Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Nonequilibrium Thermodynamics and Heat Transport at Nanoscale Read chapter Read first chapter

2016 | OriginalPaper | Chapter

2. Linear and Nonlinear Heat-Transport Equations

Authors : Antonio Sellitto, Vito Antonio Cimmelli, David Jou

Published in: Mesoscopic Theories of Heat Transport in Nanosystems

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Nanotechnology, like biotechnology and information technology, nowadays is a growing industry with the potential to greatly change the world. Engineering of nanosystems rapidly developed in recent years, and it actually allows to design and develop mechanical, optical and electronic devices, the characteristic sizes of which may be of the order of tens of nanometers. Nowadays, nanotechnology is also fighting its way in medicine, offering some exciting possibilities which few years ago were only imagined. Devices operating on nanometer length scale always provide new challenges, especially regarding their thermo-mechanical properties, and researchers face great challenges in thermal management and analysis under the extreme conditions.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
previous chapter Nonequilibrium Thermodynamics and Heat Transport at Nanoscale
next chapter Mesoscopic Description of Boundary Effects and Effective Thermal Conductivity in Nanosystems: Phonon Hydrodynamics
Literature
1.
Ackerman, C.C., Bertman, B., Fairbank, H.A., Guyer, R.A.: Second sound in solid helium. Phys. Rev. Lett. 16, 789–791 (1966)CrossRef
2.
3.
Banach, Z., Larecki, W.: Nine-moment phonon hydrodynamics based on the modified Grad-type approach: formulation. J. Phys. A: Math. Gen. 37, 9805–9829 (2004)CrossRefMATHMathSciNet
4.
Banach, Z., Larecki, W.: Nine-moment phonon hydrodynamics based on the maximum-entropy closure: one-dimensional flow. J. Phys. A: Math. Gen. 38, 8781–8802 (2005)CrossRefMATHMathSciNet
5.
Banach, Z., Larecki, W.: Chapman-Enskog method for a phonon gas with finite heat flux. J. Phys. A: Math. Gen. 41, 375502 (18 pp.) (2008)
6.
Barletta, A., Zanchini, E.: Unsteady heat conduction by internal-energy waves in solids. Phys. Rev. B 55, 14208 (5 pp.) (1997)
7.
Boltzmann, L.: Leçons sur la Théorie des Gaz. Gauthier-Villars, Paris (1902)
8.
Bubnov, V.A.: Wave concepts in the theory of heat. Int. J. Heat Mass Transf. 19, 175–184 (1976)CrossRefMATH
9.
Cahill, D.C., Ford, W.K., Goodson, K.E., Mahan, G.D., Majumdar, A., Maris, H.J., Merlin, R., Phillpot, S.R.: Nanoscale thermal transport. J. Appl. Phys. 93, 793–818 (2003)CrossRef
10.
Cao, B.-Y., Guo, Z.-Y.: Equation of motion of a phonon gas and non-Fourier heat conduction. J. Appl. Phys. 102, 053503 (6 pp.) (2007)
11.
Casas-Vázquez, J., Jou, D.: Nonequilibrium equations of state and thermal waves. Acta Physiol. Hung. 66, 99–115 (1989)
12.
13.
Cattaneo, C.: Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. C. R. Acad. Sc. 247, 431–433 (1958)MathSciNet
14.
Chen, G.: Ballistic-diffusion equations for transient heat conduction from nano to macroscales. J. Heat Transf. - T. ASME 124, 320–328 (2001)CrossRef
15.
Chen, G.: Ballistic-diffusive heat-conduction equations. Phys. Rev. Lett. 86, 2297–2300 (2001)CrossRef
