A hollow sphere of inner radius R1 and outer radius R2 respectively is covered with a layer of insulator having an outer radius R3. Obtain an expression for the rate of heat transfer through the isolated sphere, in terms of the radii, the thermal conductivities, the heat transfer coefficients, and the temperatures of the interior and medium surrounding the sphere.

Also determine an expression for the critical radius of isolation, in terms of the thermal conductivities of the insulation and the coefficient of the insulating outer surface and the fluid surrounding it.