CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application claims priority benefit from U.S. Provisional Patent Application No. 62/210,367, entitled “WORK MECHANISMS FOR DIRECTLY-COUPLED MICRO COMPRESSOR-EXPANDER THERMAL UNITS,” filed Aug. 26, 2015; which, to the extent not inconsistent with the disclosure herein, is incorporated by reference. 
    
    
     BACKGROUND 
     Refrigeration and liquefaction cycles with gas as the working fluid and sometimes also the process gas have been known since about 1900 and are well described in the technical literature. Essentially all of the these cycles operate on the principle of compressing a working gas, transferring the heat of compression to a heat sink, cooling the gas in a recuperative or regenerative heat exchanger, further cooling of the gas via either isenthalpic or isentropic expansion, transferring a thermal load into the working gas from a heat source, warming the lower pressure gas back to near the temperature of the compressor, and repeating the cycle. In cycles such as the Linde cycle, the cooled high-pressure gas is expanded isenthalphically in a Joule-Thomson valve with no work recovery. Cycles with no work recovery generally have low thermodynamic efficiency relative to the minimum work required to pump heat from a colder source to a warmer heat sink. The primary reason for such low efficiency is a fundamental limitation of poor heat transfer during rapid compression of a gas; rather than being isothermal, the process is adiabatic or nearly so via polytropic compression. This inefficiency causes significantly more work input per unit mass flow than the ideal isothermal process. Without recovery of any of this work input during a refrigeration cycle, the ratio of the cooling power to the rate of work input is much lower than the ideal ratio, i.e., low relative thermodynamic efficiency (e.g., a few percent out of 100%). 
     To improve refrigerator efficiency, gas expanders were invented whereby precooled high-pressure working gas is expanded isentropically from higher pressure to lower pressure with corresponding work production plus larger cooling effect. In refrigeration cycles that recover work of expansion to offset some input work of compression, the thermodynamic efficiency increases. Tagauchi et al. in U.S. Pat. No. 5,737,924 and Saho et al. in U.S. Pat. No. 5,152,147 describe use of regeneration to help recover some of the thermal energy of expansion of a portion of the working gas stream. Kolbinger describes an assembly of two rotary engines to form a compressor-expander with no discussion of recovery of work in U.S. Pat. No. 5,309,716. An electromagnetic apparatus to produce linear motion in a macro-structure device is described by Denne in U.S. Pat. No. 6,462,439, and a micro electro-mechanical system for providing cooling with compression and expansion spaces separated by a regenerator in a Stirling cycle without direct work recovery is described by Tsai et al. in U.S. Pat. No. 6,272,866. An array of refrigeration elements is disclosed by Reid et al., in U.S. Pat. No. 6,332,323. The refrigeration elements are combined to form a highly efficient active gas regenerative refrigerator. Refrigeration elements configured into an appropriate array of dual opposing thermal regenerators in an active regenerative refrigerator simultaneously enable the feature to alternatively provide active heating or cooling to reciprocating heat transfer fluid that flows over the outside surfaces of the refrigeration elements. The active heating or cooling in the opposite ends of small hermetic refrigeration elements can be caused by driving a sealed piston back and forth in each refrigeration element. The drive mechanisms contemplated in the &#39;323 patent are by electromagnetic, pneumatic, or other means but few details are given. The array of refrigeration elements is configured to enable reciprocating heat transfer fluid motion, as in conventional passive regenerators in regenerative cycle refrigerators such as the Stirling, Gifford McMahon, or pulse-tube cryocoolers, but in active regenerative refrigerator, the heat transfer fluid is separate from the working fluid, and the heat transfer fluid is not compressed or expanded during its cycle, other than as required for flow through the refrigeration element array and external heat exchanger. 
     A small proof-of-concept active gas regenerative refrigerator was successfully built and initially tested with the support of a NASA Phase I small business innovation research SBIR award (J. A. Barclay, M. A. Barclay, W. Jakobsen, and M. P. Skrzypkowski, NASA SBIR Phase I Final Report, 2004; “Active Gas Regenerative Liquefier”; Contract No. NNJO4JC25C). Approximately 200 identical small stainless steel tubes were assembled into a rectangular array of tubes, each with a micro-regenerator and a common pressure wave means for all tubes in parallel. Initial results from the first lab prototype proved the active end of the tubes did heat and cool upon compression or expansion, respectively, and that the active gas regenerative concept was valid. 
