# Thermal transfer calculation

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!)

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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