Response function for transient heat conduction in semi-infinite media

The concept of response function is used to investigate the transient heat conduction in a semi-infinite medium, when the temperature of the surface is an arbitrary function of time. First, the response of the system to a unit step function of surface temperature T_s(t) is obtained. The temperature distribution in the medium corresponding to arbitrary surface temperature T_s(t) is then expressed in terms of the response function and convolution integral. The analytical solution of the convolution integral is obtained for a periodic variation of T_s(t). The resulting expression for T(x, t) is identical with that obtained by the usual periodic analysis. Numerical computations are carried out to investigate the dependence of the accuracy of numerical results on the upper limit of time (t_o) in the convolution integral.     

Wall effects in heat conduction through a heterogeneous material

It is shown that wall effects may significantly alter heat flow through heterogeneous material. These effects can be accurately modelled using the concept of the apparent wall heat transfer coefficient α_w. A method for exact evaluation of α_w is proposed and is used for testing of the ad hoc formulae available in literature. The formula proposed by Kubie (1987) gives the smallest errors and is recommended for use in simplified heat transfer calculations.     

Exclusion of oscillations in heterogeneous and bi-composite media thermal conduction

Analysis of Fourier heat conduction in heterogeneous and bi-composite media (e.g. porous media, fluid suspensions, etc.) subject to Lack of Local Thermal Equilibrium (LaLotheq) reveals a condition for thermal oscillations and resonance to be possible. This paper shows that this condition cannot be fulfilled because of physical constraints leading to the exclusion of thermal waves and resonance.     

Analytical investigations for heat conduction problems in anisotropic thin-layer media with embedded heat sources

This article develops the analytical rigorous solution of a fundamental problem of heat conduction in anisotropic media. The steady-state temperature and heat flux fields in a thin-layer medium with anisotropic properties subjected to concentrated embedded heat sources or prescribed temperature on the surface are analyzed. A linear coordinate transformation is used to transform anisotropic thin-layer problems into equivalent isotropic problems without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, exact closed-form solutions of the specific problems are presented in series forms. The complete solutions of heat conduction problems for the thin-layer medium consist only of the simplest solutions for an infinite homogeneous medium with concentrated heat sources. The numerical results of the temperature and heat flux distributions are provided in full-field configurations.     

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.     

Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media

The singular boundary method (SBM) is a recent strong-form meshless boundary collocation method. Like the method of fundamental solutions (MFS), the SBM uses the fundamental solution of the governing differential equation of interest as the basis function and is mathematically simple, truly meshless, accurate, and easy-to-program. Unlike the MFS, the SBM, however, uses the concept of the origin intensity factor to isolate the singularity of the fundamental solutions and overcomes the fictitious boundary issue which has long perplexed the MFS. This study makes the first attempt to apply the SBM to steady-state heat conduction in three-dimensional (3D) anisotropic materials. Five benchmark numerical examples demonstrate that the SBM is accurate, convergent, stable, and computationally efficient in solving this kind of problems.     

General formulation and analysis of hyperbolic heat conduction in composite media

Some recent experimental results show the existence of reflections of thermal waves at the interface of dissimilar materials in superfluid helium. In light of these results, a theoretical investigation of thermal waves in composite is provided to give a theoretical foundation to the observed phenomenon. A general one-dimensional temperature and heat flux formulation for hyperbolic heat conduction in a composite medium is presented. Also, the general solution, based on the flux formulation, is developed for the standard three orthogonal coordinate systems. Unlike classical parabolic heat conduction, heat conduction based on the modified Fourier's law produces non-separable field equations for both the temperature and flux and therefore standard analytical techniques cannot be applied in these situations. In order to alleviate this difficulty, a generalized finite integral transform technique is proposed in the flux domain and a general solution is developed for the standard three orthogonal coordinate systems. The general solution is applied to the case of a two-region slab with a pulsed volumetric source and insulated exterior surfaces which displays the unusual and controversial nature associated with heat conduction based on the modified Fourier's law in composite regions.     

