A hollow sphere with inner and outer radii of R1 and R2, respectively, is covered with a layer of insulation having an outer radius of R3. Derive an expression for the rate of heat transfer through the insulated sphere in terms of the radii, the thermal conductivities, the heat transfer coefficients, and the temperatures of the interior and the surrounding medium of the sphere
GIVEN
• An insulated hollow sphere
• Radii
? Inner surface of the sphere = R1
? Outer surface of the sphere = R2
? Outer surface of the insulation = R3 FIND
• Expression for the rate of heat transfer
ASSUMPTIONS
• Steady state heat transfer
• Conduction in the radial direction only
• Constant thermal conductivities
SKETCH
Let
The thermal circuit for the sphere is shown below
The individual resistances are
The rate of heat transfer is
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