A thermodynamic apparatus, such as a Stirling engine or a Vuilleumier heat pump, has a heat exchanger in which energy is exchanged between a working fluid and an exhaust gas stream. On top of the cylinder of the thermodynamic apparatus is a dome-shaped section. By incorporating the heat exchanger within the dome, the flow paths can be simplified, the number of separate components reduced, and overall weight reduced. Flow passages for the working fluid are embedded in the dome. Channels for the exhaust gases are formed in an outer surface. The passages and the channels are helically arranged, one clockwise and one counter clockwise. The dome can be cast with a core for the casting fabricated via three-dimensional printing. In some embodiments, the dome is made of fiber-reinforced material.