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It is my understanding that of all three states of matter, gases have the lowest thermal conductivity.

If this is the case, why are solid materials, such as fiberglass and polyurethane, used for this purpose?

Would it not make the most sense to simply fill the space with air?

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Background

The trade off is between conduction and convection. Heat conduction $\dot{q}_k$ (W) is set as

$$ \dot{q}_k = k A \frac{\Delta T}{w} $$

where $k$ is thermal conductivity (W/m K), $A$ is area (m$^2$), $\Delta T$ is temperature difference ($^o$K), and $w$ is the width (m). Heat convection $\dot{q}_h$ is set as

$$ \dot{q}_h = h A \Delta T $$

where $h$ is convection coefficient (W/m$^2$ K). As the thickness of the gap $w$ increases, the rate of conduction decreases. At zero thickness, the convection coefficient $h$ is zero. As $w$ increases, $h$ increases to a value indicative of a natural convection system.

Armed with this information, you can look up values of $k$ for air and $h$ for natural convection in air. Derive a ratio as

$$\frac{\dot{q}_h}{\dot{q}_k} = \frac{hw}{k}$$

Use this to find the value of $w$ where convection dominates conduction. Any air gap thinner than this will be OK because convection will be lower than conduction, and conduction is low in any case. Any air gap thicker will be controlled by convection.

This presumes that air is not "blowing" in the gap. At that point, $h$ can increase by one or two orders of magnitude. By example, what was acceptable for natural convection as a 1cm air gap becomes acceptable perhaps only below 0.1mm as an air gap.

Examples

By reference with $k_{air} \approx$ 0.02 W/m K as per this reference, we can find as per this reference

Free Convection (low): $h \approx 0.5$ W/m$^2$ K $\Rightarrow w \approx 2$ cm

Free Convection (high): $h \approx 500$ W/m$^2$ K $\Rightarrow w \approx 40~\mu$m

Maintaining a low free convection state in a vertical gap that is 2 cm wide is difficult. Once the air at the top of the gap becomes even slightly hotter than the air at the bottom, a convection cell is established. The air rotates from bottom to top of the cell, and the convection coefficient goes up by an order of magnitude already.

A Case for an Exception?

Modern homes with brick exteriors have an air gap between the brick and the panel insulation.

https://www.carsondunlop.com/inspection/blog/brick-houses-solid-masonry-vs-brick-veneer/

https://www.finehomebuilding.com/2013/03/06/best-practices-methods-for-installing-brick-or-stone-veneer

This is to avoid the case where moisture will collect at the junction of brick-touching-wall. It also allows the bricks to dry properly when they become wet.

The air gap is a modest thermal insulator. However, when the sun shines on the brick, free convection cells can form in the gap. This is one reason why thermal insulation is still used between the brick+air gap and the interior house walls.

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Insulation is used to "fill the space" with something other than air, because air can conduct thermal activity via convection. Convection in this application means that the air on the outside (cold in this example) will fall to the bottom of the air space, forcing the warmer air on the inside upward and outward, causing it to become colder.

Such movement can be observed in a simple manner. Inflate a mylar helium ballon and attach sufficient weight to create neutral buoyancy. As the balloon travels to a colder or warmer location, it will rise or fall depending on the temperature in the area where the neutral state was created. This is because it is now in the convection airflow.

Insulation reduces this convection by breaking up the air into smaller packets and surrounding it with fiberglass or other material. It is more difficult for sufficient convection (and conduction) to occur over a specific time.

If the temperature in an area does not change appreciably (inside heating), thermal equilibrium can be expected. An example would be a shed or outbuilding in a very cold environment. Even insulated, the inside temperature can be expected to match the outside temperature. This would be mitigated by some form of interior heating, but also by solar energy (sunshine) on the building walls.

Vacuum is a much more efficient insulator, hence the dewar flasks commonly called Thermos bottles. Today's steel insulated mugs, originally manufactured by Yeti use a vacuum within a double walled sealed construction. Even though there is conduction via the steel that is common to the inside and outside, the internal vacuum removes the convection aspect. Don't drop one and break the seal, however. The cooling capacity drops from 24 hours to only a few!

EDIT: I neglected to add the aspect of conduction as an additional factor in insulation. The outside material uses conduction to transfer the heat inside to the outside, otherwise there would be no problem with convection in the air.

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