Effective Thermal Conductivity of Composite Solids
Section 9.6 from BSL
Spherical Inclusions in a Continuous Solid Phase
Non-spherical Inclusions in a Continuous Solid Phase
Solids Containing Gas Pockets
Cylindrical Ducts Filled with Granular Materials
Notation:
keff—effective thermal conductivity
k1—thermal conductivity of embedded
material
k0—thermal conductivity of continuous
phase
f—volume
fraction of embedded material
The effective thermal conductivity for a composite solid depends on the geometry assumed for the situation. Following are different approximations used for various geometries:
Spherical Inclusions in a Continuous
Solid Phase:
Maxwell’s Derivation:
·
Models heterogeneous solids as a material made
of spheres of k0 embedded in a continuous phase of k1.
·
Assumes spheres do not interact thermally (small
volume fraction f)
Figure 1:
Spherical inclusion with a small volume
fraction
Rayleigh’s
Derivation:
·
Models heterogeneous solids for large volume fraction
f as spheres
embedded in the intersections of a cubic lattice
Figure 2:
Spherical inclusions with a
large volume fraction
Note:
The interaction between spheres is found to be small even when the Rayleigh
derivation is used. Thus, the simpler
derivation from Maxwell is generally used to simplify the calculations.
Non-spherical Inclusions in a Continuous Solid Phase:
Raleigh’s
Derivation:
·
For square arrays of long cylinders parallel to
z axis
·
The composite solid is anisotropic (the thermal
conductivity is not the same in all directions)
Figure 3: Long cylinders parallel to
z-axis
Comparison of Experimental
Data with Non-spherical Inclusions Model
Complex non-spherical
approximation:
where
·
Models simple unconsolidated granular beds as
complex non-spherical inclusions in a continuous solid phase
·
gk are “shape factors” for granules
Figure 4:
Complex nonspherical
inclusions
Solids Containing Gas Pockets:
·
Assumes thermal radiation important
·
Used for parallel planar fissures perpendicular
to the direction of heat conduction
·
s
is the Stefan-Boltzmann constant and L is the total thickness of the material
in direction of heat conduction
Cylindrical Ducts Filled With Granular Materials Through Which a Fluid Is Flowing:
·
keff,zz
and keff,rr
denote thermal conductivities in axial and radial directions, respectively
·
Used for highly turbulent flow
·
Thermal conductivity is highly anisotropic
·
v0 is superficial velocity, Dp
diameter of particle