Consider a slab made of two walls separated by air. Why do we need insulation material between the two walls. Air thermal conductivity is lower than most thermal conductivities of insulating material and convection cannot be an issue in the enclosed volume: hot air rises, so what? it won't go any further than the top of the cavity.
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You can think of thermal conductivity as a measure of how readily heat will flow through the material while it is stationary. The low thermal conductivity of air means that it takes a long time for heat to diffuse through an air pocket.
If the air is permitted to move, however, this intuition goes out the window. The air in contact with one wall gets warm and rises, and the resulting circulation causes it to be brought into contact with the other wall. In this way, the heat doesn't need to diffuse through the air, as it's being transported by bulk air flow.
Insulating materials such as blown fiberglass (or a wool sweater) are good insulators precisely because they trap many small pockets of air, which shuts down convection and forces the heat to flow diffusively. Once there's no convection, the low thermal conductivity of the air pockets makes the material a good insulator. You're right that the thermal conductivity of the trapping material is usually higher than the thermal conductivity of the air itself, but that's the (fairly modest) price we have to pay for killing the convection.
Albatross
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If the air gap between the walls is wider than approximately 0.5 inches, the warm wall will heat the air, causing it to rise. The cold wall will cool the air, causing it to fall. This will set up a circulating air flow between the walls which transfers heat across the gap to a greater degree than expected. Insulation between the walls is a barrier to air circulation, which somewhat decreases the heat transfer across the gap.
Note that this is the exact reason that panes in double paned windows are spaced approximately 0.5 inches apart. At that spacing, the rising air and falling air in the gap between panes "fight" each other, and prevent air circulation from forming.
David White
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Complementing the excellent answers on convection; some heat can also transfer through radiation, if the internal surfaces of the two walls are not treated to counter this. Both the warm and the cold wall will radiate, but at different rates, and this results in an equalizing heat flow.
A quick calculation using [1] gives that a wall at 25 C will radiate 85 W/m2 to a wall at 0 C, if the walls have an arbitrary emissivity of 0.64.
Putting some IR-blocking material in-between the walls decreases this mode of heat transfer.
[1] https://www.engineeringtoolbox.com/radiation-heat-transfer-d_431.html
Anders Petersson
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To take full account of the low thermal conductivity of air, it needs to be stationary.
