Revisiting the optimal shape of cooling fins: A one-dimensional analytical study using optimality criteria

TL;DR

This study analytically investigates the optimal shape of cooling fins using a one-dimensional heat conduction model, deriving conditions for optimality and revealing that optimal fins balance conduction and convection at a Biot number of 1.

Contribution

It introduces an optimality condition approach for fin shape optimization and demonstrates the equivalence of different optimization criteria, highlighting the Biot number's significance.

Findings

Optimal fins have a Biot number of 1.

Stationarity of the Lagrangian yields the optimality condition.

Minimizing root temperature and maximizing heat transfer are equivalent.

Abstract

This paper revisits the optimal shape problem of a single cooling fin using a one-dimensional heat conduction equation with convection boundary conditions. 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 root temperature for a prescribed thermal power, rather than maximising the heat transfer rate for a constant root 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.