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 Tw, while the bottom surface of the sample is maintained at a much lower temperature Te by circulating coolant through the sample container. The sample surface is diffuse and gray with an emissivity ??. Its temperature Ts is measured optically.

(a) Neglecting convection effects, obtain an expression from which the sample thermal conductivity may be evaluated in terms of measured and known quantities (Tw, Ts, Tc, ??, L). The measurements are made under steady-state conditions. If Tw = 1400 K, Ts = 1000 K, ?s = 0.85, L = 0.015 m, and Te = 300 K, what is the sample thermal conductivity?

(b) If W = 0.10 m and the coolant is water with a flow rate of mc = 0.1 kg/s, is it reasonable to assume a uniform bottom surface temperature Tc?

T- 1400 K Furnace T,= 1000 K, e, = 0.85 Sample, k T = 300 K Coolant, n