Page 297 - Bird R.B. Transport phenomena

§9.6 Effective Thermal Conductivity of Composite Solids 281

pure metals at 0°C and changes but little with temperatures above 0°C, increases of
10-20% per 1000°C being typical. At very low temperatures (-269.4°C for mercury) met-
als become superconductors of electricity but not of heat, and L thus varies strongly with
temperature near the superconducting region. Equation 9.5-1 is of limited use for alloys,
since L varies strongly with composition and, in some cases, with temperature.
The success of Eq. 9.5-1 for pure metals is due to the fact that free electrons are the
major heat carriers in pure metals. The equation is not suitable for nonmetals, in which
the concentration of free electrons is so low that energy transport by molecular motion
predominates.

§9,6 EFFECTIVE THERMAL CONDUCTIVITY
OF COMPOSITE SOLIDS

Up to this point we have discussed homogeneous materials. Now we turn our attention
briefly to the thermal conductivity of two-phase solids—one solid phase dispersed in a
second solid phase, or solids containing pores, such as granular materials, sintered met-
als, and plastic foams. A complete description of the heat transport through such materi-
als is clearly extremely complicated. However, for steady conduction these materials can
be regarded as homogeneous materials with an effective thermal conductivity k , and the
e{{
temperature and heat flux components are reinterpreted as the analogous quantities av-
eraged over a volume that is large with respect to the scale of the heterogeneity but small
with respect to the overall dimensions of the heat conduction system.
The first major contribution to the estimation of the conductivity of heterogeneous
solids was by Maxwell. 1 He considered a material made of spheres of thermal conductiv-
ity fcj embedded in a continuous solid phase with thermal conductivity k . The volume
0
fraction ф of embedded spheres is taken to be sufficiently small that the spheres do not
"interact" thermally; that is, one needs to consider only the thermal conduction in a large
medium containing only one embedded sphere. Then by means of a surprisingly simple
derivation, Maxwell showed that for small volume fraction ф

- ф
[ ^ 0 ,
(see Problems 11B.8 and 11C.5).
For large volume fraction ф, Rayleigh 2 showed that, if the spheres are located at the in-
tersections of a cubic lattice, the thermal conductivity of the composite is given by
к а Ъф
-г* = 1 + ~, ; ~, ; (9-6-2)
1.569 1.10/3
*, - *b / ^ \3k, - 4k
t
Comparison of this result with Eq. 9.6-1 shows that the interaction between the spheres
is small, even at ф = \тт, the maximum possible value of ф for the cubic lattice arrange-
ment. Therefore the simpler result of Maxwell is often used, and the effects of nonuni-
form sphere distribution are usually neglected.

1 Maxwell's derivation was for electrical conductivity, but the same arguments apply for thermal
conductivity. See J. C. Maxwell, A Treatise on Electricity and Magnetism, Oxford University Press, 3rd
edition (1891, reprinted 1998), Vol. 1, §314; H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids,
Clarendon Press, Oxford, 2nd edition (1959), p. 428.
2 J. W. Strutt (Lord Rayleigh), Phil. Mag. (5), 34, 431-502 (1892).