Brian Vick

Propagation and reflection of thermal waves in a finite medium

Abstract For situations involving extremely short times following the start of transients or for very low temperatures near absolute zero, the classical diffusion theory of heat conduction breaks down since the wave nature of thermal energy transport becomes dominant. In this work, analytical solutions are developed for the hyperbolic heat conduction equation describing the wave nature of thermal energy transport in a finite slab with insulated boundaries subjected to a volumetric energy source in the medium. The exact analytical solutions developed for the temperature field and heat flux show that the release of a concentrated pulse of energy gives rise to a severe thermal wave front which travels through the medium at a finite propagation speed, dissipating energy in its wake and reflecting from the insulated surfaces.

ON THE NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION

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 MacCormacks 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.

Hyperbolic heat conduction with surface radiation

Abstract A numerical approach is used to solve hyperbolic heat conduction in a semi-infinite medium where the boundary at x = 0 is subjected to a surface heat flux and dissipates heat by radiation into an ambient at temperature T ∞ . The results are compared with those obtained from the standard parabolic heat conduction equation.

Flux formulation 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.

General formulation and analysis of hyperbolic heat conduction in composite media

Abstract Some recent experimental results show the existence of reflections of thermal waves at the interface of dissimilar materials in superfluid helium. In light of these results, a theoretical investigation of thermal waves in composite is provided to give a theoretical foundation to the observed phenomenon. A general one-dimensional temperature and heat flux formulation for hyperbolic heat conduction in a composite medium is presented. Also, the general solution, based on the flux formulation, is developed for the standard three orthogonal coordinate systems. Unlike classical parabolic heat conduction, heat conduction based on the modified Fouriers law produces non-separable field equations for both the temperature and flux and therefore standard analytical techniques cannot be applied in these situations. In order to alleviate this difficulty, a generalized finite integral transform technique is proposed in the flux domain and a general solution is developed for the standard three orthogonal coordinate systems. The general solution is applied to the case of a two-region slab with a pulsed volumetric source and insulated exterior surfaces which displays the unusual and controversial nature associated with heat conduction based on the modified Fouriers law in composite regions.

Hyperbolic heat conduction with temperature‐dependent thermal conductivity

Hyperbolic heat conduction in a semi‐infinite slab with temperature‐dependent thermal conductivity is studied numerically, and the results are compared with those obtained from the classical parabolic equation for the following cases: (a) constant applied temperature at x=0.0, (b) constant applied heat flux at x=0.0, and (c) a pulsed heat source released instanteously at t=0.0 in the region 0.0≤x≤Δx adjacent to an insulated boundary. In addition to changing the temperature profiles, the nonlinear thermal conductivity also altered the speed of the thermal front. An increase in the thermal conductivity increased the wave speed, while a decrease in the thermal conductivity decreased the wave speed.

Non-Fourier effects on transient temperature resulting from periodic on-off heat flux

Abstract The transient temperatures resulting from a periodic on-off heat flux boundary condition have many applications, including, among others, the sintering of catalysts frequently found during coke burnoff, and the use of laser pulses for annealing of semiconductors. In such situations, the duration of the pulses is so small (i.e. picosecond-nanosecond) that the classical heat diffusion phenomenon breaks down and the wave nature of energy propagation characterized by the hyperbolic heat conduction equation governs the temperature distribution in the medium. In this work, an explicit analytic solution is presented for a linear transient heat conduction problem in a semi-infinite medium subjected to a periodic on-off type heat flux at the boundary x = 0 by solving the hyperbolic heat conduction equation. The non-linear case allowing for the added effect of surface radiation into an external ambient is studied numerically.

A basic theoretical study of the temperature rise in sliding contact with multiple contacts

Abstract The objectives of this paper are to develop a theoretical solution for the temperature rise due to sliding contact between surfaces with multiple, interacting asperities and to use this solution to examine the effects of the important contact area and system parameters. A solution based on the Greens function method is developed for the basic problem of two half-space regions in sliding contact with any arbitrarily specified arrangement of rectangular asperities. Studies are conducted to demonstrate the effects of the contact area parameters, namely the number, size, spacing and orientation of the contacts, as well as sliding velocity. Results indicate that the contact temperatures are extremely sensitive to the number and relative spacing between contacts, where subdivision of a single contact into separated pieces significantly reduces the contact temperature rises. The orientation of the contacts relative to the sliding direction is shown to have only a small influence on temperature. The shape of the contacts also has only a small influence, except in the case of contact patches with large aspect ratios where significantly lower surface temperatures can occur. Sliding speed is shown to be extremely important in that increased speed causes both higher temperature levels and greater interaction between contacts due to the convective effect. The current paper is intended to describe the basic solution methodology for calculating temperature rises due to multiple, interacting contacts and to show some fundamental trends for a selected set of regularly arranged contact area distributions.

The Question of Thermal Waves in Heterogeneous and Biological Materials

In the 1990s, there were two experimental studies that sparked a renewed interest in thermal wave behavior at the macroscale level. Both reported thermal relaxation times of 10 s or higher. However, no further experimental evidence of this behavior has been reported. Due to the extreme significance of these findings, the objectives of this study were to try to reproduce these earlier studies and offer an explanation for the outcome. These two previous studies, one using heterogeneous materials and one using bologna, were repeated following the experimental protocol provided in the studies as closely and as practically as possible. In both cases, the temperature response to a specified boundary condition was recorded. The results from the first set of experiments suggested that the thermal relaxation times presented in the previous study were actually the thermal lag expected from applying Fouriers law, taking into account the uncertainty of the temperature sensor. In the second set of experiments, unlike the distinct jumps in temperature found previously, no indication of wave behavior was found. Here, the explanation for the previous results was more difficult to ascertain. Possible explanations include problems with either the experimental protocol or the temperature sensors used.

Theoretical surface temperatures generated from sliding contact of pure metallic elements

Abstract Surface temperatures and thermal effects produced in tribological processes are important not only in influencing possible mechanisms of friction, wear, and lubricant film failure but also in initiating protective film-formation. As part of a continuing combined theoretical and experimental study of surface temperatures generated by friction, the fundamental Greens function approach has been applied to a number of pure metallic elements to compare and discuss their predicted behavior in A-on-A sliding contact. Assuming a single area of real contact, calculated ratios of surface temperature rise to coefficient of friction plotted against area of contact, velocity and load on a logarithmic scale are presented and summarized for several pure metallic elements in the first transition series of the Periodic Table (e.g., Ti, V, Cr, Mn, Fe, Co, Ni) as well as members in connecting groups, e.g., Cr, Mo, and W in Group VIa and Cu, Ag, and Au in Group Ib. These include metals which are tribologically difficult to machine and use (e.g., Ti), common elements in bearing steels (e.g., Fe, Cr), and metals useful in reducing friction or wear when applied as thin surface coatings (e.g., Ag, Au). The results of this comparison are interesting and surprising. They may add to our understanding of why some metals are very “difficult” in a tribological sense while others provide benefits in controlling friction and/or wear.