Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux

Presented is the analysis of non-Fourier effect in a hollow sphere exposed to a periodic boundary heat flux. The problem is studied by deriving an analytical solution of the hyperbolic heat conduction equation. Using the obtained analytical expression, the temperature profiles at outer and inner surfaces of the sphere are evaluated for various thermal relaxation times. By comparing the results of non-Fourier model with those obtained from Fourier heat conduction equation, the transition process from parabolic model to hyperbolic one is shown. The phase difference and amplitude ratio of boundary surfaces are calculated as functions of the thermal relaxation time and the results are depicted graphically.     

DPL Model Analysis of Non—Fourier Heat Conduction Restricted by Continuous Boundary Interface

IntroductionAs widely known, the hahonal Fourier law isbased on a large quantity of regular heat transfer (i.e. thethermal bine scale is comparatively lOng and the heatflux density is comparatively small) experiments and it'sjust a phenomenological descriphon of regular thermalProcesses. The Fourier law itself mpes an infinitespeed of Propagation of thermal distUrbance, indicatingthat a local change in tempera~ causes aninstantaneous per'tUrbation in the temperatore at eachPOint in the medi…     

Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue

In this article, the parabolic (Pennes bioheat equation) and hyperbolic (thermal wave) bioheat transfer models for constant, periodic and pulse train heat flux boundary conditions are solved analytically by applying the Laplace transform method for skin as a semi-infinite and finite domain. The bioheat transfer analysis with transient heat flux on skin tissue has only been studied by Pennes equation for a semi-infinite domain. For modeling heat transfer in short duration of an initial transient, or when the propagation speed of the thermal wave is finite, there are major differences between the results of parabolic and hyperbolic heat transfer equations. The non-Fourier bioheat transfer equation describes the thermal behavior in the biological tissues better than Fourier equation. The outcome of transient heat flux condition shows that by penetrating into the depths beneath the skin subjected to heat, the amplitude of temperature response decreases significantly. The blood perfusion rate can be predicted using the phase shift between the surface temperature and transient surface heat flux. The thermal damage of the skin is studied by applying both the parabolic and hyperbolic bioheat transfer equations.     

Fourier and non-Fourier heat conduction analysis in the absorber plates of a flat-plate solar collector

This paper presents an analytical analysis of both Fourier and non-Fourier heat conduction in the absorber plates of a flat-plate solar collector. Separation of variables was employed to develop the model. For the analysis, a repetitive heat transfer module was used for the solution of parabolic and hyperbolic equations. From the practical point of view, two types of boundary conditions were separately chosen. A numerical technique based on the finite difference method was employed to determine the temperature for validation purposes. A comparative investigation was carried out to understand the requirements for use of the non-Fourier heat conduction model easily. A significant difference in the temperatures obtained from the Fourier and non-Fourier models was observed for lower values of the Fourier number and higher values of the Vernotte number. Finally, the effect of the boundary conditions on the Fourier and non-Fourier heat transfer was demonstrated.     

RAPID TRANSIENT HEAT CONDUCTION IN MULTILAYER MATERIALS WITH PULSED HEATING BOUNDARY

ABSTRACT

Rapid transient heat conduction in multilayer materials under pulsed heating is solved numerically based on a hyperbolic heat conduction equation and taking into consideration the non-Fourier heat conduction effects. An implicit difference scheme is presented and a stability analysis conducted, which shows that the implicit scheme for the hyperbolic equation is stable. The code is validated by comparing the numerical results with an existing exact solution, and the physically unrealistic conditions placed on the time and space increments are identified. Using the validated model, the numerical solution of thermal wave propagation in multilayer materials is presented. By analyzing the results, the necessary conditions for observing non-Fourier phenomena in the laboratory can be inferred. The results are also compared with the numerical results from the parabolic heat conduction equation. The difference between them is clearly apparent, and this comparison provides new insight for the management of thermal issues in high-energy equipment. The results also illustrate the time scale required for metal films to establish equilibrium in energy transport, which makes it possible to determine a priori the time response and the measurement accuracy of metal film, thermal-resistant thermometers.     

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.     

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.     

Fractional heat conduction in a thin hollow circular disk and associated thermal deflection

The time nonlocal generalization of the classical Fourier law with the “Long-tail” power kernel can be interpreted in terms of fractional calculus and leads to the time fractional heat conduction equation. The solution to the fractional heat conduction equation under a Dirichlet boundary condition with zero temperature and the physical Neumann boundary condition with zero heat flux are obtained by integral transform. Thermal deflection has been investigated in the context of fractional-order heat conduction by quasi-static approach for a thin hollow circular disk. The numerical results for temperature distribution and thermal deflection using thermal moment are computed and represented graphically for copper material.     

