2.5 Thermodynamic Engines
81
from V 1 to 5V 1 . Corresponding calculations utilizing the ideal gas equation
of state give pressures P 2 0.3150P 1 , P 3 0.6300P 1 , P 4 = 2P 1 ,
and temperatures T high 1.260T low , T min 0.6300T low . The Otto cycle
(V 1 , P 1 , T low ) → (2V 1 , P 2 , T max ) → (2V 1 , P 3 , T high ) → (V 1 , P 4 , 2T low ) →
(V 1 , P 1 , T low ) thus becomes (V 1 , P 1 , T low ) → (2V 1 , 0.3150P 1 , 0.6300T low ) →
(2V 1 , 0.6300P 1 , 1.260T low ) → (V 1 , 2P 1 , 2T low ) → (V 1 , P 1 , T low ). Notice that in
this CCW Otto cycle, heat Q import =
3
2 Nk B (T high − T low ) = 0.630
3
2 Nk B T low
is imported from the higher-temperature heat reservoir, while heat Q export =
3
2 Nk B (T max − T low ) =
3
2 Nk B T low is exported into the lower-temperature heat
reservoir. Not only is this the reverse of the manner in which a refrigeration cycle
typically operates, it also requires work to be done on the system in order that
Q export > Q import .
Because this Otto cycle starts at a point (V 1 , P 1 ) on the T low heat reservoir
isotherm, the corresponding Carnot cycle must evolve along an adiabat from
(V 1 , P 1 , T low ) to a point (V
2 , P
2 , T high ), which requires the working substance to
move from right to left along the adiabat. It then imports heat from the T high -
reservoir as it expands isothermally to a point (V
3 , P
3 , T high ). Similarly, the system
moves from left to right along a second adiabat to arrive at (V
4 , P
4 , T low ) and finally
moves from right to left as it compresses to return to the starting point along the
T low -isotherm. The corresponding Carnot cycle is thus a CW Carnot cycle, so that
this CCW Otto cycle cannot be a refrigeration cycle. Indeed, we see that this CCW
Otto cycle not only imports heat from the T high -reservoir but it also requires external
work to be done on the system in order to export the heat into the T low -reservoir. It
has been aptly, and somewhat sarcastically, referred to [16] as a ‘cold pump’!
A calculation of the coefficient of performance for this CCW Otto cycle
employing Eq. (2.5.58) gives
ε
CCW
COP =
(1.260 − 0.6300)T low
(2 − 1.260 − 1 + 0.6300)T low
1.703 ,
while a formal calculation of the coefficient of performance for the equivalent
corresponding Carnot cycle can be carried out based upon Eq. (2.5.19) as
ε
Carnot
COP =
Q import
Q export − Q import
=
1.260T low
(1 − 1.260)T low
≈ −4.85 .
That this is a meaningless value provides yet another indication that this CCW Otto
cycle cannot be a refrigeration cycle.
We see that four criteria, namely, that
(a) the cycle contains two adiabats in order to enable the working substance to
achieve both a maximal temperature T max that exceeds the temperature T high
81
from V 1 to 5V 1 . Corresponding calculations utilizing the ideal gas equation
of state give pressures P 2 0.3150P 1 , P 3 0.6300P 1 , P 4 = 2P 1 ,
and temperatures T high 1.260T low , T min 0.6300T low . The Otto cycle
(V 1 , P 1 , T low ) → (2V 1 , P 2 , T max ) → (2V 1 , P 3 , T high ) → (V 1 , P 4 , 2T low ) →
(V 1 , P 1 , T low ) thus becomes (V 1 , P 1 , T low ) → (2V 1 , 0.3150P 1 , 0.6300T low ) →
(2V 1 , 0.6300P 1 , 1.260T low ) → (V 1 , 2P 1 , 2T low ) → (V 1 , P 1 , T low ). Notice that in
this CCW Otto cycle, heat Q import =
3
2 Nk B (T high − T low ) = 0.630
3
2 Nk B T low
is imported from the higher-temperature heat reservoir, while heat Q export =
3
2 Nk B (T max − T low ) =
3
2 Nk B T low is exported into the lower-temperature heat
reservoir. Not only is this the reverse of the manner in which a refrigeration cycle
typically operates, it also requires work to be done on the system in order that
Q export > Q import .
Because this Otto cycle starts at a point (V 1 , P 1 ) on the T low heat reservoir
isotherm, the corresponding Carnot cycle must evolve along an adiabat from
(V 1 , P 1 , T low ) to a point (V
2 , P
2 , T high ), which requires the working substance to
move from right to left along the adiabat. It then imports heat from the T high -
reservoir as it expands isothermally to a point (V
3 , P
3 , T high ). Similarly, the system
moves from left to right along a second adiabat to arrive at (V
4 , P
4 , T low ) and finally
moves from right to left as it compresses to return to the starting point along the
T low -isotherm. The corresponding Carnot cycle is thus a CW Carnot cycle, so that
this CCW Otto cycle cannot be a refrigeration cycle. Indeed, we see that this CCW
Otto cycle not only imports heat from the T high -reservoir but it also requires external
work to be done on the system in order to export the heat into the T low -reservoir. It
has been aptly, and somewhat sarcastically, referred to [16] as a ‘cold pump’!
A calculation of the coefficient of performance for this CCW Otto cycle
employing Eq. (2.5.58) gives
ε
CCW
COP =
(1.260 − 0.6300)T low
(2 − 1.260 − 1 + 0.6300)T low
1.703 ,
while a formal calculation of the coefficient of performance for the equivalent
corresponding Carnot cycle can be carried out based upon Eq. (2.5.19) as
ε
Carnot
COP =
Q import
Q export − Q import
=
1.260T low
(1 − 1.260)T low
≈ −4.85 .
That this is a meaningless value provides yet another indication that this CCW Otto
cycle cannot be a refrigeration cycle.
We see that four criteria, namely, that
(a) the cycle contains two adiabats in order to enable the working substance to
achieve both a maximal temperature T max that exceeds the temperature T high