A 50-mm-thick air gap separates two horizontal metal plates that form the top surface of an industrial furnace. The bottom plate is at T_h = 200°C and the top plate is at T_c = 50°C. The plant operator wishes to provide insulation between the plates to minimize heat loss. The relatively hot temperatures preclude use of foamed or felt insulation materials. Evacuated insulation materials cannot be used due to the harsh industrial environment and their expense. A young engineer suggests that equally spaced, extremely thin horizontal sheets of aluminum foil may be inserted in the gap to eliminate natural convection and minimize heat loss through the air gap.

(a) Determine the convective heat flux across the gap when no insulation is in place.

(b) Determine the minimum number of foil sheets that must be inserted in the gap to eliminate free convection.

(c) Determine the conduction heat flux across the air gap with the foil sheets in place.

Gond T= 50C Aluminum Air foil sheets T= 200C