systems develop electrical or mechanical power from sources of chemical, solar, or
nuclear origin. In vapor power systems the working fluid, normally water, undergoes a phase change from
liquid to vapor, and conversely. In gas power systems, the working fluid remains a gas throughout, although
the composition normally varies owing to the introduction of a fuel and subsequent combustion.
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