     SUMMARY 
     Embodiments relate to methods and apparatuses for work input with simultaneous work recovery in a refrigeration cycle by nearly isothermal polytropic compression and synchronous nearly isothermal polytropic expansion of a working gas. Embodiments of the invention relate to a basic thermal unit of an efficient refrigerator and more particularly to active gas regenerative refrigerators utilizing an array of directly coupled micro compressor-expander units (MCEUs) with electromagnetic or pneumatic mechanisms for producing linear reciprocating motion of a piston to cause simultaneous heating or cooling by compression and expansion of a working gas within the basic thermal unit. Embodiments generally relate to fabrication of apparatuses and methods to enable work input into each micro gas compressor region coupled with simultaneous work recovery from the micro gas expander region. The combined effect of a high-performance regenerator array of micro compressor-expander units creates an efficient active gas regenerative refrigeration cycle for transferring heat from a colder thermal source to a hotter thermal sink for numerous refrigeration applications including liquefying natural gas, hydrogen, helium or other gases. 
     Various embodiments provide work recovery of compression of an equal amount of working gas on one end of a MCEU tube by a common drive piston by simultaneous expansion of an equal amount of working gas on the opposite end of the common drive piston. The net driving force to move the piston alternatively inside the MCEU tube is provided by arrangements of permanent magnets and drive coils, in one embodiment of the invention. 
     According to an embodiment, the length of thermally active sections at each end of a MCEU remains constant by using radial compression and expansion of a helium (He) working gas. This overcomes limitations of previous designs that used bellows or axial movement of the working gas with changes in the geometry of thermally active regions of the MCEU during its operation. 
     According to an embodiment, radial motion of helium gas keeps a mass of He working gas constant in each thermally active section during the MCEU cycle. This overcomes one of the disadvantages of the NASA SBIR proof-of-principle prototype referenced above, of having different thermal mass in the thermally active sections at opposite ends of a MCEU by moving more or less working helium gas into or out of each MCEU during compression and expansion steps, respectively. 
     According to an embodiment, the Biot number of a He working gas and tube walls of a MCEU (e.g. 0.125″ outer diameter Al alloy 2024 T6 tubes with 0.003″ wall) is ˜10 −3 , so tube walls in thermally active sections of the MCEU change temperature almost synchronously with the He working gas during a nominal 1 Hz cycle. The tube walls become part of the active thermal mass of each MCEU during an active gas regenerative refrigeration cycle. 
     According to an embodiment, a drive piston of a MCEU has two or more sets of small opposing Nd 2 Fe 14 B magnets that create two or more concentrated transverse magnetic flux regions perpendicular to the axis of a center section of the MCEU tube. The MCEU also includes a thin, electrically-energizable coil around the outside of the center section of the MCEU. This arrangement significantly increases the Lorenz force on the drive piston from a magnetic field generated by the coil. 
     According to an embodiment, a piston of a MCEU has two or more sets of small opposing Nd 2 Fe 14 B magnets that create two or more concentrated transverse magnetic flux regions perpendicular to the axis of a center section of the MCEU tube. The MCEU also includes a thin, annular, cylindrically-shaped permanent magnet array which is closely fitted with low-friction seals inside a hermetic tubular enclosure around the center section of the MCEU. This annular permanent magnet array is pneumatically driven back and forth by pressurized gases such as N 2  or H e  alternatively supplied to drive chambers defined in part by the tubular enclosure, via small tubes from a separate gas-supply subsystem. The transverse flux of the permanent magnets within the drive piston couples strongly with the cylindrically-shaped permanent magnet array. The strong magnetic flux coupling between the opposing magnets in the annular drive array and the magnets of the drive piston cause the drive piston to reciprocally move with the annular permanent magnet array, which simultaneously compresses and expands the working gas at respective ends of the piston during MCEU operation. 
     According to an embodiment, a hoop stress of thin-walled tubes of a MCEU array during maximum compression of a He working gas is only about ½ of the yield strength of MCEU tube materials such as Al 2024-T6. This enables good dimensional stability and good sealing in the MCEU. 
     According to an embodiment, a MCEU design enables work recovery from expansion of working gas at one end of the MCEU to offset work input to compress the working gas on an opposite end of the MCEU. 
     According to an embodiment, a magnetic drive is provided, including a hermetic pneumatic shell containing thin, cylindrical annular permanent magnets around the outer shell wall of a center section of a MCEU tube. The tube contains two or more sets of opposing permanent magnets in an axially moveable compressor/expander piston assembly within the MCEU, which increases the transverse magnetic flux and thereby increases the magnetic coupling between the permanent magnets in the piston and those in the pneumatic drive. 
