Although no reversible thermodynamic cycles exist in nature, nearly all cycles covered in textbooks are reversible. This is a review, clarification, and extension of results and concepts for quasistatic, reversible and irreversible processes and cycles, intended primarily for teachers and students. Distinctions between the latter process types are explained, with emphasis on clockwise (CW) and counterclockwise (CCW) cycles. Specific examples of each are examined, including Carnot, Kelvin and Stirling cycles. For the Stirling cycle, potentially useful task-specific efficiency measures are proposed and illustrated. Whether a cycle behaves as a traditional refrigerator or heat engine can depend on whether it is reversible or irreversible. Reversible and irreversible-quasistatic CW cycles both satisfy Carnot's inequality for thermal efficiency, η η C a r n o t . Irreversible CCW cycles with two reservoirs satisfy the coefficient of performance inequality K K C a r n o t . However, an arbitrary reversible cycle satisfies K K C a r n o t when compared with a reversible Carnot cycle operating between its maximum and minimum temperatures, a potentially counterintuitive result.

1.
R. H.
Dickerson
and
J.
Mottman
, “
Not all CCW thermodynamic cycles are refrigerators
,”
Am. J. Phys.
84
,
413
418
(
2016
).
2.
R.
Clausius
,
The Mechanical Theory of Heat
(
Macmillan and Co
,
London
,
1879
), Chap. X.
3.
See also,
W. H.
Cropper
, “
Rudolf Clausius and the road to entropy
,”
Am. J. Phys.
54
,
1068
1074
(
1986
);
E.
Pellegrino
,
E.
Ghibaudi
, and
L.
Cerruti
, “
Clausius' disgregation: A conceptual relic that sheds light on the second law
,”
Entropy
17
,
4500
4518
(
2015
).
4.

đQ, đW connote inexact differentials, infinitesimals that sum to the non-state functions Q and W. In contrast, dS=đQ/T is the exact differential of the entropy state function.

5.
J. D.
Norton
, “
Thermodynamically reversible processes in statistical physics
,”
Am. J. Phys.
85
,
135
145
(
2017
).
6.
Position-dependent temperatures and mole numbers are used in
D.
Kondepudi
and
I.
Prigogine
,
Modern Thermodynamics: From Heat Engines to Dissipative Structures
(
John Wiley
,
New York
,
2014
).
7.
See
E. G.
Mishchenko
and
P. F.
Pshenichka
, “
Reversible temperature exchange upon thermal contact
,”
Am. J. Phys.
85
,
23
29
(
2017
) for a totally different approach.
8.
W.
Thomson (Lord Kelvin)
, “
On the economy of the heating or cooling of buildings by means of currents of air
,”
Proc. Roy. Philos. Soc. Glasgow
3
,
269
269
(
1852
).
9.

The subsequent invention of vapor compression refrigeration brought refrigeration and heat pump CCW cycles into prominence.

10.
K.
Seidman
and
T. R.
Michalik
, “
The efficiency of reversible heat engines
,”
J. Chem. Ed.
68
,
208
210
(
1991
).
11.

If x=T/T+<1,ΔStotal=(5Nk/2)(x1ln(x))>0 because x – 1, the straight line tangent to ln(x) at x = 1 lies above ln(x) for x1.

12.
H. S.
Leff
and
G. L.
Jones
, “
Irreversibility, entropy production, and thermal efficiency
,”
Am. J. Phys.
43
,
973
980
(
1975
).
13.
F. L.
Curzon
and
B.
Ahlborn
, “
Efficiency of a Carnot engine at maximum power output
,”
Am. J. Phys.
43
,
22
24
(
1975
).
14.
A.
Bejan
, “
Engineering advances on finite-time thermodynamics
,”
(Letter), Am. J. Phys
62
,
11
12
(
1994
).
15.
A.
Bejan
, “
Models of power plants that generate minimum entropy while operating at maximum power
,”
Am. J. Phys
64
,
1054
1059
(
1996
);
An even older history has come to light:
A.
Vaudrey
,
F.
Lanzetta
, and
M. H. B.
Feidt
, “
Reitlinger and the origins of the efficiency at maximum power formula for heat engines
,”
J. Non. Equilib. Thermodyn.
39
,
199
203
(
2014
).
16.
H. S.
Leff
, “
Thermodynamics of combined-cycle electric power plants
,”
Am. J. Phys.
80
,
515
518
(
2012
).
17.
H. S.
Leff
, “
Thermal efficiency at maximum work output: new results for old heat engines
,”
Am. J. Phys.
55
,
602
610
(
1987
).
18.
B.
Andresen
,
Finite-time Thermodynamics
(
Physics Lab II
,
Copenhagen
,
1983
).
19.
D. V.
Schroeder
,
An Introduction to Thermal Physics
(
Addison-Wesley
,
Reading, MA
,
1999
), p.
127
.
20.
M. H.
Ross
 et al,
Efficient Use of Energy
, AIP Conf. Proc. No. 25 (
American Institute of Physics
,
New York
,
1975
), pp.
27
35
;
T. V.
Marcella
, “
Entropy production and the second law of thermodynamics: An introduction to second law analysis
,”
Am. J. Phys.
60
,
888
895
(
1992
).
21.

President Harry Truman famously said “If you can't stand the heat, get out of the kitchen.” For years I jokingly told students that President Truman really knew his thermodynamics.

22.
C. E.
Mungan
, “
Coefficient of performance of Stirling refrigerators
,”
Eur. J. Phys.
28
,
055101
(
2017
).
23.

The task-specific efficiency idea was suggested by a clever AJP referee.

24.
T.
Ehrenfest-Afanassjewa
,
Die Grundlagen der Thermodynamik
(
Brill
,
Leiden
,
1956
), p.
75
;
Believed to be out of print. See also
M. C.
Tobin
, “
Engine efficiencies and the second law of thermodynamics
,”
Am. J. Phys.
37
,
1115
1117
(
1969
).
25.

This cycle is arbitrary within the class of cycles that transfer energy (from, to) reservoirs along the upper path and (to, from) reservoirs along the lower path for (clockwise, CCW) cycles, and are separated by adiabatic paths.

26.

For reversible cycles, the maximum and minimum reservoir temperatures coincide with the arbitrary cycle's maximum and minimum working substance temperatures.

27.

One can derive Eq. (15) for the CW case writing η as a weighted average, η=jfjηjηCarnot with fj=Tin,jΔSj/Tin,ΔS, ηj=1Tout,j/Tin,j, and using Tin,j/Tout,jTin/Tout.

28.

See, for example, p. 128 of Ref. 19.

29.
The Ericsson cycle (https://en.wikipedia.org/wiki/Ericsson_cycle) has alternating isothermal and constant-pressure segments and an argument similar to that for the Stirling cycle can be made regarding compensation of energy exchanges along the two non-isothermal segments. It can also be operated with a regenerator.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.