Seven samples of ceramic materials; glasses and marbles with different mineralogy and effective conductivities were selected. These samples were labelled from P1 to P7. These materials are classified into two types of samples, rock marbles and ceramic glasses. The rocks were two greyred gneissic granite samples labelled P1 & P2 and the other five were either provided through a private communication with Dr. Vlastimil Bohac from the Institute of Physics, Slovak Academy of Sciences (SAS) or selected from the AZ materials database within the www. amazon. com website. The effective thermal conductivity (Aeff) of such composite multiphase samples with unknown isotropy requires the previous knowledge of the thermal conductivity of each phase, the volume concentrations of the phases, and the shape of inclusions of the different phases in the solid matrix.
Table 1 shows the volumetric fractions of constituents of the seven samples which were determined either by the pointcounting method for the marbles or using Xray (XRF) analysis for the ceramic glasses. The effective thermal conductivity of these samples depends on its mineralogy and the microstructure details of its constituents. Moreover, it appears that for these samples, the major influence on the effective conductivity is due to the magnitude of different conductivities of the constituents and their volumetric composition. For these homogeneous and isotropic (with unknown morphology) multiphase composite materials, we have used the EMT Model mentioned in sec 2.1 to estimate the overall conductivity of the random mixture after determining the serial and parallel conduction (Wiener) bounds. Wiener was the first one to introduce these two bounds.
According to the definitions of Wiener bounds, a dry rock sample consisting of a mixed orientation of mineral layers (parallel and serial distributions of phases), is expected to have an effective conductivity value within the these two bounds(homogeneity condition). Hashin & Shtrikman, 1962, introduced a model narrowing the range between Wiener bounds by imposing requirements on the isotropy of the medium. They derived new bounds for an effective magnetic permeability of a composite material by means of variational theorems. The HS model can be directly applied to different transport properties including thermal conductivity (Helsing & Grimvall, 1991). The corresponding equations of HS bounds for three dimension multiphase media can be expressed in terms of Upper (Xu) and Lower (XL) bounds which are functions of the highest and lowest values of Xi ‘s of the different phases and their volume fractions Pi’s. (Porfiri et. al 2008) & (Suleiman, 2010).
Marble Samples 
Density (g cm3) 
Quartz 
Alkali felds 
Plagi oclae 
Bio tite 
Chlo rite 
Musc ovite 
Epid ote 
Zir con 
Sph ene 
0/ % 

P1 
2.79 
27.20 
24.00 
35.20 
9.00 
0.80 
0.80 
2.60 
0.00 
0.40 
P2 
2.53 
22.20 
64.10 
0.00 
1.70 
0.00 
10.20 
1.40 
0.40 
0.00 
Ceramic Samples 
Density (g cm3) 
Al2O3 
SiO2 
Fe2O3 
TiO2 
CaO 
MgO 
K2O 
Na2O 
Impurity 
0/ % 

P3 
2.76 
57.83 
36.14 
0.68 
0.27 
0.12 
0.25 
3.45 
0.38 
0.41 
P4 
2.51 
47.50 
46.30 
0.90 
0.29 
0.23 
0.25 
3.36 
0.56 
0.61 
P5 
2.75 
59.98 
34.48 
0.65 
0.25 
0.16 
0.25 
3.32 
0.50 
0.41 
P6 
2.50 
41.90 
55.20 
0.80 
0.50 
0.30 
0.20 
0.50 
0.40 
0.00 
P7 
2.52 
46.00 
16.00 
0.00 
0.00 
0.00 
17.00 
10.00 
0.00 
7 (B2O3) 
Table 1. The volume fractions of the samples components 
Using the more confined limits of Hashin and Shtrikman, it is reasonable to estimate the effective thermal conductivity of the multiphase isotropic medium by taking the mean value of the upper and lower limits (Horaiet al., 1972 ), i. e.,
In general, the utility of bounds gives a narrower range to predict a good match to the real value of the effective conductivity only if the conductivities of the components are not too different in magnitude. It is possible to evaluate Aeff using the EMT model which is dependent on the existence of statistically homogeneous medium surrounding inclusions of different phases. As it was stated in sec 2.1, in this model, it is assumed that the phases with Xi’s are distributed in such a way that the material can be considered isotropic and homogeneous, then according to EMT, the effective conductivity Xeff(EMT) is determined self consistently from the formula:
r. .
(4)
y* ven(EJVll) !«1 J
This formula has been derived from the solution to the problem of dilute spherical inclusions of one phase embedded in a matrix of a second phase (Noh et al. 1991). However, equation (4) designates a symmetric representation to all phases without singling out a certain phase as dilute. The solution of this equation will have a physical meaning under certain imposed conditions such as considering eff(EMT) as continues function of the volume fraction P.
Table 3 shows the values of the calculated thermal conductivities at room temperature using effective medium theory (EMT), Hashin & Shtrikman, and Wiener bounds. The calculated values indicates that there are four out of the seven samples namely; P3, P5, P6, and P7 do not satisfy the theoretical necessary rule imposed by the parallel and serial bounds. According to this rule and by definition, the measured values should be within the limits of the parallel and serial bounds. This can be seen from the obtained values of the parallel and serial bounds.. In other words, these samples they will not satisfy the validity condition (homogeneity condition) and therefore, have been exempted from our discussion regarding the models validities. The average deviations of eff(EMT) from the measured values of these four samples are within 2829%.Therefore, for the samples P3, P5, P6, and P7 the mismatch between the theoretical models and the measured values should not be attributed to models validities. The most proper reason is that the distribution of phases within these samples is not uniform within the composite matrix.
Further work should be done to investigate the uniformity of different phases within these samples in order to estimate the values of X’s and draw any affirmative results. However,
for the other three samples namely; P1, P2 and P4 the average deviations of Xeff(EMT) from Xmeas did not exceeding 3%. It is also obvious that, the mean value of HS bounds Kff(HS) for these samples is rather good estimation to the measured (Xmeas) value at room temperature; The best approximation in this context, were obtained from the effective medium approach represented by the solution of equation (4). The selfconsistent solutions of this equation for the three samples yielded nine roots. Each solution resulted in only one positive root that has a physical meaning. The deviation of eff(EMT) from the experimental values is less than 3.3%. Thus, of all of the calculated approximations, the EMT method appears to be the best for estimating the effective value of thermal conductivity only for samples that satisfy the homogeneity condition.
sample 
The number of components 
Parallel & Serial Wiener bounds 
H S mean of U & L bounds 
The E. M. T Model 
Measured Value 