     According to an embodiment, the work required for a cycle of a MCEU array is distributed over a wide range of temperatures near the operating temperature of each MCEU of the array, rather than input in a lumped fashion as through a compressor in most conventional gas cycle refrigerators and liquefiers. 
     According to an embodiment, electronic control of each MCEU of an array is provided, so the performance of an overall active regenerator that includes the array of MCEUs can be fine-tuned during cool-down, to permit compensation for variations in thermal loads from a process stream, to accommodate o-p conversion for hydrogen, and to compensate for performance degradation during long term operation. The hermetic nature of each MCEU provides highly reliable operation. 
     According to an embodiment, entropy changes required for heat flows in a dual-regenerator design of an active gas regenerative refrigerator (AGRR) come from simultaneous compression and expansion of working gas in each MCEU of an array. Heat flow through the dual regenerators on opposite thermally active ends of the array of MCEUs comes from the coupling of individual MCEUs of the array via a reciprocating flow of heat transfer fluid. The thermodynamic cycle of each MCEU is distinct, consisting of a polytropic compression and associated temperature increase, heat transfer to the heat transfer fluid with a corresponding small temperature and pressure decrease of the compressed working gas inside the MCEU, a polytropic expansion with an associated temperature decrease, and heat transfer from the heat transfer fluid with a corresponding small temperature and pressure increase in the expanded working gas. This combination of events creates a small unique thermodynamic cycle for each MCEU with corresponding heat flows at mean temperatures, T H  and T C , and associated work input. 
     According to an embodiment, there is a recovery of compression work by direct coupling to an expansion at a slightly lower temperature in this cycle. If the heat transfer fluid through the dual regenerators is shut off, the net work input into a MCEU will drop to zero even though the working gas is being compressed and expanded on opposite ends of the MCEU (excluding frictional dissipation in the seal and Joule heating in the drive coils). This feature is difficult to do effectively in conventional gas cycle refrigerators and is one of the reasons that gross efficiencies of conventional gas refrigerators are so low relative to ideal. Turbo-expander units have been built for cryogenic Claude cycle refrigerators but the amount of work recovery is generally relatively small because the gas expansion is done at a temperature substantially different from the gas compression. Intrinsic work recovery to the extent allowed by a thermodynamic refrigeration cycle is one of the reasons that active gas regenerative refrigerators show promise of high efficiency. This is caused by the synchronous force balance in each MCEU. This very desirable feature is enabled by directly coupling the compression of the working gas at one end of each MCEU with the simultaneous expansion of the working gas at the other end of the same MCEU in identical dual regenerators. Accomplishing this coupling allows efficient distributed work input and work recovery from near ambient temperature to cryogenic temperatures as low as ˜4 K. By using this novel concept the net required work input for a given thermal load is reduced substantially no matter what the temperature span of the refrigerator or liquefier is. To the knowledge of the inventors, this input of “distributed net work” is unique among gas refrigerators. 
     According to an embodiment, the thermal mass of each active end of a MCEU of an array in dual regenerators are similar and provide the desirable feature of thermally-balanced regenerators, even with heat capacity variations of tubing material, piston material, drive mechanism, and working gas as a function of temperature. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A and 1B  illustrate the basic structure of a micro compressor-expander unit (MCEU), according to an embodiment, with a moveable drive piston coupling compression and expansion of a working gas in opposite end sections of the MCEU, with the piston in, respectively, a neutral position and a position at one extreme of movement. 
         FIGS. 2A and 2B  illustrate, respectively, the idealized pressure vs. volume, and pressure vs. temperature cycles of the working gas within one thermally active end section of a MCEU, according to an embodiment. 
         FIG. 3  illustrates the relative work input in a complete cycle for the working gas in one end section of a MCEU, according to an embodiment. 
         FIG. 4  illustrates the entropy-temperature diagram for the cycle of the working gas in the thermally active end sections of a MCEU, according to an embodiment. 
         FIG. 5  shows a calculated P-T diagram for an ideal MCEU gas cycle near 100 K with instantaneous heat transfer during compression/expansion within an active gas regenerative refrigerator (AGRR) cycle, according to an embodiment. 
         FIG. 6  illustrates details of a piston structure of a MCEU, according to an embodiment, with two sets of opposing permanent magnets, with a magnetic coupler, to create a stronger transverse flux, compared to a single permanent magnet. 
         FIG. 7  shows key elements of a pneumatically-driven MCEU design, according to an embodiment, with a moveable annular permanent magnet shell around a center section of the MCEU. 
