Macro‐ to Microscale Heat Transfer

Tzou, D. Y.

doi:10.1002/9781118818275

Cited by 378 publications

(332 citation statements)

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“…It is checked that, after substituting in (5) the expressions of and given by (4) and (6), respectively, and eliminating in (4) the time derivative ∂ t e via the energy conservation law, = −∇ • , one obtains for the entropy production…”

Section: A General Heat Transport Equation In Terms Of High Order Heamentioning

confidence: 99%

“…These observations justify the need to generalize Fourier's law. This can be achieved by various ways, for instance, via the resolution of Boltzmann's equation [3], making use of the dual time approach [4] or by computer simulations [5]. Here we follow a different route based on Extended Irreversible Thermodynamics (EIT) [6] whose main characteristic is to upgrade the heat flux and higher order heat fluxes to the rank of independent variables.…”

Section: Introductionmentioning

confidence: 99%

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General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

Machrafi

Lebon

2016

Applied Mathematics Letters

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A generalized heat transport equation applicable at small length and short time scales is proposed. It is based on extended irreversible thermodynamics with an infinite number of highorder heat fluxes selected as state variables. Extensions of Fick's and Ohm's laws are also formulated. As a numerical illustration, heat conduction in a rigid body subject to fixed and oscillatory temperature boundary conditions is discussed. KeywordsBallistic heat transport; Small length-systems; High-frequency; Extended irreversible thermodynamics IntroductionThe increasing interest in nano-technology has led to new insights in the study of heat transport. It is well known that heat transfer at micro and nanoscale behaves differently from that at macroscales [1,2]. At small length scales, transport of heat in complex systems is best quantified by means of the so-called Knudsen number ≡ ℓ/ with ℓ denoting the mean free path of the heat carriers, namely phonons, and , the characteristic dimension of the system under study. The Knudsen number becomes typically comparable or larger than one for micro-and nano-systems, in which case heat transport is referred to as ballistic, i.e. dominated by phonon collisions with the walls. If the Knudsen number is much smaller than one, heat transport is simply diffusive, i.e. dominated by phonon-phonon collisions inside the system and described by Fourier's law. For small-length systems, as well as for high-frequency processes, Fourier's law is no longer valid. Another drawback associated to Fourier's law is that it implies that temperature spreads infinitely fast through the whole body, which is physically unsustainable. These observations justify the need to generalize Fourier's law. This can be achieved by various ways, for instance, via the resolution of Boltzmann's equation [3], making use of the dual time approach [4] or by computer simulations [5]. Here we follow a different route based on Extended Irreversible Thermodynamics (EIT) [6] whose main characteristic is to upgrade the heat flux and higher order heat fluxes to the rank of independent variables. In most applications, for convenience, the analysis is restricted by taking heat flux as single extra variable. In this letter, we go one step further by selecting an infinite number of higher order fluxes. The use of a large number of flux variables finds its justification in the recent progress of nanotechnology and high-frequency processes. The procedure described in this paper is by no means limited to heat transfer but can be easily generalized to other transport phenomena, as electrical conduction and matter diffusion. As a numerical application, we consider the problem of heat conduction in a one-dimensional rigid body of length L at rest, subject to two different kinds of boundary conditions, made explicit in Section 5. The theoretical model for the general heat transport equation is presented in the next Section 2.

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Section: A General Heat Transport Equation In Terms Of High Order Heamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

Machrafi

Lebon

2016

Applied Mathematics Letters

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“…Three such approaches are the phonon hydrodynamics [5][6][7], the thermomass theory [12][13][14], and the dualphase-lag model [15,16]. All these models consider the heat carriers as a fluid, whose hydrodynamic-like equations of motion describe the heat transport.…”

Section: Heat Transport Equations Beyond the Fourier Law And Hydrodynmentioning

confidence: 99%

“…However, the research in this field is still in progress, since the physics of thermoelectric nanomaterials presents several points which are not wellunderstood, as for instance the exact role played by nonlocal and nonlinear effects [15,22,30,68,69]. In the classical form, thermoelectric effects are described by the following constitutive equations q = −λ∇T + Π + µ e z e I (36a)…”

Section: Heat Conduction In Thermoelectric Systemsmentioning

confidence: 99%

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Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

Jou¹,

Cimmelli

2016

Communications in Applied and Industrial Mathematics

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A Lagging Model for Describing Drawdown Induced by a Constant‐Rate Pumping in a Leaky Confined Aquifer

Lin

Yeh

2017

Water Resources Research

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This study proposes a generalized Darcy's law with considering phase lags in both the water flux and drawdown gradient to develop a lagging flow model for describing drawdown induced by constant‐rate pumping (CRP) in a leaky confined aquifer. The present model has a mathematical formulation similar to the dual‐porosity model. The Laplace‐domain solution of the model with the effect of wellbore storage is derived by the Laplace transform method. The time‐domain solution for the case of neglecting the wellbore storage and well radius is developed by the use of Laplace transform and Weber transform. The results of sensitivity analysis based on the solution indicate that the drawdown is very sensitive to the change in each of the transmissivity and storativity. Also, a study for the lagging effect on the drawdown indicates that its influence is significant associated with the lag times. The present solution is also employed to analyze a data set taken from a CRP test conducted in a fractured aquifer in South Dakota, USA. The results show the prediction of this new solution with considering the phase lags has very good fit to the field data, especially at early pumping time. In addition, the phase lags seem to have a scale effect as indicated in the results. In other words, the lagging behavior is positively correlated with the observed distance in the Madison aquifer.

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Macro‐ to Microscale Heat Transfer

Cited by 378 publications

References 0 publications

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

A Lagging Model for Describing Drawdown Induced by a Constant‐Rate Pumping in a Leaky Confined Aquifer

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