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WOLFRAM|DEMONSTRATIONS PROJECT

Temperature Propagation in a Thin Steel Rod

​
temperature (left)
10
temperature (right)
10
time (min)
0.0001
environment temperature
10
bar00
bar01
The temperature contour problem
u
t
=
2
a
u
xx
-h(u-
u
0
)
has been recast for a "long" stainless steel 316 rod with lateral heat exchange, under the initial distributions
u(x,0)=(12
πϵ
)
-
2
x
4ϵ
e
(K) (Gaussian distribution represented by bar00 on the Demonstration chart) and
u(x,0)=300+(273-300)sin(πx/l)
(K) (represented by bar01, likewise). The factors
a
and
h
encode information related to the conduction and diffusion coefficients of the rod, under the parametrization
c
1
=
h
l/a
,
c
2
=
2
l
/
2
a
, while
u
0
labels the environment temperature. Particular values of
l=1
m (length) and
p=1
inch (perimeter) were chosen. Boundaries and environment temperatures are kept uniform and controllable in range with a maximum value of 473 K at both extremes, while the transient regime is mapped on real time for the first four hours. Factor is adjusted for an effective distribution of the external field localized around the rod's center with 10% deviation.
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