EX1: Infinite Slab
Suppose an infinite slab of material, bounded by planes
and
, has a
constant thermal conductivity
. If the faces are
maintained for a long time at temperatures
and
respectively, what is the temperature distribution inside
the slab? What are the direction and magnitude of the heat
flux at each point?
Since the boundary (face) temperatures do not vary with
time, it is reasonable to assume that the temperature
within the slab is also independent of time (a steady
state). Furthermore, since on each face the temperature is
independent of the other space variables,
and
, we assume that within the slab
is independent of
and
. In other words,
, so that
Equation
4
and from equation (2.1),
the heat flux vector
is also in the x-direction (perpendicular to the faces). Therefore
Equation
5
where
is the
-component (and only component) of the heat flux vector.
Since depends only on
so do and
. Thus we may write:
Equation
6
In fact, it may be shown that:
Equation
7
throughout the slab.
Click here for the proof!
The relations
and
give
Equation
10
While
is constant, its value is not yet known.
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