Minimum entropy production principle (MEPP) is an important variational principle for the evolution of systems to nonequilibrium stationary state. However, its restricted validity in the domain of Onsager's linear theory requires an inverse temperature square-dependent thermal conductivity for heat conduction problems. A previous derivative principle of MEPP still limits to constant thermal conductivity case. Therefore, the present work aims to generalize the MEPP to remove these nonphysical limitations. A new dissipation potential is proposed, the minimum of which thus corresponds to the stationary state with no restriction on thermal conductivity. We give both rigorous theoretical verification of the new extremum principle and systematic numerical demonstration through 1D transient heat conduction with different kinds of temperature dependence of the thermal conductivity. The results show that the new principle remains always valid while MEPP and its derivative principle fail beyond their scopes of validity. The present work promotes a clear understanding of the existing thermodynamic extremum principles and proposes a new one for stationary state in nonlinear heat transport.

References

References
1.
Prigogine
,
I.
,
1980
,
From Being to Becoming Time and Complexity in the Physical Sciences
,
W. H. Freeman
,
San Francisco, CA
.
2.
Jou
,
D.
,
Casas-Vázquez
,
J.
, and
Criado-Sancho
,
M.
,
2011
,
Thermodynamics of Fluids Under Flow
, 2nd ed.,
Springer
,
New York
.
3.
Grmela
,
M.
, and
Öttinger
,
H. C.
,
1997
, “
Dynamics and Thermodynamics of Complex Fluids. I. Development of a General Formalism
,”
Phys. Rev. E
,
56
(
6
), pp.
6620
6632
.
4.
Liu
,
L. H.
, and
Chu
,
S. X.
,
2006
, “
On the Entropy Generation Formula of Radiation Heat Transfer Processes
,”
ASME J. Heat Transfer
,
128
(
5
), pp.
504
506
.
5.
Cimmelli
,
V. A.
,
Sellitto
,
A.
, and
Jou
,
D.
,
2010
, “
Nonlinear Evolution and Stability of the Heat Flow in Nanosystems: Beyond Linear Phonon Hydrodynamics
,”
Phys. Rev. B
,
82
(
18
), p.
184302
.
6.
Sellitto
,
A.
, and
Cimmelli
,
V. A.
,
2012
, “
A Continuum Approach to Thermomass Theory
,”
ASME J. Heat Transfer
,
134
(
11
), p.
112402
.
7.
Guo
,
Y.
, and
Wang
,
M.
,
2015
, “
Phonon Hydrodynamics and Its Applications in Nanoscale Heat Transport
,”
Phys. Rep.
,
595
, pp.
1
44
.
8.
Guo
,
Y.
, and
Wang
,
M.
,
2016
, “
Thermodynamic Analysis of Gas Flow and Heat Transfer in Microchannels
,”
Int. J. Heat Mass Transfer
,
103
, pp.
773
782
.
9.
Müller
,
I.
,
2007
,
A History of Thermodynamics
(The Doctrine of Energy and Entropy),
Springer
,
Heidelberg
.
10.
Onsager
,
L.
,
1931
, “
Reciprocal Relations in Irreversible Thermodynamics I
,”
Phys. Rev.
,
37
(
4
), pp.
405
426
.
11.
Onsager
,
L.
,
1931
, “
Reciprocal Relations in Irreversible Thermodynamics II
,”
Phys. Rev.
,
38
(
12
), pp.
2265
2279
.
12.
Prigogine
,
I.
,
1947
,
Etude Thermodynamique des Processus Irréversibles
,
Desoer
,
Paris, France
.
13.
Prigogine
,
I.
,
1955
,
Thermodynamics of Irreversible Processes
,
Charles C Thomas
, Springfield, IL.
14.
Klein
,
M. J.
, and
Meijer
,
P. H.
,
1954
, “
Principle of Minimum Entropy Production
,”
Phys. Rev.
,
96
(
2
), p.
250
.
15.
De Groot
,
S. R.
, and
Mazur
,
P.
,
1962
,
Non-Equilibrium Thermodynamics
,
Dover Publications
,
New York
.
16.
Glansdorff
,
P.
, and
Prigogine
,
I.
,
1971
,
Thermodynamic Theory of Structure, Stability and Fluctuations
,
Wiley
,
London
.
