Contrary to what Dickerson and Mottmann [Am. J. Phys. 84, 413–418 (2016)] state, the temperatures at which a refrigerator's working fluid absorbs heat need not always lie below those at which it expels heat; nor must a refrigerator's thermodynamic cycle have two adiabats. Moreover, what Dickerson and Mottmann call a “comparison Carnot cycle” cannot always be defined. These conclusions are illustrated here using a counter-clockwise Stirling cycle without regeneration. A refrigerator's cold reservoir can absorb some heat and its hot reservoir can expel some heat, so long as the net heat flow is still out of the cold reservoir and into the hot reservoir.

In a recent article in this journal,1 Dickerson and Mottmann (D&M) stressed the fact that not all counterclockwise (CCW) thermodynamic cycles represent refrigerators. However, some of the criteria that D&M suggested to be necessary for a working refrigerator are too stringent. Specifically, it is not true that the range of temperatures over which the working fluid absorbs heat must be entirely below the range of temperatures over which it expels heat; nor is it true that a refrigeration cycle must employ two adiabats. As explained below, a simple Stirling refrigerator (i.e., one employing no heat regeneration) provides a counter-example to both of these criteria.2 

The flaw in D&M's analysis is the assumption that a refrigerator's working fluid can extract heat only from the cold reservoir and reject heat only to the hot reservoir. The correct statement, instead, is that over the course of a full cycle the cold reservoir must have given up a net amount of heat (and therefore, necessarily, the hot reservoir must have absorbed a net amount of heat). Much of the analysis in D&M's paper remains valid, and it is especially important to understand their point that in many refrigeration cycles the extreme temperatures are not the same as (or even close to) the reservoir temperatures.

Consider the quasistatic CCW Stirling cycle shown in Fig. 1. The working fluid is n moles of an ideal gas, with a constant molar specific heat CV. Its volume varies from Vmin to Vmax and it is alternately in thermal contact with reservoirs at temperatures Tin and Tout (there is no regenerator). The cycle consists of four steps, with heat transfers that can be written in terms of the preceding variables and the gas constant R:

  • ab, an isothermal expansion from Vmin to Vmax with the fluid in contact with the cold reservoir, from which heat Qin*=nRTinln(Vmax/Vmin) is extracted;

  • bc, an isochoric heating from Tin to Tout with the fluid in contact with the hot reservoir, from which heat Q*=nCV(ToutTin) is extracted;

  • cd, an isothermal compression from Vmax to Vmin with the fluid in contact with the hot reservoir, to which heat Qout*=nRToutln(Vmax/Vmin) is rejected; and

  • da, an isochoric cooling from Tout to Tin with the fluid in contact with the cold reservoir, to which heat Q*=nCV(ToutTin) is rejected.

Fig. 1.

A CCW Stirling cycle. The working fluid is in contact with the cold reservoir along dab, and with the hot reservoir along bcd. The dashed lines are continuations of the two isotherms. Also sketched to the right is a CCW “comparison Carnot cycle” that utilizes the same two reservoirs.

Fig. 1.

A CCW Stirling cycle. The working fluid is in contact with the cold reservoir along dab, and with the hot reservoir along bcd. The dashed lines are continuations of the two isotherms. Also sketched to the right is a CCW “comparison Carnot cycle” that utilizes the same two reservoirs.

Close modal

Note that the reservoirs with which the fluid is in contact during the isochors bc and da are such that heat spontaneously “flows downhill,” consistently with the second law.3 

The net amount of heat extracted from the cold reservoir is Qin=Qin*Q*, while the net amount of heat rejected to the hot reservoir is Qout=Qout*Q*, so the coefficient of performance (COP) for the CCW Stirling cycle is

