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

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? 