Analytical solution of hyperbolic heat conduction equation in relation to laser short-pulse heating

In the present study, the hyperbolic heat conduction equation is derived from the Boltzmann transport equation and the analytical solution of the resulting equation appropriate to the laser short-pulse heating of a solid surface is presented. The time exponentially decaying pulse is incorporated as a volumetric heat source in the hyperbolic equation to account for the absorption of the incident laser energy. The Fourier transformation is used to simplify the hyperbolic equation and the analytical solution of the simplified equation is obtained using the Laplace transformation method. Temperature distribution in space and time are computed in steel for two laser pulse parameters. It is found that internal energy gain from the irradiated field, due to the presence of the volumetric heat source in the hyperbolic equation, results in rapid rise of temperature in the surface region during the early heating period. In addition, temperature decay is gradual in the surface region and as the depth below the surface increases beyond the absorption depth, temperature decay becomes sharp.     

Time fractional dual-phase-lag heat conduction equation

We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.     

The triple integral equations method for solving heat conduction equation

A new type of triple integral equation was used to determine a solution of nonstationary heat equation in axially symmetric cylindrical coordinates under mixed discontinuous boundary of the first and second kind conditions acted on the level surface of solid cylinder, with the aid of a Laplace transform, the solution of the given triple equations is introduced to a singular integral equation of the second kind.     

The stationary state and the heat equation for a variant of Davies' model of heat conduction

We study a variant of Davies' model of heat conduction, consisting of a chain of (classical or quantum) harmonic oscillators, whose ends are coupled to thermal reservoirs at different temperatures, and where neighboring oscillators interact via intermediate reservoirs. In the weak coupling limit, we show that a unique stationary state exists, and that a discretized heat equation holds. We give an explicit expression of the stationary state in the case of two classical oscillators. The heat equation is obtained in the hydrodynamic limit, and it is proved that it completely describes the macroscopic behavior of the model.     

On a generalization of the Gibbs equation for heat conduction

A generalized Gibbs equation for the heat conduction problem is proposed in order to take finite wave speed into account.     

An approximate analytic solution of the heat conduction equation at nanoscale

The Adomian decomposition method (ADM) and the Adomian double decomposition method (ADDM) for solving the 3D non-Fourier heat conduction equation at nanoscale based on the dual-phase-lag framework are proposed. We show that the noise terms that appear in ADM solution can be removed, if the ADDM is employed.     

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.     

Peculiarities of solutions to the heat conduction equation in fractional derivatives

Features of solutions to the heat conduction equation in fractional derivatives taking into account diffusion and convection mechanisms of heat transfer are analyzed. One-dimensional cases of infinite straight line, semi-infinite line, and the problem with zero initial conditions are considered.     

Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

In this paper exact analytical solutions for the equation that describes anomalous heat propagation in a harmonic 1D lattices are obtained. Rectangular, triangular and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to \(1/\sqrt t \). In the center of the perturbation zone the decay is proportional to 1/t. Thus, the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experimentally.     

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Effects of defects on heat conduction of graphene/hexagonal boron nitride heterointerface

We investigate the effects of point defects on the Interface Thermal Resistance (ITR) of graphene/hexagonal boron nitride (G/h-BN) heterointerface with various stacking forms by ultrafast thermal pulse method. The results reveal that the ITR of different stacking forms presents a significant downward trend with the existence of point defects. This counterintuitive behavior is attributed to the defects increase the vibration intensity of out-of-plane phonons of graphene in low-frequency region, thus enhancing the phonons coupling between graphene and h-BN layer. ITR of G/h-BN is further reduced by 50% with the defect rate increases from 0% to 5% and that is reduced by 65% with the temperature rises from 200 K to 700 K. Besides, it is found that the defective G/h-BN has thermal rectification characteristic and that is positively related to temperature and defect rate. Our study provides a practical way for the application of defects in graphene and a new approach for the design of thermal rectifier devices.     

基于分子动力学的石墨炔纳米带空位缺陷的导热特性

空位缺陷石墨炔比完整石墨炔更贴近实际材料,而空位缺陷的多样性可导致更丰富的导热特性,因此模拟各种空位缺陷对热导率的影响显得尤为重要.采用非平衡分子动力学方法,通过在纳米带长度方向上施加周期性边界条件,基于AIREBO(adaptive intermolecular reactive empirical bond order)势函数描述碳-碳原子间的相互作用,模拟了300 K时单层石墨炔纳米带乙炔链上单空位缺陷和双空位缺陷以及苯环上单空位缺陷对其热导率的影响,利用Fourier定律计算热导率.模拟结果表明,对于几十纳米尺度范围内的石墨炔纳米带热导率,1)由于声子的散射集中和声子倒逆过程增强,与完美无缺陷的石墨炔纳米带相比,空位缺陷会导致石墨炔纳米带热导率的下降;2)由于声子态密度匹配程度高低的不同,相比于乙炔链上的空位缺陷,苯环的空位缺陷对石墨炔纳米带热导率影响更大,乙炔链上空位缺陷数量对石墨炔纳米带热导率的影响明显;3)由于尺寸效应问题,随着长度增加,石墨炔纳米带热导率会相应增大.本文的研究可为在一定尺度下进行石墨炔纳米带热导率的调控问题提供参考.     

The heat equation with singular coefficients

The small time asymptotics of the kernel ofe ^?tH is defined and derived for \(H = \frac{{d^2 }}{{dx^2 }} + \frac{\kappa }{{x^2 }}\) on ?¹. Lemmas on singular asymptotics in the sense of distributions are formulated and used. The results are applied to derive an index formula on ?¹.     

A model of heat conduction

We construct a model of a chain of atoms coupled at its ends to two reservoirs at different temperatures. In a weak coupling limit the atoms obey a stochastic evolution law and have an equilibrium state with a uniform temperature gradient along the chain.     

Modified equations for heat conduction

It is shown that the form of the modifications previously suggested for the heat conduction equations when macroscopic parameters are changing rapidly may be incorrect when the heat carriers' relaxation time is wave number dependent. Even when the relaxation time is constant it is shown that for insulators the correction terms previously suggested are in error by a factor of 5.     

A semi-microscopic model of the heat conduction law

We give two very simple quantum models for the heat conduction law using a master equation approach for the probability distribution of the quantum numbers of the oscillators. The probability of interaction of the oscillators is given by the Landau-Teller formula.     

Analysis of a heat wave induced by laser radiation absorption in an optical fiber on the basis of a 2D nonstationary heat conduction equation

A heat wave resulting from the absorption of laser radiation in the core of an optical fiber is studied using a nonstationary 2D heat conduction equation. The velocity of the wave as a function of the laser intensity is determined, and the threshold intensity generating the heat wave is calculated. At high intensities, the velocity of the wave can be qualitatively described by a well-known formula from combustion theory; i.e., the velocity is shown to be proportional to the square root of the radiation intensity. The analytical threshold laser intensities closely agree with the available experimental data.