Porous Media Modeling of Microchannel Cooled Electronic Chips with Nonuniform Heating

Abstract

Microchannels are used for the cooling of electronic chips. However, the three-dimensional computational fluid dynamics modeling of the large number of channels in a full chip requires a huge number of meshes and computation time. Although porous media modeling of microchannels can significantly reduce the effort of simulation, most previous porous media models are based upon the assumption that the surface heat flux or temperature is uniform on the chip. In reality, the heat flux on the chip is usually highly nonuniform. In the present study, the porous media model considers the simultaneously developing entrance effect at the microchannel inlet and the thermally developing entrance effect due to the severe heat flux variation along the channel. Duhamel's integral is used to provide the Nusselt number distribution corresponding to the nonuniform heat flux distribution along the channel. The computing cost of this modeling method is only about 1% of the three-dimensional conjugate simulation. This porous media thermal modeling method is applied to model two full-scale electronic chips with realistic power distributions on the surfaces, and temperature maps are generated. The porous media thermal modeling offered by this study is an accurate and efficient alternative for modeling the electronic chips cooled by microchannels. Nomenclature a = wetted area per volume, m −1 c p = thermal capacity of the fluid, J∕kg · K D h = channel hydraulic diameter, m f = Fanning friction factor, τ w ∕1∕2ρV 2 H c = channel height, m H s = substrate height, m h = convective heat transfer coefficient, W∕m 2 · K K = permeability of the porous media, m 2 k 1 = thermal conductivity of solid substrate, W∕m · K k 2 = thermal conductivity of liquid coolant, W∕m · K L = channel length, m L = characteristic length scale, m L i = length of different periods in single-channel case, i 1; : : : ; 5, m Nu = Nusselt number, hL∕k 2 Po = Poiseuille number, fRe Pr = Prandtl number, c p μ∕k 2 q w = heat flux on top of the substrate, W∕m 2 q wi = heat flux of different periods in single-channel case, i 1; : : : ; 5, W∕m 2 Re = Reynolds number, ρVD h ∕μ s = fin width, m T = temperature,°C V = mean velocity of fluid flow, m∕s V D = filter velocity of porous media, m∕s W = width of modeling region, m W i = design parameters for split-flow-type design, i 1; : : : ; 3, m W inlet , W outlet = inlet and outlet width for split-flow-type design, m x, y, z = Cartesian coordinates, m Y i = design parameters for split-flow-type design, i 1; : : : ; 3, m y = dimensionless hydraulic axial distance, y∕LRe L y = dimensionless thermal axial distance, y∕LRe L Pr y th = dimensionless thermal entrance length, y th ∕LRe L Pr α = channel aspect ratio, δ∕H c < 1 Δp = pressure drop, Pa δ = channel width, m ε = porosity of the porous media κ = Hagenbach factor μ = dynamic viscosity of fluid, kg∕m · s ξ = integration variable in Duhamel integration ρ = density of fluid, kg∕m 3 Subscripts app = apparent DP = developing FD = fully developed i = local in = inlet s = solid substrate

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