Hyperbolic Heat Conduction Equation Research Articles

On measurement of the thermal relaxation time in transient thermal processes of solids

Experimental transient processes are compared with theoretical processes described by unsteady boundary value problems of the third kind with parabolic and hyperbolic heat conduction equations of a solid. Physical meaning of transfer rate is specified in the thermal relaxation formula. Thermal relaxation time of the solid is measured.

Analytical solution to non-Fourier heat conduction as a laser beam irradiating on local surface of a semi-infinite medium

The two dimensional non-Fourier heat conduction in a semi-infinite medium on which local surface is irradiated by a laser beam is analyzed. The mathematical model is based on hyperbolic heat conduction equation with thermal relaxation time and is deals with boundary condition with time step heating and uniform temperature distribution in local area of the surface. A new analytic solution to the problem is derived by using Laplace and Hankel transforms and is owing to finding analytic result of inverse Laplace transform to time domain exactly. The solution of the temperature field in semi-infinite medium is expressed as an infinite integral of known function. The proposed numerical technique to the integral is implemented. Non-Fourier effect of temperature field is shown in numerical examples. The evolution of contour plots of temperature field over time is presented by evaluating the analytic solution demonstrated significant difference between hyperbolic and classical heat conduction.

Determination of the Dynamic Stresses in an Infinite Plate on the Basis of an Exact Analytical Solution of the Hyperbolic Heat-Conduction Equation for It

With the use of an exact analytical solution of the classical hyperbolic heat-conduction equation, derived on the basis of the Maxwell–Cattaneo–Luikov relaxation formula, an exact analytical solution of the problem on the dynamic thermoelasticity of an infinite plate was obtained for the case where the outer surfaces of this plate are free of mechanical loads. It is shown that the undamped thermoelastic stresses in this plate vary spasmodically in time with periodic change in their sign. The stress jumps arising near the opposite outer surfaces of the indicated plate move along the spatial variable to its center where they superimpose, with the result that the dynamic thermal stresses in the plate double.

3-Omega Method for Thin Films with the Non-Fourier Boundary Condition

The effect of non-Fourier boundary condition on the 3-omega method for measuring the thermal conductivity of microscale thin films using the hyperbolic heat conduction equation and the Fourier equation is examined. Non-Fourier boundary condition with the Fourier equation leads to 80% error in the temperature oscillations and increases the error to 85% in the case of non-Fourier boundary condition with the hyperbolic heat conduction equation. The solution of the non-Fourier boundary condition with the hyperbolic heat conduction equation gives the most accurate thermal conductivity expression. The analysis also provides a method for determining the relaxation time of thin films.

ANALYTICAL SOLUTION OF THE PROBLEM OF NON-FOURIER HEAT CONDUCTION IN A SLAB USING THE SOLUTION STRUCTURE THEOREMS

This paper studies an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction equation can be obtained by using the fundamental mathematics. In this paper, the non-Fourier heat conduction in a slab at whose a left boundary there is a constant heat flux and at the right boundary, a constant temperature Ts = 15, has been investigated. The complicated problem is split into multiple simpler problems that in turn can be combined to obtain a solution to the original problem. The original problem is divided into five subproblems by setting the heat generation term, the initial conditions, and the boundary conditions for different values in each subproblem. All the solutions given in this paper can be easily proven by substituting them into the governing equation. The results show that the temperature will start retreating at approximately t = 2 and for t = 2 the temperature at the left boundary decreases leading to a decrease in the temperature in the domain. Also, the shape of the profiles remains nearly the same after t = 4. The solution presented in this study can be used as benchmark problems for validation of future numerical methods.

