Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition

In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss two- and three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. The solution of the proposed systems is analyzed, and it is shown that it accounts for two modes: a slow diffusive mode, and a fast advective mode. This latter additional mode makes them suitable for heat transfer phenomena on fast time-scales, such as high-frequency pulses and heat transfer in small-scale devices. We finally show that, if proper initial conditions are provided, the advective mode disappears, and the solution of the system tends asymptotically to the transient solution of the classical parabolic heat-conduction equation.

[1]  M. K. Moallemi,et al.  Experimental evidence of hyperbolic heat conduction in processed meat , 1995 .

[2]  Vladimir Pushkarev,et al.  Thermal transport phenomena in nanoparticle suspensions , 2016 .

[3]  D. Tzou,et al.  On the Wave Theory in Heat Conduction , 1994 .

[4]  Gang Chen,et al.  Applied Physics Reviews Nanoscale Thermal Transport. Ii. 2003–2012 , 2022 .

[5]  Robert A. Guyer,et al.  Thermal Conductivity, Second Sound, and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals , 1966 .

[6]  A. Compte,et al.  The generalized Cattaneo equation for the description of anomalous transport processes , 1997 .

[7]  Bahgat Sammakia,et al.  Multiscale thermal device modeling using diffusion in the Boltzmann Transport Equation , 2013 .

[8]  Haitao Qi,et al.  Transient fractional heat conduction with generalized Cattaneo model , 2014 .

[9]  Guillermo Fernández-Anaya,et al.  On generalized fractional Cattaneo’s equations , 2011 .

[10]  Dinesh Kumar,et al.  Numerical solution of non-linear dual-phase-lag bioheat transfer equation within skin tissues. , 2017, Mathematical biosciences.

[11]  Jing Liu,et al.  Nanoparticle-mediated cryosurgery for tumor therapy. , 2018, Nanomedicine : nanotechnology, biology, and medicine.

[12]  Wenshan Cai,et al.  3D self-assembly of aluminium nanoparticles for plasmon-enhanced solar desalination , 2016, Nature Photonics.

[13]  R. Bracewell.  The Fourier transform. , 1989, Scientific American.

[14]  Konstantin V. Zhukovsky,et al.  Operational Approach and Solutions of Hyperbolic Heat Conduction Equations , 2016, Axioms.

[15]  W. Roetzel,et al.  Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure , 2003 .

[16]  Vito Antonio Cimmelli,et al.  Non-Fourier Heat Transfer with Phonons and Electrons in a Circular Thin Layer Surrounding a Hot Nanodevice , 2015, Entropy.

[17]  Jung-Chang Wang,et al.  Analysis of thermal damage to laser irradiated tissue based on the dual-phase-lag model , 2014 .

[18]  Mauro Ferrari,et al.  Hierarchically Structured Magnetic Nanoconstructs with Enhanced Relaxivity and Cooperative Tumor Accumulation , 2014, Advanced functional materials.

[19]  Robert A. Taylor,et al.  Nanofluid-based direct absorption solar collector , 2010 .

[20]  Zhifeng Huang,et al.  Nanoparticle embedded double-layer coating for daytime radiative cooling , 2017 .

[21]  Di Rienzo,et al.  Mesoscopic Numerical Methods for Reactive Flows: Lattice Boltzmann Method and Beyond , 2012 .

[22]  Bong Jae Lee,et al.  Phonon Transport through Nanoscale Contact in Tip-Based Thermal Analysis of Nanomaterials , 2017, Nanomaterials.

[23]  Vito Antonio Cimmelli,et al.  Different Thermodynamic Theories and Different Heat Conduction Laws , 2009 .

[24]  Ingo Müller,et al.  Extended Thermodynamics: a Theory of Symmetric Hyperbolic Field Equations , 2008, Entropy.

[25]  Amir Keshavarz,et al.  Plasmon-actuated nano-assembled microshells , 2017, Scientific Reports.

[26]  R. Guyer,et al.  Solution of the Linearized Phonon Boltzmann Equation , 1966 .

[27]  J. K. Chen,et al.  A semiclassical two-temperature model for ultrafast laser heating , 2006 .

[28]  Fernando Bresme,et al.  Thermal transport across nanoparticle-fluid interfaces: the interplay of interfacial curvature and nanoparticle-fluid interactions. , 2017, Physical chemistry chemical physics : PCCP.

[29]  G. Lebon,et al.  An extended thermodynamic model of transient heat conduction at sub-continuum scales , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  G Chen,et al.  Ballistic-diffusive heat-conduction equations. , 2001, Physical review letters.

[31]  Bahgat Sammakia,et al.  A novel alternate approach for multiscale thermal transport using diffusion in the Boltzmann Transport Equation , 2011 .

[32]  Mingtian Xu,et al.  Dual-phase-lagging heat conduction based on Boltzmann transport equation , 2005 .

[33]  Mingtian Xu,et al.  A ballistic-diffusive heat conduction model extracted from Boltzmann transport equation , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  Masoud Bozorg Bigdeli,et al.  Thermal transmittance in graphene based networks for polymer matrix composites , 2017 .

[35]  Shigeo Maruyama,et al.  Non-Fourier heat conduction in a single-walled carbon nanotube: Classical molecular dynamics simulations , 2006 .

[36]  Arunn Narasimhan,et al.  Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation , 2013 .

[37]  Ali Shakouri,et al.  Nanostructured Thermoelectrics: Big Efficiency Gains from Small Features , 2010, Advanced materials.

[38]  G. Lebon,et al.  Extended irreversible thermodynamics , 1993 .

[39]  I. Müller,et al.  Rational Extended Thermodynamics , 1993 .

[40]  Cristina H. Amon,et al.  Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling , 2006 .

[41]  A. Majumdar,et al.  Nanoscale thermal transport , 2003, Journal of Applied Physics.

[42]  C. Cattaneo,et al.  Sulla Conduzione Del Calore , 2011 .