Critical thickness of an optimum extended surface characterized by uniform heat transfer coefficient

TL;DR

This study analyzes the heat transfer in periodic fin arrays considering two-dimensional conduction and fin base effects, identifying a critical Biot number and optimal fin thickness for enhanced heat transfer.

Contribution

It introduces a novel approach using boundary element and shape optimization methods to determine the critical fin thickness and Biot number for optimal heat transfer performance.

Findings

Optimal fin is infinitely thin.

Existence of a critical Biot number for heat transfer enhancement.

Rectangular fin effective if thickness < 1.64 k/h.

Abstract

We consider the heat transfer problem associated with a periodic array of extended surfaces (fins) subjected to convection heat transfer with a uniform heat transfer coefficient. Our analysis differs from the classical approach as (i) we consider two-dimensional heat conduction and (ii) the base of the fin is included in the heat transfer process. The problem is modeled as an arbitrary two-dimensional channel whose upper surface is flat and isothermal, while the lower surface has a periodic array of extensions/fins which are subjected to heat convection with a uniform heat transfer coefficient. Using the generalized Schwarz-Christoffel transformation the domain is mapped onto a straight channel where the heat conduction problem is solved using the boundary element method. The boundary element solution is subsequently used to pose a shape optimization problem, i.e. an inverse problem, where the objective function is the normalized Shape Factor and the variables of the optimization are the parameters of the Schwarz-Christoffel transformation. Numerical optimization suggests that the optimum fin is infinitely thin and that there exists a critical Biot number that characterizes whether the addition of the fin would result in an enhancement of heat transfer. The existence of a critical Biot number was investigated for the case of rectangular fins. {\bf It is concluded that a rectangular fin is effective if its thickness is less than} $1.64 k/h$, where the $h$ is the heat transfer coefficient and $k$ is the thermal conductivity. This result is independent of both the thickness of the base and the length of the fin.