# Transient Conduction in a Finite Cylinder (Cooking a Roast)

UPDATED: 12/26/2022

This module was intended for K-12 Outreach, but for the more advanced user provides plenty of opportunities to learn the fundamentals of conduction heat transfer in solids. In it we model the cooking of a roast by solving the transient heat conduction equation in a finite, axisymmetric cylindrical geometry. The solution of the governing equation takes place entirely behind the scenes. The chef can “watch” the roast heat up over time. When cooking is completed, he or she can slice the roast at positions along its length and (virtually) sample the results.  The model includes both conventional “roasting” through heat transfer at its surface and volumetric heating as occurs in a microwave oven.

Module Description

This module includes a numerical solution (the finite-volume method) for the time-dependent, transient conduction equation in a homogeneous right circular cylinder. It assumes that during the heating and cooling process there are no chemical or physical changes in the material being cooked. Of course, in cooking a roast, there are both water migration and chemical and physical changes affecting the thermophysical properties. If that were not the case, then you could cool it back to its initial temperature and you would have raw meat again! Despite only including conduction inside the roast, convection at the surface and volumetric heat generation (as in a microwave oven), the module demonstrates much of what needs to be known. One can see, for instance, how easy it is to overcook the outside before the inside even knows it is in the oven. Food scientists are known to use one-dimensional transient conduction models, e.g., that provided by HTTonedt or the Heisler Charts, for a first-order estimate of the necessary cooking times of various edibles.

The user can specify the diameter and length of the cylinder to be heated. The needed thermal properties (density, specific heat and thermal conductivity) are provided on a pop-up form for about 20 materials (See Figure 3 below). Many of those listed are inorganic materials that one would not ordinarily “cook,” but are provided so that users can explore the effects of thermal conductivity and heat capacity (the product of density and specific heat) on the conduction process. A wide variety of materials are included, included very good conductors (copper and diamond) and non-conductors (insulators).

The ambient oven air temperature and two convection coefficients, one for the sides of the roast and the other for the two ends, are inputs. The convection coefficients can be used to demonstrate the value of a “convection oven” in hastening the cooking process. The resistance to heat flow internal to the solid material (measured by its thermal conductivity and also depending on its thickness) versus the “convective” resistance at its surface is well worth studying. By setting the convective resistance at the ends to a zero value, the transient should be identical to that found with the HTTonedt infinite cylinder option. With the side convection coefficient having a null value, one can model a 1-D slab or plane wall. Eight representative cases are provided in the menu to illustrate salient features.

To give even more insight than can be had by watching the evolving contour plots of temperature, the latest version also includes superimposed heat flux vectors.   In order not to clutter the plot excessively, these vectors are shown at only 1/16 of the finite volume cells.   Especially when cooling off the roast after a period of heating, these vectors fields help explain some of the observed transient temperature patterns.

Another recent feature is the creation of a meat color spectrum, an attempt to correlate the local temperature with actual beef colors as a function of temperature:

The radiation (broiling) option is not implemented yet. One can model microwave heating by including a non-zero value for the volumetric heat addition. This heat rate is assumed to be uniformly distributed within the material. Real microwave ovens are known not to do so, although those where the item being cooked is rotated on a turntable tend to be better.

The temperature range expected must be specified by the user as it determines the palette used in the color contour plots. At any point the “chef” can turn off the heat and watch the roast cool off. Then he or she can “slice” it to see the temperature distribution at any point along the length.

Video Introduction

A five-minute video illustrating most features of the HTT_cook module may be accessed on my YouTube channel.

Software Availability

A DRAFT version of HTT_Cook module is 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 (unless installed in cloud environment). You are welcome to contact me with suspected errors and suggestions for improvements. Because this program is still under active development, this version has a “drop dead” date of January 1, 2025.

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).