Appendix C
C.1 Endoreversible Thermodynamics
“Endoreversible thermodynamics is a subset of irreversible thermodynamics aimed
at making more realistic assumptions about heat transfer than are typically made
in reversible thermodynamics. It gives an upper bound on the energy that can be
derived from a real process that is lower than that predicted by Carnot for a Carnot
cycle, and accommodates the exergy destruction occurring as heat is transferred
irreversibly.” [1].
Endoreversible thermodynamics was discovered in simultaneous work by
Novikov [2] and Chambadal, [3] although sometimes mistakenly attributed to
Curzon and Ahlborn [4]. Later, Rebhan [6], studied a Carnot engine with thermal
losses and friction. He found that the efficiency of these machines was bounded
from above by the Curzon-Ahlborn expression. Van den Broeck, [7] has shown,
in the framework of non-equilibrium thermodynamics, that the efficiency of any
thermal engine is bounded from above by η CA . Curzon and Ahlborn in the original
derivation for the efficiency of a Carnot engine with thermal losses, time appears
explicitly and this is disappointing in the framework of classical thermodynamics.
Miranda, [8] made a derivation without any explicit reference to time.
The efficiency of a heat engine is given by, see Fig. C.1
η =
W
Q H
= 1 −
Q C
Q H
(C.1)
The Carnot’s engine is reversible, see Fig. C.2, this means non-entropy production,
S tot = S E = 0.
S tot = −
Q C
T C
+
Q H
T H
+ S E = 0.then,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
157
C.1 Endoreversible Thermodynamics
“Endoreversible thermodynamics is a subset of irreversible thermodynamics aimed
at making more realistic assumptions about heat transfer than are typically made
in reversible thermodynamics. It gives an upper bound on the energy that can be
derived from a real process that is lower than that predicted by Carnot for a Carnot
cycle, and accommodates the exergy destruction occurring as heat is transferred
irreversibly.” [1].
Endoreversible thermodynamics was discovered in simultaneous work by
Novikov [2] and Chambadal, [3] although sometimes mistakenly attributed to
Curzon and Ahlborn [4]. Later, Rebhan [6], studied a Carnot engine with thermal
losses and friction. He found that the efficiency of these machines was bounded
from above by the Curzon-Ahlborn expression. Van den Broeck, [7] has shown,
in the framework of non-equilibrium thermodynamics, that the efficiency of any
thermal engine is bounded from above by η CA . Curzon and Ahlborn in the original
derivation for the efficiency of a Carnot engine with thermal losses, time appears
explicitly and this is disappointing in the framework of classical thermodynamics.
Miranda, [8] made a derivation without any explicit reference to time.
The efficiency of a heat engine is given by, see Fig. C.1
η =
W
Q H
= 1 −
Q C
Q H
(C.1)
The Carnot’s engine is reversible, see Fig. C.2, this means non-entropy production,
S tot = S E = 0.
S tot = −
Q C
T C
+
Q H
T H
+ S E = 0.then,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
157