By using the thermomass-theory approach, the temperature in a thin layer heated by a hot spot is derived in steady states. It is shown that an anomalous temperature profile, which seems to be at odds with the fundamental laws of continuum physics, may occur. The compatibility of this situation with second law of thermodynamics is analyzed in view of the concept of flux limiter.

References

1.
Fourier
,
J.
,
1878
,
The Analytical Theory of Heat
,
Cambridge University Press
,
Cambridge, UK
.
2.
Boltzmann
,
L.
,
1902
,
Leçons sur la théorie des gaz
,
Gauthier-Villars
,
Paris
.
3.
Hill
,
T. L.
,
1994
,
Thermodynamics of Small Systems
,
Dover
,
New York
.
4.
Tzou
,
D. Y.
,
1997
,
Macro to Micro-Scale Heat Transfer. The Lagging Behaviour
,
Taylor and Francis
,
New York
.
5.
Müller
,
I.
, and
Ruggeri
,
T.
,
1998
,
Rational Extended Thermodynamics
,
Springer-Verlag
,
Berlin
.
6.
Chen
,
G.
,
2005
,
Nanoscale Energy Transport and Conversion—A Parallel Treatment of Electrons, Molecules, Phonons, and Photons
,
Oxford University Press
,
Oxford
.
7.
Lebon
,
G.
,
Jou
,
D.
, and
Casas-Vázquez
,
J.
,
2008
,
Understanding Nonequilibrium Thermodynamics
,
Springer
,
Berlin
.
8.
Ferry
,
D. K.
, and
Goodnick
,
S. M.
,
2009
,
Transport in Nanostructures
, 2nd ed.,
Cambridge University Press
,
Cambridge, UK
.
9.
Jou
,
D.
,
Casas-Vázquez
,
J.
, and
Lebon
,
G.
,
2010
,
Extended Irreversible Thermodynamics
, 4th rev. ed.,
Springer
,
Berlin
.
10.
Cattaneo
,
C.
,
1948
, “
Sulla conduzione del calore
,”
Atti Semin. Mat. Fis. Univ. Modena
,
3
, pp.
83
101
.
11.
Morse
,
P. M.
, and
Feshbach
,
H.
,
1953
,
Methods of Theoretical Physics
,
McGraw-Hill
,
New York
.
12.
Vernotte
,
P.
,
1958
, “
Les paradoxes de la théorie continue de léquation de la chaleur
,”
Comput. Rend.
,
246
, pp.
3154
3155
.
13.
Guyer
,
R. A.
, and
Krumhansl
,
J. A.
,
1966
, “
Solution of the Linearized Phonon Boltzmann Equation
,”
Phys. Rev.
,
148
, pp.
766
778
.10.1103/PhysRev.148.766
14.
Guyer
,
R. A.
, and
Krumhansl
,
J. A.
,
1966
, “
Thermal Conductivity, Second Sound and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals
,”
Phys. Rev.
,
148
, pp.
778
788
.10.1103/PhysRev.148.778
15.
Chen
,
G.
,
2001
, “
Ballistic-Diffusive Heat-Conduction Equations
,”
Phys. Rev. Lett.
,
86
, pp.
2297
2300
.10.1103/PhysRevLett.86.2297
16.
Cimmelli
,
V. A.
,
Sellitto
,
A.
, and
Jou
,
D.
,
2009
, “
Nonlocal Effects and Second Sound in a Nonequilibrium Steady State
,”
Phys. Rev. B
,
79
, p.
014303
.10.1103/PhysRevB.79.014303
17.
Cimmelli
,
V. A.
,
Sellitto
,
A.
, and
Jou
,
D.
,
2010
, “
Nonequilibrium Temperatures, Heat Waves, and Nonlinear Heat Transport Equations
,”
Phys. Rev. B
,
81
,
054301
.10.1103/PhysRevB.81.054301
18.
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
, p.
184302
.10.1103/PhysRevB.82.184302
19.
