Last updated:
8/27/2012
One-Dimensional,
Transient Conduction
(Replace
those Heisler Charts!)
(FREE, NEW (8/27/2012) 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 a half-century 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.
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. A static sample of the entirely new VB.net
module is presented in the graphic below. This update will be included in the Heat
Transfer Today.
Click on
the image for an enlarged view (GIF at 112k
bytes).
The user may also select
explicit or implicit differencing -- or a weighted average of the two. When
the former is chosen and the timestep 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, Version 4.1.3.0 of
HTTonedt now includes a graphical “tour,” where, as a function of Fourier
number (time) and Biot number (internal conductive to surface convective
resistance), the user can select any of several sample cases to watch. These analytical means (Heisler charts and
one-term solutions, lumped capacitance, semi-infinite solid solutions), each of
which has a particular (and limited) range of applicability, may be used to
verify the results given by this finite-volume solution.
Also embedded in this latest
edition are five PowerPoint presentations including one dealing with 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 own code
for cases not covered by this module.
In order to view these slide shows, the user must have PowerPoint or the
free PowerPoint Viewer installed on his or her computer. When you click on the menu item, a small
window will appear. Double click on that
and in a few seconds the PowerPoint slides will show full screen.
Software Availability
Copies of our 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.
Back to Heat Transfer Today Main Page
Back to R. J. Ribando Home Page