An opaque, diffuse, gray (200 mm by 200 mm) plate with an emissivity of 0.8 is placed over the opening of a furnace and is known to be at 400 K at a certain instant. The bottom of the furnace having the same dimensions as the plate is black and operates at 1000 K. The side-walls of the furnace are well insulated. The top of the plate is exposed to ambient air with a convection coefficient of 25 W/m² ? K and to large surroundings. The air and surroundings are each at 300 K.

(a) Evaluate the net radiative heat transfer to the bottom surface of the plate.

(b) If the plate has mass and specific heat of 2 kg and 900 J/kg ? K, respectively, what will be the change in temperature of the plate with time, dT_p/dt? Assume convection to the bottom surface of the plate to be negligible.

(c) Extending the analysis of part (b), generate a plot of the change in temperature of the plate with time, dT_p/dt, as a function of the plate temperature for 350 Tp 900 K and all other conditions remaining the same. What is the steady-state temperature of the plate?

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T= 300 K T- 300 K h = 25 Wim-K Air Plate, 400 K, E = 0.8 -Insulated sidewalls 200 mm Furnace bottom, 1000 K