EX3: Non-Constant Conductivity I
5: Non-Constant Conductivity I (Example 3)
Consider the same geometry as in : an infinite slab of material lies between the planes
and
, and these plane boundaries are kept at constant temperatures
and
respectively. The conductivity
, however, is not assumed to be constant.
A more realistic assumption is that
depends on the temperature
; the graphs of several k-vs-T
relations, for various materials, appear in Figure 6.
Figure 6:
Specifically, let us assume that
is a linear
function of
, which can be a good approximation over a
short range of temperatures. Thus
Equation
43
where, ,
are given constants (). The sign of
depends on the material and the temperature range as seen in figure 6.
We seek a steady state temperature distribution and the
rate of heat flow at each point in the slab. As in Example 1, because
the boundary surfaces and the boundary values of
are independent of
and , the temperature distribution within the slab will also be independent of
and . Thus , and
. Therefore the heat flux vector has only an x-component,
.
Does actually depend on ? An
argument like that in Example 1 shows that, in spite of the
non-constant character of the conductivity
,
is constant; heat flows at the same rate through every plane
between the boundary planes.
Now substituting
given by (5.1) and
in Fourier's law of heat conduction,
we obtain the differential equation
Equation
44
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