64
2 Macroscopic Thermodynamics
= 1 −
T 4
comp V 3
T 4
expn V 2
.
As both V 3 and V 2 lie on the same adiabat, we know that T 3
comp V 3 = T 3
expn V 2 , so
that η max simplifies further to
η max = 1 −
T comp
T expn
,
which is precisely the same result obtained for an ideal classical gas.
2.5.2 Reverse Carnot Engine (Refrigeration Cycle)
The Carnot cycle illustrated in Fig. 2.2 for a classical ideal gas proceeds in a
clockwise (CW) direction in the P V -plane through the sequence of (P V )-points
1 → 2 → 3 → 4 → 1, and may thus be referred to as a CW Carnot cycle.
A (Carnot) heat engine operates between two heat reservoirs, one at a higher
temperature, T high , from which it imports heat Q import (in accordance with Fig. 2.1),
performs work (−W ) on its surroundings, and exports heat −Q export (with Q export >
0) into the reservoir at temperature T low .
If we examine the outcome of the reverse of this CW Carnot cycle, with the
first step starting from point 2 in Fig. 2.2a, then proceeding in the counterclockwise
(CCW) direction through the cycle 2 → 1 → 4 → 3 → 2, we see that it
corresponds to a reversal of the three directional arrows associated with Q import , W ,
and Q export in Fig. 2.1. This ‘reverse Carnot engine’ will thus import heat Q import
from the reservoir at temperature T low and, driven by work W done on the working
substance, export heat −Q export to the reservoir at temperature T high . Each CCW
cycle will thus transfer heat energy from the lower-temperature heat reservoir to
the hotter-temperature heat reservoir. A CCW Carnot cycle is therefore termed a
refrigeration cycle.
A device that employs a working fluid to import heat from a low-temperature
heat reservoir (which, as implied by the term ‘reservoir’, may be assumed to have
an infinite capacity) and, coupled with work done on the working fluid, export
heat to a high-temperature heat reservoir, is termed a heat pump. In particular,
should the high-temperature heat reservoir be replaced by a finite-mass system
(such as a building) as the ‘reservoir’, this process provides the thermodynamic
basis for commercial heat pumps, whereby an electrical device provides the work
input needed to export heat from the low-temperature heat reservoir (essentially
the surroundings of the building) to the high-temperature ‘reservoir’. If, however,
the low-temperature reservoir is replaced by a finite-mass system as the ‘reservoir’
then, following a (sufficiently large) number of completed refrigeration cycles, that
2 Macroscopic Thermodynamics
= 1 −
T 4
comp V 3
T 4
expn V 2
.
As both V 3 and V 2 lie on the same adiabat, we know that T 3
comp V 3 = T 3
expn V 2 , so
that η max simplifies further to
η max = 1 −
T comp
T expn
,
which is precisely the same result obtained for an ideal classical gas.
2.5.2 Reverse Carnot Engine (Refrigeration Cycle)
The Carnot cycle illustrated in Fig. 2.2 for a classical ideal gas proceeds in a
clockwise (CW) direction in the P V -plane through the sequence of (P V )-points
1 → 2 → 3 → 4 → 1, and may thus be referred to as a CW Carnot cycle.
A (Carnot) heat engine operates between two heat reservoirs, one at a higher
temperature, T high , from which it imports heat Q import (in accordance with Fig. 2.1),
performs work (−W ) on its surroundings, and exports heat −Q export (with Q export >
0) into the reservoir at temperature T low .
If we examine the outcome of the reverse of this CW Carnot cycle, with the
first step starting from point 2 in Fig. 2.2a, then proceeding in the counterclockwise
(CCW) direction through the cycle 2 → 1 → 4 → 3 → 2, we see that it
corresponds to a reversal of the three directional arrows associated with Q import , W ,
and Q export in Fig. 2.1. This ‘reverse Carnot engine’ will thus import heat Q import
from the reservoir at temperature T low and, driven by work W done on the working
substance, export heat −Q export to the reservoir at temperature T high . Each CCW
cycle will thus transfer heat energy from the lower-temperature heat reservoir to
the hotter-temperature heat reservoir. A CCW Carnot cycle is therefore termed a
refrigeration cycle.
A device that employs a working fluid to import heat from a low-temperature
heat reservoir (which, as implied by the term ‘reservoir’, may be assumed to have
an infinite capacity) and, coupled with work done on the working fluid, export
heat to a high-temperature heat reservoir, is termed a heat pump. In particular,
should the high-temperature heat reservoir be replaced by a finite-mass system
(such as a building) as the ‘reservoir’, this process provides the thermodynamic
basis for commercial heat pumps, whereby an electrical device provides the work
input needed to export heat from the low-temperature heat reservoir (essentially
the surroundings of the building) to the high-temperature ‘reservoir’. If, however,
the low-temperature reservoir is replaced by a finite-mass system as the ‘reservoir’
then, following a (sufficiently large) number of completed refrigeration cycles, that