Revisiting the optimal thickness profile of cooling fins: A one-dimensional analytical study using optimality conditions

This paper revisits the optimal thickness profile problem of a single cooling fin using a one-dimensional heat conduction equation with a convection boundary condition. Firstly, in contrast to previous works, we apply an approach using optimality conditions based on requiring stationarity of the Lagrangian functional of the optimisation problem. This yields an optimality condition basis for the commonly touted constant temperature gradient condition. Secondly, we seek to minimise the base temperature for a prescribed thermal power, rather than maximising the heat transfer rate for a constant base temperature as previous works. The optimal solution is shown to be fully equivalent for the two, which may seem obvious but to our knowledge has not been shown directly before. Lastly, it is shown that optimal cooling fins have a Biot number of 1, exhibiting perfect balance between conductive and convective resistances.