17.
Jaynes
,
E. T.
,
1980
, “
The Minimum Entropy Production Principle
,”
Annu. Rev. Phys. Chem.
,
31
(
1
), pp.
579
601
.
18.
Kondepudi
,
D.
, and
Prigogine
,
I.
,
1998
,
Modern Thermodynamics: From Heat Engines to Dissipative Structures
,
Wiley
,
New York
.
19.
Fischer
,
F.
,
Svoboda
,
J.
, and
Petryk
,
H.
,
2014
, “
Thermodynamic Extremal Principles for Irreversible Processes in Materials Science
,”
Acta Mater.
,
67
, pp.
1
20
.
20.
Vujanovic
,
B.
, and
Djukic
,
D.
,
1972
, “
On One Variational Principle of Hamilton's Type for Nonlinear Heat Transfer Problem
,”
Int. J. Heat Mass Transfer
,
15
(
5
), pp.
1111
1123
.
21.
Ván
,
P.
, and
Muschik
,
W.
,
1995
, “
Structure of Variational Principles in Nonequilibrium Thermodynamics
,”
Phys. Rev. E
,
52
(
4
), pp.
3584
3590
.
22.
Lucia
,
U.
,
2013
, “
Stationary Open Systems: A Brief Review on Contemporary Theories on Irreversibility
,”
Physica A
,
392
(
5
), pp.
1051
1062
.
23.
Cimmelli
,
V. A.
,
Jou
,
D.
,
Ruggeri
,
T.
, and
Ván
,
P.
,
2014
, “
Entropy Principle and Recent Results in Non-Equilibrium Theories
,”
Entropy
,
16
(
3
), pp.
1756
1807
.
24.
Bejan
,
A.
,
1979
, “
A Study of Entropy Generation in Fundamental Convective Heat Transfer
,”
ASME J. Heat Transfer
,
101
(
4
), pp.
718
725
.
25.
Herwig
,
H.
,
2012
, “
The Role of Entropy Generation in Momentum and Heat Transfer
,”
ASME J. Heat Transfer
,
134
(
3
), p.
031003
.
26.
Martyushev
,
L.
, and
Seleznev
,
V.
,
2006
, “
Maximum Entropy Production Principle in Physics, Chemistry and Biology
,”
Phys. Rep.
,
426
(
1
), pp.
1
45
.
27.
Lebon
,
G.
, and
Dauby
,
P.
,
1990
, “
Heat Transport in Dielectric Crystals at Low Temperature: A Variational Formulation Based on Extended Irreversible Thermodynamics
,”
Phys. Rev. A
,
42
(
8
), pp.
4710
4715
.
28.
Jou
,
D.
,
Lebon
,
G.
, and
Criado-Sancho
,
M.
,
2010
, “
Variational Principles for Thermal Transport in Nanosystems With Heat Slip Flow
,”
Phys. Rev. E
,
82
(
3
), p.
031128
.
29.
Bergman
,
T.
,
Lavine
,
A.
,
Incorpera
,
F.
, and
DeWitt
,
D.
,
2011
,
Fundamentals of Heat and Mass Transfer
,
Wiley
,
Hoboken, NJ
.
30.
Lampinen
,
M.
,
1990
, “
A Problem of the Principle of Minimum Entropy Production
,”
J. Non-Equilib. Thermodyn.
,
15
(
4
), pp.
397
402
.
31.
Barbera
,
E.
,
1999
, “
On the Principle of Minimal Entropy Production for Navier–Stokes-Fourier Fluids
,”
Continuum Mech. Thermodyn.
,
11
(
5
), pp.
327
330
.
32.
Martyushev
,
L.
,
Nazarova
,
A.
, and
Seleznev
,
V.
,
2007
, “
On the Problem of the Minimum Entropy Production in the Nonequilibrium Stationary State
,”
J. Phys. A: Math. Theor.
,
40
(
3
), pp.
371
380
.
33.
Bertola
,
V.
, and
Cafaro
,
E.
,
2008
, “
A Critical Analysis of the Minimum Entropy Production Theorem and Its Application to Heat and Fluid Flow
,”
Int. J. Heat Mass Transfer
,
51
(7–8), pp.
1907
1912
.
34.
Zullo
,
F.
