non-Fourier Heat Transport Research Articles

Thermodynamic costs of temperature stabilization in logically irreversible computation

Abstract In recent years, great efforts are devoted to reducing the work cost of the bit operation, but it is still unclear whether these efforts are sufficient for resolving the temperature stabilization problem in computation. By combining information thermodynamics and a generalized constitutive model which can describe Fourier heat conduction as well as non-Fourier heat transport with nonlocal effects, we here unveil two types of the thermodynamic costs in the temperature stabilization problem. Each type imposes an upper bound on the amount of bits operated per unit time per unit volume, which will eventually limit the speed of the bit operation. The first type arises from the first and second laws of thermodynamics, which is independent of the boundary condition and can be circumvented in Fourier heat conduction. The other type is traceable to the third law of thermodynamics, which will vary with the boundary condition and is ineluctable in Fourier heat conduction. These thermodynamic costs show that reducing the work cost of the bit operation is insufficient for resolving the temperature stabilization problem in computation unless the work cost vanishes.

Discrete heat conduction equation: Dispersion analysis and continuous limits

The dispersion relation in ω-k space has been derived and analyzed for discrete heat conduction equation (DE). The DE is inherently nonlocal both in time and space and can be used to describe non-Fourier heat transport occurring on ultrashort time and length scales. An important distinction between the DE and the continuum description is that the frequencies and wave numbers available to the discrete system are limited, whereas in the continuum description there are no such limitations. Analytical expressions for the attenuation distance, group and phase velocities have been obtained and analyzed as functions of frequency. In the continuum limit, depending on the basic invariant of the continualization procedure, the dispersion relation for the DE reduces to either dispersion relation for the classical Fourier parabolic equation (PE) or to dispersion relation for the hyperbolic heat equation (HE). However, at the moderate and high frequencies the spectrum of the DE differs significantly from the spectra of the PE and HE. This implies that the continuum-based approaches, including both the PE and HE, unable to describe high-frequency and short-wavelength processes and the DE is preferable. This work provides a relatively simple analytical tool to interpret the transient wave-like behavior of heat transport on ultrashort space and time scales.

Transient effects in the coupling of thermal radiation and non-Fourier heat transport at the nano-scale

In this work, we study thermalization between two bodies separated by a vacuum gap by coupling the non-Fourier behavior of the materials with the radiative heat transfer in the near-field. Unlike the diffusion-type temperature profile, in non-Fourier materials, the temperature behaves as a wave, changing the thermalization process. Due to the temperature profile induced by the coupling with conduction, we show that the radiative heat flux exchanged between the two bodies differs from the Fourier case and exhibits transient temperature effects at the onset of the thermalization process. These results have important implications in nanoscale thermal management, near-field solid-state cooling, and nanoscale energy conversion.

Lattice Boltzmann scheme for hydrodynamic equation of phonon transport

In this work, a lattice Boltzmann scheme is developed for numerical solution of the phonon Boltzmann equation under Callaway's dual relaxation model in the hydrodynamic limit. Through a Chapman-Enskog expansion to the lattice Boltzmann equation with the resistive scattering term as an equivalent source term, we recover a phonon hydrodynamic equation which is reduced to the Guyer-Krumhansl heat transport equation and the Fourier's law in the limit of dominant normal scattering and dominant resistive scattering respectively. The transition of heat transport from diffusive regime to hydrodynamic regime is modeled extensively by the present numerical scheme, which produces results in good agreement with the benchmark solutions and experimental measurement. Two well-known phonon hydrodynamic phenomena including the phonon Poiseuille flow and second sound propagation are well captured by the lattice Boltzmann scheme. This work will promote the numerical modeling and deeper understanding of the non-Fourier heat transport induced by phonon normal scattering.

Non-Fourier heat transport across 1D nano film between thermal reservoirs with different boundary resistances

We describe the nonlocal heat transport across a thin film in contact with two thermal baths using discrete variable model. The effective thermal conductivities , the boundary temperature jumps, the steady-state heat flux, and the internal temperature gradient are calculated analytically as functions of the film size and the boundary resistances. As the system size decreases, all the parameters demonstrate the transition from diffusive (Fourier) heat conduction when the scattering occurs mainly inside the film to ballistic transport when scattering occurs at the boundaries, while inside the film there is no scattering and the temperature profile is flat. We show that when the boundary resistance increases, the temperature profile inside the film also tends to be flat, but in this case the scattering, though small, occurs inside the film. The predicted effective thermal conductivity is consistent with classical continuous theories and available experimental data. • We consider heat transport across 1D nano film. • Discrete variable model has been used. • Effective thermal conductivity is size and resistance dependent. • Transition to ballistic regime is described.