NON-FOURIER HEAT CONDUCTION PHENOMENA IN POROUS MATERIAL HEATED BY MICROSECOND LASER PULSE

Experiments on porous material heated by a microsecond laser pulse and the corresponding theoretical analysis are carried out. Some non-Fourier heat conduction phenomena are observed in the experimental sample. The experimental results indicate that only if the thermal disturbance is strong enough (i.e., the pulse duration is short enough and the pulse heat flux is great enough) is it possible to observe apparent non-Fourier heat conduction phenomenon in the sample, and evident non-Fourier heat conduction phenomenon can only exist in a very limited region around the thermal disturbance position. The hyperbolic heat conduction (HHC) equation and the dual-phase lag (DPL) model are employed, respectively, to describe the non-Fourier heat condution process happening in the experimental sample, and the finite-difference method (FDM) is used to solve them numerically. The numerical solutions show that both the HHC equation and the DPL model can predict the non-Fourier heat conduction phenomenon emerging in the experimental sample qualitatively. Moreover, if τ_q and τ_T are assumed to have suitable values, the theoretical result of the DPL model is more agreeable to the experimental result.     

A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium

Many non-equilibrium heat conduction processes can be described by the macroscopic dual-phase lag model (DPL model). In this paper, a numerical method, which combines the dual reciprocity boundary element method (DRBEM) with Laplace transforms, is constructed to solve such mathematical equation. It is used to simulate the non-Fourier phenomenon of heat conduction in a single-phase medium, then numerically predict the differences between the thermal diffusion, the thermal wave and the non-Fourier heat conduction under different boundary conditions including pulse for one- and two-dimensional problems. In order to check this numerical method's reliability, the numerical solutions are still compared with two known analytical solutions.     

Lattice Boltzmann simulation of non-Fourier heat conduction with phase change

The lattice Boltzmann method (LBM) combined with the enthalpy method is a very effective method to solve the solid–liquid phase transition problem. However, when the heat flux is very high or the time of the process is in the same order of magnitude as the relaxation time, it is necessary to consider the non-Fourier effect in heat conduction. At this time, whether the LBM-BGK format based on Bhatnagar-Gross-Krook (BGK) approximation is still valid remains to be discussed. In this paper, the hyperbolic lattice Boltzmann method (HLBM) is combined with the enthalpy method to solve the non-Fourier solid–liquid phase change problem. By solving the non-Fourier heat conduction problem and the Fourier solid–liquid phase change problem, the numerical solution is compared with the analytical solution to verify the accuracy of the algorithm. The effect of different relaxation times on the solid–liquid phase transition is analyzed. In addition, the effect of changes in thermal diffusivity due to state changes on the non-Fourier solid–liquid phase transition is discussed.     

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.     

An axisymmetric bioheat transfer analysis through a unified model

A unified model is developed for the analysis of heat transfer (radiation and non-Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating-conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two-dimensional axisymmetric geometry. The equation of transfer and the coupled non-Fourier conduction-radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri-diagonal-matrix-algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation- conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non-Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction-radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.     

A NUMERICAL SCHEME FOR NON-FOURIER HEAT CONDUCTION,PART II: TWO-DIMENSIONAL PROBLEM FORMULATION AND VERIFICATION

A flux-splitting algorithm based on the Godunov numerical scheme developed for the solution of the one-dimensional non-Fourier heat conduction equation by Yeung and Lam [1] is extended for the investigation of thermal wave propagation in rectangular media. The derivation of the solution method and the stability criteria are presented in detail. Physical problems subjected to various boundary conditions (e.g., first, second, and third kinds) can be studied with the numerical scheme. A comparison of the exact solution with the one calculated by the proposed procedure is presented to confirm the validity of the numerical procedure. The numerical scheme is applicable for the study of short-pulse heating in advanced materials, microstructures, thin films, semiconductor devices, and superconductors.     

Thermodynamic stability analysis of non-Fourier model-based bio-heat transfer in laser-irradiated cylindrical-shaped multilayered biological tissue

The present research focuses on examining the thermic response of living tissue in the form of a triple-layered cylindrical structure when subjected to laser light and the compatibility analysis of non-Fourier heat transfer with thermodynamics second law. The temperature field in the triple-layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non-Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non-Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.     

