Graphic courtesy of Dr. Israel Urieli of Ohio University.

In order to construct a Stirling Engine, we really need to understand the basic functioning of the gas circuit (heater, cooler and regenerator). But the Gas Circuit is wrapped in a cloud of virtually impenetrable fact and fiction as evidenced by the noise level of the Stirling engine discussion sites.   From the early Schmidt equation which somehow ultimately  indicates that the optimum heat exchanger volume is zero (no heat exchanger at all) to the concepts of Scaling and Similarity which shows that the scaled configuration may contain hundreds and sometimes even thousands of tubes, the optimum gas circuit is very difficult to pinpoint empirically or mathmatically.

As there are no valves directing the flow of the working fluid and the mechanics of the piston motion are reasonably simple,it is amazing that the actual computation of the thermodynamics of the working fluid in this process is so very complicated.  Many of the components are complex approximations made on a very non intuitive basis.  That is to say that unlike the internal combustion engine, the Stirling engine can not be designed at one size by a formula and scaled to the desired size.(At least not without adding the concepts of Similarity to the mix).   The design of a real world Stirling engine must begin with an estimate and end with an approximation.  Even though we can design precisely the exact engine we wish to build only with great dificulty, we can make certain reasonable estimates of the gross performance we wish to achieve and then through scaling and similarity principles turn this into a design which has a good chance at succeeding.  The success or failure of the final hardware will depend upon the care and consideration taken to avoid three classes of defects in the engine.
1.) Friction in all its forms. 2.) Thermal Shorts in all guises.  3.) Inadequate heat exchanger performance  This is in addition to the care paid to the creation of the gas circuit itself.

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