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.     

Thermodynamically consistent description of heat conduction with finite speed of heat propagation

In this work, governing equations for heat conduction with finite speed of heat propagation are derived directly from classical thermodynamics. For a one-dimensional flow of heat, the developed governing equation is linear and of parabolic type. In a three dimensional case, the system of nonlinear equations is formulated.Analytical solutions of the equations for one-dimensional flow of heat are obtained, and their analysis shows characteristic features of heat propagation with finite speed, being fully consistent with classical thermodynamics.     

Microscopic vs. Macroscopic Hyperbolic Heat Conduction: Validation Criterion under the Effect of a Moving Heating Source

The validity of using the microscopic hyperbolic heat conduction model under the effect of a moving heating source is investigated. In macroscopic heat conduction, the solid lattice and the electron gas have the same temperature, while in the microscopic heat conduction model, the solid lattice and electron gas do not have the same temperature. Two configurations are considered: the finite and the semi-infinite domains. For each configuration, two types of thermal boundary conditions are considered: the isothermal and the insulated types. Four parameters are found to control the thermal behavior of the considered problem: the dimensionless heating source speed, heat capacity ratio, dimensionless relaxation time, and dimensionless plate thickness for the finite domain configuration. It is found that using the microscopic hyperbolic heat conduction model instead of the hyperbolic macroscopic model is essential when the dimensionless speed of the source is greater than 0.01. The heat capacity ratio is found to have an insignificant effect on the domain thermal behavior, and this is true for small values of the relaxation time. However, the deviation between the microscopic and macroscopic models increases as the dimensionless plate thickness for the finite domain decreases. The deviation between the two models is significant within the very early stage of time.     

Approximate analyses of Fourier and non-Fourier heat conduction models by the variational principles based on Laplace transforms

Approximate analysis is a major application of variational principles for heat conduction. Recently, O’Toole’s variational principle for Fourier’s law has been extended to non-Fourier heat conduction models, which are applied to approximate analyses based on the Rayleigh–Ritz method. Suitable trial functions satisfying boundary conditions are sought, and then substituted into the variational principles to obtain the undetermined coefficients. From the inverse Laplace transforms, the approximate solutions are obtained. Examples are provided for 1D problems for different heat conduction models. The largest calculation errors are one or two orders of magnitude smaller than the equilibrium temperature, which will tend to be zero.     

Analysis of a problem arising in porous thermoelasticity of type II

In this article, we consider, from the numerical point of view, a linear thermo-porous-elastic model. The heat conduction is assumed to be of type II. The mechanical problem is written as a coupled system of three hyperbolic partial differential equations for the displacements, the porosity and the thermal displacements. Then, its variational formulation is derived, which is written as a coupled system of three hyperbolic linear variational equations in terms of the velocity, the porous speed and the temperature. An existence and uniqueness result, as well as an energy decay property, is recalled. The fully discrete approximation of the aforementioned problem is introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. A stability property is proved, from which the energy decay of the discrete energy is deduced. Then, a priori error estimates are obtained, from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution.     

有限速率过程对活塞式斯特林发动机性能的影响

本文讨论在给定热源温度和压缩比的情况下,过程进行的速率有限,并受热传导不可逆影响的内可逆活塞式斯特林发动机的最优性能,导出以理想气体或范德瓦尔斯气体为工质的斯特林发动机的最大输出功率与热效率的关系,以及最大热效率与输出功率的关系,并推出了一些新的有限时间热力学的性能界限。     

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

Abstract

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.     

Conduction based calibration of handmade platinum thin film heat transfer gauges for transient measurements

Heat transfer measurement using thin film gauges (TFG) is the most prevalently used technique for determination of surface heat flux. They are best suited for short duration transient surface temperature measurements and typically used in the applications where convection is a dominant mode of heat transfer such as gas turbine engines, high speed flights etc. However, in few interdisciplinary research areas, there are practical issues and difficulties in exposing the gauges for convection based measurements. These present investigations are aimed at exploring the possibility of using thin film gauges for short duration conduction based transient measurements with pure conduction mode of heat transfer. A simple calibration set-up has been used to supply known heat flux of different magnitudes to the thin film gauges that are fabricated in-house with platinum as sensing element and pyrex as an insulating substrate. Experimentally recorded temperature signals from the gauges are compared with simulated temperature histories obtained through finite element analysis. Convoluted integral of one-dimensional heat conduction equation is used to predict the surface heat flux and compared with input heat loads. The presently developed calibration setup is seen to be very useful for conduction based measurements of thin film gauges.     

Thermal propagation in solids due to surface laser pulsation and oscillation

Most studies of heat transfer pertaining to a planar medium subjected to time-varying and spatially-decaying laser incidence along with external surface cooling are based on the diffusion theory (parabolic heat conduction equation), an approximation that implies a non-physical infinite speed of energy transport. In this study, temperature distributions within one-dimensional plates subjected to the aforementioned heating and cooling conditions, but with thermal energy propagation at a finite speed, are presented. Incident energy that is partially absorbed at the external surface is transferred through the plate by conduction, while the remaining energy is convectively cooled to the environment. The present investigation will examine three different time characteristics of the incident heat sources which include a continuously operating, a pulsed and an oscillatory laser source. The temperature results were obtained by using a simple and concise finite-difference algorithm based on the Godunov scheme, a method developed for the solution of resulting characteristic equations that govern thermal wave propagations within the solid.     

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.     

