non-Fourier Heat Conduction Problems Research Articles

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

In this paper we present an extension of the formulation given in (EABE, 37 (2013) 868–883) to direct and inverse heat conduction problems in layered materials. Expressing the numerical solution as a series of exponential basis functions (EBFs) defined in space and time, is the main idea of the presented method. We shall show that the use of the EBFs provides a versatile tool for the solution of a variety of problems such as the classical heat diffusion with internal heat sources and the inverse problems in layered media with perfect and imperfect contacts. To this end, we choose a proper set of EBFs satisfying the governing differential equation and interface conditions. Then a collocation technique is employed to satisfy both initial and boundary conditions. The proposed formulation can also be easily extended to the solution of non-Fourier heat conduction problems or the analysis of thermal wave propagation. The capability of the presented method is investigated in the solution of some direct and inverse heat conduction problems including a boundary identification problem and backward heat conduction problems. Issues pertaining to short time solution and penetration time, in inverse heat conduction problems, are addressed in the problems solved.

Application of solution structure theorems to Cattaneo–Vernotte heat conduction equation with non-homogeneous boundary conditions

In this study, a non-Fourier heat conduction problem formulated using the Cattaneo–Vernotte (C–V) model with non-homogeneous boundary conditions is solved with the superposition principle in conjunction with solution structure theorems. It is well known that the aforementioned analytical method is not suitable for such a class of thermal problems. However, by performing a functional transformation, the original non-homogeneous partial differential equation governing the physical problem can be cast into a new form such that it consists of a homogeneous part and an additional auxiliary function. As a result, the modified homogeneous governing equation can then be solved with solution structure theorems for temperatures inside a finite planar medium. The methodology provides a convenient, accurate, and efficient solution to the C–V heat conduction equation with non-homogeneous boundary conditions.

General expression for entropy production in transport processes based on the thermomass model

The entropy production in classical irreversible thermodynamics is expressed as a bilinear form of generalized (driving) forces and conjugated (driven) fluxes, which suffers from the arbitrary decomposition of the forces and the fluxes, and the possible negative entropy production in non-Fourier heat conduction problems (heat waves). This paper presents a general form of the entropy production for heat conduction based on the thermomass model, which is the product of the friction force and the drift velocity of the thermomass divided by the temperature; it holds true for both Fourier and non-Fourier heat conduction. Then a generalization of the entropy production is given for other kinds of linear and nonlinear transport processes. The general expression for entropy production is consistent with that given by extended irreversible thermodynamics, where the system entropy depends not only on the classical variables, but also on the dissipative fluxes, for example, the heat flux in heat conduction problems.

A New Spectral-Finite Volume Approach in Non-Fourier Heat Conduction Problems With Periodic Surface Disturbances

In this study, a simple spectral-finite volume approach for hyperbolic heat conduction problems under periodic surface temperature is presented. In this approach, by choosing only three frequencies from a continuum frequency spectrum of the periodic temperature field, the time dependent governing equation is transformed into the steady state one in the frequency domain. Then, using the finite volume technique, temperature field in the frequency domain for each wave number is obtained. Finally, by transforming back the result to the time domain, the temperature field in the time domain would be obtained. This new method has been validated against some published results and a good agreement has been found. Despite the simplicity of the present method, it is able to accurately predict the temperature distribution in the periodic steady state portion of non-Fourier heat conduction problems subjected to periodic surface temperature.

Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is generated at the base of the fin, which moves toward the tip of the fin and is reflected from the tip. The current study of various parameters on the thermal shock location shows that relaxation time has a great influence on the temperature distribution on the fin. An unsteady boundary condition in the base fin caused the shock, which is generated continuously from the base and has interacted with the other reflected thermal shocks. Results of the current study show that the hyperbolic heat conduction equation can violate the second thermodynamic law under some unsteady boundary conditions.

Application of solution structure theorem to non-Fourier heat conduction problems: Analytical approach

This study presents 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 (HHC) equation can be obtained using fundamental mathematics. The method is applied to investigate two one-dimensional HHC problems with different boundary and initial conditions to study the effect of spatially-decaying and time-varying laser incidence at the surface. The methodology provides a convenient, accurate, and efficient solution to the HHC equation, which is applicable to a variety of HHC analyses for various engineering applications. The examples presented in this study can be used as benchmark problems for future numerical method validations.