16.
Chen, G.: Nanoscale Energy Transport and Conversion - A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. Oxford University Press, Oxford (2005)
17.
18.
Cimmelli, V.A.: An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys. 48, 113510 (13 pp.) (2007)
19.
Cimmelli, V.A.: Different thermodynamic theories and different heat conduction laws. J. Non-Equilib. Thermodyn. 34, 299–333 (2009)CrossRefMATH
20.
Cimmelli, V.A., Frischmuth, K.: Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction. Physica B 400, 257–265 (2007)CrossRef
21.
Cimmelli, V.A., Sellitto, A., Triani, V.: A new thermodynamic framework for second-grade Korteweg-type viscous fluids. J. Math. Phys. 50, 053101 (16 pp.) (2009)
22.
Cimmelli, V.A., Sellitto, A., Jou, D.: Nonlocal effects and second sound in a nonequilibrium steady state. Phys. Rev. B 79, 014303 (13 pp.) (2009)
23.
Cimmelli, V.A., Sellitto, A., Jou, D.: Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations. Phys. Rev. B 81, 054301 (9 pp.) (2010)
24.
Cimmelli, V.A., Sellitto, A., Jou, D.: Nonlinear evolution and stability of the heat flow in nanosystems: beyond linear phonon hydrodynamics. Phys. Rev. B 82, 184302 (9 pp.) (2010)
25.
Cimmelli, V.A., Jou, D., Ruggeri, T., Ván, P.: Entropy principle and recent results in non-equilibrium theories. Entropy 16, 1756–1807 (2014)CrossRefMathSciNet
26.
Coleman, B.D., Fabrizio, M., Owen, D.R.: On the thermodynamics of second sound in dielectric crystals. Arch. Ration. Mech. Anal. 80, 135–158 (1982)CrossRefMATHMathSciNet
27.
De Tomas, C., Cantarero, A., Lopeandia, A.F., Alvarez, F.X.: From kinetic to collective behavior in thermal transport on semiconductors and semiconductor nanostructures. J. Appl. Phys. 115, 164314 (2014)CrossRef
28.
De Tomas, C., Cantarero, A., Lopeandia, A.F., Alvarez, F.X.: Thermal conductivity of group-IV semiconductors from a kinetic-collective model. Proc R. Soc. A 470, 20140371 (12 pp.) (2014)
29.
Dong, Y., Cao, B.-Y., Guo, Z.-Y.: Generalized heat conduction laws based on thermomass theory and phonon hydrodynamics. J. Appl. Phys. 110, 063504 (6 pp.) (2011)
30.
Dong, Y., Cao, B.-Y., Guo, Z.-Y.: General expression for entropy production in transport processes based on the thermomass model. Phys. Rev. E 85, 061107 (8 pp.) (2012)
31.
Dong, Y., Cao, B.-Y., Guo, Z.-Y.: Temperature in nonequilibrium states and non-Fourier heat conduction. Phys. Rev. E 87, 032150 (8 pp.) (2013)
32.
33.
Ferry, D.K., Goodnick, S.M.: Transport in Nanostructures, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRef
34.
35.
Fourier, J.: The Analytical Theory of Heat. Cambridge University Press, Cambridge (1878)MATH
36.
Fryer, M.J., Struchtrup, H.: Moment model and boundary conditions for energy transport in the phonon gas. Contin. Mech. Thermodyn. 26, 593–618 (2014)CrossRefMathSciNet
37.
Grmela, M., Lebon, G., Dauby, P.C., Bousmina, M.: Ballistic-diffusive heat conduction at nanoscale: GENERIC approach. Phys. Lett. A 339, 237–245 (2005)CrossRefMATH
38.
Guyer, R.A., Krumhansl, J.A.: Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, 766–778 (1966)CrossRef
39.