Accessing temperature waves: A dispersion relation perspective

In order to account for non-Fourier heat transport, occurring on short time and length scales, the often-praised Dual-Phase-Lag (DPL) model was conceived, introducing a causality relation between the onset of heat flux and the temperature gradient. The most prominent aspect of the first-order DPL model is the prediction of wave-like temperature propagation, the detection of which still remains elusive. Among the challenges to make further progress is the capability to disentangle the intertwining of the parameters affecting wave-like behaviour. This work contributes to the quest, providing a straightforward, easy-to-adopt, analytical mean to inspect the optimal conditions to observe temperature wave oscillations. The complex-valued dispersion relation for the temperature scalar field is investigated for the case of a localised temperature pulse in space, and for the case of a forced temperature oscillation in time. A modal quality factor is introduced showing that, for the case of the temperature gradient preceding the heat flux, the material acts as a bandpass filter for the temperature wave. The bandpass filter characteristics are accessed in terms of the relevant delay times entering the DPL model. The optimal region in parameters space is discussed in a variety of systems, covering nine and twelve decades in space and time-scale respectively. The here presented approach is of interest for the design of nanoscale thermal devices operating on ultra-fast and ultra-short time scales, a scenario here addressed for the case of quantum materials and graphite.

Evaluation of Non-Fourier Heat Transfer on Temperature Evolution in an Aquifer Thermal Energy Storage System

The heat transport in fractured porous rocks is classically modeled employing the advection–dispersion equation (ADE). However, the nature of heat transfer in fractured reservoir rocks may not be represented by the effective medium properties when the ADE formulation is adopted. In this study, a modified mathematical model describing non-Fourier heat transport in an aquifer thermal energy storage (ATES) system is proposed employing the fractional calculus theory. This mathematical model incorporates the effect of heat losses to the surrounding impermeable rock formations. The Laplace transformation method is applied to derive the semi-analytical solutions describing the dimensionless temperature evolution in the confined aquifer, and the surrounding impermeable rocks (i.e., underlying and overlying rocks). Detailed parametric studies are performed to investigate the role of the introduced parameters, i.e., the fractional order of differentiation, generalized friction coefficient and aquifer pseudo-effective thermal conductivity on the propagation of heat within the ATES system. Computations performed on the derived solutions demonstrate that the temperature profiles in the confined aquifer and the surrounding rocks are influenced by the magnitude of the respective fractional exponents. In addition, observation of the temperature profiles within the thermally perturbed zones demonstrates that larger values of the fractional order of differentiation lead to efficient heat transfer within the ATES system. Furthermore, analysis of the results indicates that the impact of the aquifer pseudo-effective thermal conductivity on the temperature propagation in the ATES system is limited to the aquifer only. The derived solutions will find widespread application in designing and simulating the heat injection performance in an ATES system and assessing the influence of non-Fourier heat transport and geological parameters on temperature transients through porous media.

Phonon and electron temperature and non-Fourier heat transport in thin layers

We present a thermodynamic model of heat conductor which allows for different temperatures of phonons and electrons. This model is applied to calculate the steady-state radial temperature profile in a circular thin layer. The compatibility of the obtained temperature profiles with the second law of thermodynamics is investigated in view of the requirement of positive entropy production and of a nonlocal constitutive equation for the entropy flux.

Laser Heating of the Core-Shell Nanowires

AbstractThe induced thermal stress in a heating process is an important parameter to be known and controlled in the magnetization process of core-shell nanowires. This paper analyses the stress produced by a laser heating source placed at one end of a core-shell type structure. The thermal field was computed with the non-Fourier heat transport equation using a finite element method (FEM) implemented in Comsol Multiphysics. The internal stresses are essentially due to thermal gradients and different expansion characteristics of core and shell materials. The stress values were computed using the thermo elastic formalism and are depending on the laser beam parameters (spot size, power etc.) and system characteristics (dimensions, thermal characteristics). Stresses in the GPa range were estimated and consequently we find that the magnetic state of the system can be influenced significantly. A shell material as the glass which is a good thermal insulator induces in the magnetic core, the smaller stresses and consequently the smaller magnetoelastic energy. These results lead to a better understanding of the switching process in the magnetic materials.

