How to estimate the maximum temperature between heating a cooling curve?

Question

I have a thermodynamics/mathematical problem I was hoping somebody could help me with.

I am simulating the heating and cooling process of several thermal masses connecting through thermal resistances. In the first phase of the simulation there are a heat sources in two of the masses that heat the system. The heat equation is solved with a finite difference/explicit method and the result is an exponential curve that reaches a maximum temperature at some time. However, before this temperature is reached, the heat sources are turned off or reduced, so a cooling process sets in.

What I am aiming for is a way to estimate the maximum temperature (without solving the heat equation for the next several time steps) in the phase between the heating and cooling processes, so where the system swings over into the cooling phase. I'm guessing this should be a function of the slope of the temperature curve right before the heat sources are turned off/reduced, volumetric flow rate of the coolant and, in the case the heat source is only reduced, the reduced heat source.

Does anybody have an idea how to tackle this problem?

Thanks in advance!

Answer 1

I don't think your question has a closed form solution for more than two thermal masses and one heat source, if you can reduce it to that. With one reservoir (no thermal gradients within each reservoir) it's the familiar first-order differential equation. For two reservoirs, one heated by the other, it's a linear ordinary second order differential equation (DE). The second reservoir goes through a maximum because it's getting heat from a first reservoir that is warmer than it is, but at the same time is cooling down. These 2nd order linear DE's have a closed-form solution for T(t) in the second reservoir that you can look up in any book on DE's. It's some ugly integral that you have to evaluate, but it's generally possible.

Once you write it down, you can then differentiate with respect to time and set equal to zero to get the maximal temp at the top of your curve where dT/dt = 0. With systems much more difficult that that, you're stuck doing a numerical computer model.

Answer 2

As an estimate could it be as simple as applying a cooling correction of the type described in the link below?

http://www.nuffieldfoundation.org/practical-physics/cooling-corrections