Consider parallel surfaces that are opaque to thermal radiation. The horizontal surfaces are separated by a gap of thickness \(L=0.12 \mathrm{~m}\). The top surface is at temperature \(T_{2}\) and is coated with a high emissivity paint, \(\varepsilon \approx 1\). The bottom surface is at temperature \(T_{1}\) and is characterized by:

\[\begin{array}{ll}\varepsilon_{\lambda}=0.1 & 0 \leq \lambda \leq 10 \mu \mathrm{m} \\\varepsilon_{\lambda}=0.8 & \lambda>10 \mu \mathrm{m}\end{array}\]

(a) The enclosure formed by the surfaces is filled with air at atmospheric pressure with \(T_{1}=500 \mathrm{~K}\) and \(T_{2}=300 \mathrm{~K}\). Determine the heat flux across the enclosure.

(b) The enclosure is filled with liquid water at an average temperature of \(\bar{T}=300 \mathrm{~K}\). There is a \(5 \mathrm{~K}\) temperature difference across the enclosure with the bottom surface being hot. Determine the heat flux across the enclosure.