Thermal wave scattering in functionally graded materials containing a spherical inclusion

In this paper, based on the law of non-Fourier heat conduction, using the wave function expansion method, the thermal wave scattering and temperature distributions of semi-infinite functionally graded materials containing a spherical inclusion were presented. According to the hyperbolic equation of heat conduction, a general solution of scattered fields of thermal waves was obtained. In addition, a matrix formulation to determine mode coefficients of scattered waves were used. Taking into account the engineering background, the incidence of thermal waves excited by the periodically modulated laser and the subsurface defect was treated as a spherical inclusion in the modeling. Numerical simulation was graphically presented and analyzed. It is shown that the influence of the spherical inclusion on the thermal wave scattering and temperature distributions in functionally graded materials are related to the depth of buried spherical inclusion, the nonhomogeneous parameters and the thermal relaxation time. The paper is expected to provide data references of the inverse problem for infrared thermal wave nondestructive evaluation of functionally graded materials.

Transient Thermoelastic Analysis of a Solid Cylinder Containing a Circumferential Crack Using the C–V Heat Conduction Model

This article presents the transient thermoelastic analysis in a long solid cylinder with a circumferential crack using the C–V heat conduction theory. The outer surface of the cylinder is subjected to a sudden temperature change. The Laplace transform technique is adopted to solve the one-dimensional hyperbolic heat conduction equation, and the axial thermal stress is obtained for the un-cracked cylinder in the Laplace domain. Then this axial thermal stress with a minus sign is applied to the crack surface to form a mixed boundary value problem in the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier and Hankel transforms to solve the mode I crack problem. The transient thermal stress intensity factors are obtained by solving the singular integral equation numerically. The influences of thermal relaxation time, crack geometry, and Biot's number upon transient temperature distributions, axial stress fields, and stress intensity factors are analyzed.

Effect of thermal wave propagation on thermoelastic behavior of functionally graded materials in a slab symmetrically surface heated using analytical modeling

Design and development of FGMs as the heat treatable materials for high-temperature environments with thermal protection require understanding of exact temperature and thermal stress distribution in the transient state. This information is primary tool in the design and optimization of the devices for failure prevention. Frequently FGMs are used in many applications that presumably produce thermal energy transport via wave propagation. In this study, transient non-Fourier temperature and associated thermal stresses in a functionally graded slab symmetrically heated on both sides are determined. Hyperbolic heat conduction equation in terms of heat flux is used for obtaining temperature profile. Separation of variables scheme based on the variation of parameters is implemented to solve the non-homogenous thermoelastic problem in which any arbitrary temperature distribution can be employed. Physical properties are assumed to vary exponentially in the media and the problem is analyzed for different mechanical boundary conditions. Furthermore effect of the material inhomogeneity, thermal relaxation time and the Fourier number on the stress distribution, temperature variation and jumps is investigated. Parallel computation is used to present the fully converged numerical results. Findings indicate the non-Fourier heat conduction has significant influence on the dynamic temperature and stress field. Based on the results it is suggested that in the design of FG structures against failure under thermal loading and heat treatability, the hyperbolic model is more appropriate than the classical Fourier model.

Influence of inclusion in functionally graded materials on the surface temperature distributions

In this paper, based on the hyperbolic equation of heat conduction, utilizing the image method and the wave function expansion method, the temperature distributions on the surface of a functionally graded material(FGM) containing a cylindrical inclusion are investigated. According to the model of thermal waves, a general solution of scattered fields of thermal waves is obtained. Effects of different physical parameters (such as the depth of buried inclusion, the thermal conductivity, the thermal diffusion length, the thermal diffusivity, and the thermal relaxation time) on the distribution of temperature are analyzed. The thermal waves are excited on the surface of the FGM by a periodically modulated laser. A cylindrical defect is taken as an inclusion under a thermal conduction condition. Results are expected to provide calculation methods and reference data for infrared thermal wave nondestructive evaluation of an FGM and the inverse problems in mathematical physics.

An exact analytical solution of non-Fourier thermal stress in cylindrical shell under periodic boundary condition

Article history: Received March 6, 2014 Accepted 23 August 2014 Available online 24 August 2014 This paper presents a perfect analytical solution of the hyperbolic asymmetric heat conduction equation and the related thermal displacement equation within a long hollow cylinder (plain strain condition) exposed to a harmonic boundary condition. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The standard method of separation of variables is used for solving the problem with time-independent boundary conditions and the Duhamel integral is used for applying the time-dependency. The results show the wave behavior of Non-Fourier thermal stresses and higher oscillation amplitude in comparison with Fourier one. The developed analytic answer can be applied for modeling cylindrical shell of nuclear rod and can be applied as a benchmark to validate the other numerical solutions. © 2014 Growing Science Ltd. All rights reserved.

