The Geometry of LTD Sterling Engines

What makes a LTD Sterling engine different from a conventional flame-heated or high temperature differential engine is simply the size and shape of the displacer unit. LTD engines favor a short displacer with a large area and having a short stroke. The reasons for this come from the basic physics underlying all heat engines.

There is a definite relationship between the compression ratio of an engine and the temperatures between which it can operate. Although experiments and trial and error hinted at the qualitative character of this connection, the exact and explicit mathematical relationship was not discovered until 1985. This relationship dictates that engines operating at low temperature differentials must have low compression ratios. It is an inescapable consequence of the laws of physics governing the wonderful world we were given. Thus we know for sure now that we are not making LTD engines with low compression ratios just because the first one happened to be made that way and it worked. The fact is that it simply cannot be done any other way.

Now given that the compression ratio must be low if the temperature differential is low, this implies that the volume swept out by the displacer must be relatively large compared to that of the piston for a LTD Stirling engine. For a LTD Stirling engine to run satisfactory on the heat of a hand, its displacer must sweep out a volume about 50 times that of the piston. The displacer swept volume equals the displacer stroke times its area, and the area is proportional to the square of the diameter. Hence the best way to get a large volume out of reasonable size is with a large diameter and short stroke.

The additional role of the displacer in any Stirling engine is as an insulator between the hot and cold sides of the engine. The displacer must thermally isolate the hot air section from the cold section. Since heat conduction through the displacer represents a loss efficiency of the engine.

The rate of heat conduction through a solid is directly proportional to the temperature difference at each end and inversely proportional to its length. Therefore a LTD engine does not require a Long displacer to keep conduction losses down to a reasonable level. Moreover, Styrofoam is an excellent material for making displacers in LTD engines and a very short displacer made from this material will suffice. Hence short displacers are favored for LTD engines. This is also consistent with a short stroke.

The physics of heat transfer also favors a large diameter Displacer for LTD engines, in fact, the larger, the better.  Consider an engine with a displacer chamber that is large enough to allow the engine to run.   This means the four step cycle that was described above is being repeatedly carried out within the engine. This includes the heat flow from the hot plate to the engine air, and then later in the cycle from the engine air to the cold plate.  The rate of heat transfer between the surface of the plate and the engine air next to it is directly proportional to the area of the plate and to the temperature difference between the plate and the air.   If the plate were to be made larger in diarneter, the active area would be greater so the rate of heat transfer would increase. This would allow the engine to run faster than before. Therefore, from the point of view of heat transfer, the larger the plate diameter, the better. But there is an important practical consideration that prevents us from making really huge round displacers with ultra tiny strokes.   This is the difficulty of keeping a large diameter displacer flat enough and squares enough to its rod to permit the displacer to come up close against the plates.   The larger the displacer, the closer it must approach the plates to limit the dead space.

From the above discussion, it should be clear now why LTD engines have the shape they do, and more generally, how the geometry of any Stirling engine matches the temperature differential; that it can best work between.