Last updated:
4/9/2014

One-Dimensional,
Transient Conduction

(Replace
those Heisler Charts!)

**(FREE, NEW (4/9/2014) DOWNLOAD BELOW!)**

The mathematical description
of transient heat conduction yields a second-order, parabolic,
partial-differential equation. When applied to regular geometries such as
infinite cylinders, spheres, and planar walls of small thickness, the equation
is simplified to one having a single spatial dimension. With
specification of an initial condition and two boundary conditions, the equation
can be solved using separation of variables -- leading to an analytical expression
for temperature distribution in the form of an infinite series. The
time-honored Heisler charts were generated 67 years ago using a one-term
approximation to the series, and have been used widely ever since for 1-D,
transient-conduction applications.

**Module Description**

In our software module,
HTTonedt, we take a more fundamental numerical approach by computing a
finite-volume (FVM) solution to the transient, one-dimensional heat equation as
applied to planar walls, infinite cylinders and spheres -- i.e., the three
geometries for which the Heisler Charts are used. A single algorithm is used
for all three, and there is no need for Bessel or other transcendental
functions! This module should not be
described as an “electronic” Heisler Chart; rather it is a modern, numerical
solution for the same problem that allows the user to watch the entire
transient process on his or her monitor.

The user must specify the
surface Biot number, the initial temperature distribution, and a completion
criterion. The latter criteria can be based on the overall elapsed time
or the desired temperature at some particular location in the solid (much as
with the Heisler charts, one can use two of the non-dimensional centerline
temperature, Biot and Fourier number to find the third). The initial temperature distribution can be
set as uniform (corresponding to the Heisler Charts), or the user may specify
an initial (steady) temperature distribution corresponding to uniform volumetric
heating (which is turned off when this transient begins).

The numerical solution is
then performed quickly using the specified time increments, with a bar-chart
display of temperature distribution (to emphasize the discretization used in
this numerical solution) plotted at the end of each time step. The entire
transient evolution is presented to the user in animated form. The user
also has the option of plotting the entire temperature history at predetermined
points. Unlike the Heisler Charts, which, because they are based on a one-term
approximation of the infinite series, are not valid for short transients, this
numerical solution may be used for small Fourier numbers (i.e., short time
solutions when one might otherwise use the semi-infinite medium solution). The module is also applicable for low Biot
numbers (where the lumped capacitance model is usually employed); one will
simply find that the temperature distribution in the solid is flat.

The user may also select
explicit or implicit differencing -- or a weighted average of the two. When
the former is chosen and the time step limit exceeded, the user can watch a
dramatic display of a numerical instability. The Help topics included in
HTTonedt provide thorough discussions covering nearly all aspects of numerical
solution techniques for parabolic partial differential equations.

For those accustomed to the
traditional analytical solutions of transient problems, HTTonedt includes a
graphical “tour,” where, as a function of Fourier number (non-dimensional time)
and Biot number (internal conductive to surface convective resistance), the
user can select any of several sample cases to watch. These analytical methods (Heisler charts and
one-term solutions, lumped capacitance, semi-infinite solid solutions), each
have a particular (and limited) range of applicability, but may be used to
verify the results given by this finite-volume solution.

Inputs on the main form seen
above are non-dimensional, i.e., Fourier and Biot numbers. To facilitate their computation another form
is set up for input in *dimensional *form.
That form is seen here:

Beginning with Version 4.6
another form that includes the thermal properties of about 20 representative
materials is also available. Those
properties (conductivity, density and specific heat) may be automatically
imported from the form seen below into the form seen above.

Also embedded in this latest
edition are five presentations including one covering the “lumped capacitance”
method, one covering analytical solutions for one-dimensional bodies (including
the Heisler Charts) and a third covering various aspects of the finite-volume
method. The FVM runs behind the scenes in this module and allows a single
numerical algorithm to cover all three one-dimensional geometries. The information in this presentation will
help the user develop his or her code for cases not covered by this
module.

**Software Availability**

The HTTonedt module

**Notice to International Users **(in those countries where decimal points
(periods) are used instead of commas to break up long numbers): If, after you
have installed this module, it does not work properly, then in the
International Setting of the Windows Control Panel, please change the language
to English (US).

**Reference**

Complete documentation of the
algorithm and interface (much of which also appears in the included “help”
files) may be found in Ribando, R.J. and O'Leary, G.W., "A Teaching Module
for One-Dimensional, Transient Conduction", *Computer Applications in
Engineering Education*, Vol. 6, pp. 41-51, 1998.

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