EX4: Non-Constant Conductivity II
6: Non-Constant Conductivity II (Example 4)
Consider again radial flow in the cylindrical geometry studied in .
There the conductivity was taken to be constant. Now we make the more
realistic assumption that conductivity varies with temperature.
Specifically let
Equation
75
as in the previous discussion. [Note that the parameter
in equation (5.1) is
replaced here by
.]
A cylindrical shell of infinite length and circular cross-section is
under consideration, with inner radius
and outer radius
. The inner and outer cylindrical surfaces are assumed
to have steady uniform temperatures
and
, respectively.
We wish to find how the steady state temperature
varies
within the shell. In practice it is important also to determine the
rate at which heat flows from one boundary surface to the other. As in
previous examples we shall formulate and solve an appropriate
differential equation subject to the boundary conditions.
As in example 2, heat flows radially, at a steady rate (since the
boundary temperatures,
and
, are unchanging). If
, say, heat flows outward.
We consider a unit (one - meter) length of the infinite
cylinder. Within this tube let
denote a cylindrical surface of radius
. At all points of the temperature
is the same; thus depends only on
: . Similarly ,
the rate of heat flow in watts per square meter (directed radially,
perpendicular to
) is uniform over
. The area of is
, so heat flows through
at the rate of watts.
Because this is a steady state process, the rate of heat flow through
, has the same value for all
(). Let denote this value. Thus
Equation
76
is a constant, independent of
.
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