Tzou
,
D. Y.
,
2011
, “
Nonlocal Behavior in Phonon Transport
,”
Int. J. Heat Mass Transfer
,
54
, pp.
475
481
.10.1016/j.ijheatmasstransfer.2010.09.022
20.
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2007
, “
Equation of Motion of a Phonon Gas and Non-Fourier Heat Conduction
,”
J. Appl. Phys.
,
102
, p.
053503
.10.1063/1.2775215
21.
Tzou
,
D. Y.
, and
Guo
,
Z.-Y.
,
2010
, “
Nonlocal Behavior in Thermal Lagging
,”
Int. J. Therm. Sci.
,
49
, pp.
1133
1137
.10.1016/j.ijthermalsci.2010.01.022
22.
Dong
,
Y.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2011
, “
Generalized Heat Conduction Laws Based on Thermomass Theory and Phonon Hydrodynamics
,”
J. Appl. Phys.
,
110
, p.
063504
.10.1063/1.3634113
23.
Dong
,
Y.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2012
, “
General Expression for Entropy Production in Transport Processes Based on the Thermomass Model
,”
Phys. Rev. E
,
85
, p.
061107
.10.1103/PhysRevE.85.061107
24.
Dong
,
Y.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2013
, “
Temperature in Nonequilibrium States and Non-Fourier Heat Conduction
,”
Phys. Rev. E
,
87
, p.
032150
.10.1103/PhysRevE.87.032150
25.
Wang
,
M.
,
Yang
,
N.
, and
Guo
,
Z.-Y.
,
2011
, “
Non-Fourier Heat Conductions in Nanomaterials
,”
J. Appl. Phys.
,
110
, p.
064310
.10.1063/1.3634078
26.
Sellitto
,
A.
, and
Cimmelli
,
V. A.
,
2012
, “
A Continuum Approach to Thermomass Theory
,”
ASME J. Heat Transfer
,
134
, p.
112402
.10.1115/1.4006956
27.
Balandin
,
A. A.
,
Ghosh
,
S.
,
Baoand
,
W.
,
Calizo
,
I.
,
Teweldebrhan
,
D.
,
Miao
,
F.
, and
Lau
,
C.-N.
,
2008
, “
Superior Thermal Conductivity of Single-Layer Graphene
,”
Nano Lett.
,
8
(
3
), pp.
902
907
.10.1021/nl0731872
28.
Liboff
,
R. L.
,
1990
,
Kinetic Theory (Classical, Quantum, and Relativistic Descriptions)
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
29.
Ziman
,
J. M.
,
2001
,
Electrons and Phonons
,
Oxford University Press
,
Oxford
.
30.
Alvarez
,
F. X.
,
Jou
,
D.
, and
Sellitto
,
A.
,
2009
, “
Phonon Hydrodynamics and Phonon-Boundary Scattering in Nanosystems
,”
J. Appl. Phys.
,
105
, p.
014317
.10.1063/1.3056136
31.
Wang
,
M.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2010
, “
General Heat Conduction Equations Based on the Thermomass Theory
,”
Front. Heat Mass Transfer
,
1
, p.
013004
.10.5098/hmt.v1.1.3004
32.
Sellitto
,
A.
,
Cimmelli
,
V. A.
, and
Jou
,
D.
,
2012
, “
Analysis of Three Nonlinear Effects in a Continuum Approach to Heat Transport in Nanosystems
,”
Physica D
,
241
, pp.
1344
1350
.10.1016/j.physd.2012.04.008
33.
Levermore
,
C. D.
,
1984
, “
Relating Eddington Factors to Flux Limiters
,”
J. Quantum Spectrosc. Radiat. Transfer
,
31
, pp.
149
160
.10.1016/0022-4073(84)90112-2
34.
Anile
,
A. M.
,
Pennisi
,
S.
, and
Sammartino
,
M.
,
1991
, “
A Thermodynamical Approach to Eddington Factors
,”
J. Math. Phys.
,
32
, pp.
544
550
.10.1063/1.529391
35.
