Fourier's Law Of Heat Conduction Research Articles

Mathematical Models and Numerical Solutions of Liquid-Solid and Solid-Liquid Phase Change

This paper presents numerical simulations of liquid-solid and solid-liquid phase change processes using mathematical models in Lagrangian and Eulerian descriptions. The mathematical models are derived by assuming a smooth interface or transition region between the solid and liquid phases in which the specific heat, density, thermal conductivity, and latent heat of fusion are continuous and differentiable functions of temperature. In the derivations of the mathematical models we assume the matter to be homogeneous, isotropic, and incompressible in all phases. The change in volume due to change in density during phase transition is neglected in all mathematical models considered in this paper. This paper describes various approaches of deriving mathematical models that incorporate phase transition physics in various ways, hence results in different mathematical models. In the present work we only consider the following two types of mathematical models: (i) We assume the velocity field to be zero i.e. no flow assumption, and free boundaries i.e. zero stress field in all phases. Under these assumptions the mathematical models reduce to first law of thermodynamics i.e. the energy equation, a nonlinear diffusion equation in temperature if we assume Fourier heat conduction law relating temperature gradient to the heat vector. These mathematical models are invariant of the type of description i.e. Lagrangian or Eulerian due to absence of velocities and stress field. (ii) The second class of mathematical models are derived with the assumption that stress field and velocity field are nonzero in the fluid region but in the solid region stress field is assumed constant and the velocity field is assumed zero. In the transition region the stress field and the velocity field transition in a continuous and differentiable manner from nonzero at the liquid state to constant and zero in the solid state based on temperature in the transition zone. Both of these models are consistent with the principles of continuum mechanics, hence provide correct interaction between the regions and are shown to work well in the numerical simulations of phase transition applications with flow. Details of other mathematical models, problems associated with them, and their limitations are also discussed in this paper. Numerical solutions of phase transition model problems in R 1 and R 2 are presented using these two types of mathematical models. Numerical solutions are obtained using h; p; k space-time finite element processes based on residual functional for an increment of time with time marching in which variationally consistent space-time integral forms ensure unconditionally stable computations during the entire evolution.

Nonlinear Waves in Solid Continua with Finite Deformation

This work considers initiation of nonlinear waves, their propagation, reflection, and their interactions in thermoelastic solids and thermoviscoelastic solids with and without memory. The conservation and balance laws constituting the mathematical models as well as the constitutive theories are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green’s strain tensor and their material derivatives [1]. Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. Numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals and the non-linear equations resulting from the first variation of the residual functional are solved using Newton’s Linear Method with line search. Space-time local approximations are considered in higher order scalar product spaces that permit desired order of global differentiability in space and time. Computed results for non-linear wave propagation, reflection, and interaction are compared with linear wave propagation to demonstrate significant differences between the two, the importance of the nonlinear wave propagation over linear wave propagation as well as to illustrate the meritorious features of the mathematical models and the space-time variationally consistent space-time finite element process with time marching in obtaining the numerical solutions of the evolutions.

The Bresse system in thermoelasticity

In this paper, we consider the Bresse system coupled with the Fourier law of heat conduction. We prove that the decay rate of the solution is very slow. In fact, we show that the L2‐norm of the solution decays with the rate of (1 + t)−1/12 similar to the one obtained for the Timoshenko system. In addition, we found that the wave speed of the first two equations still control the decay rate of the solution with respect to the regularity of the initial data. This seems to be the first result dealing with the behavior of the Cauchy problem in the Bresse–Fourier model. Copyright © 2014 John Wiley & Sons, Ltd.

Nonintegrability and the Fourier heat conduction law.

We study in momentum-conserving systems, how nonintegrable dynamics may affect thermal transport properties. As illustrating examples, two one-dimensional (1D) diatomic chains, representing 1D fluids and lattices, respectively, are numerically investigated. In both models, the two species of atoms are assigned two different masses and are arranged alternatively. The systems are nonintegrable unless the mass ratio is one. We find that when the mass ratio is slightly different from one, the heat conductivity may keep significantly unchanged over a certain range of the system size and as the mass ratio tends to one, this range may expand rapidly. These results establish a new connection between the macroscopic thermal transport properties and the underlying dynamics.

