Chapter 13, Problem 13.94P

An opaque, diffuse, gray $\left(200\text{mm×}200\text{mm}\right)$ 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 sidewalls of the furnace are well insulated. The top of the plate is exposed to ambient air with a convection coefficient of $25{\text{W/m}}^2\cdot \text{K}$ and to large surroundings. The air and surroundings are each at 300 K.

(a) Evaluate the net radiative heat transfer lo the bottom surface of the plate.
(b) If the plate has mass and specific heat of 2 kg and $900\text{J/kg}\cdot \text{K}$ , respectively, what will be the change in temperature of the plate with time, $d{T}_p/dt$ ? 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, $d{T}_p/dt$ , as a function of the plate temperature for $350\le T_p\le 900\text{K}$ and all other conditions remaining the same. What is the steady-state temperature of the plate?