Wang
,
M.
, and
Guo
,
Z.-Y.
,
2010
, “
Understanding of Temperature and Size Dependences of Effective Thermal Conductivity of Nanotubes
,”
Phys. Lett. A
,
374
, pp.
4312
4315
.10.1016/j.physleta.2010.08.058
36.
Wang
,
H.-D.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2010
, “
Heat Flow Choking in Carbon Nanotubes
,”
Int. J. Heat Mass Transfer
,
53
, pp.
1796
1800
.10.1016/j.ijheatmasstransfer.2010.01.010
37.
Dong
,
Y.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
,
2014
, “
Size Dependent Thermal Conductivity of Si Nanosystems Based on Phonon Gas Dynamics
,”
Physica E
,
56
, pp.
256
262
.10.1016/j.physe.2013.10.006
38.
Li
,
D.
,
Wu
,
Y.
,
Kim
,
P.
,
Shi
,
L.
,
Yang
,
P.
, and
Majumdar
,
A.
,
2003
, “
Thermal Conductivity of Individual Silicon Nanowires
,”
Appl. Phys. Lett.
,
83
,
2934
.10.1063/1.1616981
39.
Torii
,
S.
, and
Yang
,
W.-J.
,
2005
, “
Heat Transfer Mechanisms in Thin Film With Laser Heat Source
,”
Int. J. Heat Mass Transfer
,
48
, pp.
537
544
.10.1016/j.ijheatmasstransfer.2004.09.011
40.
Luzzi
,
R.
,
Vasconcellos
,
A. R.
, and
Galvão Ramos
,
J.
,
2002
,
Predictive Statistical Mechanics: A Nonequilibrium Ensemble Formalism (Fundamental Theories of Physics)
,
Kluwer Academic Publishers
,
Dordrecht
, The Netherlands.
41.
Zhang
,
Z. M.
,
2007
,
Nano/Microscale Heat Transfer
,
McGraw-Hill
,
New York
.
42.
Sellitto
,
A.
,
Jou
,
D.
, and
Bafaluy
,
J.
,
2012
, “
Nonlocal Effects in Radial Heat Transport in Silicon Thin Layers and Graphene Sheets
,”
Proc. R. Soc., London, Sect. A
,
468
, pp.
1217
1229
.10.1098/rspa.2011.0584
43.
Sellitto
,
A.
,
Cimmelli
,
V. A.
, and
Jou
,
D.
,
2013
, “
Entropy Flux and Anomalous Axial Heat Transport at the Nanoscale
,”
Phys. Rev. B
,
87
, p.
054302
.10.1103/PhysRevB.87.054302
44.
Yin
,
M. T.
, and
Cohen
,
M. L.
,
1992
, “
Theory of Lattice-Dynamical Properties of Solids: Application to Si and Ge
,”
Phys. Rev. B
,
26
, pp.
3259
3272
.10.1103/PhysRevB.26.3259
45.
Balandin
,
A.
, and
Wang
,
K. L.
,
1998
, “
Significant Decrease of the Lattice Thermal Conductivity due to Phonon Confinement in a Free-Standing Semiconductor Quantum Well
,”
Phys. Rev. B
,
58
, pp.
1544
1549
.10.1103/PhysRevB.58.1544
46.
Dong
,
Y.
, and
Guo
,
Z.-Y.
,
2011
, “
Entropy Analyses for Hyperbolic Heat Conduction Based on the Thermomass Model
,”
Int. J. Heat Mass Transfer
,
54
, pp.
1924
1929
.10.1016/j.ijheatmasstransfer.2011.01.011
47.
Sellitto
,
A.
, and
Alvarez
,
F. X.
,
2012
, “
Non-Fourier Heat Removal From Hot Nanosystems Through Graphene Layer
,”
NanoMMTA
,
1
, pp.
38
47
.
48.
Mott
,
N.
,
1990
,
Metal Insulator Transitions
, 2nd ed.,
Taylor and Francis
,
London
.
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