Conductivity Of Heterogeneous Media Research Articles

A heat conduction equation for heterogeneous media and its connections to various known equations

We derive the governing equation of the macroscopically average temperature field in a heterogeneous medium. Although the constituents of the heterogeneous medium obey Fourier's law, the average temperature field is governed by a spatiotemporally nonlocal integro-differential formulation. Then, we find that the average fluctuation of the temperature in the matrix region of the heterogeneous medium plays a key role in determining the various degenerate forms of the governing equation. Depending on the value of this average fluctuation, the governing equation can degenerate into the well-known equations in the literature, including the Jeffreys-type equation, Nunziato equation, Gurtin and Pipkin equation, peridynamic formulation, and dual-phase-lag (DPL) equation. These connections establish the relations between the coefficients in these equations with the properties of the constituents of a heterogeneous medium, and provide guidelines for their applications to the prediction of heat conduction in heterogeneous media.

Numerical investigation of heat conduction in heterogeneous media with a discrete element method approach

Composite materials have been widely used across industry sectors. However, they are characterized by a variability of thermal conductivity with the architecture and manufacturing processes. Hence, thermal transfer in composite materials requires an improved fundamental understanding. From numerical purposes, the Finite Element Method (FEM) seems a robust method to study heat transfer in composite material. However, it does not establish a high-fidelity with the real life of material since it is difficult to take into account potential damage matrix or interface imperfection by this method. As such, this paper discusses the development of numerical approach based on the Discrete Element Method (DEM) to study heat transfer in composite materials. For that purpose, we consider a hybrid lattice model based on the equivalence between a particulate domain and a continuous medium. Several works have used the DEM to study heat transfer in a homogeneous and continuous medium. Through this contribution, we aim to extend this approach to investigate composite materials. The model is then validated in terms of temperature by comparison with numerical and experimental results through several applications. Furthermore, a special care taken in the evaluation of the heat flux density fields. To the knowledge of the authors, previous works did not interest to the examination of heat flux density when using the DEM. Indeed, this sensitive to the packing configuration and consequently always heterogeneous even if there typically homogeneous. To overcome this problem, an original smoothing technique called Halo is proposed and discussed in this work. Results exhibit the relevance of the proposed approach to evaluate both temperature and heat flux density fields with a good degree of precision compared with th FEM, the Fast Fourier Transformer (FFT) based method and experimental data.

Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through interfaces, which is mediated by a suitable Laplace-Beltrami type equation.

Transient three-dimensional heat conduction in heterogeneous media: Integral transforms and single domain formulation

A hybrid numerical-analytical methodology based on integral transforms is employed for solving transient three-dimensional heat conduction problems in heterogeneous media comprising arbitrarily space variable thermophysical properties, including configurations with multiple subdomains of different materials and geometries. The Generalized Integral Transform Technique (GITT) is used to solve for both the temperature distribution and the corresponding eigenvalue problem accounting for the spatially variable coefficients. A previously introduced single domain reformulation strategy is adopted when multiple subdomains are involved, so as to rewrite the heat conduction equation in terms of coefficients with abrupt variations in the spatial coordinates. The hybrid numerical-analytical approach is demonstrated for three classes of problems in which the thermophysical properties undergo significant variations throughout the domain, such as: (i) FGMs (Functionally Graded Materials) with three-dimensional space variable thermophysical properties; (ii) composite medium with inclusions of different geometries and thermophysical properties; (iii) a multi-scale/multi-material heat conduction problem in a IGBT (Insulated Gate Bipolar Transistor) module. In each considered application, both the temperature eigenfunction expansions and the associated eigenvalue problem solutions are critically analyzed in terms of convergence rates and co-verified with purely numerical solutions.

Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace–Beltrami operator

We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace–Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.

