Different from a fully developed continuous flow in an annular pipe, a liquid plug moving in an annular pipe generates two toroidal vortexes, which cause radial transport that enhances heat transfer. The fully developed heat transfer of the concentric plug is studied for three types of thermal wall conditions: inner-flux, outer-flux, and isothermal. The fully developed heat transfer of the continuous flow is analytically solved for the same thermal wall conditions. The comparison of heat transfer is made between the plug and the continuous flow. Two heat transfer mechanisms, boundary layer transport and radial transport, are considered to explain the heat transfer enhancement. If the radial transport is weak, the plug shows similar heat transfer performance to the continuous flow, which is dominated by the boundary layer transport. The heat transfer enhancement relies on the radial transport, which increases with decreasing the inner radius, decreasing the plug length, and increasing the Peclet number.