High order numerical simulation of non-Fourier heat conduction: An application of numerical Laplace transform inversion

Non-Fourier heat conduction phenomenon in a finite slab with insulated boundaries is investigated in the present paper. Since solving the hyperbolic heat conduction equation analytically requires considerable effort, a new high-order numerical approach has been implemented to achieve comparable exactitude. This method solves the considered equation in Laplace space and numerical inversion is employed with the intention of transformation to temporal domain. In order to examine numerical accuracy of this method, Dirac delta heat flux is applied to the assumed medium and results were compared with those of the analytical solution. It was observed that numerical values follow exact ones, at least up to the seventh order of accuracy. In addition, Step and Triangular heat pulses in the medium were studied to reveal temporal and spatial non-Fourier heat conduction characteristics. It was found that in large values of Ve number, for various kinds of heat fluxes carrying the same amount of energy, temperature distribution varies conspicuously through the medium; nevertheless, at each pass of heat wave, a specific point experiences a definite rise of temperature regardless of the type of heat flux provided that the same conditions are present.

Investigation Into Pulse Laser Heating of Nanoscale Au Film Using Dual-Phase-Lag Model

In this study the thermal field is presented for pulse laser processing of nanoscale Au films. Fourier law is inadequate for describing the heat conduction in nanoscale process due to the boundary scattering and the finite relaxation time of heat carriers. In the regime where the particle description of electrons and phonons is valid, the Boltzmann equation is the most accurate option to model heat transfer in such problems. However, solving the Boltzmann equation is generally difficult due to involving three spatial, three momentums and one time. Dual-phase-lag (DPL) model is averaged over the momentum space and thus involves only spatial coordinates plus time, as in the Fourier equation. Therefore this paper utilizes the dual-phase-lag (DPL) model with scattering boundary condition to study the temperature field for laser processing of nanometer-sized thin films instead of Boltzmann equation. The results obtained from the dual-phase-lag heat conduction model, hyperbolic and parabolic heat conduction equations were compared with the available experimental data to validate the compatibility of the thermal models for analyzing the heat transfer in nanoscale thin film irradiated by laser. The temperature history at different locations of the thin film and the effects of boundary phonon scattering on the normalized temperature were also discussed.

Numerical investigation of fast precooling of a cylindrical food product based on the hyperbolic heat conduction model

PurposeIn this study, the purpose was to introduce two‐dimensional hyperbolic heat conduction equations in order to simulate the fast precooling process of a cylindrically shaped food product with internal heat generation. A modified model for internal heat generation due to respiration in the food product was proposed to take the effect of relaxation time into account. The obtained governing equations were solved numerically using an efficient finite difference technique. The influence of Biot number and heat generation parameters on thermal characteristics was examined and discussed. The results based on hyperbolic model were compared with the classical parabolic heat diffusion model. The present numerical code was validated via comparison with analytical solution and a good agreement was found.Design/methodology/approachThe obtained governing equations were solved numerically using an efficient finite difference technique.FindingsThe influence of Biot number and heat generation parameters on thermal characteristics was examined and discussed. The results based on hyperbolic model were compared with the classical parabolic heat diffusion model. The present numerical code was validated via comparison with analytical solution and a good agreement was found.Originality/valueTwo‐dimensional analysis of fast precooling of cylindrical food product based on hyperbolic heat conduction model has not been investigated yet.

Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time

Differential-difference heat conduction and diffusion models and equations with a finite relaxation time are described that give a finite disturbance propagation rate. The modified Biot-Fourier law with delay is used for the heat flux, which leads to the differential-difference heat-conduction equation $$\left. {\frac{{\partial T}} {{\partial t}}} \right|_{t + \tau } = a\Delta T, $$ where the left-hand side is calculated at t + τ (τ is the relaxation time) and the right-hand side is calculated, as usual, at t (without a time shift). At τ = 0, the differential-difference heat-conduction equation turns into the classical parabolic heat-conduction equation; if the left-hand side is expanded into a series in τ and two main terms of expansion are retained, we obtain the Cattaneo-Vernotte hyperbolic heat-conduction equation. An exact solution to the differential-difference heat-conduction equation is derived for a one-dimensional problem without the initial conditions with an arbitrary periodic boundary condition. The approximate solution is constructed to the general three-dimensional initial-boundary value problem of heat propagation with a finite relaxation time in a bounded domain with arbitrary initial heat distribution and the boundary condition of the third kind. It is shown that the differential-difference model makes it possible to derive the Oldroyd-type differential heat-conduction model.

Splitting schemes for hyperbolic heat conduction equation

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat fluxes. In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions. Unconditional stability of locally one-dimensional splitting schemes is established. New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.

Solution and analysis of non-Fourier heat conduction in a plane slab under arbitrary periodic surface thermal disturbance

In this paper, the non-Fourier heat conduction in a plane slab under arbitrary periodic surface thermal disturbance is solved analytically. Hyperbolic heat conduction equation is employed to describe this problem involving high-rate change of temperature. Firstly, when the plane slab surface is subjected to a sudden heat flux change or a harmonic heat flux change, the analytic solution of this problem is found by using the separation of variables method and Duhamel’s principle. On this basis, when the plane slab surface is subjected to an arbitrary periodic heat flux change, the analytic solution of temperature field is obtained by using the Fourier series and the principle of superposition. Using the obtained analytical solution, the temperature profiles of the plane slab are analyzed, and the differences between the temperature response obtained by using non-Fourier heat conduction model and that obtained by using Fourier model are discussed. This solution can be applied to more realistic periodic boundary conditions in technology.

Thermal Analysis of Non-linear Convective–Radiative Hyperbolic Lumped Systems with Simultaneous Variation of Temperature-Dependent Specific Heat and Surface Emissivity by MsDTM and BPES

With the advent of temperatures near absolute zero, it is often claimed that at very low temperatures the effect of thermal wave propagation must be included by the hyperbolic heat conduction equation (HHCE). In this paper the non-linear convective–radiative HHCE is investigated. Opposite to common numerical analyses, analytical expressions are obtained for the temperature variations by the multi-step differential transformation method. Some conclusions about alteration of the specific heat of the material, temperature steeping, and Vernotte number have been formulated.

Analysis of dual-phase-lag heat conduction in short-pulse laser heating of metals with a hybrid method

In this study, the dual phase lag (DPL) heat transfer model is applied to investigate the transient heat transfer in a thin metal film exposed to short-pulse laser heating. An efficient numerical scheme involving the hybrid application of the Laplace transform and control volume methods in conjunction with hyperbolic shape functions is used to solve the hyperbolic heat conduction equation in the linearized form of DPL model. The transformed nodal temperatures are inverted to the physical quantities by using numerical inversion of the Laplace transform. Comparison between the numerical results and the analytic solution for a short-pulse laser heating with the Gaussian temporal profile evidences the accuracy of the present numerical results. Effect of different phase lags values of the heat flux and the temperature gradient on the behavior of heat transfer is also investigated. The results show that the phase lag of the heat flux tends to induce thermal waves with sharp wave-fronts separating heated and unheated zones in the metal film, while the phase lag of the temperature gradient destroys the waveforms and increases the thermally disturbed zone.

Large thermal transport phase lagging improves thermoelectric efficiency

A thermoelectric (TE) generator driven by fast pulse heating is theoretically analyzed using the hyperbolic transient heat conduction equation. Results show that heat leaking in the transient TE generator can be reduced due to the lagging behavior of non-Fourier heat conduction. The total energy conversion efficiency of the transient TE generator is found to exceed that of normal steady-state operation when the dimensionless non-Fourier thermal transport relaxation time, scaled by length squared over thermal diffusivity, is larger than one. These findings may emerge as a new way to improve TE efficiency.

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.