Stirling cycle machines (both engines and coolers) are categorized as alpha, beta, or gamma configurations. Those, and how they work, are described below. The new delta configuration introduced by SI is described in a separate delta machine tab.
STIRLING MACHINE CATEGORIES
A quick Google search reveals many explanations of Stirling machine categories and how a Stirling engine works. Most are hobbyist-oriented websites, some of which offer reasonable explanations. There is a dearth of serious websites that delve into this subject, so we are providing a relatively rigorous description here for the benefit of those who seek a more complete understanding. Stirling cycle machines (both engines and coolers) are categorized as alpha, beta, or gamma configurations. Those, and how they work, are described below. The new delta configuration introduced by SI is described in a separate delta machine tab. The three traditional cycle types can further be classified as kinematic or free piston. Most Stirling machines from their invention in 1816 to the present are kinematic, where the strokes and phase angles of the pistons and/or displacers are dictated by mechanical linkages such as connecting rods and crankshafts. Free-piston Stirling (FPS) machines were conceived in 1962, and after a few decades of development are being used in a variety of specialized applications. Kinematic machines, with an oil lubricated crankcase, have fundamental life and reliability limitations and require periodic maintenance due to various issues, particularly oil vapors leaking past seals to contaminate heat exchangers. FPS machines that use gas bearings or flexure bearings with clearance seals require no lubrication. They can be hermetically sealed and operate for decades with no maintenance or degradation when properly designed. They are mechanically simple but dynamically complex, as also described separately.
HOW THE STIRLING CYCLE WORKS
Figure 1 is a schematic illustration of an ideal beta cycle Stirling engine that is useful for explaining the basics of Stirling cycle operation. This explanation is for a heat engine, and then the same schematic with a reverse sequence is also used to explain a Stirling refrigerator or cryocooler. A beta Stirling machine has a displacer piston and a power piston with the same diameters that operate within a single cylinder. The upper displacer piston is driven by a relatively thin drive rod that passes through the piston and piston rod. These two rods can be connected to a kinematic mechanism that imposes selected strokes and a typically 90-degree phase angle between the two motions, or they can terminate in an average pressure buffer region, with a linear alternator attached to the power piston, for FPS operation. The displacer motion is very lightly damped because its function is simply to shuttle the working gas back and forth through the heat exchanger circuit, shown as an appendage to the left of the displacer. The heat exchanger circuit consists of the upper heat acceptor or heater portion where heat is continually supplied at the maximum cycle temperature and a lower heat rejector or cooler section where waste cycle heat is continually rejected at the minimum cycle temperature to an external ambient temperature heat exchanger. The very critical regenerator heat exchanger, an insightful key component in Stirling’s original patent, is typically a very fine porous wire mesh that is located between the heater region and the cooler region, and where the cycle hot to cold temperature drop occurs. A similar temperature gradient occurs along the displacer wall and the displacer cylinder wall, which are as thin as practical to minimize parasitic heat losses. As the working gas is shuttled from the hot end to the cold end, its heat capacity is transferred to the regenerator. When the gas moves back from the cold end to the hot end later in the cycle, a well-designed regenerator will return over 99% of that heat to the working gas.
Figure 1 - Schematic representation of ideal beta Stirling cycle operation
Displacer reciprocation within its cylinder does not change the gas volume (except for a generally negligible swept volume of the displacer rod), but piston motion changes the working volume, which is defined as all the gas above the piston upper face. Numbered corner points on the ideal Stirling cycle pressure-volume (P-V) work diagram in Figure 1 correlate with the four piston and displacer positions illustrated. Position 1 starts with both the piston and displacer at top dead center, which corresponds to minimum gas volume above the piston and minimum gas in the hot end or expansion space. The piston is held fixed as the displacer moves down to position 2 so there is no volume change, but the pressure rises dramatically as the displacer shuttles gas from the cold compression space to the hot expansion space. The piston then executes an isothermal (constant temperature) expansion of the hot gas to deliver P-V work equal to the area under the expansion curve as it moves to position 3. The gas then cools as the displacer shuttles it from the hot space to the cold space while moving to position 4, this time with the piston held fixed at maximum cycle volume. The cycle is closed as the piston moves back to position 1 and compresses the cold gas at the low cycle temperature. Isothermal P-V work corresponding to the area under the curve from position 4 to position 1 has to be delivered to the piston in order to compress the working gas. The net cyclic work output is the difference between the hot expansion work and the cold compression work, which is the area enclosed by the 1-2-3-4-1 diagram. Some of the hot expansion work is stored in a kinematic flywheel or a FPS spring to enable the cold compression work to be returned to the cycle.
For a Stirling cooler, the cycle is traversed in the opposite direction of 1-4-3-2-1. External work from a motor compresses the gas from position 3 to position 2. This heats the gas, but that heat is rejected to ambient in the rejector to minimize gas temperature increase. The isothermal expansion from 1 to 4 absorbs heat from the expansion (blue) space region (Red) and reduces its temperature. The object to be cooled is thermally connected to that expansion space acceptor with insulation surrounding it. A well-designed Stirling cryocooler can cool from ambient to around 30 °K (-405 °F) in a single stage with a high fraction of Carnot efficiency relative to other cycles operating between given temperatures.
