5

Forces on displacer

-Line ol"displacement

Forces on displacer

Line of displacement L

Fig. 11.6. Dynamics of the Harwell ftee-piston electric generator.

Fig. 11.6. Dynamics of the Harwell ftee-piston electric generator.

A combination of spring linkage and differential area can provide a strong displacer drive capable of operating the engine at higher frequency with more massive displacéis, Fig. 11.8 shows an arrangement used on larger Sunpowcr designs, in which the displacer is sprung to the cylinder in order to resonate the displacer mass, and a differential area is used to provide a force overcoming displacer damping. The sum of damping and force on the displacer differential area brings the force diagram on the line of displacement, and a sprint», of sufficient strength to provide the necessary resultant force for resonance is then added. Similarly, a spring Ky of sufficient stiffness to resonate the piston is added to the spring component of the working space pressure wave in order to permit piston resonance.

For purposes of simplicity, the slight additional effects of the moving cylinder have not been shown. If it is desired to do so, these effects may be added in the manner shown previously in Fig. 1 1.6.

A drive method intermediate between Figs. 11.7 and 11.8 which is sometimes useful is shown in Fig. 11.9. Here a spring K2 between piston and displacer assists to resonate the displacer. The fact that neither the pressure nor the spring force are co-linear with displacer displacement results in rather large work flows from the displacer to the piston. In practice this results in hysteresis losses in the gas spring K. and reduced system thermal efficiency. Engines made in this way have large rod

Pressure nnil (Jisplnccmcnl vcctors

Pressure nnil (Jisplnccmcnl vcctors

Displacer forces

Line of displacement

Displacer forces

FPiAR-Ac)

FPiAR-Ac)

Line of displacement Fr

Fig. 1 I.v. Dynamics of an engine with a common spring in bounce space.

Fig. 1 I.v. Dynamics of an engine with a common spring in bounce space.

diameters compared to the spring to cylinder models of Fig. 11.8 if operated at the same conditions.

Double-acting free-piston engines

The force diagrams of Fig. 11.10 show the possibility of double-acting free-piston engines. A three piston engine is illustrated but larger numbers of pistons are possible, as well as different connections between cylinders, as a study of the figure will immediately suggest. The three piston double-acting machine was originally suggested by Walkert as a candidate for three-phase power generation.

As the figure shows, there must be a large phase angle between the piston and its associated hot space pressure or the force diagram will not result in positive work. In fact, for the three piston engine, this phase t private communication

Position and pressure vectors

of displacement

Displucer forces

Position and pressure vectors

Displucer forces of displacement

Piston forces

Fio. 1 1.8. Dynamics of an engine in which the displacer is sprung to the cylinder.

angle must exceed 30° or the resultant of the two pressure forces will lie on or below the line of displacement and hence have a negative work component.

This situation may be avoided by making the cold piston area smaller than the hot piston area, as would naturally be the case with a power rod leaving the cold space. With the rod area subtracted, /■",.„ is a smaller force and the force diagram may have net positive work with less pressure phase lag.

Constraints such as the one cited above, resulting from the necessary symmetry of double-acting piston arrangements, make them less flexible and less responsive to optimization procedures than piston-displacer machines. The merit of the double-acting free-piston is the same as its kinematic twin—higher power density and fewer moving components.

Variations of geometrical arrangements

Besides the arrangements already discussed, there seems to exist an almost endless array of variations. For example, in Fig. 11.11 are shown a

FREE-PISTON STIRLING ENGINES CXAXP%

Displacement and pressure vectors

Displacement and pressure vectors

Displacer forces

Line of displacement

Fisión forces

Fisión forces

I.ine of displacement

Fit». 11.'.). Dynamics of an engine in which the displacer is sprung to the piston.

I.ine of displacement

Fit». 11.'.). Dynamics of an engine in which the displacer is sprung to the piston.

number of displacer geometries, each completely equivalent to the others. This is by no means an exhaustive list and the reader may doubtless add more after brief reflection. It is the task of the designer to weigh the merits of many possible configurations for his particular applications, taking into account the realities of mechanical design thermal distortion, leakage, wear, gas spring losses, centering, control, power modulation, fouling with wear particles, cost, ease of manufacture, alignment, operating stability, startup characteristics, and so on.

A host of multi-cylinder arrangements are also possible: heat pumps, cooling engines, heat-driven cooling engines and other combinations. Again the designer must choose with care. The authoi prefers to slick to the rule cited by Professor Egon Orowon—'Never try something complex until you have failed with something simple'. In this case, if one piston will do, should one be tempted to try two or live?

qtl Qt±j

Position and jirc.ssurc vectors

Position and jirc.ssurc vectors

Forces on piston 1 Case for zero work. AE=AC, 30° Phase lap of pressure

Case for 3(T phase lag, ar>ac, work produced C, (work force) -

Fig. 11.10. Dynamics of a double-acting machine with three pistons.

Other useful arrangements, and other methods of dynamic analysis are given by Benson (1977b), Martini (1975a), Martini ct al. (1977). and Ranch (1975).

computp.r simulation op free-piston dynamics

Figs. 11.12 and 11.13 are a much simplified analogue computer representation of the dynamic model of Fig. I 1.5. If the chief aim is a study of dynamics rather than thermodynamics, the analogue computer is a marvellously easy and, thanks to modern electronics, cheap way to study free-piston engine dynamics. It is a quite simple matter to construct such a special purpose analogue from readily-available components and with il to study the free-piston engine. By manipulating potentiometers representing the dynamic components, the student may quickly develop a feel for the machine and he may be able to get good values for the required springs, areas, and loads necessary to achieve desired operation.

Fio. 11.11. Seven equivalent displacer gconicUies.

With the acklilion oí non-linear elements and empirically derived correction factors in the analogue, the modelling may be made an excellent representation of a real machine, including accurate pressure drop values and component collision phenomena.

The digital computer may. of course, be used to do precisely the same thing as an analogue with greater versatility, but in most cases, less opportunity for the operator to observe instantly the results of his changes. The IBM Continuous System Modelling Program (CSMP) is a particularly useful tool for digital simulation of analogue circuits, quickly learned by those familiar with analogue computers or problems in vibration (Beale 1969).

Hir-FtMOOYNAMIC AND DYNAMIC OPTIMIZATION

'l"he previous discussion was aimed only at an illustration of free-piston engine dynamics. The complete design task must start with a

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