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It's generally accepted that a reflective surface, when used to insulate a system by reflecting radiant energy back to it, must have an air gap in front of it. Why exactly is this, and what determines the size of an acceptable air gap? Values of 1/2 - 3/4 inch are normally quoted.

The explanations that I've found are simplistic, and run along the lines of "radiant heat can only exist in a vacuum or in air, and once the the air gap is removed, there can be no radiant heat". There's all sorts of this stuff on Youtube, basically saying that with no air gap, you get only conduction, and no radiation.

This is clearly wrong, if only because a reflective surface behind glass (ie. not air) would still effectively reflect energy back through the glass.

A practical system (in my case) might be a sheet of plywood forming a wall to a living area, which is at 20C. There might be 1/2" of air behind the plywood, and then a reflective Aluminium surface. The plywood is at 20C, and radiates long wavelengths to the reflector. The reflector sends most of this back, where it is re-absorbed in the plywood. But what happens as the 1/2" gap is reduced to nothing? Eventually the Al and the plywood touch. The plywood still radiates, the Al still reflects. Nothing appears to have changed, except that the Al is now part of the radiating system.

So is the right answer that there is no change at all in the reflected energy, but the system is no longer insulated, because the outside of the reflector is now at 20C and radiating more? In which case, where does the magic 1/2" come from? Maybe 1/2" of air is considered to be thermally conductive 'enough'?

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At the point where the two sheets of material are touching, the primary mode of heat conduction between them stops being radiative and instead becomes conductive. Conduction is an effective means of transferring thermal energy, and so to prevent it, the sheets must have an air space between them.

To radiate heat, the "source" sheet will be warmer than the air next to it, and it is inevitable that this air will eventually warm up too. Once it does, it will be set into convective motion and heat will be conveyed across the air gap between the sheets by convection in addition to radiation. This can be postponed by spacing the sheets far apart but a "thick" wall like this is not practical. The recommended spacing between the sheets thus represents a practicality-driven compromise.

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