2.5 Thermodynamic Engines
65
finite-mass system will have been cooled to a lower temperature, and will therefore
have been refrigerated.
As the internal energy U is a state function, we know that U ≡ 0 for a
closed thermodynamic cycle, so that by the First Law (essentially the conservation
of energy), we have
U (cycle) ≡ 0 = W + Q
CCW
import − Q
CCW
export ,
with W , Q CCW
import , Q CCW
export all positive, or
Q
CCW
export = W + Q
CCW
import .
(2.5.17)
Were we to attempt to evaluate an efficiency η CCW using Eq. (2.5.13) as the
ratio of Q CCW
export to Q CCW
import , we would obtain a negative value for it, which is
meaningless (see also Problem 21). For this reason, we need to identify a meaningful
indicator of the effectiveness of a CCW (or refrigeration) cycle. Such an indicator
is afforded by the ratio of the heat, Q CCW
import , imported from the lower-temperature
reservoir to the external work input that, according to the First Law, is given by the
difference between Q CCW
export and Q CCW
import . This ratio is referred to as the ‘coefficient
of performance’, which we shall designate ε COP . Thus, ε COP is defined in general
for a refrigeration cycle as
ε COP ≡
Q CCW
import
W
=
Q CCW
import
Q CCW
export − Q CCW
import
.
(2.5.18)
For an ideal gas working system, we may compute Q CCW
import from the isothermal
expansion step as
Q
CCW
import ≡ Q 4→3 (gas) = Nk B T low ln
V 3
V 4
(2.5.19a)
and Q CCW
export from the isothermal compression step via
Q 2→1 (gas) = Nk B T high ln
V 1
V 2
≡ −Q
CCW
export .
(2.5.19b)
We have already seen via Eq. (2.5.11) that the two adiabats of the Carnot cycle
require the volumes V i (i = 1, . . . , 4) to be such that the condition V 2 V 4 = V 1 V 3 is
met.
More specifically, in terms of our expressions for Q CCW
import and Q CCW
export for an ideal
gas working substance and a reverse Carnot cycle, we see that ε COP is given by
65
finite-mass system will have been cooled to a lower temperature, and will therefore
have been refrigerated.
As the internal energy U is a state function, we know that U ≡ 0 for a
closed thermodynamic cycle, so that by the First Law (essentially the conservation
of energy), we have
U (cycle) ≡ 0 = W + Q
CCW
import − Q
CCW
export ,
with W , Q CCW
import , Q CCW
export all positive, or
Q
CCW
export = W + Q
CCW
import .
(2.5.17)
Were we to attempt to evaluate an efficiency η CCW using Eq. (2.5.13) as the
ratio of Q CCW
export to Q CCW
import , we would obtain a negative value for it, which is
meaningless (see also Problem 21). For this reason, we need to identify a meaningful
indicator of the effectiveness of a CCW (or refrigeration) cycle. Such an indicator
is afforded by the ratio of the heat, Q CCW
import , imported from the lower-temperature
reservoir to the external work input that, according to the First Law, is given by the
difference between Q CCW
export and Q CCW
import . This ratio is referred to as the ‘coefficient
of performance’, which we shall designate ε COP . Thus, ε COP is defined in general
for a refrigeration cycle as
ε COP ≡
Q CCW
import
W
=
Q CCW
import
Q CCW
export − Q CCW
import
.
(2.5.18)
For an ideal gas working system, we may compute Q CCW
import from the isothermal
expansion step as
Q
CCW
import ≡ Q 4→3 (gas) = Nk B T low ln
V 3
V 4
(2.5.19a)
and Q CCW
export from the isothermal compression step via
Q 2→1 (gas) = Nk B T high ln
V 1
V 2
≡ −Q
CCW
export .
(2.5.19b)
We have already seen via Eq. (2.5.11) that the two adiabats of the Carnot cycle
require the volumes V i (i = 1, . . . , 4) to be such that the condition V 2 V 4 = V 1 V 3 is
met.
More specifically, in terms of our expressions for Q CCW
import and Q CCW
export for an ideal
gas working substance and a reverse Carnot cycle, we see that ε COP is given by