The mathematical formulation of a new device of counterflow heaters or coolers in concentric circular tubes with uniform wall temperature and external refluxes is developed and the analytical solution has been investigated with the use of an orthogonal expansion technique by expanding eigenfunctions in terms of power series. Comparisons of enhancement in heat-transfer efficiency are made with single-pass operations of the same size (without an inner tube inserted). Considerable improvement in the heat-transfer efficiency for large Graetz numbers is obtainable by introducing the recycle-effect concept in designing such double-pass operations. Analytical results show that suitable adjustment of impermeable-sheet location could have a substantially improving the heat-transfer efficiency. The increment of power consumption has been also discussed.