,
2016
, “
Entropy Production in the Theory of Heat Conduction in Solids
,”
Entropy
,
18
(
3
), p.
87
.
35.
Semenov
,
A. M.
,
2002
, “
Numerical Simulation of the Dynamics of Entropy Production in a One-Dimensional Unsteady, Thermally Inhomogeneous System
,”
High Temp.
,
40
(
2
), pp.
320
321
.
36.
Danielewicz-Ferchmin
,
I.
, and
Ferchmin
,
A. R.
,
2000
, “
A Check of Prigogine's Theorem of Minimum Entropy Production in a Rod in a Nonequilibrium Stationary State
,”
Am. J. Phys.
,
68
(
10
), pp.
962
965
.
37.
Palffy-Muhoray
,
P.
,
2001
, “
Comment on ‘A Check of Prigogine's Theorem of Minimum Entropy Production in a Rod in a Nonequilibrium Stationary State’ by Irena Danielewicz-Ferchmin and A. Ryszard Ferchmin [Am. J. Phys., 68(10), pp. 962–965 (2000)]
,”
Am. J. Phys.
,
69
(
7
), pp.
825
826
.
38.
Hoover
,
W. G.
,
2002
, “
Note on ‘Comment on “A Check of Prigogine's Theorem of Minimum Entropy Production in a Rod in a Nonequilibrium Stationary State,” by Irena Danielewicz-Ferchmin and A. Ryszard Ferchmin [Am. J. Phys., 68(10), 962–965 (2000)],’ by Peter Palffy-Muhoray [Am. J. Phys., 69(7), 825–826 (2001)]
,”
Am. J. Phys.
,
70
(
4
), pp.
452
454
.
39.
Glansdorff
,
P.
, and
Prigogine
,
I.
,
1964
, “
On a General Evolution Criterion in Macroscopic Physics
,”
Physica
,
30
(
2
), pp.
351
374
.
40.
Gyarmati
,
I.
,
1970
,
Non-Equilibrium Thermodynamics
(Field Theory and Variational Principles),
Springer-Verlag
,
Berlin
.
41.
Kiss
,
E.
,
1994
, “
On the Validity of the Principle of Minimum Entropy Production
,”
Period. Polytech. Chem. Eng.
,
38
(
3–4
), pp.
183
197
.
42.
Kiss
,
E.
,
1997
, “
On the Principle of Minimum Entropy Production in Quasilinear Case and Its Connection to Statistical Mechanics
,”
Period. Polytech. Chem. Eng.
,
41
(
2
), pp.
205
211
.
43.
Suzuki
,
M.
,
2013
, “
Irreversibility and Entropy Production in Transport Phenomena, III—Principle of Minimum Integrated Entropy Production Including Nonlinear Responses
,”
Physica A
,
392
(
2
), pp.
314
325
.
44.
Rosen
,
P.
,
1953
, “
On Variational Principles for Irreversible Processes
,”
J. Chem. Phys.
,
21
(
7
), pp.
1220
1221
.
45.
Richardson
,
I. W.
,
1969
, “
On the Principle of Minimum Entropy Production
,”
Biophys. J.
,
9
(
2
), pp.
265
267
.
46.
Pleshanov
,
A.
,
2001
, “
On the Extremal Principles in the Theory of Irreversible Processes
,”
Doklady Phys.
,
46
(
11
), pp.
765
769
.
47.
Jou
,
D.
,
Carlomagno
,
I.
, and
Cimmelli
,
V.
,
2015
, “
A Thermodynamic Model for Heat Transport and Thermal Wave Propagation in Graded Systems
,”
Physica E
,
73
, pp.
242
249
.
48.
Guo
,
Y.
, and
Wang
,
M.
,
2016
, “
Lattice Boltzmann Modeling of Phonon Transport
,”
J. Comput. Phys.
,
315
, pp.
1
15
.
49.
Hahn
,
D. W.
, and
Özisik
,
M. N.
,
2012
,
Heat Conduction
, 3rd ed.,
Wiley
,
Hoboken, NJ
.
50.
Vadasz
,
P.
,
2010
, “
Analytical Solution to Nonlinear Thermal Diffusion: Kirchhoff Versus Cole–Hopf Transformations
,”
ASME J. Heat Transfer
,
132
(
12
), p.
121302
.
You do not currently have access to this content.