Steady heat pipe solutions with capillarity and conduction effects included are considered. A single layer model is studied as a singular perturbation problem, with capillary boundary layers in the temperature-saturation phase plane leading to new insights and results. in the geothermal context, with bottom heating and heat flow that is dominated by convection, phase plane trajectories of temperature vs saturation typically track zero capillarity (gravity-driven) solutions when they exist, one liquid-dominated and one vapor-dominated. The way in which choice of boundary conditions selects from these solutions is studied. Capillary boundary layers, not encountered in previous studies, play an important role in this selection process. Maximum possible lengths of heat pipes are calculated in cases where it has previously been speculated that the lengths may be unbounded.