An enthalpy-based lattice Boltzmann formulation for unsteady convection-diffusion heat transfer problems in heterogeneous media

In this paper, we propose a direct extension of a previous work presented by Hamila et al. [1 R. Hamila, M. Nouri, S. Ben Nasrallah, and P. Perré, Int. J. Heat Mass Transfer, vol. 100, pp. 728–736, 2016.[Crossref], [Web of Science ®] , [Google Scholar]] dealing with the simulation of conjugate heat transfer by conduction in heterogeneous media. In [1 R. Hamila, M. Nouri, S. Ben Nasrallah, and P. Perré, Int. J. Heat Mass Transfer, vol. 100, pp. 728–736, 2016.[Crossref], [Web of Science ®] , [Google Scholar]] a novel enthalpy-based lattice Boltzmann (LB) formulation was successfully simulated in several conjugate heat transfer problems by conduction. We propose testing this enthalpic LB formulation in solving convection-diffusion heat transfer problems in heterogeneous media. The main idea of this formulation is to introduce an extra source term, avoiding any additional treatment of the distribution functions at the interface. Continuity of temperature and normal heat flux at the interface is satisfied automatically. The performance of the present method is successfully validated by comparison to the control volume methods (CVMs) solutions of several heat convection-diffusion problems in heterogeneous media.     

New derivations of the fundamental solution for heat conduction problems in three-dimensional general anisotropic media

This work presents two new methods to derive the fundamental solution for three-dimensional heat transfer problems in the general anisotropic media. Initially, the basic integral equations used in the definition of the general anisotropic fundamental solution are revisited. We show the relationship between three, two and one-dimensional integral definitions, either by purely algebraic manipulation as well as through Fourier and Radon transforms. Two of these forms are used to derive the fundamental solutions for the general anisotropic media. The first method gives the solution analytically for which the solution for the orthotropic case agrees with the well known result obtained by the domain mapping, while the fundamental solution for the general anisotropic media is new. The second method expresses the solution by a line integral over a semi-circle. The advantages and disadvantages of the two methods are discussed with numerical examples.     

A study on generalized thermoelasticity theory based on non-local heat conduction model with dual-phase-lag

Non-local continuum theory helps to analyze the influence of all the points of the body at a material point. Involvement of non-local factor, i.e., size effect in heat conduction theory enhances the microscopic effects at a macroscopic level. The present work is concerned with the generalized thermoelasticity theory based on the recently introduced non-local heat conduction model with dual-phase-lag effects by Tzou and Guo. We formulate the generalized governing equations for this non-local heat conduction model and investigate a one-dimensional elastic half-space problem. Danilovskaya’s problem is taken, i.e., we assume that thermal shock is applied at the traction free boundary of the half-space. Laplace transformation is used to solve the problem and numerical method is applied to solve the problem by finding Laplace inversion through the Stehfest method. Various graphs are plotted to analyze the effects of different parameters and to mark the variation of this non-local model with previously established models.     

An analytic approach to the unsteady heat conduction processes in one-dimensional composite media

The transient heat conduction problems in one-dimensional multi-layer solids are usually solved applying conventional techniques based on Vodicka's approach. However, if the thermal diffusivity of each layer is retained on the side of the heat conduction equation modified from the application of the separation-of-variables method where the time-dependent function is collected, then the modified heat conduction equation by itself represents a transparent statement of the physical phenomena involved. This `natural' choice so simplifies unsteady heat conduction analysis of composite media that thermal response computation reduces to a matter of relatively simple mathematics when compared with traditional techniques heretofore employed.     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

A technique for uncertainty analysis for inverse heat conduction problems

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the uncertainty analysis. An example calculation is presented. Parameter importance factors are defined and computed for the example problem; the volumetric heat capacity is the dominant parameter and an explanation is offered. Thoughts are presented on extending the analysis to include the non-linear problem of temperature dependent properties.     

