Thermal transfer calculation

Question

Say I have a flat rectangular bar of aluminum with a known volume. Sitting directly on top of that piece of aluminum is a rectangular circuit board made of FR4 (standard circuit board material). I can raise and lower the temperature of the aluminum very rapidly and I can monitor the temperature of the aluminum instantaneously but I cannot monitor the temperature of the circuit board directly. We can assume that the entire aluminum piece heats and cools evenly as does the circuit board.

If I know that I want to raise the temperature of the circuit board from 30°C to 200°C linearly in 90 seconds what calculations would I need to do to determine how fast i need to raise the temperature of the aluminum.

( I am tagging this incorrectly because there isn't a good tag for it and I can't add tags yet!)

Answer 1

I don't think there's enough information to solve the problem. You write that you "cannot monitor the temperature of the circuit board directly", but you also don't describe any way of monitoring it indirectly, so it seems you can't monitor it at all. There's also no information about the efficiency of heat transport between the aluminum and the circuit board. If you can really assume that the entire aluminum and the entire circuit board heat and cool evenly (which seems like a bad approximation to me if they're in contact), and if you can also assume that heat is transferred only through heat transport (and not e.g. through radiation reflecting off a case), then the problem is characterized by a single number quantifying the efficiency of heat transport between the aluminum and the circuit board -- this is the constant of proportionality between the rate of change of the temperature of the circuit board and the temperature difference. (This is assuming, as you seem to be applying, that you fully control the temperature of the aluminum, so the heat loss of the aluminum to the circuit board doesn't have to be taken into account. Otherwise we'd need a second quantity, like the ratio of the heat capacities of the two bodies.) You're going to have to know that constant of proportionality to be able to calculate anything; if you do know it, the rest should be a relatively simple exercise in ordinary differential equations.

Answer 2

Given your assumption that the circuit board is isothermal and so is the aluminum, the heat transfer across the interface will be proportional to the temperature difference between them. This means if you want to raise the circuit board temperature linearly with time (and no heat leaks out of the board) you need to maintain a constant temperature difference between the board and the heat source. If you insulate the sides or have the board and aluminum large compared to the thickness you have a one-dimensional problem. You have $\frac{dT_{board}}{dt}=\frac{CdQ}{dt}=Ck(T_{sink}-T_{board})$ where C is the heat capacity of the board and k is the thermal transfer constant