# What's the simplest, first order way to calculate the temperature change in this scenario?

I am trying to model this experiment in a simple (i.e., relatively analytic, non-computational) way.

I have a little square of a thin film (10s of nanometers thin) of material A, on a relatively thick (1mm) and large area ($(10mm)^2$) slab of material B. Now, periodically, for some duty cycle, let's say 50% to make it easy -- I start dumping energy into this square of material A, at a power P. The rest of the duty cycle, there is no energy entering A. The environment around B is held constant at some temperature, let's say room temperature to make it simple.

Assuming we've been running this experiment for many cycles, so that it's in steady state, I want to find out what is the temperature at the beginning and end of the energy pulse.

Intuitively, I think I know what's happening in this scenario. When the pulse starts, energy starts heating up material A, according to the equation $\Delta Q=Cm\Delta T$. This causes a temperature gradient at the interface between A and B, so heat starts transferring from A to B (I think the movement of heat is a relatively slow molecular process, so the speed of the pulse will affect things). I think it's probably safe to assume that B reaches some steady state temperature with the environment, and probably because it's so big, we can assume that it stays about the same temperature during the pulse length.

Okay, so then it seems kind of straightforward at first: the heat transferred will be proportional to the AB interface area, and the difference between $T_A$ (changing during the pulse) and $T_B$ (which we're assuming is constant during the pulse). So it'll be something like $\dot Q_{AB} \propto A(T_A(t)-T_B)t$. Then you'll look at the total heat change (energy of the pulse minus energy lost to B), then apply $\Delta Q=Cm\Delta T$, finding $T_A$ as a function of time. During the time the pulse is off, it will only be losing energy to B.

Right now I'm running into the problem of figuring out a more specific form for $Q_{AB}$. The obvious step seems to be applying Fourier's Law, except that at my interface, the gradient is infinite. It seems like the equation in this article for Thermal contact conductance just assumes that it's linear for a short region, and if you take the limit as those two regions go to 0, you get the nice form $Q=h_c A \Delta T$. However, trying to find values for $h_c$ goes down a long rabbit hole of interface chemistry, surface roughness, and pressure.

It really seems like this should be a pretty straightforward problem, theoretically. Am I missing something?

• You are probably missing that for this scenario thermal conduction may not even be classical and you may have to use a phonon model to get it right. Classical thermal conduction works at infinite velocity, while the actual effect is limited by the velocity of the fastest phonons in the material. – CuriousOne Feb 13 '16 at 6:33
• This is a doable problem, but it isn't as easy as you may think. I assume you wish to assume that there is intimate contact between the square and the underlying plate, with no heat transfer resistance at the interface, correct? Would you also be willing to replace the square with a circle/cylinder? Would you be willing to replace the underlying plate with a semi-infinite slab? Certainly, the first step would be to solve for the average temperature profile that you would have with half the power on all the time. – Chet Miller Feb 13 '16 at 12:54