An experimental and analytical study of the effect of cold compression on the thermophysical properties of a granular medium

Based on the previous literature, very few models have described the thermal behavior of granular media or powders as a function of the mechanical stresses to which they are subjected. In recent years, many researchers have been interested in establishing laws that can express the relationship between the apparent thermal conductivity and the mechanical behavior of granular media. The present paper seeks to present a simple model that describes the variation of the apparent thermal conductivity of a granular medium as a function of the mechanical stress. One of the main objectives of this paper is to produce a tool for calculating the thermal conductivity of heterogeneous media, especially that of granular media. For the resolution of the problem, it was decided to use an experimental method recently developed in the laboratory and which is due to be the basis of our calculations. This method is called the hot rod method. It was initially developed to evaluate the damage to a soil subject to superficial heat shock (fires, burns). The results show that for short times and a distance between two measurement points large enough, the 2D transfer can be reduced to a 1D transfer which for long times is hybridized to a hot wire transfer. The modeling of thermal transfers within the environment makes it possible to know the temperature field of soil under the effect of a thermal accident.

Numerical study with OpenFOAM on heat conduction problems in heterogeneous media

OpenFOAM was used to numerically investigate heat conduction in heterogeneous media. The main advantage of this platform is that it offers an open flexible interface with great convenience for mesh generation and solver modification. Cases of 1D and 2D heat conduction problems in media with different levels of heterogeneity were considered to check the performance of OpenFOAM. The results were compared with references and other commercial software and showed good accuracy. Note that all of the processes, including the mesh generation, numerical simulation, and post-processing, relied on free open-source codes.

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

ABSTRACTIn this paper, we propose a direct extension of a previous work presented by Hamila et al. [1] dealing with the simulation of conjugate heat transfer by conduction in heterogeneous media. In [1] 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.

An extension of the Hill–Mandel principle for transient heat conduction in heterogeneous media with heat generation incorporating finite RVE thermal inertia effects

SummaryIn this work, we propose a homogenization formulation to model transient heat conduction in heterogeneous media that takes into account thermal inertia contributions, which arise from a finite description of the microscale. Rewriting the variational form of the transient heat conduction problem and making use of key assumptions, we arrive at a mathematical formulation that suggests an extension of the Hill–Mandel principle when considering non‐null heat flux divergence in the representative volume element (RVE). Along the manuscript, we highlight that the main results of the proposed formulation are in agreement with recent advances in the field of computational homogenization applied to transient mechanical and heat flow problems. The proposed extension of the Hill–Mandel principle contributes to the understanding of the microscale thermal inertia effects incorporation into the multiscale framework. We also present the calculations needed for implementing the model and numerical results, which give support to the theoretical model developed. The numerical results highlight the importance of considering full transient aspects when dealing with multiscale heat conduction in heterogeneous media which are subjected to high thermal gradients. Copyright © 2016 John Wiley & Sons, Ltd.

Enhanced convergence of eigenfunction expansions in convection-diffusion with multiscale space variable coefficients

ABSTRACTA convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.

Enthalpic lattice Boltzmann formulation for unsteady heat conduction in heterogeneous media

In this paper, we propose an enthalpic lattice Boltzmann formulation for heat transfer in heterogeneous media. This formulation leads to the appearance of a source term, which introduces the jump conditions at the interface between two phases or components, with different thermal properties. It conserves conductive heat flux, which makes it suitable for modeling heat conduction in multiphase and multi-component systems. This approach is simple to implement and can be easily applied to simulate heat transfer with complex and time dependent interfaces, efficiently. It does not require any specific treatment dependent on interface topology. It is also independent on the choice of lattice. This approach is validated by comparisons with analytical and numerical results of several test problems with flat and curved interfaces. Both 2- and 3-dimensional conduction transfer problems were simulated. The obtained results show excellent agreements with reference solutions.

