Prob. $1.($Semi$-$circular furnace$)$

A long furnace with a semi$-$circular top wall is used to heat treat sheet metal products is comprised of three zones. The heating zone $(1)$ is a ceramic plate which is kept at $1600$ K using gas burners. The load zone $(2)$ consists of sheet metal products, assumed to be black surfaces, which are to be maintained at $500$ K $.$ The top wall is made of insulating bricks.

$($a$)$ First assume all surfaces to be black, neglect convection, and determine the heat rate that must be supplied by gas burners and the temperature of the insulating brick surface at steady state conditions.

$($b$)$ Now consider surface emissivities as noted in the figure below. Determine the rate at which energy must be supplied by gas burners and the temperature of the insulating surface.