Transient Heat Conduction with Temperature-Dependent Thermal Conductivity

Consider transient heat conduction in a finite slab with temperature-dependent thermal conductivity. The phenomenon is governed by the following dimensionless equation:
where is the dimensionless thermal diffusivity and is an empirical parameter to be chosen by the user. The variables , , and are the dimensionless time, temperature, and position. The initial condition and boundary conditions are: , , and .
This Demonstration plots the dimensionless temperature versus the dimensionless position in the slab for , , , and (the orange, green, red, and cyan curves, respectively). The colored dots correspond to the solution using orthogonal collocation. You can specify the number of collocation points for the calculation as well as the functional dependency for the thermal diffusivity through the parameter . The colored curves correspond to the dimensionless temperature obtained using the Mathematica built-in function NDSolve. The solid curves correspond to ; the dashed curves correspond to , and the dotted curves are obtained using . The calculations show that for small values of the linearized form for the thermal diffusivity is adequate for short time periods, but less so for large time periods.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] S. H. Lin, "Transient Heat Conduction with Temperature-Dependent Thermal Conductivity by the Orthogonal Collocation Method," Letters in Heat and Mass Transfer, 5(1), 1978 pp. 29–39.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+