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Heat Diffusion

[screen shot]

If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web


This model simulates transient and steady-state temperature distribution of a thin plate. The View shows a square thin plate as viewed from above. The plate is thermally isolated on the two faces parallel to the view such that heat can flow only in and out from the perimeter of the plate and not into or out of the world. Heat is kept constant at the edges.

As the simulation runs, heat is transmitted from warmer parts of the plate to cooler parts of the plate as shown by the varying color of the plate. Therefore, the temperature of the plate begins to change immediately and possibly differently at different locations, gradually converging to a stable state. Overall, the temperature distribution over the plate is a function of time and location. In addition to this simple use of the model, you are encouraged to control various paramaters, such as the temperature of each edge edge of the plate and of the center of the plate before--and even while--the model is running.


Heat diffuses ("spreads") at different rates through different media. These rates can be determined and are called the Thermal Diffusivity of the material. The Greek letter alpha is often associated with this value. The diffusivity of a material does not change based on how much of the material there is; it is always the same.

Below is a table containing several different materials with different diffusivity rates. See that wood (bottom row) has a lower heat diffusivity than, say, iron. This means that it takes a longer for heat to spread through a wooden object than an iron one. That is one reason why the handles of iron saucepans are often wooden and not the other way round.

Think of a marble table with iron legs that has just been put out in the sun in a street-side cafe. Which material part of the table do you expect will warm up faster? The model allows you to change thermal diffusivity of the plate in two ways. You can directly change the value of ALPHA to any value you like, or you can indirectly change ALPHA by selecting a material.

Thermal diffusivity of selected materials

<table border> <tr><th>Material<th>Thermal diffusivity<br>(alpha cm*cm/s) <tr><td>Wood (Maple)<td>0.00128 <tr><td>Stone (Marble)<td>0.0120 <tr><td>Iron<td>0.2034 <tr><td>Aluminum<td>0.8418 <tr><td>Silver<td>1.7004 </table>

How the diffusion is modeled

Initialize the plate and edges to have temperatures that equal their respective slider values. Each time through the GO procedure, diffuse the heat on each patch in the following way. Have each patch set its current temperature to the sum of the 4 neighbors' old temperature times a constant based on ALPHA plus a weighted version of the patch's old temperature (the updated temperature is calculated by using a Forward Euler Method). Then the edges are set back to the specified values and the old temperature is updated to the current temperature. Finally, the plate is redrawn.


There are five temperature sliders which enable users to set four fixed edge temperatures and one initial plate temperature: -- TOP-TEMP - Top edge temperature -- BOTTOM-TEMP - Bottom edge temperature -- INITIAL-PLATE-TEMP - Initial plate temperature -- LEFT-TEMP - Left edge temperature -- RIGHT-TEMP - Right edge temperature

Note, if all of these are the same temperature, the model will fail to SETUP because there is no heat differential.

There are two sliders that govern the thermal diffusivity of the plate: -- MATERIAL-TYPE - The value of the chooser is that of the above chart. You must press UPDATE ALPHA for this to change the value of ALPHA. -- ALPHA - The alpha constant of thermal diffusivity

There are four buttons with the following functions: -- SETUP - Initializes the model -- GO - Runs the simulation indefinitely -- GO ONCE - Runs the simulation for 1 time step -- UPDATE ALPHA - press this if you want to set ALPHA to a preset value based on a material selected by the MATERIAL-TYPE chooser

The TIME monitor shows how many time steps the model has gone through.


How does the equilibrium temperature distribution vary for different edge temperature settings?

Notice how an equilibrium (the steady-state condition) is reached.

Keep track of the units:

<table border> <tr><th>Variables<th>Units <tr><td>time<td>0.1 second <tr><td>temperature<td>degrees Celsius <tr><td>length<td>centimeters <tr><td>diffusivity<td>square centimeters per second </table>


Set the parameters on the temperature sliders. Pick a value for ALPHA (or pick MATERIAL-TYPE and press UPDATE ALPHA). After you have changed all the sliders to values you like, press Setup followed by GO or GO ONCE.

Try different materials to observe the heat transfer speed. How does this compare to physical experiments?

Try the following sample settings: - Top:100, Bottom:0, Left:0, Right:0 - Top:0, Bottom:100, Left:100, Right:100 - Top:0, Bottom:66, Left:99, Right:33 - Top:25, Bottom:25, Left:100, Right:0


This model simulates a classic partial differential equation problem (that of heat diffusion). The thin square plate is a typical example, and the simplest model of the behavior. Try changing the shape or thickness of the plate (e.g. a circular or elliptical plate), or adding a hole in the center (the plate would then be a slice of a torus, a doughnut-shaped geometric object).

Add a slider to alter this thickness.

Try modeling derivative or combined boundary conditions.


Checkout the other Materials Science models in the library


If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

Please cite the NetLogo software as:


Copyright 1998 Uri Wilensky.


This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at

This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2001.

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