This ideal cycle description where the piston and displacer move intermittently and independently is not realized in practice. These components reciprocate continuously with near-sinusoidal motion and the piston motion lagging behind the displacer by typically about 90 degrees. The true work diagram is then a distorted oval within the ideal work diagram that just touches the ideal isothermal and constant volume phases as shown by the gray area. The net cycle work is then reduced to the area enclosed by the oval rather than the complete work diagram. It may sound confusing as stated earlier that heat is continually added at the acceptor and rejected at the rejector while those actions occur intermittently when executing the cycle. The statement is effectively accurate however since the cycle executes at a typical 30-60 Hz rate that is far faster than any perceptible change in the thermal inertia of the acceptor and rejector. Another critical element not fully addressed in the above discussion is the function of the regenerator. Typical regenerator fibers are on the order of 20 microns in diameter so there is a large surface area in the regenerator fibers, even with a typical porosity of 80-90%. As the working gas moves from the hot end to the cold end it deposits heat into the regenerator fibers with a very small temperature difference between the gas and fiber at any given axial position/temperature, only to similarly recover that heat a half-cycle later. The regenerator fibers have a far greater heat capacity than the gas passing through it so a well-designed regenerator will have an effectiveness of 99% or more, corresponding to the percentage of heat deposited that is recovered each cycle. Robert Stirling’s innovation of using a regenerator was every bit as critical as the concept of the machine itself.
Figure 2 illustrates the three traditional Stirling machine configurations with the component functions analogous to those illustrated in Figure 1. The operational description of a gamma Stirling machine is virtually identical to the beta description above. The basic functional difference is that the gamma piston and displacer are in separate cylinders with a gas connecting duct between the top of the piston and the bottom of the displacer. Advantages of the gamma configuration include a reduction in constraints as the piston and displacer can have independently optimized strokes and diameters, can be configured in a variety of ways such as dual opposed pistons for vibration balancing, and can avoid the complexity and precision of running the displacer rod through the piston and piston rod. Beta advantages include the potential for more compact machines and improved power density by avoiding the cycle dead volume associated with the connecting duct and with separate end clearances for both the piston and displacer.
Figure 2 - Schematic comparison of three basic Stirling configurations
Alpha Configuration
Beta Configuration
Gamma Configuration
Figure 3 - Four-cylinder DAA schematic configuration
The alpha Stirling configuration at top left in Figure 2 is fundamentally different from the beta and gamma versions in that it uses two power pistons and no displacer piston. In practice, the hot expansion piston includes an insulating hot cap and associated thin cylinder wall not shown in the schematic (but illustrated in the delta Stirling discussion below) that minimizes the heat leak down the cylinder wall and keeps the seal portion of the piston in the cold region. With respect to the Stirling cycle pressure-volume diagram in Figure 1, position 1 is represented by the hot piston at top dead center (TDC) and the cold piston at or near mid-stroke. The constant volume phase from 1 to 2 has the cold piston moving up to TDC while the hot piston moves down to mid-stroke to heat the gas by shuttling it from the cold region to the hot region. The hot isothermal expansion stroke from 2 to 3 has the hot piston moving down to bottom dead center (BDC) while the cold piston remains fixed at TDC. The phase 3-4 constant volume cooling occurs as the pistons again move in unison with the hot piston ending at TDC and the cold piston at BDC. The final isothermal compression from 4 to 1 occurs as the cold piston moves from BDC to mid-stroke and the hot piston remains at TDC. Again, in real machines the pistons actually move with near sinusoidal motion and a constant phase lag to produce a distorted oval P-V work diagram that is contained within the boundaries of the ideal cycle. The thermodynamic processes that take place between the two pistons of an alpha machine constitute one Stirling cycle, as do the entire beta and gamma machines. Alpha Stirling machines have a greater pressure swings and higher density than beta or gamma machines because they do not have the cycle dead volume associated with the displacer swept volume.
A special version of the alpha configuration is the double-acting alpha (DAA) shown in a 4-cylinder configuration in Figure 3. With the daisy chain connection of the hot end of one piston to the cold end of the next piston through a heat exchanger circuit, each piston functions as an expansion piston for one Stirling cycle and a compression piston for the adjacent cycle, resulting in four Stirling cycles in a single machine. Four-cylinder DAA concepts were patented by C. L. F. Franchot (Hargreaves, 1991) p. 21 and C. W. Siemens (Hargreaves, 1991) p. 43 in the 19th century but it was re-invented, or at least reduced to practice, by Rinia at Philips in the 1940’s. it is generally referred to as a Siemens or Rinia configuration. This type of engine offers many advantages and has been used for most modern high-capacity and high-performance kinematic Stirling engines. A DAA FPS machine was not generally considered to be feasible until (USA Patent No. 7,134,279 B2, 2006) and (White M. A., 2005). The latter described a feasibility demonstration using a modified WhisperGen™ kinematic engine in which the kinematic mechanism was replaced by four linear alternators connected directly to the four piston rods. That demonstration definitively proved the feasibility of DAA FPS machines. Subsequent DOE contracts with Infinia demonstrated both engine and cryocooler 3-cylinder prototypes that were fully successful in proving the innovative elements of the concept, but both had limited performance due to poor implementation of the heat exchanger manifolds and interconnecting duct flow patterns. The delta Stirling machines provide an optimized 3-cylinder DAA FPS topology that enables full realization of their benefits. They are described in more detail in the next subsection.
