80
2 Macroscopic Thermodynamics
The Carnot cycle corresponding to the CCW Otto cycle of Fig. 5a must operate
between the same two heat reservoirs (at temperatures T high , T low ). It must start
from (V 1 , P 1 , T high ) (the starting point for the CCW Otto cycle), then undergo an
adiabatic expansion to (V
2 , P
2 , T low ) in step 1, followed by an isothermal expansion
to (V
3 , P
3 , T low ) in step 2, during which the system takes in heat Q import from
the heat reservoir at temperature T low , then undergoes an adiabatic compression to
(V
4 , P
4 , T high ) in step 3, and finally undergoes an isothermal compression to return
the system to its starting point (V 1 , P 1 , T high ), during which the system exports heat
Q export into the heat reservoir at temperature T high . This sequence of steps clearly
describes a CCW Carnot cycle that has an associated coefficient of performance
ε COP given by Eq. (2.5.20).
Example 2.5 A counterclockwise Otto refrigeration cycle.
Let us consider a CCW Otto cycle [16] involving an ideal gas working fluid
operating between a pair of heat reservoirs having temperatures T high and T low .
We shall examine a CCW Otto cycle that begins with volume V 1 and pressure P 1
at temperature T high (the hotter heat reservoir), and for which T max = 2T high , while
V 2 = 5V 1 . The resultant Otto CCW cycle sequence of four steps is illustrated in
Fig. 5a. It may also be represented as the cycle (V 1 , P 1 , T high ) → (5V 1 , P 2 , T min ) →
(5V 1 , P 3 , T low ) → (V 1 , P 4 , 2T high ) → (V 1 , P 1 , T high ). This Otto cycle leads to (see
Problem 20) pressures P 2 0.06840P 1 , P 3 0.13680P 1 , P 4 = 2P 1 , and to
temperatures T low 0.6840T high and T min 0.3420T high =
1
2 T low .
We may utilize these results to evaluate the coefficient of performance, ε COP , for
this CCW Otto cycle as
ε
Otto
COP =
T low − T min
T max − T high − (T low − T min )
= 0.5198
or ε Otto
COP ≈ 0.52. The corresponding value for a CCW Carnot cycle operating
between the same two heat reservoirs is given via Eq. (2.5.20) as
ε
Carnot
COP =
T low
T high − T low
= 2.165 .
Upon noting that, as has been emphasized by Dickerson and Mottmann [16], it is
a corollary of the Second Law that any refrigeration cycle operating between two
heat reservoirs must have a coefficient of performance that is smaller than that for
a Carnot CCW cycle operating between the same two heat reservoirs, it becomes
clear that this CCW Otto cycle does indeed operate as a refrigeration cycle.
Let us now consider a CCW Otto cycle that starts from (V 1 , P 1 ) at
temperature T low , rather than at T high . By analogy with the CCW Otto cycle
considered in Example 2.5, let us fix T max to be 2T low but, in this case,
undergoes an expansion from volume V 1 to volume V 2 = 2V 1 rather than
2 Macroscopic Thermodynamics
The Carnot cycle corresponding to the CCW Otto cycle of Fig. 5a must operate
between the same two heat reservoirs (at temperatures T high , T low ). It must start
from (V 1 , P 1 , T high ) (the starting point for the CCW Otto cycle), then undergo an
adiabatic expansion to (V
2 , P
2 , T low ) in step 1, followed by an isothermal expansion
to (V
3 , P
3 , T low ) in step 2, during which the system takes in heat Q import from
the heat reservoir at temperature T low , then undergoes an adiabatic compression to
(V
4 , P
4 , T high ) in step 3, and finally undergoes an isothermal compression to return
the system to its starting point (V 1 , P 1 , T high ), during which the system exports heat
Q export into the heat reservoir at temperature T high . This sequence of steps clearly
describes a CCW Carnot cycle that has an associated coefficient of performance
ε COP given by Eq. (2.5.20).
Example 2.5 A counterclockwise Otto refrigeration cycle.
Let us consider a CCW Otto cycle [16] involving an ideal gas working fluid
operating between a pair of heat reservoirs having temperatures T high and T low .
We shall examine a CCW Otto cycle that begins with volume V 1 and pressure P 1
at temperature T high (the hotter heat reservoir), and for which T max = 2T high , while
V 2 = 5V 1 . The resultant Otto CCW cycle sequence of four steps is illustrated in
Fig. 5a. It may also be represented as the cycle (V 1 , P 1 , T high ) → (5V 1 , P 2 , T min ) →
(5V 1 , P 3 , T low ) → (V 1 , P 4 , 2T high ) → (V 1 , P 1 , T high ). This Otto cycle leads to (see
Problem 20) pressures P 2 0.06840P 1 , P 3 0.13680P 1 , P 4 = 2P 1 , and to
temperatures T low 0.6840T high and T min 0.3420T high =
1
2 T low .
We may utilize these results to evaluate the coefficient of performance, ε COP , for
this CCW Otto cycle as
ε
Otto
COP =
T low − T min
T max − T high − (T low − T min )
= 0.5198
or ε Otto
COP ≈ 0.52. The corresponding value for a CCW Carnot cycle operating
between the same two heat reservoirs is given via Eq. (2.5.20) as
ε
Carnot
COP =
T low
T high − T low
= 2.165 .
Upon noting that, as has been emphasized by Dickerson and Mottmann [16], it is
a corollary of the Second Law that any refrigeration cycle operating between two
heat reservoirs must have a coefficient of performance that is smaller than that for
a Carnot CCW cycle operating between the same two heat reservoirs, it becomes
clear that this CCW Otto cycle does indeed operate as a refrigeration cycle.
Let us now consider a CCW Otto cycle that starts from (V 1 , P 1 ) at
temperature T low , rather than at T high . By analogy with the CCW Otto cycle
considered in Example 2.5, let us fix T max to be 2T low but, in this case,
undergoes an expansion from volume V 1 to volume V 2 = 2V 1 rather than