Modern Engineering Thermodynamics

7.5 The Absolute Temperature Scale 213
If we now follow Carnot’s lead and presume that the thermal efficiency of a reversible heat engine (η T ) rev
depends only on the absolute temperatures of the thermal reservoirs, then we can write
ðη T Þ rev
= 1 − jQ
outj
= 1 − jQ
Lj
Q in
Q H
or
rev
1 − ðη T Þ rev
= jQ
Lj T
= f L
Q H rev
T H
rev
(7.10)
where f( ) is an unknown function that eventually is used to define the absolute temperature scale, and the subscripts
L and H refertothelow-andhigh-temperature reservoirs, respectively.
Now consider the two reversible heat engines connected in series shown in Figure 7.6. The thermal efficiency of
each of the individual reversible heat engines is determined from an analysis of systems A and B individually.
The thermal efficiency of the engine in system A is 5
and that in system B is
ðη T Þ A
= W A
= 1 − jQ 2j
Q 1 Q 1
ðη T Þ B
= W B
= 1 − jQ 3j
Q 2 Q 2
T
= 1 − f 2
T 1
T
= 1 − f 3
T 2
Now, if we include both engines inside the system boundary, as in system C of Figure 7.6, then we have W C =
W A + W B , and utilizing the previous results, we can write
System
C
System
A
System
B
ðη T Þ C
= W C
= W A + W B ð
= Q 1 − jQ 2 jÞ+ ðjQ 2 j − jQ 3 jÞ
Q 1 Q 1 Q 1
= 1 − jQ
2j
+ jQ
2j − jQ 3 j jQ2 j
Q 1 jQ 2 j
High-temperature
thermal source
at temperature T 1
Q 1
Cyclic heat
engine A
(reversible)
Cyclic heat
engine B
(reversible)
= 1 − f
= 1 − f
T 2
T 1
T 2
T 1
+ 1 − f
T
f 3
T 2
(W out ) = W A
Q 2 at temperature T 2
Q 3
Low-temperature
thermal sink
at temperature T 3
FIGURE 7.6
Two reversible heat engines connected in series.
(W out ) = W B
T 3
T 2
f
Q 1
T 2
T 1
(7.11)
We can also compute the heat engine thermal efficiency
of system C as
ðη T Þ C
= W C
Q 1
= Q 1 − jQ 3 j
Q 1
= 1 − jQ 3j
Q 1
T
= 1 − f 3
T 1
(7.12)
Comparing Eqs. (7.11) and (7.12), we conclude that the
following functional relation must hold for the unknown
temperature function, f():
f
T 3
T 1
≡ f
T 2
f
T 1
T 3
T 2
(7.13)
Many common functions do not satisfy this equation.
For example,
sin T
3
≠ sin T
2
sin T 3
T 1 T 1 T 2
log T
3
≠ log T
2
log T 3
T 1 T 1 T 2
exp T
3
≠ exp T
2
exp T 3
T 1 T 1 T 2
5 Since these engines are defined at the outset to be reversible, the rev subscript on the η T , , Q, and W terms in these equations has
been dropped for simplicity. This subscript reappears in the equations at the end of this analysis.