Heat and mass transfer in polygonal micro heat pipes under small imposed temperature differences

Abstract

Micro heat pipes have been used to cool microelectronic devices, but their heat transfer coefficients are low compared with those of conventional heat pipes. We model heat and mass transfer in triangular, square, hexagonal, and rectangular micro heat pipes under small imposed temperature differences. A micro heat pipe is a closed microchannel filled with a wetting liquid and a long vapor bubble. When a temperature difference is applied across a micro heat pipe, the equilibrium vapor pressure at the hot end is higher than that at the cold end, and the difference drives a vapor flow. As the vapor moves, the vapor pressure at the hot end drops below the saturation pressure. This pressure drop induces continuous evaporation from the interface. We solve for the evaporation rate in the limit the evaporation number E→∞, and find that the liquid evaporates mainly in a boundary layer at the contact line. An analytic solution is obtained for the leading-order evaporation rate. Since the pipe is slender and the imposed temperature difference small, the heat and mass transfer along the pipe is skew-symmetric about the mid point of the pipe. Hence, we only need to focus on the heated half of the pipe. Furthermore, because the pipe and bubble are long, the coupled vapor and liquid flows along the pipe are predominantly uni-directional, and the heat transfer by vapor flow and by conduction in the liquid and wall are essentially one-dimensional. Thus, we find analytic solutions for the temperature profile and vapor and liquid pressure distributions along the pipe. Two dimensionless numbers emerge from the momentum and energy equations: the heat-pipe number, H, which is the ratio of heat transfer by vapor flow to conductive heat transfer in the liquid and pipe wall, and the evaporation exponent S, which controls the evaporation gradient along the pipe. In the limit H→0 or S→0, conduction in the liquid and wall dominates. When H→∞ and S→∞, vapor-flow heat transfer dominates and a thermal boundary layer appears at the hot end, the thickness of which scales as S-1L, where L is the half-length of the pipe. A similar boundary layer exists at the cold end. Outside the boundary layers, the temperature is uniform. These regions correspond to the evaporating, adiabatic, and condensing regions commonly observed in conventional heat pipes and are absent in most micro heat pipes leading to their low heat transfer coefficients. We also find a dimensionless optimal pipe length Sm=SmH for maximum evaporative heat transfer. Thus, our model suggests that micro heat pipes should be designed with H≫1 and S=Sm. We calculate H and S for four published micro-heat-pipe experiments, and find encouraging support for our design criterion.