KStirling=QinW=QinQoutQin=Qin*Q*Qout*Qin*=(1Q*Qin*)KCarnot,
(1)

where W is the net work delivered to the fluid in a cycle and KCarnot=Tin/(ToutTin) is the COP for a CCW Carnot cycle operating between the same reservoir temperatures (also sketched in Fig. 1). Note also that Tout>Tin entails Qout*>Qin* and that, since Qout* and Qin* obey the Carnot-like relation Qout*/Tout=Qin*/Tin, the net entropy produced in a single cycle can be written

ΔSStirling=(1Tin1Tout)Q*.
(2)

This last result clearly shows that irreversibility in a quasistatic Stirling cycle arises from heat exchanges over finite temperature differences, and corresponds to a net transfer of heat from Tout to Tin. This effect can be viewed as the equivalent of a heat leak between the two reservoirs and would vanish if the Stirling refrigerator used a regenerator (so that Q* would not have to be moved from the hot to the cold reservoir).

As a refrigerator, the CCW Stirling cycle has to effectively extract heat from the cold reservoir Tin (i.e., the “freezer”) while rejecting heat to the hot reservoir Tout (i.e., the “kitchen”). Hence Qin > 0, or Qin*>Q*, must apply, which implies Qout*>Q*, or Qout > 0, ensuring not only that KStirling > 0 but also that KStirling < KCarnot, in accordance with the second law. This CCW Stirling cycle represents a refrigerator where neither are the temperatures over which the fluid absorbs heat entirely below the temperatures over which it rejects heat, nor are there any adiabats, thus contradicting D&M.

If Q*>Qout* instead of Qin*>Q*, so Q*>Qin* also, then Qin < 0 and Qout < 0 both follow and the CCW Stirling cycle no longer represents a refrigerator but is instead what D&M refer to as a “cold pump,” a device that uses work to help move heat from a hot to a cold reservoir.4 

In the intermediate case Qout*Q*Qin*, so Qin ≤ 0 and Qout ≥ 0, the CCW Stirling cycle describes neither a refrigerator nor a “cold pump”, but a device that takes work and delivers heat to both reservoirs (or to a single reservoir if one of the equalities applies). This device, which converts work integrally into heat (i.e., a heater), can be named a “Joule pump” in memory of Joule's famous paddle-wheel experiment.5,6 D&M's statement that all CCW cycles can be divided into two categories, refrigerators and “cold pumps”, is thus contradicted, as has been reported previously.2 Note that the CCW Stirling “Joule pump” does not admit what D&M call a “comparison Carnot cycle,” as none of the reservoirs has heat extracted from it in this mode of operation (i.e., there is no freezer). The usefulness of a comparison Carnot cycle to distinguish between the different types of devices represented by CCW cycles is thus limited, inasmuch as the net heat flows to and from the reservoirs have to be worked out (which immediately identifies the device) before one can ascertain the direction of the “comparison Carnot cycle”.

Figure 2 summarizes the three possible modes of operation of a CCW Stirling cycle, i.e., Qin*>Q* (refrigerator or heat pump), Q*>Qout* (“cold pump”), and Qout*Q*Qin* (“Joule pump”).7 All three modes are consistent with the first and second laws, and all three can be achieved through different choices of the temperature ratio Tout/Tin and the expansion ratio Vmax/Vmin. For example, if Tout=2Tin and CV = 5R/2, the cycle represents a refrigerator if ln(Vmax/Vmin)>5/2, a “cold pump” if ln(Vmax/Vmin)<5/4, and a “Joule pump” at intermediate values of the expansion ratio. All three cases are represented by a diagram similar to that of Fig. 1, with varying isotherm locations and lengths. Note that, with KStirling ≤ 0 for “cold pumps” and “Joule pumps”, the COP condition 0 < KStirling < KCarnot provides a solid, quantitative criterion to discriminate between Stirling refrigerators and the other CCW Stirling cycles. Moreover, a similar analysis, with CV replaced by CP = CV + R, can be applied to the CCW Ericsson cycle (formed by two isotherms and two isobars).

Fig. 2.

Three possible modes of operation of a CCW Stirling cycle: (a) a refrigerator or heat pump, (b) a “cold pump”, and (c) a “Joule pump”.