OS1405 傾斜機能中空球の熱応力に及ぼす非フーリエ効果 : 第1報,FGM中空球の非フーリエ熱伝導問題の解析解(OS14-1 圧電材料・セラミックス・傾斜機能材料の弾性数理解析,OS-14 弾性数理解析の発展と新展開1)

OS1405 傾斜機能中空球の熱応力に及ぼす非フーリエ効果 : 第1報,FGM中空球の非フーリエ熱伝導問題の解析解(OS14-1 圧電材料・セラミックス・傾斜機能材料の弾性数理解析,OS-14 弾性数理解析の発展と新展開1)

Effect of spatial variation of thermal conductivity on non-fourier heat conduction in a finite slab

The non-Fourier heat conduction problem in a finite slab is studied analytically. Dependence of thermal conductivity on space has been considered. The Laplace transform method is used to remove the time-dependent terms in the governing equation and the boundary conditions. The hyperbolic heat conduction (HHC) equation has been solved by employing trial solution method and collocation optimization criterion. Results show that the space-dependent thermal conductivity strongly affects the temperature distribution. A temperature peak on the insulated wall of the slab has been observed due to linear variation of thermal conductivity. It has been shown that the magnitude of the temperature peak increases with increasing the dimensionless relaxation time. To validate the approach, the results have been compared with the analytical solution obtained for a special case which shows a good agreement.

Uniform-Parameter Spline Method for Solving Non-Fourier Heat Conduction Problems

From the theoretical basis of the parameter spline method, this study establishes a simple computing procedure to solve multidimensional non-Fourier heat conduction problems. Compared with the ordinary finite-difference method and the traditional cubic spline method, this method is similar in computing process, but the numerical results increase by two-order accuracy without adding computer load. It also can solve for numeric oscillations due to discontinuous wavefront edge points during conduction.

Soret Effects on Non-Fourier Heat and Non-Fickian Mass Diffusion Transfer in a Slab

In this article, non-Fickian and non-Fourier models were used for the rapid heat and mass transfer problem in a slab. The boundaries are convective and insulated sides, and the solution takes the Soret effect into consideration. The Laplace transform method is used to obtain a local-analytic solution in the transformed domain and then inverted to obtain the results in physical quantities. The results show that the temperature and concentration curves possess discontinuities when the relaxation-time in the non-Fourier heat conduction and non-Fickian mass diffusion problem is considered. These discontinuities are caused by wave propagation. Propagated waves are reflected at the insulated side and grow weaker as time passes. When considering the Soret effect for mass transfer, the intensity of concentration rise in the discontinuities will increase.

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.

Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique

In this paper, a non-Fourier heat conduction problem is analyzed by employing newly developed theory. Application of conventional numerical schemes leads to strong oscillations of the results around discontinuities in solution domain. To overcome this difficulty the variational formulation of the Laplace-transformed hyperbolic heat conduction equation is developed. The results were used for evaluation of parameters used in approximate transformed temperature profiles. To validate the approach the results were compared with the exact analytical solution solved at special case and with an approach previously reported in the literature. Both showed a close agreement with the proposed approach.

The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.

Estimating the boundary condition in a 3D inverse hyperbolic heat conduction problem

This study is intended to provide an inverse method for estimating the unknown boundary condition T( x, 0, z, t) in a 3D non-Fourier heat conduction problem. In this study, finite-difference methods are employed to discretize the problem domain, and then a linear inverse model is constructed to identify the unknown boundary condition. The matrix forms of the original differential governing equations are rearranged in order that the unknown conditions can be represented explicitly, and the linear least-squares method is then adopted to determine their solution. The results show that the proposed method is capable of generating precise solutions using just a small number of measuring points. Furthermore, it is indicated that the method is capable of providing good numerical approximations even where measurement errors are present. The phenomenon of complicated reflection and interaction of thermal waves reflects the fact that the inverse non-Fourier heat conduction problem is different from the inverse Fourier heat conduction problem. In contrast to traditional approaches, the proposed inverse analysis method: requires no prior information regarding the functional form of the unknown quantities. In addition, only one iteration is necessary for calculation, and no initial guess is required. Furthermore, the existence and uniqueness of the solutions can be easily identified.

