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 simplifies to one having a single spatial dimension. With specification of an initial condition and two boundary conditions, the mathematician uses separation of variables to solve the equation. That process leads to an expression for temperature distribution in the form of an infinite series. The time-honored Heisler charts use a one-term approximation to the series and present the results in graphical form. Heat transfer practitioners have used these charts widely for the last 75 years.
Module Description
The calculation
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. We apply it to planar walls, infinite cylinders and spheres, i.e., the three geometries for which the Heisler Charts are used. We use a single algorithm for all three, and there is no need for Bessel or other transcendental functions! This module is NOT 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 screen.
Inputs
The user specifies the surface Biot number, the initial temperature distribution, and a completion criterion. The algorithm can use either the overall elapsed time or the desired temperature at some particular location in the solid for the latter. Much as with the Heisler charts, the user can use two of the non-dimensional centerline temperature, Biot and Fourier number to find the third. The user can set the initial temperature distribution as uniform corresponding to the Heisler Charts. Alternately, they may specify an initial (steady) temperature distribution corresponding to uniform volumetric heating. (The model turns that heating off when this transient begins).
Output/Display
The numerical solution is then performed quickly using the specified time increments. We have chosen a bar-chart display of temperature distribution to emphasize the discretization used in this numerical solution. The display shows the entire transient evolution in animated form. The user also has the option of plotting the entire temperature history at predetermined points. Recall that the Heisler Charts are not valid for short transients. In contrast, 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 simply finds 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 sees a dramatic display of a numerical instability. The HTTonedt Help topics provide thorough discussions covering nearly all aspects of numerical solution techniques for parabolic partial differential equations.
The “Tour”
For those accustomed to the traditional analytical solutions of transient conduction problems, HTTonedt includes a graphical “tour.” The user selects, as a function of Fourier number (non-dimensional time) and Biot number (internal conductive to surface convective resistance), 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. They 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 we provide another form for input in dimensional form. That form is seen here:
Property Data
HTTonedt includes another form providing thermal properties of about 20 representative materials. The user may import those properties (conductivity, density and specific heat) from that form (seen below) into the input form seen above. The user simply tabs to the appropriate material, clicks “Select.”
We include five presentations within the help files of this module. One covers the “lumped capacitance” method and one covers analytical solutions for one-dimensional bodies (including the Heisler Charts). A third covers various aspects of the finite-volume method. The information in this presentation will help the user develop his or her code for cases not covered by this module.
Virtual Laboratory
Jorge Navalho has created a virtual laboratory based on the HTTonedt module. You can download it here.
YouTube VIDEO INTRODUCTION TO THE HTTonedt MODULE (3:21)
Software Availability
The HTTonedt module is 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 that.
Notice to International Users (in those countries where decimal points (periods) are used instead of commas to break up long numbers): If this module does not work properly, then please change the language setting to English (US).
Reference
The interested user can find complete documentation of the algorithm and interface (much of which also appears in the included “help” files) 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.