A universal constitutive equation between the heat flux vector and the temperature gradient is proposed to cover the fundamental behaviors of diffusion (macroscopic in both space and time), wave (macroscopic in space but microscopic in time), phonon-electron interactions (microscopic in both space and time), and pure phonon scattering The model is generalized from the dual-phase-lag concept accounting for the laging behavior in the high-rate response While the phase lag of the heat flux captures the small-scale response in time, the phase lag of the temperature gradient captures the small-scale response in space The universal form of the energy equation facilitates identifications of the physical parameters governing the transition from one mechanism (such as diffusion or wave) to another (the phonon-electron interaction)

A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales

The generalized lagging response in small-scale and high-rate heating

This work examines the lagging behavior for heat transport in small-scale and fast-transient processes. The experimental result by Qiu et al. for the femtosecond transient response in gold films and that by Bertman and Sandiford for the temperature pulse traveling through superfluid liquid helium are re-examined with emphasis on the lagging behavior. The model employing the phase-lag concept provides as competent or even better results when describing the special response observed in these experiments.

Experimental support for the lagging behavior in heat propagation

Comparative studies on the well-known Pennes' equation and the newly developed thermal wave model of bioheat transfer (TWMBT) were performed to investigate the wave like behaviors of bioheat transfer occurred in thermal injury of biological bodies. The one-dimensional TWMBT in a finite medium was solved using separation of variables and the analytical solution showed distinctive wave behaviors of bioheat transfer in skin subjected to instantaneous heating. The finite difference method was used to simulate and study practical problems involved in burn injuries in which skin was stratified as three layers with various thermal physical properties. Deviations between the TWMBT and the traditional Pennes' equation imply that, for high flux heating with extremely short duration (i.e., flash fire), the TWMBT which accounts for finite thermal wave propagation may provide realistic predictions on burn evaluation. A general heat flux criterion has been established to determine when the thermal wave propagation dominates the principal heat transfer process and the TWMBT can be used for tissue temperature prediction and burn evaluation. A preliminary interpretation on the mechanisms of the wave like behaviors of heat transfer in living tissues was conducted. The application of thermal wave theory can also be possibly extended to other medical problems which involve instantaneous heating or cooling.

New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating

Non-Fourier analysis of skin biothermomechanics

Propagation and reflection of thermal waves in a finite medium

Non-Fourier heat conduction is governed by the hyperbolic heat conduction equation, which involves the wave nature of thermal energy transport. In such cases, energy propagates in the medium as a wave with sharp discontinuities at the wave front. Difficulties encountered in the numerical solution of such problems include, among others, numerical oscillations and the representation of sharp discontinuities with good resolution at the wave front. In this work it is shown that a numerical technique based on MacCormack's predictor-corrector scheme can be used to handle discontinuities at the wave front with high resolution and little oscillation. Numerical predictions are compared with the exact analytic solutions for a wide variety of strict test conditions. For all the cases considered, the present numerical scheme remains stable and produces high resolution at the sharp wave front.

On the numerical solution of hyperbolic heat conduction

The development of the general flux formulation for heat conduction based on the modified Fourier’s law is presented. This new formulation produces a hyperbolic vector equation in heat flux which is more convenient to use for analysis in situations involving specified flux conditions than the standard temperature formulation. The recovery of the temperature distribution is obtained through integration of the energy conservation law with respect to time. The Green’s function approach is utilized to develop a general solution for hyperbolic heat conduction in a finite medium. The utility of the flux formulation and the unusual nature of heat conduction based on the hyperbolic formulation are demonstrated by developing analytical expressions for the heat flux and temperature distributions in a finite slab exposed to a pulsed surface heat flux.

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Flux formulation of hyperbolic heat conduction

Etude de la nature ondulatoire de la propagation de la chaleur dans un milieu semi-infini contenant des sources volumiques d'energie par resolution de l'equation hyperbolique de la conduction thermique. Etablissement d'expressions analytiques pour les repartitions de temperature et de flux thermique. Demonstration de la vitesse finie de propagation d'une onde thermique avec decroissance exponentielle

Brian Vick

Papers

Propagation and reflection of thermal waves in a finite medium

On the numerical solution of hyperbolic heat conduction

Flux formulation of hyperbolic heat conduction

Growth and decay of a thermal pulse predicted by the hyperbolic heat conduction equation

Hyperbolic heat conduction with surface radiation