Final answer:
The efficiency of a fin with constant cross-sectional area A is derived by balancing the conductive heat transfer within the fin and the convective heat transfer from the fin to its surroundings.
Step-by-step explanation:
The efficiency of a fin with constant cross-sectional area A can be derived by creating a balance between the heat transfer through conduction within the fin and the heat transfer by convection from the fin to the surrounding environment. The rate at which heat is conducted through the fin can be modeled using Fourier's law, and it is proportional to the temperature gradient along the fin. Conversely, the rate at which heat is convected from the fin into the ambient air is given by Newton's law of cooling and is proportional to the temperature difference between the fin and the surrounding air.
For a fin with a constant cross-sectional area, the thermal efficiency is defined as the ratio of the actual heat transfer from the fin to the maximum possible heat transfer if the entire fin were at the base temperature. This involves solving the differential equation that arises from equating the conduction heat transfer rate to the convection heat transfer rate, subject to the appropriate boundary conditions. The solution to this differential equation provides the temperature distribution, which can then be integrated to find the total heat transfer and, subsequently, the efficiency of the fin.