A Two-Temperature Model for the Analysis of Passive Thermal Control Systems

Passive control of steady and unsteady thermal loads using effective thermal conductivity enhancers, such as metal foams, internal fins and metal filler particles, is being explored for a variety of electronics applications. The interstices are filled with air, phase change materials, or other fluids. Local thermal equilibrium between the solid filler and the matrix is not ensured in such systems since their thermal diffusivities are frequently very different. The use of a single volume-averaged energy equation for both the phases cannot be justified in such situations. A two-medium approach is used in the present work to account for the local thermal non-equilibrium. Separate energy equations are written for the solid and fluid respectively, and are closed using a steady-state interphase heat transfer coefficient between the two phases. A general momentum equation which includes the Brinkman-Forchheimer extension to Darcy flow is employed. The resulting equations are solved implicitly using a fully transient method on fixed orthogonal co-located finite volumes. Unsteady natural convection in a metal-foam filled cavity is computed. The influence of various parameters such as the ratios of solid-to-fluid thermal conductivities and heat capacities, Rayleigh number, Prandtl number and Darcy number on the thermal and flow fields is investigated. The results illustrate that local thermal equilibrium is not assured, either during the transient or at steady state for the range of parameters considered. Furthermore, even if the steady-state solid-to-fluid temperature differences are small, large temperature differences are seen during the unsteady response.