Numerical simulation of thermal wave propagation during laser processing of thin films

A numerical model is developed to represent the thermal wave propagation during ultrashort pulsed laser processing of thin films. The model developed is based on the solution of non-Fourier heat conduction problem with temperature and thermal flux delays using discontinuous finite-element method. The mathematical formulation is described and computational procedures are given. The computer model is validated using the analytical solution for one-dimensional (1-D) thermal wave equations. Numerical simulations are performed to study the thermal wave propagation in a GaAs thin film exposed to ultrashort laser pulses. A wavelike behavior of the thermal signal propagation is observed, and the diffusive effect of the time relaxation in the temperature gradient is calculated and discussed. The thermal behavior of thin films under laser radiation is also studied as a function of various process parameters including pulse duration, laser pulse shapes and characteristic times of heat fluxes.

Some non-Fourier heat conduction characters under pulsed inlet conditions

Through simulating one- and two-dimensional non-Fourier heat conduction problems under different pulsed inlet conditions, this paper numerically predicts some different non-Fourier heat conduction characters arose from different pulse types and different pulse frequencies. Meanwhile, the differences among thermal wave, non-Fourier and Fourier heat conduction are also showed.

An inverse non-Fourier heat conduction problem approach for estimating the boundary condition in electronic device

This study is intended to provide an inverse method for estimating the unknown boundary condition T(0, y, t) in a non-Fourier heat conduction electronic device. In this study, finite-difference methods are employed to discretize the problem domain, and then a linear inverse model is constructed to identify the unknown boundary condition. The present approach is to rearrange the matrix forms of the differential governing equations and to estimate the unknown conditions. Then, the linear least-squares method is adopted to obtain the solution. The results show that one measuring point is sufficient to estimate the unknown boundary condition T(0, y, t) without measurement errors. When considering the measurement errors, the magnitudes of the discrepancies in the boundary condition T(0, y, t) are directly proportional to the size of measurement errors. Due to the complicated reflection and interaction of the thermal waves, this phenomenon reflects the fact that the inverse non-Fourier heat conduction problem is different from the inverse Fourier heat conduction problem. In contrast to the traditional approach, the advantage of applying this method in inverse analysis is that no prior information is needed on the functional form of the unknown quantities. In addition, no initial guess is required and the calculation can be done in only one iteration.

A numerical solution of two-dimensional hyperbolic heat conduction with non-linear boundary conditions

An explicit TVD scheme is used for two-dimensional non-Fourier heat conduction problems in the general coordinate system with both convection and radiation boundary conditions. The hyperbolic heat flux model is used to simulate the non-Fourier heat conduction. Because of the wave nature of hyperbolic equations, characteristics are used to find the unknown value (either heat flux or temperature) on the boundaries. For convective boundary the unknown temperature is calculated explicitly; for radiation boundary Newton's iteration method is applied to find the boundary temperature. Results of numerical examples agree with the physical expectations indicating that the present approach can be used for modeling non-Fourier heat conduction with complex boundary conditions.

An Approximate Spectral Method for Accurate Solution of Fourier and Non-Fourier Heat Conduction Problems

A new form of the pseudo-spectral method is presented. The method is theoretically simple yet robust enough to produce very accurate solutions to hyperbolic and parabolic PDE's while avoiding the effects of Gibb's phenomenon. Moreover the method uses a relatively small amount of computational memory. The method is based on the observation that an analytical function may be well represented in a set of small neighborhoods that share common boundaries, called sub-domains, by low order Chebyshev polynomials. A collocation solution scheme is used in each sub-domain to march facilitate a time. Throughout this process the Chebyshev expansion coefficients of the highest order terms are monitored. If these coefficients grow beyond a specified small size, the new sub-domains are then redefined so that the function is again well represented by Chebyshev polynomial expansions. An approach for the determination of computational sub-domains of the physical domain for the special case of a discontinuous function is discussed. The strategy for solving the PDE's is presented. The method is then applied to Fourier and non-Fourier heat conduction problems