Math 326, Feb 26 -
Heat
Propagation
We do the heat model from section 3.5. Heat propagates from
one piece of a surface to the next, proportionally to the difference in
the temperatures of the two sections. The proportionality
constant depends on how well that particular material conducts
heat. We start with a one-dimensional model, heat in a rod.
The rod is divided into 9 pieces, including the end pieces whose
temperature is fixed. The interior pieces have temperatures which
vary according to the following formula:
Tn = Tn-1 + a*(Tleft-Tn-1)
+ a*(Tright-Tn-1)
In two dimensions, each piece will transfer heat with its neighbors
which now number four. 2-D uses macros, which is a bit involved
to describe here, so we'll discuss it together in class.
Additional Questions
- Hot Spots: In the
2-D model, try fixing 3 or 4 hot spots within the plate, whose
temperatures are fixed. This time, allow the outside edge
temperature to vary. Note the edge pieces will have fewer
neighbors, so their update formula will be different.
- Varying the Source:
In the 1-D, try giving one end of the rod an oscillating temperature
instead of fixed temperature. Instead of using a fixed
temperature of 60 for the left end, trying using ( = 60 +
20*SIN(2*PI()*TIME/16) ). The sine term will cause the left
end temperature to oscillate around the base temperature of 60, from 40
to 80 and back every 16 timesteps.