DELTA STIRLING MACHINES
Other than a few hypothetical or research engines, all previous DAA engines have been kinematic and used four cylinders with parallel piston axes arranged either in a straight line or a square pattern. The square pattern is almost universally used to avoid the asymmetry of one long connecting duct between the end cylinders. When Infinia developed their 3-cylinder DAA FPS engine and cryocooler, the parallel piston axes were located perpendicular to the vertices of an equilateral triangle. They used the conventional Infinia annular heat exchangers and a stepped piston with the central portion ported to one cycle and the annular portion ported to a different cycle. This resulted in a compact topology but led to significant flow path asymmetries and undesirable circumferential temperature gradients around the heat exchangers. These factors reduced efficiency and limited piston strokes due to thermal strain induced by the circumferential temperature gradients that impacted tight clearances in the concentric clearance seals. However, the critical innovative objective of achieving symmetric 3-cylinder operation with near-kinematic engine stability performed flawlessly and exceeded expectations. For example, the 9-kW engine maintained precise 120-degree phasing between the pistons all the way down to the 10-W output level as it coasted down to a minimal hot end temperature after shutting off the heat source.
The delta configuration is the culmination of many topological iterations that sought to produce more symmetric flow patterns with simple heat exchangers and straightforward manifolding between them. The result is that the three cylinder axes in a delta Stirling machine align with the sides of an equilateral triangle rather than being perpendicular to the vertices. As illustrated in Figure 4, a piston with a clearance seal is deployed on each end of each linear alternator. This piston/cylinder topology is essentially identical to the production-line approach used very successfully by Qnergy for over a thousand engines, the basic difference being the implementation of a piston on each end of the linear alternator instead of one end. While delta engines are described with three cylinders, strictly speaking any DAA FPS machine with pistons on each end of each motor/alternator with piston axes in one or two parallel planes is considered a delta machine. Sage analyses indicated that any number of cylinders from three to six can be designed to achieve similar performance levels. Three- and six-cylinder versions make the most sense because they use native 3-phase power and can be fully balanced. Six-cylinder versions are inherently balanced, as are two properly coupled 3-cylinder machines.
The cold heat exchanger/regenerator module is directly in line with the compression piston on one end of the alternator, with an expansion piston and hot cap on the other end. The hot heat exchanger (or heater) is deployed between the regenerator on one end of one alternator and the expansion piston with a hot cap on the other end of a second alternator. The thin wall hot cap with internal radiation shields reciprocates with a relatively large clearance between it and a thin cylinder wall. This approach is analogous to a displacer in a beta or gamma engine and is functionally identical to the hot cap used in all alpha engines. The function is to minimize heat leak between the hot end and cold end in a similar manner to the thin wall cylinder around the regenerator. The heater tubes that accept heat input and enable helium working fluid transfer back and forth between the hot cap associated with one linear alternator and the regenerator in the adjacent alternator are schematically illustrated. In this example, a shroud around the heater region contains sodium (or NaK) vapor heated by nuclear, solar or a combustion heat source that condenses on the heater tubes to provide the engine heat input. A first laboratory prototype would most likely be heated by electric cartridge heaters immersed in the Na or NaK pool to enable precise heat input measurements. In a variant, the heater tubes can be configured to interface directly with a combustor heat source.
Figure 4 - Cut-away drawing of delta Stirling engine conceptual design
Figure 5 - Perspective view of conceptual delta prototype lab generator with two delta engines integrated for full balancing and a sodium pool boiler with an electric heat source
A perspective view of a complete fully balanced delta Stirling generator is shown in Figure 5. The imbalance in a delta machine does not consist of either a typical linear oscillation or rotational oscillation about an axis. Instead, the entire unit traces a small circle of a magnitude that depends on the moving masses, their amplitude, and the entire system mass. It has been demonstrated that over 90% of this motion can be eliminated with suitable passive balancers. The imbalance is projected to be fully eliminated with two properly coupled delta machines as illustrated in the 3-D perspective view of Figure 5. Not shown are a coolant pump and fan-cooled radiator to reject cycle waste heat to the environment. To balance this generator, the upper delta engine is turned “upside down”relative to the lower engine and the heaters are aligned for convenience in connecting the heat source vapor space ducting. The upper and lower engines are rigidly coupled together and allowance is made to accommodate thermal expansion and contraction that occurs during heatup and cooldown.