Friction factors and nusselt numbers in microchannels with superhydrophobic walls

The thermal management of electronics is becoming an increasing concern as industry continues to simultaneously push performance while shrinking the size of electronic devices. Microchannel cooling is a promising technology to accommodate the heat dissipation rates and associated fluxes projected for future generations of electronics while also satisfying the need for a reduced footprint to accommodate ever-shrinking device sizes. One shortfall of microchannel cooling, however, is the large pressure drop associated with pumping liquids through microchannels, i.e., channels in which the smallest dimension is between about 1 micron and 1 mm. Superhydrophobic surfaces combine roughness features with low surface energy coatings to create materials with substantially decreased wettability and drag resistance in laminar flows and represent a promising technology for reducing the flow resistance of microchannels. The presence of an (insulating) air layer that is trapped within the superhydrophobic surface, and which separates the microchannel wall from the working fluid, gives rise to a low shear-stress region responsible for the observed reduction in flow resistance. There have been a limited number of studies on the fluid mechanics in superhydrophobic microchannels and, to our knowledge, heat transfer has not been examined. Quantifying the trade-off between the enhanced heat transfer due to pressure drop reduction versus the insulating characteristics of the air layer is of paramount importance for determining the viability of superhyfordrophobic surfaces as a technology for enhancing microchannel heat transfer. In this work we compute friction factors and Nusselt numbers for the fully-developed (with respect to energy and momentum) flow of a fluid in a parallel-plane microchannel with different heat flux and momentum boundary conditions at the upper and lower channel walls. Two approximations are taken for modeling the superhydrophobic microchannel. In the first case we study the single-phase flow of a fluid in a microchannel where one or both microchannel walls is assumed to be superhydrophobic and where the superhydrophobicity is modeled via application of Navier's slip model at the microchannel wall. Solutions for the velocity profiles are then employed to calculate theoretical friction factors and Nusselt numbers for the constant heat flux condition. This analysis is then extended to examine the implications on the thermal resistance of a superhydrophobic surface due to the presence of a purely conductive air layer. In the second case we model the fluid flow in the presence of a recirculating air layer that separates the fluid from the microchannel wall. In this instance the low-viscosity air layer gives rise to apparent fluid slip for the working fluid which is dependant on the thickness of the air layer and the viscosity ratio of the two working fluids. This case represents an upper apparent-slip limit as the characteristic spacing of the surface roughness becomes large relative to the channel height and air-layer thickness.