Differential Equations, Part Two

After I finished part one, I got to thinking about it and realized several things.

1. It isn't a pure math problem, it is more of an applied mechanical engineering problem.
2. Although my niece says it is a steady state situation, it isn't. The metal plate in question starts out at a temperature of zero, heat is applied to one edge, so the plate heats up to a point where it eventually reaches equilibrium, which is steady state. So it starts in a steady state and it ends in a steady state, but in between the temperature is changing.
3. The problem specifies we have a heat source on one edge, and heat sinks on the other three edges, but it doesn't say anything about the faces (back and front) of the plate. In this case we can presume that the faces of the plate are perfectly insulated so no heat is gained or lost from radiation. In practical terms, this is reasonable. You apply a torch to one edge of a plate and the plate will heat up. Any heat lost to the air or through radiation will be insignificant compared to the amount of heat being conducted away by the metal plate itself. In reality these losses may not be insignificant, but one thing at a time, so we are just going to pretend they don't matter in this case. After we figure out this part we can build a more sophisticated model.
4. We don't know how thick the metal plate is, but it probably doesn't matter. The thicker the plate, the more material there is available to conduct heat, but there is also more material to heat, so these two balance each other out. If our heat source (or sink) had a limited capacity, then this would make a difference, but this is a thought experiment, so we don't have to worry about little details like this.
5. The problem doesn't specify whether the temperature of zero is absolute, Celsius or Fahrenheit. It may not make any difference, other than the range of temperatures we find.
6. Neither the material, nor the specific heat, nor the heat conductivity of  the material are specified. They are not necessary. Any formula that is going to calculate the actual temperature of the plate will use constants for these values. Plug in the right constant, turn the crank, and you will get the correct answer.

I think what they are looking for is a what the temperature of any point on the plate after it has reached equilibrium. Let's start with a simpler example. If you have a metal rod with a heat source on one end, a heat sink on the other, and it is insulated along it's length, eventually it will reach thermal equilibrium. Heat will be flowing in one end, and flowing out the other, but the temperature of the rod will not be changing. The heated end will be hot, the other end will be cold, and I think we can safely say that the temperature will increase uniformly along it's length from the cold end to the hot end. If we graph the temperature of the rod against it's length we will get a straight line: the temperature gradient is linear.

Let's make it a little more complicated. Say we have a disk with a heat source being applied to the center and a heat sink all around the edge. In this case as you move away from the center, the temperature will start out dropping quickly, and then more slowly as it approaches the edge. Or maybe it will start out dropping slowly and then more quickly as you get farther from the center. This is because the farther you get from the center, there is more material to suck away the heat. In any case, it is not going to be linear.

With the rectangular plate described in our problem, we have two complications. One is that the heat source is not applied to a single spot, but all along one edge, and it isn't applied evenly. The other is the shape of the plate. As you go from the top (heated) edge to the bottom, you might expect the temperature gradient to be linear. While the faces are insulated, as in our example with the rod, the sides are not, they are connected to heat sinks. So the closer you are to one edge, the faster your temperature is going to drop on the way from the top edge to the bottom. The closer you are to the center, the closer your temperature gradient will be to linear. See the graph I included in Part 1.

I don't know how to calculate that, but if I stew on it for a couple of days, I may get an idea.

I graphed the top edge temperature. Clicking on the caption will take you to the spreadsheet I used to make it.

Graph of Top Edge Temperature

Since the source temperature drops below zero, I think we can assume that the zero specified in the problem is not absolute zero, but rather a more conventional one. Also, the teacher who crafted this problem is a crafty devil. If you followed my logic in analyzing the problem, and didn't bother to check out the formula for the source temperature, you might be surprised when your temperature gradients all point the wrong way.

P.S. Here's the formula I used in the spreadsheet for cell B3:

=-2*sin(PI()*A3/$C$1)+sin(3*PI()*A3/$C$1)

The 3 in the A3 reflects the row the formula is on. Formulas on different rows will have a different number.
I added this explanation because I didn't want to mess with the spreadsheet anymore. I am not sure this is the best way to explain this.

Update October 2016 replace missing graph.