A reader who has seen the density cancel for viscosity and conduction will object. If k does not depend on density, why is a gas such a poor conductor? The cancellation applies to compressing or thinning the same gas. It does not apply when you compare a gas with a liquid or solid, which work differently. Still air has a thermal conductivity of about 0.026 W/(m·K), against about 0.6 for water, about 1 for glass, about 50 for carbon steel, and about 400 for copper.[1] In a gas, energy moves only when a molecule travels. Each molecule makes a trip of about one mean free path, roughly 200 molecular diameters in air, before handing off its excess in a collision, and there are more than a thousand times fewer molecules per unit volume to hold energy than in water (about 1,400).[1] In a liquid or solid, atoms are in contact, so a vibration passes straight to a neighbor with no free flight.
In a solid, atoms stay in place and energy moves as vibrations handed from neighbor to neighbor, and by free electrons in metals. A gas conducts only because the molecules themselves travel.
A vacuum does better still, and here the mean free path decides it. The density cancellation requires the mean free path to be much shorter than the gap. Take a gap of one centimeter. At atmospheric pressure the mean free path is about 7 × 10−6 cm, and λ rises as 1/pressure, so the two become equal near 1 Pa, roughly a hundred-thousandth of an atmosphere. Below that, molecules cross the gap without colliding, each bringing its energy straight from the hot wall to the cold wall. The number crossing now falls in proportion to the pressure, so the conductivity falls with it, and the density-independence you just derived no longer holds. In a perfect vacuum there are no molecules and no conduction at all. Only radiation remains, plus conduction through whatever holds the walls apart. That is why a thermos flask is evacuated between its walls and its surfaces are silvered to reflect radiation.
Clothing that traps air works for the same reason as still air, with one extra condition: the air must stay still. Moving air transports energy in bulk by convection, which beats molecular conduction by a wide margin. Wool, fleece, fur, and down divide the air into pockets too small to circulate, so only slow molecular conduction through the gas is left. The fibers are thin, tangled, and a small fraction of the volume, which gives energy long, poor paths, and they also scatter radiation. The result is a conductivity only a little above that of still air, multiplied by thickness. Wet clothing fails because water fills the pockets, and water conducts more than twenty times better than air (about 23).[1]
References
[1] Air and water at 300 K and 1 atm: 0.0264 and 0.610 W/(m·K), computed with CoolProp 8.0.0 from E. W. Lemmon and R. T. Jacobsen, International Journal of Thermophysics 25, 21–69 (2004), and M. L. Huber et al., “New International Formulation for the Thermal Conductivity of H₂O,” Journal of Physical and Chemical Reference Data 41, 033102 (2012). The ratio of molecules per unit volume, about 1,400, is computed from the densities of air and liquid water at the same conditions. Solids at about 300 K: F. P. Incropera, D. P. DeWitt, T. L. Bergman and A. S. Lavine, Fundamentals of Heat and Mass Transfer (Wiley), Appendix A. Glass (about 1 to 1.4) and carbon steel (about 43 to 64) vary with composition, and stainless steel is about 15 W/(m·K).
END