Chapter 12, Problem 12.95P

A procedure for measuring the thermal conductivity of solids at elevated temperatures involves placement of a sample at the bottom of a large furnace. The sample is of thickness L and is placed in a square container of width W on a side. The sides are well insulated. The walls of the cavity are maintained at ${\text{T}}_w$ , while the bot- tom surface of the sample is maintained at a much lower temperature $T_c$ by circulating coolant through the sample container. The sample surface is diffuse and gray with an emissivity ${\epsilon }_s$ . Its temperature $T_s$ is measured optically.

1. Neglecting convection effects, obtain an expression from which the sample thermal conductivity may be evaluated in terms of measured and known quantities ${\text{(T}}_w{T}_s{T}_c{\epsilon }_s,L)$ . The measurements are made under steady-state conditions. If $T_w=1400K,T_s=1000K,{\epsilon }_s=0.85,L=0.015m{\text{ and T}}_c=300K$, what is the sample thermal conductivity?
2. If $W=0.10m$ and the coolant is water with a flow rate of $m_c=0.1kg/s$ , is it reasonable to assume a uniform bottom surface temperature $T_c$?