Reiterated homogenization applied to heat conduction in heterogeneous media with multiple spatial scales and perfect thermal contact between the phases

Several types of heterogeneous media with multiple spatial scales presently offer good potential to improve upon more traditional materials used in heat transfer and other engineering applications. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. In this paper, the strong-form Fourier heat conduction problem in such media is formulated using the method of reiterated homogenization. The constituent phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter \(\varepsilon\). The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter \(\varepsilon\). The technique leads to two pairs of local and homogenized problems, linked by effective coefficients. In this manner the phenomenon description at the smallest scale is seen to affect the medium macroscale, or effective, behavior, which is the main interest in engineering. To facilitate the physical understanding of the derived sub-problems, an analytical solution is obtained for the heat conduction problem in a laminated binary composite. The present formulation shall serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

An interface integral formulation of heat energy calculation of steady state heat conduction in heterogeneous media

This paper presents a new boundary integral formulation of the heat energy calculation of steady state heat conduction in heterogeneous media using two methods, i.e. one which is based on some formulations from Hatta and Taya (1986) who investigated equivalent inclusion method, and another one which is closely similar to the method from Christensen (1979) who studied elastic inclusion problem. For the generalized self-consistent scheme (GSCS) model, the corresponding heat energy formulation is also given. Based on the obtained interface heat energy formulation, the boundary element method together with the Maxwell homogeneous scheme are used to calculate the effective heat conduction of 2D and 3D heterogeneous steady state heat conduction media.

Influence of particle-matrix interface, temperature, and agglomeration on heat conduction in dispersions

A combination of the effective medium and the phonon approaches is used to investigate heat conduction in heterogeneous media composed of a homogeneous matrix in which spherical particles of micro and nanosizes are dispersed. In particular, we explore the effect of different types of scattering on the particle-matrix interface, temperature dependence of the effective heat conduction coefficient, and the effect of various degrees of agglomeration of the particles. Predictions calculated explicitly for Si nanoparticles dispersed in Ge matrix agree with available Monte Carlo simulations. Our predictions show that the higher is the temperature the lower is the heat conductivity and the smaller is the influence of the details of the particle-matrix interactions. As for the influence of the agglomeration, we predict both decrease and increase of the heat conduction depending on the degree of the agglomeration.

Polymer growth in solution and the dynamic epidemic model: dimensionality and critical exponents

Recent interest in solutions of the static and dynamic epidemic models has led us to reconsider some features of the critical exponents of these models. In particular, our starting point is that the known values of such exponents take different values depending on dimensionality d, for dynamic epidemics, depending also on the looped nature of the chain, but the universality is regained for d > 1. According to this superuniversality property for d > 1, we could predict that the critical exponents γ and ν for dynamic epidemics on a square lattice should be 43/18 and 4/3, respectively. Furthermore, from numerical results in literature, we predict the critical exponents for the Potts model in three dimensions and the percolation exponents (q = 1) for d = 1, 2, 3, 4, 5 and d ≥ 6. Other areas of relevance beyond that in the title embrace conduction in heterogeneous media as well as a description of the spreading of a fluid in a medium possessing mobile impurities.

Integral Transforms and Bayesian Inference in the Identification of Variable Thermal Conductivity in Two-Phase Dispersed Systems

This work illustrates the use of Bayesian inference in the estimation of spatially variable thermal conductivity for one-dimensional heat conduction in heterogeneous media, such as particle-filled composites and other two-phase dispersed systems, by employing a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings algorithm. The direct problem solution is obtained analytically via integral transforms, and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field, which are desirable features for the implementation of the inverse analysis. Instead of seeking the function estimation in the form of a sequence of local values for the thermal conductivity, an alternative approach is proposed here, which is based on the eigenfunction expansion of the thermal conductivity itself. Then, the unknown parameters become the corresponding series coefficients. Simulated temperatures obtained via integral transforms are used in the inverse analysis. From the prescription of the concentration distribution of the dispersed phase, available correlations for the thermal conductivity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the so-called inverse crime. Both Gaussian and noninformative uniform distributions were used as priors for comparison purposes. In addition, alternative correlations for the thermal conductivity that yield different predictions are also employed as Gaussian priors for the algorithm in order to test the inverse analysis robustness.

