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 published over 75 years ago using a one-term approximation to the series and have been used widely ever since for 1-D, transient-conduction applications.
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 conduction 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. The user simply tabs to the appropriate material, clicks “Select” and the data is inserted on 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.
A virtual laboratory by Jorge Navalho based on the HTTonedt module may be downloaded here.
The HTTonedt module is as available as freeware. If you are up to date with your Windows updates, this executable should work directly without any installation needed. It will not work on Apple computers. Your virus checker may balk at running this program, so you’ll have to override it.
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).
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.