(Solution) - A metal rod of length 2L diameter D and thermal conductivity -(2025 Original AI-Free Solution)

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A metal rod of length 2L, diameter D, and thermal conductivity k is inserted into a perfectly insulating wall, exposing one-half of its length to an air stream that is of temperature T? and provides a convection coefficient h at the surface of the rod. An electromagnetic field induces volumetric energy generation at a uniform rate q within the embedded portion of the rod.




(a) Derive an expression for the steady-state temperature Tb at the base of the exposed half of the rod.
The exposed region may be approximated as a very long fin. (b) Derive an expression for the steady-state temperature To at the end of the embedded half of the rod.
(c) Using numerical values provided in the schematic, plot the temperature distribution in the rod and describe key features of the distribution. Does the rod behave as a very long fin?