Page 298 - Bird R.B. Transport phenomena

282 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

For nonspherical inclusions, however, Eq. 9.6-1 does require modification. Thus for
square arrays of long cylinders parallel to the z axis, Rayleigh 2 showed that the zz com-
ponent of the thermal conductivity tensor к is

and the other two components are

9 6 4
!T = -jr = 1+ -— — г т - ^ ( - - )

That is, the composite solid containing aligned embedded cylinders is anisotropic. The
2
effective thermal conductivity tensor has been computed up to О(ф ) for a medium con-
taining spheroidal inclusions. 3
For complex nonspherical inclusions, often encountered in practice, no exact treatment
4 5 6
is possible, but some approximate relations are available. ' ' For simple unconsolidated
granular beds the following expression has proven successful:
*eff_(l - Ф) + аф(к /к )
1
0
k (1 -</>) + аф (9.6-5)
0
in which

The g k are "shape factors'' for the granules of the medium/ and they must satisfy g^ +
§2 + £з = 1- For spheres gi = g 2 = g 3 = \, and Eq. 9.6-5 reduces to Eq. 9.6-1. For unconsol-
idated soils, 5 #! = g 2 = I and g 3 = |. The structure of consolidated porous beds—for ex-
ample, sandstones—is considerably more complex. Some success is claimed for
4 6 8
predicting the effective conductivity of such substances, ' ' but the generality of the
methods is not yet known.
9
For solids containing gas pockets, thermal radiation (see Chapter 16) may be impor-
tant. The special case of parallel planar fissures perpendicular to the direction of heat
conduction is particularly important for high-temperature insulation. For such systems it
may be shown that

(9.6-7)

where a is the Stefan-Boltzmann constant, k is the thermal conductivity of the gas, and
A
L is the total thickness of the material in the direction of the heat conduction. A modifica-
tion of this equation for fissures of other shapes and orientations is available. 7

3
S.-Y. Lu and S. Kim, AIChE Journal, 36, 927-938 (1990).
V. I. Odelevskii, /. Tech. Phys. (USSR), 24, 667 and 697 (1954); F. Euler, /. Appl. Phys., 28,1342-1346
4
(1957).
5
D. A. de Vries, Mededelingen van de Landbouwhogeschool te Wageningen, (1952); see also Ref. 6 and
D. A. de Vries, Chapter 7 in Physics of Plant Environment, W. R. van Wijk, ed., Wiley, New York (1963).
W. Woodside and J. H. Messmer, /. Appl. Phys., 32,1688-1699,1699-1706 (1961).
6
A. L. Loeb, /. Amer. Ceramic Soc, 37, 96-99 (1954).
7
8
Sh. N. Plyat, Soviet Physics JETP, 2, 2588-2589 (1957).
M. Jakob, Heat Transfer, Wiley, New York (1959), Vol. 1, §6.5.
9