Relativistic heat conduction

The hyperbolic heat conduction equation (HHCE), which acknowledges the finite speed of heat propagation, is based on microscopic evidence from the kinetic theory and statistical mechanics. However, it was argued that the HHCE could violate the second law of thermodynamics. This paper shows that a HHCE-like equation (RHCE) can be derived directly from the theory of relativity, as a direct consequence of space-time duality, without any consideration of the microstructure of the heat-conducting medium. This approach results in an alternative expression for the heat flux vector that is more compatible with the second law. Therefore, the RHCE brings the classical field theory of heat conduction into agreement with other branches of modern physics.     

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.     

Intrinsic verification methods in linear heat conduction

Verification of the codes that provide numerical heat transfer solutions obtained by finite difference and other methods is important. One way to verify these solutions is to compare the values with exact solutions. However, these exact solutions should also be verified. Fortunately, intrinsic verification methods are possible. Intrinsic verification utilizes at least two independent exact solutions to obtain accurate numerical values. Three different types of intrinsic verification for transient and steady state heat conduction are developed and illustrated by examples.     

Transient heat fluxes measurement analysis from platinum based thin film gauges in open and closed cavities

Platinum thin film gauges (PTFGs) measure heat fluxes in the applications involving very short duration of the heating environment. Heat transfer measurement is the frequently used technique for measuring the surface heat flux using thin film gauges. The present investigation has been focused on the design and manufacturing methods for heat transfer gauge, their stability, and dynamic calibrations in certain situations where the heat load suddenly build up. PTFGs measure heat fluxes in heating environments applications during the very duration. The measurement for heat transfer is a technique used often with thin film gauges to measure the surface heat flux. The convection devices are regarded as the best measuring units in short-term transient temperature measurement applications and are usually used when the heat transfer mode is dominant means gas turbine engines, high speed aircraft, etc. In addition to that, there are many difficulties incurred for convection based measurement practically and few interdisciplinary research fields. A convective heat load is provided with a hot air gun to get the temperature signal. By using thin film gauge through present investigations, it is very ambitious to explore the possibility of short term conduction based transient measurements with pure conduction heat transfer mode. A simple experimental set up is used to supply the thin film gauges with heat flux, which is manually manufactured with platinum as a sensing material and quartz as a substrate material. The body's nose tip to high speed flow is expected to be the maximum heat transfer at the stagnation point. The stagnation point probes are fabricated for PTFGs, and baking material is quartz. The recorded temperature histories are compared with the experimentally recorded temperature signals from the gauges through the finite element method. The heat flux forecast was configured by using the one dimension thermal conduction equation convolution integral and by comparison with the heat input loads. This study reveals the ability of PTFGs to be used for a short period.     

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…     

Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams

In this work, the thermal effect on the buckling response of the axially functionally graded (AFG) nanobeams is studied based on the nonlocal thermoelasticity theory. Size effects of elastic deformation and heat conduction are considered simultaneously. Non-uniform distribution of temperature along the longitudinal direction of the AFG nanobeams is taken into account and determined by the nonlocal heat conductive law. Equations of motion and the corresponding boundary conditions are derived with the aid of the variational principle within the sinusoidal shear deformation theory and the nonlocal thermoelasticity theory. Ritz method is used to obtain the solutions for the thermal buckling response of the AFG nanobeams with various boundary conditions. Numerical results addressing the significance of the AFG index, the nonlocal parameters of elasticity and heat conduction, and the transverse shear deformation on the buckling behavior are displayed. It is found that, in addition to the nonlocal effect of elasticity, the nonlocal heat conduction plays an important role in analyzing the thermal–mechanical behaviors of the FG nanostructures.     

Improvement of a chemical heat pump by computer‐aided design

This paper deals with an optimization problem about thermal exchanges between a reactive porous environment and two networks of heat pipes together enclosed in a chemical heat pump reactor. Because all physico‐chemical features of the reactive matter are fixed (actually the reactor contains a patented chemical mixture), the only way to optimise the quality of these exchanges, is to find an optimal location within the reactor for the heat pipes using ozone friendly fluid as the working fluid. Two numerical approaches, the first quantitative by the finite element method and the second based on the heat conduction hyperbolic theory, enabled us to obtain the same optimization results. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

SOLUTION OF TWO-DIMENSIONAL HYPERBOLIC HEAT CONDUCTION BY HIGH-RESOLUTION NUMERICAL METHODS

Studies of hyperbolic heat conduction have so far been limited mostly to one-dimensional frameworks. For two-dimensional problems, the reflection and interaction of oblique thermal waves and complicated geometries present a challenge. This paper describes a numerical solution of two-dimensional hyperbolic heat conduction by high-resolution schemes. First, the governing equations are transformed from Cartesian coordinates into generalized curvilinear coordinates. Then the dependent variables are cast in a characteristic form that decouples the original system equation into scalar equations. Two-dimensional high-resolution numerical schemes, suck as total variational diminishing ( TVD) are built up by forming symmetrical products of one-dimensional difference operators on each individual wave. Three examples are used to demonstrate the unique feature of complicated interaction of two-dimensional thermal waves.     

多孔介质导热的分形模型

多孔介质中热量传递与多孔介质内部的几何结构有密切的关系，讨论了多孔介质的分形结构和相关的分形维数，利用能量方程，导出了分形维数为D的有限尺度多孔介质中的广义热传导方程，在此基础上，假定热量在多孔介质中的传导路线也是一种分形结构，提出了一个筒化的多孔介质并联通道分形导热模型，求出了基于分形理论的多孔介质有效导热系数表达式。