Numerical Study of Laser Cutting of Titanium Alloy (TI-6Al-4V)

A finite element model has been developed utilizing the commercial ANSYS software to determine the temperature distribution & heat affected zone in laser cutting of titanium alloy (Ti-6%Al-4%V).The main aim of this paper is to determine the temperature reached in the vicinity of the location where the laser interaction with the work piece material takes place. This model is based on Fourier law of heat conduction and Gaussian distribution of laser. Temperature dependent thermal and mechanical properties of material are assumed in this analysis. Result obtained are then analyzed and thermal effect is then understood for the effective control of the process.

Analysis of Hydraulic Prop High Frequency Induction Cladding Temperature Field

The temperature field of mobile column is simulated, with the theory of high frequency induction heating and fourier heat conduction law , to analyze the change law of the temperature field with time, with different cladding parameters. The results show that the induction heating speed is mainly affected by power frequency of high frequency induction heating and cladding current. The higher frequency induction heating power is or the larger the cladding current is, the the faster the surface temperature of hydraulic prop is rising, the lower the inside-wall termperature and the less influenced the matrix is. The effect and the efficiency of cladding is comparatively superior when cladding frequency is 250KHz and current is 1160A.

Numerical Study of Temperature and Stress Fields in Laser Cutting of Aluminium Alloy Sheet

Due to thermal nature of laser cutting, high temperature and thermal stresses are developed at the cut edge that affects finally the cut edge quality. This paper aims for developing a numerical simulation model to predict the temperature and residual stresses in the laser cutting of Aluminium alloy (Al-2024).The temperature and stress fields developed in cut section are predicted numerically using ANSYS finite element code. For the analysis, Fourier law of heat conduction and Gaussian distribution of a laser beam are considered. Temperature dependent thermo-physical properties of the material are used in numerical simulation. It is found that high temperature gradient exists at laser irradiated spot which results in high thermal stresses across the cut section. Also it has been found that the maximum temperature obtained during laser cutting is reduced with increasing laser scanning velocity. Stress distribution results shows that stresses attains low values at laser irradiated spot because of the reduction of thermal expansion coefficient with increasing temperature. The comparison of results based on numerical simulation with the mathematical model shows good agreements.

An anomalous wave-like kinetic energy transport in graphene nanoribbons at high heat flux

Most investigations of thermal transport in graphene focused on low heat fluxes when the Fourier heat conduction law is valid. Here thermal transport in armchair graphene nanoribbons was investigated at high heat flux by non-equilibrium molecular dynamics simulations. Besides the energy transported through the Fourier heat conduction, an anomalous wave-like kinetic energy transport was observed. By comparing the two energy transportation paths, we find that the latter constitutes a considerable amount of the total energy and efficiently transports the energy at high heat flux. It means that the heat transfer efficiency in graphene would be greatly improved through wave kinetics. The frequency of the wave (low frequency phonon) would decrease and the amplitude of vibration would increase when the length of the graphene increases. This finding would be useful for thermal analysis when is graphene used as a heat dissipation material.

The structure of shock and interphase layers for a heat conducting Maxwellian rate-type approach to solid–solid phase transitions

We consider a thermoelastic model for phase transforming materials which can adequately describe the evolution with respect to the temperature of the hysteresis loop both in compression and tension tests. The specificity of this model is that the Gruneisen coefficient changes its sign. The model is augmented by considering a dissipative mechanism governed by a Maxwellian rate-type constitutive equation that can describe stress relaxation phenomena toward equilibrium between phases. Existence and uniqueness of traveling wave solutions are investigated. One derives that the admissibility condition induced by the Maxwellian rate-type approach, coupled or not with Fourier heat conduction law is related to the chord criterion with respect to the Hugoniot locus. We investigate the structure of profile layers, and we focus on their thermodynamic properties. The influence of the exothermic or endothermic character of phase transitions on the inner structure of interphase layers is captured. A phenomenon of temperature overshoot/undershoot with respect to the front state temperature and Hugoniot back state temperature inside an interphase layer is revealed.

One-Dimensional Temperature and Stress Distributions Associated with Hyperbolic Heat Conduction

The classical Fourier heat conduction law gives sufficient accuracy for many practical engineering applications. However, it cannot exactly reflect the real physical mechanism of heat conduction by highly-varying thermal load, at very low temperatures, or at nanoscale. The hyperbolic heat conduction equation can better explain heat conduction in solids. However, such equation is very difficult to solve. This paper studies the temperature field for a 1-D plate and a 1-D cylinder. The associated thermal stress for a 1-D plate is also given. Only in these simple situations, closed form solutions are possible and are given. The results can be used in the future analysis of thermal shock cracking and reliability analysis of materials in modern science and technology.

Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect

We consider vibrations modeled by the standard linear solid model of viscoelasticity which are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. We show that the exponential stability under the Fourier law of heat conduction holds. In order to obtain the asymptotic behaviour we use multiplier techniques.

Beyond the Fourier heat conduction law and the thermal no-slip boundary condition

The problem of heat slip flow along solid walls is investigated within the framework of modern thermodynamics. The underlying idea is to elevate the heat flux at the boundary to the status of independent variable. General boundary conditions are obtained from the constraint imposed by the second law of thermodynamics expressing that the rate of entropy production is non-negative. In parallel, evolution equations for the heat flux inside the bulk of the system are also formulated.

A Low-Frequency Wave Motion Mechanism Enables Efficient Energy Transport in Carbon Nanotubes at High Heat Fluxes

The great majority of investigations of thermal transport in carbon nanotubes (CNTs) in the open literature focus on low heat fluxes, that is, in the regime of validity of the Fourier heat conduction law. In this paper, by performing nonequilibrium molecular dynamics simulations we investigated thermal transport in a single-walled CNT bridging two Si slabs under constant high heat flux. An anomalous wave-like kinetic energy profile was observed, and a previously unexplored, wave-dominated energy transport mechanism is identified for high heat fluxes in CNTs, originated from excited low frequency transverse acoustic waves. The transported energy, in terms of a one-dimensional low frequency mechanical wave, is quantified as a function of the total heat flux applied and is compared to the energy transported by traditional Fourier heat conduction. The results show that the low frequency wave actually overtakes traditional Fourier heat conduction and efficiently transports the energy at high heat flux. Our findings reveal an important new mechanism for high heat flux energy transport in low-dimensional nanostructures, such as one-dimensional (1-D) nanotubes and nanowires, which could be very relevant to high heat flux dissipation such as in micro/nanoelectronics applications.

A model for the effective thermal conductivity of metal-nonmetal particulate composites

The effective thermal conductivity of particulate composites with oriented spheroidal metallic particles embedded in a dielectric matrix is analyzed under the framework of the two-temperature model of heat conduction. The obtained analytical results show that the effective thermal conductivity depends strongly on (1) the relative size of the particle inclusions with respect to the electron-phonon coupling length and (2) the ratio between the electron and phonon thermal conductivities. The effect of the electron-phonon coupling inside metallic particles is expressed by the reduction of the composite thermal conductivity with respect to its corresponding values obtained for an infinite electron-phonon coupling factor, where the analysis could be established based on the Fourier law of heat conduction. It is shown that the composite thermal conductivity has upper and lower bounds, which are determined by the particle size in comparison with the electron-phonon coupling length. The generalized model for spheroidal particles is then used to analyze the thermal conductivity for limiting cases on the particle shape including spheres, cylinders, and flat plates. For perfect electron-phonon coupling, the proposed model reduces to various previously-reported results. This study shows that the particle size dependence of the thermal conductivity of metal-nonmetal composites appears not only through the interfacial thermal resistance but also by means of the electron-phonon coupling. The results of this work could be useful for guiding the design of particulate composites with spheroidal metallic inclusions from macro/micro- to nanoscales.

Electron-beam induced abnormal expansion in a silica-shelled gallium microball-nanotube structure (Retracted Article)

Under electron-beam irradiation of heteroshape-heteroscale structure of silica-shelled Ga microball-nanotube, an abnormally large and fast volume expansion of liquid Ga is observed. First, we analyze the processes and phenomena about experiment, and the abnormal expansion process can be regarded as a quasi-static process. Then in the framework of quasi-static thermodynamics, according to the Fourier heat conduction law the relative volume variation with temperature is further quantitatively discussed, and the relative expansion rate and expansion coefficient in system are obtained. At the same time it is found that abnormal expansion coefficient of system under electron-beam irradiation is 5-9 times the general thermal expansion coefficient. Finally, it is pointed out that the abnormal expansion is due to the gallium atom ionization effect and the retention effect resulting from a few electrons retaining in the liquid gallium system under electron-beam irradiation. In essence, both ionization effect and the retention effect make the particle densities of liquid systems increased dramatically, resulting in volume expansion abnormally large and fast of liquid Ga.

Fractional order theory in thermoelastic solid with three-phase lag heat transfer

In this work, the field equations of the linear theory of thermoelasticity have been constructed in the context of a new consideration of Fourier law of heat conduction with time-fractional order and three-phase lag. A uniqueness and reciprocity theorems are proved. One-dimensional application for a half-space of elastic material in the presence of heat sources has been solved using Laplace transform and state space techniques Ezzat (Canad J Phys Rev 86:1241–1250, 2008). According to the numerical results and its graphs, conclusion about the new theory has been established.

