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Heat Diffusion.nlogo
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 wooden, and not the other way round. Also, 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
Material Thermal diffusivity
(alpha cm*cm/s)
Wood (Maple) 0.00128
Stone (Marble) 0.0120
Iron 0.2034
Aluminum 0.8418
Silver 1.7004
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. (For those interested, 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. Then 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
-- IN-PLATE-TEMP - Initial plate temperature
-- LEFT-TEMP - Left edge temperature
-- RIGHT-TEMP - Right edge temperature
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.
Set the paramters 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.
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:
Variables Units
--------- -----
time 0.1 second
temperature degrees Celsius
length centimeters
diffusivity square centimeters per second
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.