Analysis of Non-Fourier Conduction-Radiation Heat Transfer in a Cylindrical Enclosure

This article deals with the analysis of non-Fourier conduction and radiation heat transfer in a participating medium contained between 1-D concentric cylinders. The conducting-radiating medium is radiatively absorbing, emitting, and scattering. The non-Fourier effect is analyzed by suddenly perturbing the temperatures of the concentric cylinders. With radiative information computed using the finite volume method, the finite difference method is used to solve the hyperbolic energy equation. Effects of various parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the emissivity and the radius ratio are studied on the temporal evolution of temperature field in the medium. These parameters have been found to significantly influence the temporal temperature field, and non-Fourier effects are captured well. For non-Fourier conduction and Fourier conduction–radiation cases, results have been benchmarked against those available in the literature. A good comparison has been observed. In case of non-Fourier conduction-radiation, for some sample cases, the steady-state temperature distributions have been compared against those available in the literature. Results have been found to agree well.     

Combined mode conduction and radiation heat transfer in a spherical geometry with non-Fourier effect

This article deals with the analysis of combined mode non-Fourier conduction and radiation heat transfer in a concentric spherical enclosure containing a conducting–radiating medium. The finite volume method (FVM) has been employed to calculate the volumetric radiative information and also to solve the governing energy equation, which is of hyperbolic nature. The non-Fourier effect which manifests in the form of a sharp discontinuity in the temporal temperature distribution and propagates with a finite speed has been investigated. As time progress, the discontinuity in the temperature distribution decays and in the steady-state, results with and without non-Fourier effect are the same. Detailed study of the effect of various parameters such as the extinction coefficient, the scattering albedo, the conduction radiation parameter, the emissivity and the anisotropy factor has been carried out. Results of the present work have been compared with the steady-state response of the combined mode Fourier conduction–radiation problems available in literature. Results have been found to agree well.     

“Eigen-periodic”-in-space surface heating in conduction with application to conductivity measurement of thin films

A two-dimensional heat conduction problem in Cartesian coordinates subject to a periodic-in-space boundary condition is analyzed by the Green’s functions approach. It is pointed out that when the frequency of the spatial periodic heating equates one of the natural frequencies (eigenvalues) of the system, the solution of the 2D heat conduction problem can be written down very simply as the product of the periodic surface condition (termed the “eigen-periodic”) by the solution of a 1D fin problem along the nonhomogeneous direction. This result suggests a novel and simple algebraic equation for determining the thermal conductivity of thin films placed on substrates under steady state conditions. High space frequencies of the sinusoidal heating, larger than the deviation frequency, are used to make negligible the thermal deviation effects due to the presence of the substrate.     

Transient hygrothermoelastic response in a porous cylinder subjected to ramp-type heat-moisture loading

Abstract

The classical theory of heat conduction (Fourier theory) predicts an infinite speed for thermal disturbance propagation, which is physically unrealistic. By extending the classical Fourier heat conduction and Fick’s diffusion, this article develops hyperbolic diffusion/heat conduction laws with phase lags of heat/moisture flux to simulate coupled heat-moisture diffusion-propagation behavior with the Defour and Soret effects. A porous cylinder subjected to a ramp-type heating and humidifying at the surface is studied. The Laplace transform is used to obtain a closed-form solution of the temperature, moisture, displacements and stresses in the cylinder. Numerical results are calculated via the inversion of the Laplace transform. Obtained results show that the thermal/moisture relaxation time or phase lag plays a significant role in affecting transient hygrothermoelastic field. For a non-vanishing phase lag, non-Fourier and non-Fickian effects exist and hygrothermal waves have finite propagation speeds. The influences of the phase lag of heat/moisture flux and ramp-type time parameter on the transient response of hygrothermoelastic field are presented graphically. A comparison of the numerical results based on the classical model and the present one is made. The non-Fourier heat conduction and non-Fickian diffusion can effectively avoid the shortcomings induced by the classical Fourier and Fick laws.     

Finite-Volume Solution of a Hyperbolic Two-Step Conduction Equation

The finite-volume method is developed for the solution of two-step hyperbolic conduction equations and applied to fast transient non-Fourier conduction problems in which two energy carriers are not in thermal equilibrium. Numerical methodology has been formulated to be applied to heterogeneous materials with temperature-dependent properties. Application to ultrafast laser heating of thin composite metals shows favorable results in comparison with the existing data.