Hyperbolic Heat-Conduction Equation—A Solution for the Semi-Infinite Body Problem

Abstract

This theoretical paper determines the effect of the propagation velocity of heat on the temperature and heat-flux distribution in a semi-infinite body due to a step change in temperature at the surface. The solution yields a maximum but finite heat flux under the conditions of a step change. This is contrary to the infinite value predicted by the error function solution to the Fourier transient conduction equation. In addition, assuming convection is conduction limited, an upper limit for convective heat transfer coefficients is postulated.

An alternative discretization and solution procedure for the dual phase-lag equation

“…(2), despite presence of the third-order mixed derivative term, because there are now no delays. In particular, complete well-posed problems for the HHCE are also well posed for (3). Furthermore, we see that when τ T = 0 the HHCE is recovered, and if both τ T and τ q are zero the usual parabolic heat equation results.…”

“…Associated with this is the fact that the parabolic character of the heat equation obtained from Fourier's law implies that heat flow starts (stops) simultaneously with appearance (disappearance) of a temperature gradient, thus violating the causality principle which states that two causally correlated events cannot happen at the same time; rather, the cause must precede the effect, as noted by Cimmelli [1]. In order to ensure finite propagation of thermal disturbances a hyperbolic heat conduction equation (HHCE) was proposed at least as early as the studies of Luikov [2] and Baumeister and Hamil [3]. We remark that this equation is of the same form as that termed the "telegraph equation" in the mathematics literature (see essentially any intermediate PDE text).…”

Hyperbolic heat conduction in two semi-infinite bodies in contact

“…In such cases the results predicted by the use of the classical parabolic Fourier heat equation are far from the experimental results (see for instance the survey papers [1] and [2]). Then a more accurate heat transfer theory is needed.…”

“…Then a more accurate heat transfer theory is needed. The easier alternative is to use the modified Fourier law (1) q(x, t) + τ ∂q(x, t) ∂t = −k∇T (x, t) which states that heat flux does not begin at the instant t when the temperature gradient is calculated, but at time t + τ , where τ is an assumed constant material characteristic which is called the thermal relaxation time. This assumption gives rise (for homogeneous materials) to the hyperbolic differential heat conduction equation…”

Inertia effect analysis of a half‐plane with an induced crack under thermal loading using hyperbolic heat conduction

“…The sources of these phenomena include the non-Fourier heat conduction effect, the inertia effect etc. It was reported that in such situations the Fourier heat conduction model becomes inaccurate [1]. Also, the inertia forces are not negligible and important stress waves can be induced [2].…”