Modelling Thermal Conductivity in Heterogeneous Media with the Finite Element Method

Three-dimensional finite element simulations were developed to predict the effective thermal conductivity of theoretical composite materials having complex structures. The models simulated a steady-state thermal conductivity measurement device performing measurements on theoretical materials with varying structures. The structure of a composite was considered to be composed of some simplified basic models. When the geometry, orientation type and number of dispersion are specified, the computer randomly generated the position and orientation for each dispersion and created the geometrical model and finite element mesh. The effective thermal conductivity of the theoretical composite was calculated using this method and compared to the values obtained by simple effective thermal conductivity models methods. The influence of some factors such as the volume fraction and the ratio of the thermal conductivities of the heterogeneities and the surrounding material on the effective thermal conductivity is discussed.

The flux-correct Green element formulation for linear, nonlinear heat transport in heterogeneous media

The singular integral equation, obtained from applying Green's identity to the stationary part of the linear diffusion equation operator (the Laplace operator), is implemented in an element-by-element fashion for the solution of nonlinear heat conduction in heterogeneous media so that both the temperature and its normal directional distribution (the flux) are calculated directly at every nodal point. This formulation, referred to as the flux Green element method (flux GEM), is used to solve linear and nonlinear transient heat transfer problems in two-dimensional heterogeneous domains. The flux-based Green element formulation takes advantage of the ability of the boundary element theory to correctly calculate the fluxes so that the solution in each element is complete in the sense that any solution (temperature or its directional derivative) required at any other point other than the grid points is achieved by implementing the integrations only in the element in which the point is located. One apparent drawback of the formulation is increased number of degrees of freedom (unknowns) that are generated, but this is compensated by the fact that high accuracy is achieved with coarse discretization. Furthermore, this paper presents a novel way of addressing the closure problem associated with having more unknowns than discretized equations at internal nodes. Here, an additional compatibility equation is presented for the internal normal directional derivatives of the temperature. This equation has a universal appeal in the sense that it applies not only to this heat equation but to any situation where such directional derivatives exist at a point. For the nonlinear transient heat conduction, the Picard and Newton–Raphson algorithms are applied in linearizing the discrete equations. In all examples examined in this paper, the Newton–Raphson algorithm shows superior convergence characteristics. The flux GEM is applied to four examples, and in all cases, the flux GEM achieves comparable accuracy with between 20% and 33% of the grid of the earlier Green element formulation which differences the internal fluxes.

Differential replacement procedure and bounds on conductivities of heterogeneous media with graded anisotropic cylindrical fibres

The effective conductivities of heterogeneous media containing either discretely suspended cylindrical fibres with a graded anisotropic interphase or graded anisotropic fibres, whose conductivity in the radial direction is different from that in the tangential direction, are obtained by two methods in this Letter. First, a differential replacement procedure (DRP) based on an energy equivalency condition is presented to replace the graded anisotropic constituents by equivalent homogeneous cylindrical fibres. This allows many approximate schemes to be used to predict the effective conductivities of the heterogeneous media containing graded anisotropic constituents from the conductivity of the equivalent homogeneous cylindrical fibres. Second, the optimized upper and lower bounds on the effective conductivities of these heterogeneous media are presented by introducing comparison materials. It is shown that the DRP predictions are within these bounds for the considered media.

Bounds on effective conductivities of heterogeneous media with graded constituents

The bounds on the effective conductivities of heterogeneous media containing discretely suspended particles with a graded interphase or graded particles are presented through introducing comparison materials. The microstructures of the comparison materials are different from those of the considered heterogeneous media. The comparison materials provide a means of obtaining optimized narrow bounds. The effective conductivities of these composites are also predicted using the composite sphere assemblage (CSA) model. It is shown that the CSA predictions are within the appropriate bounds for the considered media.