# 5-layered electric wire heat dissipation process

A wire is made by 5 identical extremely thin layers of the same material, as shown in the picture.

The layers are wrapped in an electrically insulating material, which is kept at a constant temperature. The thermal capacity of the insulator and resistance of the material are constant no matter the temperature. They are all connected in parallel to a generator and thus generate heat. The heat can only exit from the top and bottom faces of the wire. Each one of the wires can transfer heat to its neighboring ones and to the ambient when possible. The temperatures of each layer are given experimentally for 3 different values of the current. $$T_B=T_D$$ and $$T_E=T_A$$. Furthermore, $$T_C>T_B=T_D>T_A=T_E$$.

The total heat dissipated from A and E is proportional to the difference in temperature between C and the external ambient, why?

This is not a complete problem, but only a step that I couldn't understand; the rest of it is pretty trivial.

The reasoning of the given solution is:

It is intuitive that the heat heat transferred from C to B (and D) is proportional to the temperature difference between C and B (and D) $$\Delta T_{CB}=T_C-T_B$$. The heat that goes from B (or D) to A (or E) is again proportional to their difference in temperatures $$\Delta T_{BA}=T_B-T_A$$ and the same applies to A (or E) and the external ambient, with $$\Delta T_{Aext}=T_A-T_{ext}$$ (Every calculation is obviously still valid for symmetrical temperatures). Summing up those differences we get that the total heat dissipation process ultimately depends on the temperature difference between C and the ambient.

The step in bold is the thing I'm not able to understand. Why do we need to sum all the different heats? We're not dealing with the total dissipation but summing up the single heat transfers from a wire to the next. I don't see how this sum would help. Thanks for your time and effort.

## 1 Answer

I believe the conceptual problem lies not with you but with the solution description. It seems that the intended meaning is "Considering the coupled heat transfer and temperature differences between each layer, the total heat dissipation process depends in part on the temperature difference between C and the ambient temperature." But the part about literally adding the temperature differences seems to be bad math; to my knowledge, no standard analysis method performs this step.

In fact, the total heat dissipation process depends on a great many parameters, although the assumptions of symmetry and identical materials simplify things somewhat. Consider layer B: by Newton's Law of cooling, the heat input is $$h(T_C-T_B)$$ (where $$h$$ is some heat transfer coefficient), the heat output is $$h(T_B-T_A)$$, and the power dissipated is, say, $$P$$. Thus, an energy balance at equilibrium gives $$h(T_C-2T_B+T_A)+P=0.$$

The energy balance for layer C (well, half of layer C, applying symmetry) is $$-h(T_C-T_B)+\frac{P}{2}=0.$$

The energy balance for layer A is $$h(T_B-2T_A+T_\mathrm{ext})+P=0.$$

These three equations in three unknowns ($$T_C$$, $$T_B$$, and $$T_A$$) can be solved to determine the temperature, and yes, $$T_C$$ (or, alternatively, $$T_C-T_\mathrm{ext}$$) affects this solution. But as you note, there's no point in the analysis where temperature differences are simply added together.

• If you had the complete solution it would be obvious that it's not a description problem. Evidently, the solution's author did make a mistake in the reasoning (fortunately it's not an official solution). Now the problem gets much more difficult, I'll have some fun with it. Thanks :) Aug 16, 2021 at 16:53