Heat and mass transfer in a cylindrical heat pipe with a circular-capillary wick under small imposed temperature differences

Abstract

Heat pipes are efficient in transferring heat and have been applied in various thermal systems. Previous models of heat pipes use the heat rate through the pipe as an input parameter, and therefore lack predictive capabilities. Here, we demonstrate, using a simple heat pipe, that if the evaporation and condensing kinetics are properly modeled, then the heat rate is predicted. We consider a cylindrical heat pipe with the inner wall lined with a circular-capillary wick. The capillaries are filled with a partially wetting liquid, and the center of the pipe is filled with its vapor. Initially, the heat pipe is at temperature T0 and the system is under thermodynamic equilibrium. Then, one end of the pipe is heated to T0 + ΔT, while the other end cooled to T0 − ΔT, and the system reaches a steady state. The equilibrium vapor pressure at the hot end is higher than that at the cold end, and this pressure difference drives a vapor flow. As the vapor moves, the vapor pressure at the hot end drops below the equilibrium vapor pressure which induces continuous evaporation from circular pores on the wick surface. At the cold end, the vapor pressure exceeds the equilibrium vapor pressure so that the vapor condenses and releases the latent heat. The condensate moves back to the hot end through the capillaries in the wick to complete a cycle. We assume that the pore size is infinitesimal compared with the pipe dimensions. Thus, pore-level events can be treated separately from pipe-level events. The evaporation rate in each pore is solved in the limit the evaporation number E→∞, and an analytic leading-order solution is obtained, assuming ΔT/T0 ≪ 1. The evaporation rate is incorporated into vapor-flow and energy-balance equations along the pipe. Two dimensionless numbers emerge from these equations: the heat pipe number, H, which is the ratio of heat transfer by vapor flow to conductive heat transfer in the liquid and wall, and the evaporation exponent, S, which controls the evaporation gradient along the pipe. We find that vapor-flow heat transfer dominates in heat pipes and H ≫ S ≫ 1. Under these conditions, the non-dimensionalized heat rate through the insulated pipe is found to be simply S. Analytic solutions are also obtained for the pipe temperature and all the other variables. For maximum evaporative heat transfer, we find an optimal pipe length for fixed pipe cross-sectional dimensions, and an optimal wick thickness for a fixed pipe length. These optimal pipe length and wick thickness can help to improve the design of heat pipes and are found for the first time.