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We consider the heat transfer problem associated with a periodic array ofextended surfaces (fins) subjected to convection heat transfer with a uniformheat transfer coefficient. Our analysis differs from the classical approach as(i) we consider two-dimensional heat conduction and (ii) the base of the fin isincluded in the heat transfer process. The problem is modeled as an arbitrarytwo-dimensional channel whose upper surface is flat and isothermal, while thelower surface has a periodic array of extensions/fins which are subjected toheat convection with a uniform heat transfer coefficient. Using the generalizedSchwarz-Christoffel transformation the domain is mapped onto a straight channelwhere the heat conduction problem is solved using the boundary element method.The boundary element solution is subsequently used to pose a shape optimizationproblem, i.e. an inverse problem, where the objective function is thenormalized Shape Factor and the variables of the optimization are theparameters of the Schwarz-Christoffel transformation. Numerical optimizationsuggests that the optimum fin is infinitely thin and that there exists acritical Biot number that characterizes whether the addition of the fin wouldresult in an enhancement of heat transfer. The existence of a critical Biotnumber was investigated for the case of rectangular fins. {\bf It is concludedthat 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 thermalconductivity. This result is independent of both the thickness of the base andthe length of the fin.

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