Fig. 2.

Three possible modes of operation of a CCW Stirling cycle: (a) a refrigerator or heat pump, (b) a “cold pump”, and (c) a “Joule pump”.

Close modal

CCW cycles with two adiabats or two isotherms can represent refrigerators, examples of the former being the CCW Otto, Diesel, Brayton, and Atkinson cycles, and of the latter the CCW Stirling and Ericsson cycles.2 However, neither adiabats nor isotherms are a necessity for refrigeration. In fact, there is no need for the expansion process ab in Fig. 1 to be isothermal, as long as the temperature of the working fluid does not go above Tin and the amount of heat đQ absorbed by the fluid at each point is never negative, thus ensuring heat does “flow downhill” and from the “freezer” into the fluid. For n moles of an ideal gas with constant specific-heat ratio γ = CP/CV, these two conditions imply PV/nRTin and dP/dV ≥ −γP/V for the corresponding curve P(V). The last inequality means that the slope of such a P(V) curve at each point never lies below that of the adiabat through the same point, and stems from imposing8–11 

đQ=nCVdT+PdV=CVR(γPdV+VdP)0.
(3)

A similar discussion applies, mutatis mutandi, to the compression process cd, during which PV/nRTout and dP/dV ≥ −γP/V must simultaneously hold, if heat is indeed to be transferred from the fluid to the “kitchen”.12 

Each of the two curves replacing, slightly below or above, the isotherms in such a modified CCW Stirling cycle has their endpoints at the same temperature, so the respective change in internal energy is zero and the work delivered or received by the ideal gas during the process must equal the heat absorbed from, or expelled to the cold or hot reservoir, respectively. For the expansion curve lying below the Tin isotherm, but joining it at the endpoints, the heat extracted from the “freezer” (the area under the ab curve) is less than Qin* in the CCW Stirling cycle, while the total work input (the area enclosed by the cycle) is larger, thus leading to a COP for the modified cycle smaller than the original KStirling. Although it is less efficient than the original CCW Stirling cycle, the modified CCW cycle just described is indeed a refrigerator, but has neither adiabats (once again contradicting D&M) nor isotherms. For instance, the isotherms in Fig. 1 can be replaced with stepwise, stair-like P(V) curves formed by a succession of alternating short adiabats and isochors.

Refrigeration (or heat pumping) requires that, in a cyclic manner, a working fluid be put in thermal contact with a “freezer” and a “kitchen”, from and to which, respectively, it must extract and give off heat. Suppose the fluid has just finished expelling heat to the “kitchen” and must come subsequently into contact with the “freezer”; there are many ways of doing this, as long as it is guaranteed that the fluid does absorb some heat Qin* from the “freezer” somewhere along the process. For the rest of the process the “freezer” may eventually draw heat from the fluid in some quantity Qcold* (Qcold*=Q* in Fig. 1), provided Qcold*<Qin* so the net heat rejected to the fluid is Qin=Qin*Qcold*>0. Similarly when, after extracting heat from the “freezer”, the fluid comes into contact with the “kitchen”: during this process the fluid must effectively expel some heat Qout* to the “kitchen”, which may also give off some heat Qhot* (Qhot*=Q* in Fig. 1), as long as the net amount of heat absorbed from the fluid obeys Qout=Qout*Qhot*>0. In addition, it must be ensured that, when Qin* and Qout* are exchanged between fluid and reservoirs, they are transferred in the appropriate direction, implying for the corresponding thermodynamic processes dP/dV ≥ −γP/V, in the case of an ideal gas.

