The use of relative inverse thermal admittance for the characterization and optimization of fin-wall assemblies

The concept of relative inverse thermal admittance applied to the convective fin-wall assembly optimization of longitudinal rectangular fins under 2-D heat conduction is presented in this work. Since heat transfer at the fin tip is taken into account, it is not always possible to optimize the above cited geometry. This is relevant in optimization processes and because of this has been displayed in several graphs. Here, different values for convective conditions at the fin and wall surfaces are used and the influence of the hw/hf ratio in optimum geometry is determined. The fin effectiveness is used as the fundamental parameter to prove that the fin is fulfilling the objective of increasing heat dissipation. Once the optimum thickness has been obtained, the Biot number is easily calculated and the fin effectiveness for an isolated fin and the fin-wall assembly can be determined graphically. The optimization process is carried out through a set of universal graphs in which the range of parameters covers most of the practical cases a designer will find. The concept of relative inverse thermal admittance is applied in a general form and emerges as an easy used tool for optimizing fin-wall assemblies.

Optimizing a Functionally Graded Metal-Matrix Heat Sink Through Growth of a Constructal Tree of Convective Fins

Optimal material utilization in metal-matrix heat sink is investigated using constructal design (CD) in combination with fin theory to develop a constructal tree of optimally shaped convective fins. The structure is developed through systematic growth of constructs, consisting initially of a single convective fin enveloped in a convective medium. Increasingly complex convective fin structures are created and optimized at each level of complexity to determine optimal fin shapes, aspect ratios, and fin-base thickness ratios. One result of the optimized structures is a functional grading of porosity. The porosity increases as a function of distance from the heated surface in a manner ranging from linear to a power function of distance with exponent of about 2. The degree of nonlinearity in this distribution varies depending on the volume of the heat sink and average packing density and approaches a parabolic shape for large volume. For small volume, porosity approaches a linear function of distance. Thus, a parabolic (or least-material) fin shape at each construct level would not necessarily be optimal. Significant improvements in total heat transfer, up to 55% for the cases considered in this work, were observed when the fin shape is part of the optimization in a constructal tree of convective fins. The results of this work will lead to better understanding of the role played by the porosity distribution in a metal-matrix heat sink and may be applied to the analysis, optimization, and design of more effective heat sinks for electronics cooling and related areas.

APPLICATION OF THE NEW SPECTRAL HOMOTOPY ANALYSIS METHOD (SHAM) IN THE NON-LINEAR HEAT CONDUCTION AND CONVECTIVE FIN PROBLEM WITH VARIABLE THERMAL CONDUCTIVITY

In this work, we demonstrate the efficiency of the newly developed spectral homotopy analysis method (SHAM) in solving non-linear heat transfer equations. We demonstrate the applicability of the method by solving the problem of steady conduction in a slab and the convective fin equation with variable thermal conductivity. New closed form explicit analytic solutions of the governing non-linear equations are obtained and compared with the SHAM results and numerical solutions. The results reveal that the new SHAM approach is very accurate and efficient and converges much faster than the standard homotopy analysis method.