Finite thermal convection of non-Fourier fluids

The thermal convection is examined theoretically for fluids possessing significant thermal relaxation time, characterizing the response of the heat flux and the temperature gradient to changes in one another. Fourier's law breaks down for such fluids. Non-Fourier heat transport occurs in a wide range of applications, including superfluid helium, fluids subjected to rapid heating, and strongly confined fluids. The parallels between non-Fourier fluids and viscoelastic polymeric solutions are established. For viscoelastic fluids, the constitutive equation for stress must be frame invariant, a condition that must also hold for the constitutive equation for heat flux. The stability of conduction state is first reviewed, and the convection state is obtained using a low-order dynamical system approach. The stability of steady convection is analysed, and the Nusselt number is obtained as function of the Rayleigh and Cattaneo numbers.

Non-Fourier effects in macro- and micro-scale non-isothermal flow of liquids and gases. Review

Non-Fourier heat transfer is reviewed and examined for fluids possessing significant thermal relaxation and retardation times characterizing the response of the heat flux and the temperature gradient to changes in one another. Non-Fourier heat transport occurs in a wide range of applications, including superfluid helium, fluids subjected to rapid heating, and strongly confined fluids. For nanofluids both the relaxation and retardation times are expressed in terms of nanoparticle concentration and suspension properties. The parallels between non-Fourier fluids and viscoelastic polymeric solutions are established. For viscoelastic fluids, the constitutive equation for stress must be frame invariant, a condition that must also hold for the constitutive equation for heat flux. This survey examines a variety of conditions and flow configurations under which non-Fourier effects may be significant in the non-isothermal flow of both gases and liquids.

Thermal characterization of granular materials using a thermal-wave resonant cavity under the dual-phase lag model of heat conduction

The process of heat transport in granular materials has generated a great deal of controversy. It has been claimed that the process is affected not only by the thermal conductivity, diffusivity and heat capacity, but also additional parameters in the form of time-phase lags must be considered. These quantities permit to take into account the thermal inertia and the micro-structural interactions of the media in such a way that they establish the non-simultaneity between the temperature and the heat flux. A highly successful model that takes into account these effects is known as the dual-phase lag model of heat conduction. It constitutes an approach that generalizes and overcomes the limitations of the classical Fourier law of heat transport. One of the most sensitive techniques for measuring thermal properties is the thermal-wave resonant cavity, which is formed by three layers. The one in the middle is semi solid, liquid or gas, whose thickness can be changed moving one of the external layers. In order to study the material in the middle, a modulated heat source is applied to one of the external layers, and the changes of temperature are registered at the surface of any of the external layers. This methodology has provided high accuracy results for the thermal properties of liquids, gases and nanofluids in the context of Fourier heat diffusion equation. However results for granular materials using this methodology are scarce and the role of the phase lags in heat transport has not been fully explored. In this work, the theoretical basis for the development of a thermal-wave resonant cavity based on dual-phase lag model is studied. It is shown that this system could be used to measure not only the thermal diffusivity but also the time-phase lags of granular materials, by performing a suitable thickness scan of the cavity. It is shown that the results obtained can be a highly useful in the development of experimental methodologies revealing the possibility of non-Fourier heat transport and how the thermal characterization of granular materials can be performed.

Application of the time discontinuous Galerkin finite element method to heat wave simulation

This paper deals with the numerical simulation of heat wave propagation in the medium subjected to different kinds of heat source, particularly heat impulse. The discontinuous Galerkin finite element method (DGFEM) proposed for the stress wave propagation in solids [X.K. Li, D.M. Yao, R.W. Lewis, A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media. Int. J. Numer. Meth. Eng. 57 (2003) 1775–1800] is extended to numerically solve for the non-Fourier heat transport equation constructed according to the CV model [C. Cattaneo, A form of heat-conduction equation which eliminates the paradox of instantaneous propagation, Compute Rendus 247 (1958) 431–433; P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compute Rendus 246 (1958) 3154–3155]. Temperature and its time-derivative are chosen as primitive variables defined at each FE node. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite’s polynomial) and linear interpolations for both temperature and its time-derivative, respectively, in the time domain is particularly proposed. As a consequence the continuity of temperature at each discrete time instant is exactly ensured, whereas discontinuity of the time-derivative of temperature at discrete time levels remains. Numerical results illustrate good performance of the present method in the numerical simulation of heat wave propagation in eliminating spurious numerical oscillations and in providing more accurate solutions in the time domain.