SUMMARYIn this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell-centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second-order accurate estimate of the gradient, along with a first-order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods.
A numerical study has been undertaken to explore the details of forced convection heat transfer in finned aluminum foam heat sinks. Calculations are made using a finite-volume computational fluid dynamics (CFD) code that solves for the flow and heat transfer in conjugate fluid/porous/solid domains. The results indicate that using unfinned blocks of porous aluminum results in low convective heat transfer due to the relatively low effective thermal conductivity of the porous aluminum. The addition of aluminum fins to the heat sink significantly enhances the heat transfer with only a moderate pressure drop penalty. The convective enhancement is maximized when thermal boundary layers between adjacent fins merge together and become nearly developed for much of the length of the heat sink. It is found that the heat transfer enhancement is due to increased heat entrainment into the aluminum foam by conduction. A model for the equivalent conductivity of the finned/foam heat sinks is developed using extended surface theory. This model is used to explain the heat transfer enhancement as an increase in equivalent conductivity of the device. The model is also shown to predict the heat transfer for various heat sink geometries based on a single CFD calculation to find the equivalent conductivity of the device. This model will find utility in characterizing heat sinks and in allowing for quick assessments of the effect of varying heat sink properties.
Experiments and computations are presented to quantify the convective heat transfer and the hydraulic loss that is obtained by forcing water through blocks of graphitic foam (GF) heated from one side. Experiments have been conducted in a small-scale water tunnel instrumented to measure the pressure drop and the temperature rise of water passing through the foam and the base temperature and heat flux into the foam block. The experimental data were then used to calibrate a thermal non-equilibrium finite-volume model to facilitate comparisons between GF and aluminum foam. Comparisons of the pressure drop indicate that both normal and compressed aluminum foams are significantly more permeable than GF. Results of the heat transfer indicate that the maximum possible heat dissipation from a given surface is reached using very thin layers of aluminum foam due to the inability of the foam to entrain heat into its internal structure. In contrast, graphitic foam is able to entrain heat deep into the foam structure due to its high extended surface efficiency and thus much more heat can be transferred from a given surface area. The higher extended surface efficiency is mainly due to the combination of moderate porosity and higher solid-phase conductivity.
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