Problem $3.$ The inner and outer surfaces of a $30-$cm$-$thick wall in summer are maintained at $27$C and $44$C$,$ respectively. The outer surface of the wall exchanges heat by radiation with surrounding surfaces at $41$C$,$ and convection with ambient air also at $41$C with a h$=8$ W$/$m$2$K $($convection heat transfer coefficient$).$ Solar radia$$tion is incident on the surface at a rate of $150$ W$/$m$2.$ If both the emissivity and the solar absorptivity of the outer surface are $0\mathrm{.}8,$ determine the thermal conductivity of the wall. $(35$ points$)$