Elsevier

Ceramics International

Volume 35, Issue 7, September 2009, Pages 2921-2926
Ceramics International

Representation of thermal conductivity of solid material with particulate inclusion

https://doi.org/10.1016/j.ceramint.2009.03.038Get rights and content

Abstract

This paper reviewed the previously proposed models of thermal conductivity (κ) for a series and parallel flow of energy input in laminated composites. These two models were coupled to derive the thermal conductivity (κa) of material with simple cubic particulate inclusion. The derived equation depends upon κ1 of inclusion, κ2 of a continuous phase and volume fraction of inclusion. The size and shape factors of inclusion are cancelled during the derivation of κa. The newly constructed κa equation explains well the measured κa for AlN particle-dispersed SiO2 system.

Introduction

Thermal conductivity is an important property of material, which is used to design a structure of assembly of functional parts or to estimate a temperature gradient in the material at a given energy flux. Fortunately many thermal conductivities of metal and ceramics are available in Chemical Handbook [1] or Metal Handbook [2]. In a previous paper [3], a wide variety of thermal conductivity (κ) of metal and ceramics is discussed with a harmonic oscillator model of lattice vibration. The theoretical approach succeeded in representing κ with atomic weight, Young's modulus (E) and density (ρ). The E and ρ values are closely related to the nature of chemical bond (metallic, covalent and ionic bonds). A very good agreement is observed between the measured and calculated κ values in the wide range from 1 to 2310 J/smK. The next interesting observation is the theoretical expression of thermal conductivity for the material with inclusion (second phase or pore). In this paper, two simple models with plates of different κ values are coupled to derive κa for the material with particulate inclusion. The derived κa can be expressed with three parameters of κ1 for inclusion, κ2 for a continuous phase and volume fraction (V) of inclusion. The dependence of κa on volume fraction of inclusion is compared with the experimentally measured κa for AlN–SiO2 system [4], [5]. The thermal diffusibility (α, m−2) of AlN–SiO2 hot-pressed at 1400–1600 °С in a N2 atmosphere was measured at room temperature at Dow Chemical International Ltd., Research & Development Laboratory, Midland, USA, and converted to a thermal conductivity using the specific heat of 0.736 (pure AlN)–0.841 (pure silica) J/gK [4].

Section snippets

Steady-state analysis of thermal conductivity

The equation of heat conduction along x direction in Fig. 1 is given by Eq. (1),Tt=α2Tx2where T is the temperature and α the thermal diffusibility (1/m2). The steady-state condition is ∂T/∂t = 0 and the temperature gradient along x direction (∂T/∂x) becomes a constant value. On the other hand, thermal conductivity (κ) is defined by Eq. (2),I=κdTdxwhere I is the flux of energy (J/sm2) input in a material with thickness L in Fig. 1. Since the temperature gradient is constant at the steady-state

Series flow of energy

Fig. 2 shows the laminated composite where a flux of energy (I) is input. At the steady-state condition (I = I1 = I2), the following equation is derived,I=κ1T1TxL1=κ2TxT2L2where Tx is the temperature at the interface between layers 1 and 2. Eq. (11) gives Tx by solving Eq. (10),Tx=(κ1/L1)T1+(κ2/L2)T2(κ1/L1)+(κ2/L2)=C1T1+C2T2Ctwhere C1, C2 and Ct represent κ1/L1, κ2/L2 and (κ1/L1) + (κ2/L2), respectively.

Substitution of Eq. (11) for Eq. (10) results in Eq. (12).I=I1=I2=C1C2CtT1T2The apparent

Thermal conduction model of material with inclusion

In chapter 3, we discussed two types of thermal conductivity of laminated composite for series and parallel flow of energy. These two models are effectively coupled to derive κa of material with particulate inclusion. Fig. 5 shows a simple cubic inclusion model with a m length in one cubic box with length 1/p m. The number (n) and volume fraction (V) of cubic inclusion in 1 m3 matrix box are related by Eq. (24),V=a3nThe number (р) of inclusion along one direction of cubic matrix is equal to n1/3

Thermal conductivity of the AlN–SiO2 system

AlN is known as an insulator of high thermal conductivity (320 J/smK) [4]. Dense AlN is produced by the liquid phase sintering with rare-earth oxide at a high temperature of 1800 °С. The κ of sintered AlN depends upon the oxygen content of starting powder and second phase at grain boundaries. On the contrary the κ of SiO2 is as low as 2 J/smK [1]. In a previous paper [4], an AlN–SiO2 composite powder was prepared by mixing an AlN powder (median size 1.4 μm) with tetraethyl orthosilicate. The

Conclusions

Thermal conductivity (κC) of the laminated composite consisting of two layers with different conductivities was analyzed for the series and parallel flow of energy. These two structure models and the derived κC equations are coupled to analyze the thermal conductivity (κa) of solid material with simple cubic particulate inclusion. The newly derived κa equation can be represented by κ1 of inclusion, κ2 of a continuous phase and volume fraction of inclusion, and was compared with the measured κa

References (6)

  • Y. Hirata, Thermal conduction model of metal and ceramics, Ceram. Int., accepted for...
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