A note on stability in dual-phase-lag heat conduction

In this note we compare two different mathematical hyperbolic models in dual-phase-lag heat conduction proposed by Tzou, and we ask for the parameter regions where stability can be expected. It is demonstrated that the parameter regions for the two lag-parameters τ_q and τ_θ are different for the two models. That is, for certain parameters, in one model stability is expected while for the other one it is known that it is not stable. The first apparent contradiction is contrasted with the fact that known values for real materials (several metals are considered here) are in a range where both models predict stability or non-stability, respectively. Still, as a conclusion, one model should be considered only in a restricted parameter region.     

A note on stability in three-phase-lag heat conduction

In this note we consider two cases in the theory of the heat conduction models with three-phase-lag. For each one we propose a suitable Lyapunov function. These functions are relevant tools which allow to study several qualitative properties. We obtain conditions on the material parameters to guarantee the exponential stability of solutions. The spectral analysis complements the results and we show that if the conditions obtained to prove the exponential stability are not satisfied, then we can obtain the instability of solutions for suitable domains. We believe that this kind of results is fundamental to clarify the applicability of the models.     

An experimental and theoretical study of the nonlinear heat conduction in dry porous media

Heat transfer in porous media is important in various engineering fields, including contaminated soil incineration. Most heat transfer models are theoretical in nature. Consequently, this study was undertaken to perform both theoretical and experimental studies of heat transfer in two different sand matrices. A mathematical model based on Fourier's law of heat conduction for a one‐dimensional system with the variable thermal conductivity was developed. The experimental part included heating sand samples placed in a small reactor within an infrared furnace. The transient temperature profiles of the sand layers were monitored by thermocouples. The bulk thermal conductivity was estimated to be linearly proportional to the temperature. The temperature profiles predicted by the model of heat conduction with a variable bulk thermal conductivity was compared by the observed temperatures in Quartz and Sea sands matrices up to 1300 K. Copyright © 1999 John Wiley & Sons, Ltd.     

The theory of heat and moisture transfer in porous media revisited

In this paper a review is presented of the present status of the theory of combined heat and moisture transfer in porous media, developed by J. R. Philip and the author in the mid-1950s. First, attention is drawn to the limitations of the theory and the assumptions underlying it. Next, attempts to test the theory by laboratory and field experiments are briefly discussed, leading to the conclusion that the usefulness of the theory in describing and analysing the experiments was proven, but that doubts remain about its predictive value. These doubts are a consequence of: (a) the limitations of the theory; (b) uncertainty about the quality of the experimental procedures and data. Remarks are made on hysteresis and its possible influence. It is concluded that experiments aimed at a study of the behaviour of nonisothermal systems subjected to hysteresis are needed. Finally, the problem of the definition and use of an apparent thermal conductivity is analysed. In the original papers two alternatives were presented. An expression for the phase average of the vapour flux density is derived. A numerical example is presented and suggestions are made concerning the proper choice between the alternatives.     

Eigenfunction expansions for transient diffusion in heterogeneous media

The Generalized Integral Transform Technique (GITT) is employed in the analytical solution of transient linear heat or mass diffusion problems in heterogeneous media. The GITT is utilized to handle the associated eigenvalue problem with arbitrarily space variable coefficients, defining an eigenfunction expansion in terms of a simpler Sturm-Liouville problem of known solution. In addition, the representation of the variable coefficients as eigenfunction expansions themselves has been proposed, considerably simplifying and accelerating the integral transformation process, while permitting the analytical evaluation of the coefficients matrices that form the transformed algebraic system. The proposed methodology is challenged in solving three different classes of diffusion problems in heterogeneous media, as illustrated for the cases of thermophysical properties with large scale variations found in heat transfer analysis of functionally graded materials (FGM), of abrupt variations in multiple layer transitions and of randomly variable physical properties in dispersed systems. The convergence behavior of the proposed expansions is then critically inspected and numerical results are presented to demonstrate the applicability of the general approach and to offer a set of reference results for potentials, eigenvalues, and related quantities.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.