n 
Xp 
Xs 
Xeff(HS) 
Xeff(EMT) 
Xmeas 

P1 
9 
3.50 
2.32 
2.94 
2.93 
2.85 
P2 
9 
3.40 
2.56 
3.02 
2.97 
2.97 
P3 
9 
2.59 
2.12 
2.38 
2.36 
3.32 
P4 
9 
2.46 
1.96 
2.21 
2.19 
2.27 
P5 
9 
2.61 
2.14 
2.39 
2.39 
3.32 
P6 
8 
2.56 
2.06 
2.33 
2.31 
5.00 
P7 
6 
8.66 
2.00 
2.56 
3.12 
1.50 
Table 3. The estimatedX’s [W/m oC] calculated using the corresponding models 
At this point, it should be noted that for these ceramic composites there are some constrains imposed upon using these models such as limiting the number of samples, limiting the analysis to room temperature data, and neglecting the anisotropy of the thermal conductivity of the samples or the components that constitute the samples. In these circumstances such constrains may introduce a degree of uncertainty in the results. However, regards of all that, it seems that assuming isotropic and homogeneous conditions for ceramic samples, may still lead to a good agreement between the values of the measured and calculated thermal conductivities of such heterogeneous samples. Furthermore, in this attempt, the purpose of applied the models for these samples, just to examine the possibilities of determining the bounds and/or estimating the effective thermal conductivity of a multiphase composite system given the volume fractions and the conductivities of the components.
In some cases where structure phase morphology (interlayer structure) details are very important an accurate knowledge of the thermal conductivity of composites could only be obtained experimentally. The experimental measurements can then be used as a probe to monitor microstructure changes.
Finally, these theoretical models can only be utilized provided the investigated samples satisfy the necessary condition that the measured values for the conductivity should be within the limits of the Wigner bounds. According to our calculations for the selected samples, the effective medium approach produced calculated values that agreed best with the measured values.
5. Conclusions
Four different models are tested to investigate the problem of determining the bounds and/or estimating the effective thermal conductivity (Aeff) of composite (multiphase) systems given the volume fractions, the conductivities of the components and porosity (for woods). Three of the tested models namely; the effective medium theory (EMT), Hashin and Shtrikman (HS) bounds, Wiener bounds were applied on the ceramic samples and the forth weighted bridgefactor model based on Wiener bounds was applied on wood as fiberbased materials or multiphase composites samples.
The effective thermal conductivity of wood at 20 oC slightly increases in both longitudinal and transverse directions. The effect of density and porosity on the thermal conductivity may be attributed to the presence of other scattering mechanisms such as voids, and cell boundaries. It seems that the porosity (conduction through voids) is the dominant influencing factor on the heat conduction in wood. Regarding the anisotropic nature of wood our results indicate that the thermal conductivity in the longitudinal direction (parallel to the grain) is greater than conductivity in the transverse direction. This is may be attributed to orientation of the molecular chains within the cell wall. The longchain linear polymers (cellulose) that comprise the cell wall, are arranged in bundles called microfibrils. These microfibrils are most closely aligned with the longitudinal axis of the cell. Our data in the longitudinal direction at 20 oC lie in the range 0.81.0. The data in the transverse direction are in the range 0.50.7. At 20 oC, our values for both directions are in agreement with literature data.
Although we did not take into account the possibility of anisotropy of thermal conductivity in the ceramic samples, it seems that, assuming isotropic and homogeneous conditions, the thermal conductivity of such samples maybe calculated from their contents with rather good accuracy. The deviation of Xeff(EMT) from the experimental values is less than 3.3%. However, the possible consequences of a large anisotropy in their thermal conductivities may not be disregarded and further investigations are needed. These theoretical models can only be utilized provided that the investigated samples satisfy the necessary condition that the measured values for the conductivity should be within the limits of the parallel and serial bounds. According to our investigations for the selected samples, the effective conductivity can be used as a probe to monitor microstructure changes in both morphological and mineralogical aspects. An extended version of this work is planned to be published in the University of Sharjah international press.
[1] Stability must be satisfied, i. e. steptime can not trespass on the critical time step