68
2 Macroscopic Thermodynamics
Q 3→4 (surr) = Nk B T 2 ln
V 3
V 4
.
(2.5.24b)
Upon substituting expressions (2.5.24) for Q rev
1→2 (system) and Q 3→4 (surr) for the
Carnot cycle into Eq. (2.5.23) for t and utilizing relation (2.5.11) between the
volumes of the system during the Carnot cycle, we obtain t as
t =
1
σ 1
T 1
T high − T 1
+
1
σ 2
T 2
T 2 − T low
Nk B ln
V 2
V 1
.
(2.5.25a)
From Eq. (2.5.10), we see that the network for a Carnot cycle for a system
consisting of an ideal classical monatomic gas and operating reversibly between two
thermal reservoirs (one characterized by temperature T 1 , the other by temperature
T 2 , with T 2 < T 1 ) is given by
W
rev (system) = Nk B (T 2 − T 1 ) ln
V 2
V 1
,
so that −W rev (system) > 0 represents the net energy output of the Carnot cycle
connected irreversibly to two thermal reservoirs, one characterized by temperature
T high > T 1 , the other characterized by temperature T low < T 2 . The total time
taken for the entire process can now be written as
t =
1
σ 1
T 1
T high − T 1
+
1
σ 2
T 2
T 2 − T low
−W rev (system)
T 1 − T 2
.
(2.5.25b)
We may now utilize expression (2.5.25b) to obtain the power exported by this
arrangement as
P ≡
−W rev (system)
t
= (T 1 − T 2 )
1
σ 1
T 1
T high − T 1
+
1
σ 2
T 2
T 2 − T low
−1
,
or
P (T 1 , T 2 ) =
σ 1 σ 2 (T 1 − T 2 )(T high − T 1 )(T 2 − T low )
T 1 σ 2 (T 2 − T low ) + T 2 σ 1 (T high − T 1 )
.
(2.5.26)
The power output from an endoreversible engine thus depends upon the operating
temperatures T 1 and T 2 of the Carnot cycle, so that we should now determine the
temperatures most appropriate for its operation. We therefore wish to determine
operating temperatures T 1,opt and T 2,opt that maximize the power output of the
Carnot cycle.
By defining two new variables x and y as [13]
x ≡ T high − T 1 , y ≡ T 2 − T low ,
(2.5.27)