$7.$ A hollow sphere made of a gray, diffuse$-$emitting material with a diameter of $20$ cm is maintained at a temperature of $500$ K$.$ The inside surface of the sphere is coated with a thin layer of a specular, absorbing material with an emissivity of $0\mathrm{.}5.$ The ambient temperature is $300$ K$,$ and the convection heat transfer coefficient is $10$ W$/$m$^2$K$.$ Determine the net radiation heat transfer rate from the sphere.