Efficiency of composite fins of variable thickness

This paper discusses the thermal calculation of composite, metallic fins of variable thickness. In the simpler case of a constant-thickness (rectangular profile), the complete procedure involves first the analytical solution of the two-dimensional, two-material conduction problem, under the form of an infinite series of orthogonal eigenfunctions. Then the limit as Bi → 0 is sought, also analytically, which simplifies the series to its first term and permits to express the fin efficiency in closed form. This limit is equivalent to the usual 1D, Murray–Gardner, or thin-fin, approximation of ordinary, single-material fins, provided that an averaged thermal conductivity is used. For variable thickness (tapered profile), no analytical solutions have been found, so that resort should be made to numerical methods. Since the adoption of dimensional parameters is advisable in that context, the paper first reviews the industrial application of composite fins and selects a comprehensive set of material pairs of interest. Subsequently, two arbitrary but representative geometries and a reasonable range of dimensions and convection coefficients are fixed, thus assembling a rather exhaustive matrix of case-studies. Numerical calculations are compared to approximate results, in order to ascertain two facts: whether a single parameter exists (thermal length) that allows an accurate prediction of fin efficiency, and whether this parameter can be expressed in terms of an averaged thermal conductivity. Well within the bounds of usual engineering accuracy, the answer to both questions is affirmative. Therefore, calculation methods of ordinary fins and composite, constant-thickness fins are shown to be applicable to the most general case. Error bounds and specific recommendations for practical problems are also given.