A constitutive equation for nano-to-macro-scale heat conduction based on the Boltzmann transport equation

A constitutive equation for heat conduction is derived from the exact solution of the Boltzmann transport equation under the relaxation time approximation. This is achieved by a series expansion on multiple space derivatives of the temperature and introducing the concept of thermal multipoles, where the thermal conductivity defined under the framework of the Fourier law of heat conduction is just the first thermal pole. It is shown that this equation generalizes the Fourier law and Cattaneo equation of heat conduction, and it depends strongly on the relative values of the length and time scales compared with the mean-free path and mean-free time of the energy carriers, respectively. In the limiting case of steady-state heat conduction, it is shown that the heat flux vector depends on a spatial scale ratio whose effects are remarkable in the micro-scale spatial domains. By applying a first-order approximation of the obtained thermal multipole expansion to the problem of transient heat conduction across a thin film and comparing the results with the predictions for the same problem using the Fourier, Cattaneo and Boltzmann transport equations, it is shown that our results could be useful in the study of the heat transport in short as well as in long scales of space and time. The common and different features of the multipole expansion compared with the Ballistic-diffusive model of heat conduction are also discussed. Special emphasis is put to the cases where the physical scales of space and time are comparable to the mean-free path and mean-free time of the energy carriers.

On the stability of the exact solutions of the dual-phase lagging model of heat conduction

The dual-phase lagging (DPL) model has been considered as one of the most promising theoretical approaches to generalize the classical Fourier law for heat conduction involving short time and space scales. Its applicability, potential, equivalences, and possible drawbacks have been discussed in the current literature. In this study, the implications of solving the exact DPL model of heat conduction in a three-dimensional bounded domain solution are explored. Based on the principle of causality, it is shown that the temperature gradient must be always the cause and the heat flux must be the effect in the process of heat transfer under the dual-phase model. This fact establishes explicitly that the single- and DPL models with different physical origins are mathematically equivalent. In addition, taking into account the properties of the Lambert W function and by requiring that the temperature remains stable, in such a way that it does not go to infinity when the time increases, it is shown that the DPL model in its exact form cannot provide a general description of the heat conduction phenomena.

Thermal Performance of a Wire-Mesh/Hollow-Glass-Sphere Composite Structure

DOI: 10.2514/1.T3609 Anexperimentalinvestigationexploringtheuseofawire-screen-mesh/hollow-glass-microspherecombinationasa thermal insulation media was conducted with three primary variables. These included the number of wire-mesh layers, the size of microsphere filler material, and temperature range. The test facility used included vertically stacked samples that were thermally and mechanically controlled (e.g., via gas bellows that controlled its vertical movement). From the temperature profile in the upper and lower samples, the value of the effective thermal conductivity was determined with use of the Fourier law of heat conduction. The number of screen mesh layers investigated were two, four, six, and eight, with each separated by a metallic liner. The filler materials included air andS15,S35,andS60HShollow-glassmicrospherestestedattemperaturesof27,57,93,and127Cwithaninterface pressure of 138 kPa (20 psi). The experimental results indicated that the number of layers was the primary factor in determining the effective thermalconductivity value andthusthe structure’sinsulation effectiveness. Increasing the number of wire-mesh layers resulted in acorresponding increase in effective thermal conductivity, whereas changes in temperature had negligible effect. The effective thermal conductivity values for the proposed structure ranged from 0.22 to 0:65 W=m-K, the lowest was for the two-layer case with air as filler material. Wire-screen-mesh insulation with air in the interstices leads to improved insulation, but the use of hollow-glass microspheres does not improve the insulation capabilities.

基于原子模型的非局域连续模型

In traditional continuum theories, constitutive relations such as generalized Hooke’s law and the Fourier law for heat conduction, are representations of macroscopic phenomenons. These theories have severe limitations as the length scale of the structure becomes close to the atomistic dimensions. The failure of the traditional continuum theories at atomistic scale can be attribute to the ignoring of the length scale induced by intrinsic non-locality of atomistic interactions. The primary objective of this work is to construct a coupled atomistic-continuum model strictly based on the atomistic model bypassing any empirical constitutive law and any local homogeneous assumption such as Cauchy-Born rule. The intrinsic non-locality of the atomistic interactions are explicitly built into the model. Inhomogeneous deformation at atomistic scale can be efficiently represented by this model.