Question

A hollow aluminum sphere, with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.15 and 0.18 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250C250^{\circ} \mathrm{C}. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of 0.12 m. The system is in a room for which the air temperature is [/math]20^{\circ} C\mathrm{C}[/math] and the convection coefficient at the outer surface of the insulation is 30W/m2K30 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}. If 80W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation?

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Schematic of aluminum sphere and Equivalent thermal circuit as follows:

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