Thermal conductivity of solids with gas pockets

Input(s)

$k_{o}$: Thermal Conductivity of Solid Phase $(\mathrm{W} / \mathrm{m} \mathrm{K})$

$k_{1}$: Thermal Conductivity of the Gas (W/m K)

Ø: Porosity (fraction)

$\mathrm{T}$: Temperature $(\mathrm{K})$

L: Length $(\mathrm{m})$

$\sigma$: Stefan-Boltzmann Constant $\left(\mathrm{kg} / \mathrm{s}^{3} \mathrm{~K}^{4}\right)$

Output(s)

$k_{\text {eff: }} \quad$ Effective Thermal Conductivity (W/m K)

Formula(s)

$$ \frac{\mathrm{k}_{\mathrm{eff}}}{\mathrm{k}_{\mathrm{o}}}=\frac{1}{1-\emptyset+\left(\left(\frac{\mathrm{k}_{1}}{\mathrm{k}_{\mathrm{o}} * \varnothing}\right)+\left(\frac{4 * \sigma * \mathrm{~T}^{3} * \mathrm{~L}}{\mathrm{k}_{\mathrm{o}}}\right)\right)^{-1}} $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 9, Page: 282.