Guyer, R.A., Krumhansl, J.A.: Thermal conductivity, second sound and phonon hydrodynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778–788 (1966)CrossRef
40.
Hill, T.L.: Thermodynamics of Small Systems. Dover, New York (1994)
41.
Jiaung, W.-S., Ho, J.-R.: Lattice-Boltzmann modeling of phonon hydrodynamics. Phys. Rev. E 6, 066710 (13 pp.) (2008)
42.
43.
Jou, D., Casas-Vázquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 4th revised edn. Springer, Berlin (2010)CrossRefMATH
44.
Jou, D., Sellitto, A., Alvarez, F.X.: Heat waves and phonon-wall collisions in nanowires. Proc. R. Soc. A 467, 2520–2533 (2011)CrossRefMathSciNet
45.
Larecki, W., Banach, Z.: Consistency of the phenomenological theories of wave-type heat transport with the hydrodynamics of a phonon gas. J. Phys. A: Math. Theor. 43, 385501 (24 pp.) (2010)
46.
Larecki, W., Banach, Z.: Influence of nonlinearity of the phonon dispersion relation in wave velocities in the four-moment maximum entropy phonon hydrodynamics. Physica D 266, 65–79 (2014)CrossRefMATHMathSciNet
47.
Larecki, W., Piekarski, S.: Phonon gas hydrodynamics based on the maximum entropy principle and the extended field theory of a rigid conductor of heat. Arch. Mech. 46, 163–190 (1991)
48.
Larecki, W., Piekarski, S.: Symmetric conservative form of low-temperature phonon gas hydrodynamics I. - Kinetic aspect of the theory. Nuovo Cimento D 13, 31–53 (1991)CrossRef
49.
Lebon, G.: Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J. Non-Equilib. Thermodyn. 39, 35–59 (2014)CrossRef
50.
Lebon, G., Jou, D., Casas-Vázquez, J., Muschik, W.: Weakly nonlocal and nonlinear heat transport in rigid solids. J. Non-Equilib. Thermodyn. 23, 176–191 (1998)CrossRefMATH
51.
Lebon, G., Jou, D., Casas-Vázquez, J.: Understanding Nonequilibrium Thermodynamics. Springer, Berlin (2008)CrossRefMATH
52.
Lebon, G., Machrafi, H., Grmela, M., Dubois, C.: An extended thermodynamic model of transient heat conduction at sub-continuum scales. Proc. R. Soc. A 467, 3241–3256 (2011)CrossRefMATHMathSciNet
53.
Levermore, C.D.: Relating Eddington factors to flux limiters. J. Quant. Spectrosc. Radiat. Transf. 31, 149–160 (1984)CrossRef
54.
Luikov, A.V., Bubnov, V.A., Soloviev, I.: On wave solutions of the heat-conduction equation. Int. J. Heat Mass Transf. 19, 245–248 (1976)CrossRef
55.
Mongioví, M.S., Jou, D.: Thermodynamical derivation of a hydrodynamical model of inhomogeneous superfluid turbulence. Phys. Rev. B 75, 024507 (14 pp.) (2007)
56.
Mongiovì, M.S., Jou, D., Sciacca, M.: Energy and temperature of superfluid turbulent vortex tangles. Phys. Rev. B 75, 214514 (10 pp.) (2007)
57.
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)CrossRefMATH
58.
Narayanamurti, V., Dynes, R.D.: Observation of second sound in bismuth. Phys. Rev. Lett. 28, 1461–1465 (1972)CrossRef
59.
Peshkov, V.: Second sound in helium II. J. Phys. USSR 8, 381–383 (1944)
60.
Peshkov, V.: Determination of the velocity of propagation of the second sound in helium II. J. Phys. USSR 10, 389–398 (1946)
61.
Reissland, J.A.: The Physics of Phonons. Wiley, London (1973)
62.
Saluto, L., Mongioví, M.S., Jou, D.: Longitudinal counterflow in turbulent liquid helium: velocity profile of the normal component. Z. Angew. Math. Phys. 65, 531–548 (2014)CrossRefMATHMathSciNet
