I have little knowledge of physics and have been unable to find a general mathematical analysis of this problem. I’ve tried to present it in a general way but the underlying principles are applicable to several practical situations.

A small spherical temperature-sensitive item is about the size of a ping-pong ball (and has radius R1). It is to be shielded from the diurnal temperature variation of its environment by one or more layers of insulating material. There can be any number of these layers and they can be made of any readily-available real-world substance. However their total thickness must not make the complete entity larger than a soccer ball (of radius R2).

The environment surrounding the soccer ball is a mass of dry free-flowing air so large that the soccer ball and its contents can be assumed to have no influence on its temperature. The temperature of the surrounding air alternates between temperatures T1 and T2 and it spends times D1 and D2 respectively at those temperatures. Typically T1 and T2 might be 0 and 50 Celsius and D1 and D2 would both be 12 hours. This cycle continues indefinitely. The soccer ball is suspended in that air by an infinitessimally thin piece of string that does not conduct heat.

Q1: What combination of (real-world) layer materials and thicknesses will minimise the diurnal temperature variation of the inner sphere?

Q2: How does the insulating strategy differ if the ‘soccer ball’ is also exposed to heat as radiation (e.g. by exposure to the sun) rather than just conduction from the surrounding air?

Q3: How does the optimum strategy change if there is a limit to the total mass (as well as the outer diameter) of the complete entity?

One aspect of the problem is whether less core temperature variation occurs with a single thick layer of one material or with a series of thinner layers whose properties might be different. They might even alternate back and forth between two different materials. For example there might be alternate insulating spheres of polystyrene foam and iron. The iron might not be such a good heat insulator but its higher specific heat would mean that it could absorb more heat energy from a surrounding foam layer before causing a temperature rise to pass on to the next layer. That alternating structure seems vaguely analogous to the situation in electronics where alternate sequence of series inductors and parallel capacitors are used to make a Low-Pass Filter. The net result is that greater high frequency attenuation is achieved than with a single inductive or capacitive element.

Update #1.

First thanks to @Chet for ‘Thermal Diffusivity’. I didn’t know that term and Wikipedia has helped my thermal vocabulary. I can see that this term is basically a measure of:

k = 1 / (thermal resistivity x thermal storage ability)

So it’s (kind of) a thermal equivalent of 1/RC in electronic filter design (except that it relates intrinsic material properties on a volumetric basis rather than the absolute amounts of resistance and storage according to the dimensions of a specific instance). Fine - I understand the analogous properties in electronics and can see how a broadly equivalent set of relationships between them might exist in the thermal world.

Thanks also to @Gert. I accept his point about the complications of spherical geometry. My ‘ping-pong ball’ and ‘soccer ball’ were a way to present the problem. It might be easier to look at heat flow between infinite parallel planes and that would reduce the heat flow analysis to one linear dimension.

But that is my only progress. Chet says ‘If the diffusivity times the time period of the oscillation ... is small compared to the radius squared of the soccer ball, the temperature perturbations ... will die out rapidly with depth.’ That would appear to be the thermal equivalent of saying (electronically) that a single-section RC filter will attenuate if its time-constant (RC product) is greater than the period of the input signal. That is true - but it doesn’t help to identify the best attenuation possible for a given cost constraint. Nor does it determine what network would achieve it.

Returning to the thermal world @Gert seems to believe that there is no ‘shell’ structure that would work better than a single thick insulating layer. He may be right and that is why this was my first question - because if there is no such structure then it would seem that finding the ‘best’ solution is simply about finding the relevant material property, and then finding that material with the best value for it.

But here’s why I’m not (yet) convinced about that. In the electronic filter world I could ask the following question: ‘If resistance costs r dollars per ohm and capacitance costs c dollars per farad then what is the maximum RC filter attenuation I can achieve at a frequency of f Hz for a spend of s dollars?’

For a single-section RC filter maximum attenuation at all frequencies occurs when RC is maximum. Elementary calculus would show how the total spend must be split between R and C to achieve that maximum. From that split we could then find the maximum possible RC product.

But above a certain signal frequency more attenuation can always be achieved by using two (or more) RC sections in series. Obviously there is then less R and C available for each section. And that will raise the filter cut-off frequency. But there always comes a frequency above which the higher filter cut-off is more than offset by the higher rate of filter roll-off. So for a high enough frequency the high-order filter always provides more attenuation. And that is true even though the later filter stages ‘load’ the earlier ones.

In the filter world my question Q1 involves the trade-off between filter order and filter cut-off frequency. Fast filter roll-off is of little use if the cut-off frequency is above the frequency to be attenuated. So when ‘cost’ is constrained a single-section filter is sometimes the best option. In filter design we can use circuit optimizers to find the best possible performance for a given constraint. And if we know the circuit equations we can use algebra too. And if the thermal world really is analogous to the filter world we ought to be able to use analogous methods. Furthermore if the thermal world is analogous then it seems to me that there may be times when a multi-layered insulation structure (e.g. alternate iron and styrofoam) could give the best performance for a given constraint (whether it limits dollars, mass or physical volume). The answer will depend on those thermal properties analogous to filter order and cut-off frequency.

  • $\begingroup$ I would start off by modeling this using a single layer of insulating material around the sphere. You are obviously going to want to use a material with low thermal diffusivity. If the diffusivity times the time period of the oscillation (1 day) is small compared to the radius squared of the soccer ball, the temperature perturbations are going to be confined to the region near the surface, and will die out rapidly with depth. You can get a feel for this by studying the flat-geometry version of the problem. $\endgroup$ May 22, 2020 at 11:34
  • $\begingroup$ I would start off by modeling this using a single layer of insulating material around the sphere. I imagine this will be mathematically very involved, unless a number approximations/simplifications are called for.Just the convective cooling of a homogeneous sphere (no insulation) isn't for the faint of heart. $\endgroup$
    – Gert
    May 22, 2020 at 13:50
  • $\begingroup$ There's little reason to believe multiple insulating 'shells' would be more 'optimal' than a single insulator, IMO. $\endgroup$
    – Gert
    May 22, 2020 at 13:51


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