The emissivity of galvanized steel sheet, a common roofing material, is 0.13 at temperatures around 300 K, while its absorptivity for solar irradiation is S 0.65. Would the neighborhood cat be comfortable walking on a roof constructed of the material on a day when GS 750 W/m2, T 16 C, and h 7 W/m2 K? Assume the bottom surface of the steel is insulated.

Total heat transfer rate is combined convection and radiation transfer rate, multiplied with absorptivity factor. From that equation you can find temperature of the roof using iteration method.  

Explanation:

Known:

steel sheet roof emissivity ∈ = 0.13

steel sheet roof absorptivity _s = 0.65

air temperature T∞ = 16°C ...= 289K

heat transfer rate = 750 W/m2

natural convection h = 7W/m2K

Total heat transfer rate is combined heat transfer rate of convection and radiation, multiplied with absorptivity factor:  

=T∞-)+∈σA(

-T∞)-∈σ(=0

2562-7-7.371*10^-9=0

Use iteration method to solve for

=350.17 K...=77.17°C

In this particular situation, cat would not be comfortable with walking on a roof since its surface temperature is 77.17°C which may even cause burns to its paws.