63.
Sciacca, M., Sellitto, A., Jou, D.: Transition to ballistic regime for heat transport in helium II. Phys. Lett. A. 378, 2471–2477 (2014)CrossRefMATHMathSciNet
64.
Sellitto, A., Cimmelli, V.A.: A continuum approach to thermomass theory. J. Heat Transf. - T. ASME 134, 112402 (6 pp.) (2012)
65.
Sellitto, A., Cimmelli, V.A.: Flux limiters in radial heat transport in silicon nanolayers. J. Heat Transf. - T. ASME 136, 071301 (6 pp.) (2014)
66.
Sellitto, A., Cimmelli, V.A., Jou, D.: Analysis of three nonlinear effects in a continuum approach to heat transport in nanosystems. Physica D 241, 1344–1350 (2012)CrossRefMATH
67.
Serdyukov, S.I.: Higher order heat and mass transfer equations and their justification in extended irreversible thermodynamics. Theor. Found. Chem. Eng. 47, 89–103 (2013)CrossRef
68.
Sharma, K.R.: Damped Wave Transport and Relaxation. Elsevier, Amsterdam (2005)
69.
70.
Taitel, Y.: On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. Int. J. Heat Mass Transf. 15, 369–371 (1972)CrossRef
71.
Triani, V., Cimmelli, V.A.: Anisotropic heat transport in rigid solids. J. Non-Equilib. Thermodyn. 37, 377–392 (2012)CrossRefMATH
72.
Tzou, D.Y.: A unified field approach for heat conduction from micro-to-macro-scales. J. Heat Transf. - T. ASME 117, 8–16 (1995)CrossRef
73.
Tzou, D.Y.: Nonlocal behavior in phonon transport. Int. J. Heat Mass Transf. 54, 475–481 (2011)CrossRefMATH
74.
Tzou, D.Y.: Longitudinal and transverse phonon transport in dielectric crystals. J. Heat Transf. - T. ASME 136, 042401 (5 pp.) (2014)
75.
Tzou, D.Y.: Macro- to Microscale Heat Transfer: The Lagging Behaviour, 2nd edn. Wiley, Chichester (2014)
76.
Tzou, D.Y., Guo, Z.-Y.: Nonlocal behavior in thermal lagging. Int. J. Therm. Sci. 49, 1133–1137 (2010)CrossRef
77.
78.
Ván, P., Fülöp, T.: Universality in heat conduction theory: weakly nonlocal thermodynamics. Ann. Phys. (Berlin) 524, 470–478 (2012)
79.
80.
Vernotte, P.: Les paradoxes de la théorie continue de l’équation de la chaleur. C. R. Acad. Sc. 246, 3154–3155 (1958)MathSciNet
81.
Volz, S. (ed.): Thermal Nanosystems and Nanomaterials (Topics in Applied Physics). Springer, Berlin (2010)
82.
Wang, M., Cao, B.-Y., Guo, Z.-Y.: General heat conduction equations based on the thermomass theory. Front. Heat Mass Transf. 1, 013004 (8 pp.) (2010)
83.
Wang, M., Yang, N., Guo, Z.-Y.: Non-Fourier heat conductions in nanomaterials. J. Appl. Phys. 110, 064310 (7 pp.) (2011)
84.
Ward, J.C., Wilks, J.: The velocity of second sound in liquid helium near the absolute zero. Philos. Mag. 42, 314–316 (1951)CrossRef
85.
Zanchini, E.: Hyperbolic heat-conduction theories and nondecreasing entropy. Phys. Rev. B 60, 991–997 (1999)CrossRef
86.
Zhang, Z.M.: Nano/Microscale Heat Transfer. McGraw-Hill, New York (2007)
87.
Metadata
Title
Linear and Nonlinear Heat-Transport Equations
Authors
Antonio Sellitto
Vito Antonio Cimmelli
David Jou
Copyright Year
2016
Book
Mesoscopic Theories of Heat Transport in Nanosystems

Print ISBN: 978-3-319-27205-4

Electronic ISBN: 978-3-319-27206-1

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-27206-1_2

Premium Partners

    Image Credits