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Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Abstract

The classical model of the Fourier’s law is known as the most common constitutive relation for thermal transport in various engineering materials. Although the Fourier’s law has been widely used in a variety of engineering application areas, there are many exceptional applications in which the Fourier’s law is questionable. This paper gathers together such applications. Accordingly, the paper is divided into two parts. The first part reviews the papers pertaining to the fundamental theory of the phase-lagging models and the analytical and numerical solution approaches. The second part wrap ups the various applications of the phase-lagging models including the biological materials, ultra-high-speed laser heating, the problems involving moving media, micro/nanoscale heat transfer, multi-layered materials, the theory of thermoelasticity, heat transfer in the material defects, the diffusion problems we call as the non-Fick models, and some other applications. It is predicted that the interest in the field of phase-lagging heat transport has grown incredibly in recent years because they show good agreement with the experiments across a wide range of length and time scales.

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Abbreviations

B :

Phase-lag ratio

C :

Volumetric heat capacity (\({\hbox {J}}\) \({\cdot }\) \({\hbox {m}^{-3}}\) \({{\cdot }}\) \({\hbox {K}^{-1}}\))

Kn :

Knudsen number

k :

Thermal conductivity (\(\hbox {W}\) \({\cdot }\) \(\hbox {m}^{-1}\) \({\cdot }\) \(\hbox {K}^{-1}\))

\(L_{\mathrm{c}}\) :

Characteristic length (m)

Q :

Volumetric heat generation (\(\hbox {W}\) \({\cdot }\) \(\hbox {m}^{-3}\))

\(\overrightarrow{q}\) :

Heat flux vector (W\({\cdot }\) \(\hbox {m}^{-2}\))

\(\overrightarrow{r}\) :

Displacement vector (m)

T :

Temperature (K)

\(T_0\) :

Initial temperature (K)

\(T_{\mathrm{c}}\) :

Characteristic temperature (K)

t :

Time (s)

\(\overrightarrow{v}\) :

Phonon group velocity (\(\hbox {m}\) \({\cdot }\) \(\hbox {s}^{-1}\))

\(\alpha \) :

Thermal diffusivity (\(\hbox {m}^2\) \({\cdot }\) \(\hbox {s}^{-1}\))

\(\lambda \) :

Mean-free-path of carriers (m)

\(\tau _T\) :

Temperature phase-lag (s)

\(\tau _q\) :

Heat flux phase-lag (s)

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Ghazanfarian, J., Shomali, Z. & Abbassi, A. Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior. Int J Thermophys 36, 1416–1467 (2015). https://doi.org/10.1007/s10765-015-1913-4

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Keywords

  • Biological materials
  • CV and DPL models
  • Fluid flow
  • Macro/nanoscale
  • Non-Fourier and non-Fick models
  • Thermoelasticity