Chapter 13, Problem 13.73P

A long cylindrical healer element of diameter $D=10\text{mm}$ , temperature $T_1=1500\text{K}$ and emissivity ${\epsilon }_1=1$ is used in a furnace. The bottom area A₂is a diffuse, gray surface with ${\epsilon }_2=0.6$ and is maintained at $T_2=500\text{K}$ . The side and top walls are fabricated from an insulating, refractory brick that is diffuse and gray with $\epsilon =0.9$ . The length of the furnace normal to the page is very large compared to the width w and height h.

Neglecting convection and treating the furnace walls as isothermal, determine the power per unit length that must be provided to the healing element to maintain steady-state conditions. Calculate the temperature of the furnace wall.