Cellular solids are materials that are highly porous with very thin, solid walls. Examples include metal foams, coral, cork, aerogels, styrofoam, etc. Because they are so porous, their thermal conductivities are generally low and so they can be used for insulation. One expression for describing the thermal conductivity of these materials is:

\[k=\frac{1}{3}\left[\left(\frac{ho}{ho_{s}}\right)+2\left(\frac{ho}{ho_{s}}\right)^{\frac{3}{2}}\right] k_{\text {solid }}+\left[1-\left(\frac{ho}{ho_{s}}\right)\right] k_{\text {gas }}\]

a. If we define the porosity as \(\left[1-\left(\frac{ho}{ho_{s}}\right)\right]\), rewrite the equation for thermal conductivity in terms of the porosity.

b. Assuming \(k_{\text {solid }}\) is representative of glass \((1.4 \mathrm{~W} / \mathrm{mK})\) and \(k_{\text {gas }}\) is representative of air \((0.026 \mathrm{~W} / \mathrm{mK})\), plot the thermal conductivity of the cellular material as a function of porosity.

c. Typical aerogel materials are about \(95 \%\) porous. What is their thermal conductivity according to your equation?