Historical background of theoretical models

Estimation of the effective thermal conductivity of composite material has been subject of many theoretical and experimental investigations. The earliest model was proposed by Maxwell (Maxwell, 1892). He derived an expression for the simplest kind of two-phase dispersion consisting of spherical particles suspended or imbedded into a continuous medium of another material neglecting the interactions between the particles. The derived expression was valid only for very low concentrations of the dispersed phase i. e. for dilute volume fraction.

For non-dilute volume fractions, The interaction between the spherical particles has a significant effect and cannot be neglected, the work of Maxwell was later followed by the work of Rayleigh in 1892 to account for these interactions. By considering a form of a simple cubic lattice of spheres in a homogeneous material and using a series expansion, Rayleigh assumed that the spherical particles as inclusions that form a cubical array, and included the interaction effect of a number of nearby spheres.

Other proposed works for non-spherical inclusions developed by other researchers such as the work of (Polder & Van Santen, 1946), (Reynolds & Hough, 1957), (Hamilton & Crosser 1962), (Rocha & Acrivos, 1974) and others. Extension of these previous works was also carried out by (McPhedran & McKenzie, 1978), and (Sangani & Acrivos, 1983). In the course of models developments Hashin, 1968 proposed a general self-consistent treatment. The treatment provides a physically realistic model of particle to particle interaction for two phase system covering the full range of the volume fraction. The self-consistent field concept is extended to include the contact resistance in the composite reinforced with coated spheres. Benveniste & Miloh (1991) and Felske et. al. (2004) employed effective medium theories to predict the effective thermal conductivity of coated short-fiber composites. Later, Samantray and co-workers (Samantray et al, 2006) proposed the correlations between the inclusions to estimate the effective thermal conductivity of two-phase materials. Recently in 2009, (Fang et. al, 2009) applied the thermal wave method to investigate the unsteady effective thermal conductivity of particular composites with a functionally graded interface. The scattering and refraction of thermal waves by a spherical particle with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium.

phase, the volume concentrations of the phases, the phase morphology (distribution), the shape of inclusions of the different phases in the solid matrix, and the fractional porosity if the medium is porous.

In this chapter, the problem of determining the bounds and/or estimating the effective thermal conductivity (Aeff) of a composite (multiphase) system has been examined. A comparison between the measured data and the results predicted by theoretical models has been made. Two different categories of composite materials have been investigated, namely wood and ceramics. In particular we investigate the effect of morphology in woods and mineralogy in ceramics.

Wood is a heterogeneous porous material with known anisotropy due to its intrinsic distribution of phase morphology (inter layers arrangement). It should be mentioned that measurements of the thermal conductivity as a function of phase morphology (grain size and orientation) indicate preferential heat conduction along the conducting chain of grains. Its low for random distribution of grains and high for layered (parallel) grains.

The phenomena of heat transfer in wood depend on the geometry of the wood, as well as porosity. Such heterogeneous medium containing the three phases, its thermal conductivity is only an apparent conductivity because it results from complex exchanges concerning simultaneous conduction in solids, and fluids (gases and/or liquids). Regarding the theoretical models to estimate wood conductivity, there are various models of heat conductivity are given in the literature, for example, see references given by Gronli (1996). In this study, we will use one of these models which given by (Kollmann & Cote, 1968). The influences of density, porosity and anisotropy on thermal transport in wood are investigated. To estimate the effective thermal conductivity (Aeff) in a fibrous wood like structure, a model (Kollmann, & Cote, 1968) based on a weighting bridge-factor (§) between two limiting parallel and serial conduction cases have been used. The model indicates an increase in the effective conductivity as the density/porosity increases/decreases which was in agreement with our investigation regarding the influence of microstructure on the heat conduction in wood. Moreover, indicates significant difference between the longitudinal and the transverse directions.

At this point, it is worth mentioning that a worldwide research and development efforts are underway to examine the potential use of a wide range of non-destructive testing (NDT) and non-destructive evaluation (NDE) technologies for evaluating wood and wood-based materials as fiber-based materials or multiphase composites —from the assessment of standing trees to in-place structures (Brashaw et al., 2009).

Ceramics on the other hand are multiphase mechanically strong, relatively non-porous material with unknown anisotropy but with known volume fractions and conductivities of phases. The thermal transport in such material depends mainly on its mineralogy (microstructure details) of its constituents. Seven heterogeneous samples of Ceramics marbles and glasses have been selected. The tested models include those of the effective medium theory (EMT) (Noh et. al 19), Hashin and Shtrikman (HS) bounds (Hashin & Shtrikman, 1962) and Wiener bounds, (Wiener, 1912). These models can be used to characterize macroscopic homogeneous and isotropic multiphase composite materials either by determining the bounds for the effective thermal conductivity and/or by estimating the overall conductivity of the random mixture. It turns out that the most suitable one of these models to estimate Aeff is the EMT model. This model can be used to characterize macroscopic homogeneous and isotropic multiphase composite materials after determining the parallel and serial bounds (Wigner bounds) of the overall conductivity of the random mixture. We used Wiener bounds, as a preliminary indicator to validate the homogeneity condition of the investigated samples, i. e. the measured values for the conductivity should be within the limits of the Wigner bounds. After a comparison between the experimental data of the effective thermal conductivity with the corresponding theoretical estimation, it turns out that the EMT model is a suitable one to estimate Xeff. This model is a mathematical model based on the homogeneity condition, which satisfies the existence of a statistically homogeneous medium that encloses inclusions of different phases. Numerical values of thermal conductivity for the samples that satisfy the homogeneity condition imposed by the effective medium theory are in best agreement with the experimentally measured ones.