The CCW Otto, Diesel, Brayton, and Atkinson cycles,2 which operate with two adiabats (as too restrictively required by D&M), correspond to putting Qcold*=Qhot*=0 and letting Qin* and Qout* be exchanged along isochors or isobars, whose slopes are, respectively, dP/dV =  and dP/dV = 0, both larger than the ideal-gas adiabat slope dP/dV = −γP/V. In particular, the CCW Otto cycle (formed by two adiabats and two isochors and studied in detail by D&M) corresponds to P(V) curves whose slopes are precisely the extreme values allowed by the condition dP/dV ≥ −γP/V. If Qcold* and Qhot* do not vanish and are transferred isochorically or isobarically, while the fluid temperature is pushed to the maximum and minimum values that are still consistent with Qin* and Qout* “flowing downhill” (which implies isothermal heat transfers at Tin and Tout), the CCW Stirling and Ericsson cycles follow.2 

The analysis by D&M1 on the ability of CCW thermodynamic cycles to describe actual refrigerators has been extended here to include cycles that do not employ two adiabats and where the temperatures at which heat is expelled by the working fluid do not lie entirely above those at which heat is absorbed. The concept of a “comparison Carnot cycle” has also been shown to be of no great use in distinguishing between the different types of devices described by CCW cycles. Basically, all that is needed for a CCW cycle to represent a refrigerator, a device that keeps the “freezer” (i.e., the reservoir from which a net amount of heat is absorbed during the cycle) cooler than the “kitchen” (i.e., the reservoir to which a net amount of heat is rejected during the cycle), is that its COP be physically meaningful (i.e., positive and not greater than the COP of the Carnot cycle operating between the same two reservoirs).

The author acknowledges R. H. Dickerson and J. Mottmann for the discussions that followed the submission of this Comment, and is indebted to the anonymous reviewers, whose criticism and suggestions have greatly improved the manuscript. J. S. Ferreira helped to prepare Figs. 1 and 2. This work received financial support from the Fundação para a Ciência e a Tecnologia (FCT, Lisboa) through project No. UID/FIS/50010/2013. The views and opinions expressed herein do not necessarily reflect those of FCT, of IST, or of their services.

1.
R. H.
Dickerson
and
J.
Mottmann
, “
Not all counterclockwise thermodynamic cycles are refrigerators
,”
Am. J. Phys.
84
,
413
418
(
2016
).
2.
The CCW Stirling cycle has also been analyzed by
M. F. F.
da Silva
, “
Some considerations about thermodynamic cycles
,”
Eur. J. Phys.
33
,
13
42
(
2012
).
3.

Within the framework of classical thermodynamics, heat “flowing downhill” includes the marginal situation of heat exchange along isothermals in which the working fluid and reservoir are at the same temperature. The only situation strictly forbidden by the second law is that of heat “flowing uphill.”

4.

A “cold pump” uses work to do something that nature does for free, so its COP |Qin|/W or |Qout|/W (whether it is used to heat the cold reservoir or cool the hot reservoir) is finite for W > 0 and becomes infinite for W = 0.

5.
F.
Reif
,
Fundamentals of Statistical and Thermal Physics
(
McGraw-Hill
,
New York
,
1981
), pp.
184
186
.
6.

A “Joule pump” merely converts work into heat, so its COP (|Qin|+Qout)/W never goes above unity.

7.

See also Fig. 4 of Ref. 2.

8.
R. H.
Dickerson
and
J.
Mottmann
, “
On the thermodynamic efficiencies of reversible cycles with sloping, straight-line processes
,”
Am. J. Phys.
62
,
558
562
(
1994
).
9.

See Fig. 7 of Ref. 8.

10.

Note the distinction between slope and steepness, respectively, dP/dV and |dP/dV|: for instance, wherever the curve P(V) replacing the isotherm a → b in Fig. 1 has a negative slope, the condition dP/dV ≥ −γP/V does not allow it to be steeper than the local adiabat, but it is steeper than the latter for positive slopes larger than γP/V.

11.

In points of the P(V) curve where the fluid temperature is not increasing, its slope must also not go above that of the local isotherm, meaning dP/dV ≤ −P/V.

12.

Note that during compression one has dV < 0, which is to be taken into account when working out the condition đQ0.