         FIGS. 8A and 8B  are schematic diagrams of an AGRR system showing the system during respective isochoric steps of a refrigeration cycle, according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, reference is made to the accompanying drawings, which form a part hereof. In the drawings, similar symbols typically identify similar components, unless context dictates otherwise. The illustrative embodiments described in the detailed description, drawings, and claims are not meant to be limiting. Other embodiments may be utilized, and other changes may be made, without departing from the spirit or scope of the subject matter presented here. 
     During the NASA SBIR project referred to above, several challenging design issues were identified which were beyond the scope of the project. Most of these issues were related to manufacturing individual refrigeration elements, each with means to synchronously drive reciprocating micro pistons in each element when the working helium gas is at sufficiently high pressures (several MPa), and at pressure ratios large enough to cause polytropic temperature changes of between 2 K and 20 K during compression or expansion. The electromagnetic-magnetic drive forces in the initial drive designs were small compared to the pressure forces on the piston from the He gas at the peak pressures in the MCEU cycle. These issues are reduced or overcome by various embodiments of the present invention. 
     A simple version of a single micro compressor-expander unit (MCEU) tube  100 , according to an embodiment, is illustrated in  FIGS. 1A and 1B , including a uniform cylindrical metal tube  102  formed into a hermetic thin shell with good mechanical strength, modest thermal mass, and reasonable thermal conductivity. This MECU has three sections; two “thermally active” end sections  104 ,  106  and a thermally static center section  108 . A moveable piston  110  at equilibrium in the center section of the MCEU  100  has an electromagnetic or pneumatic drive sufficiently strong to overcome the pressure forces on the piston. A stationary close-fitting, low-friction labyrinth seal  112  keeps the working gas in both thermally active ends of the MCEU  100  during a compression-dwell-expansion-dwell cycle. Working gas in the active sections of the MCEU  100  simultaneously executes the same thermodynamic cycle, but exactly out of phase with the cycle of the working gas at the opposite end of the MCEU  100 . The working gas can be any of a number of different gases, including, for example, helium (He). The thermally active sections in a highly efficient active gas regenerator need high specific area so the tube diameter (od) will be small (specific area for a cylindrical tube is 4/(tube od) or ˜1,200 m 2 /m 3  for a ⅛″ od tube). 
     In  FIG. 1A  the piston  110  is in its equilibrium position and the pressure of the working gas is the same in both end sections  104 ,  106  of the MCEU tube  100 . In  FIG. 1B  the piston  110  is in its right-most position, with compressed, hotter helium working gas also on the right end  106  of the MCEU  100 , and expanded, colder helium working gas on the left end  104  of the tube (the polytropic temperature changes depend on several MCEU design variables and can be ˜2 K to ˜20 K). According to an embodiment, an enhanced piston design has several components; both ends of the piston  110  that extend into the thermally active sections  104 ,  106  of the MCEU  100  are made from material with reasonably high mechanical strength, low thermal mass, and poor thermal conductivity, As described in detail below with reference to  FIGS. 6 and 7 , the central part of the moveable piston  110  contains several opposing pairs of high-strength, small, cylindrically-shaped, permanent magnets held in a thin tubular structure that moves within a thin tube of material that has a low friction coefficient (e.g. loaded Teflon or Rulon) bonded to the inner wall of the center section  108  of the MCEU tube  100 . The piston&#39;s mechanical properties enable a low-leakage, low-friction labyrinth seal  112  as the piston  110  is driven between opposite ends of the MCEU tube  100  by electromagnetic or pneumatic means. 
     According to an embodiment, the thermally active regions of the MCEU  100  enable the execution of an active gas regenerative cycle in the thermally active sections  104 ,  106  of the MCEU  100 . This cycle executed half a cycle out of phase at opposite active ends of the MCEU  100  consists of four steps; i) a polytropic compression with no transverse flow of a separate heat transfer fluid (HTF); ii) an isochoric (constant volume) step with cold-to-hot flow of HTF that causes the temperature and pressure of the compressed He working gas and the shell wall  114  in one end of the MCEU  100  to decrease by the temperature increase of the compressed end of the MCEU  100  while the HTF is heated; iii) a polytropic expansion with no HTF flow; and iv) an isochoric step with hot-to-cold flow of HTF that causes the temperature and pressure of the expanded He working gas in same end of the MCEU  100  and the shell wall  114  in the thermally active regions  104 ,  106  of the MCEU  100  to increase while the HTF is cooled. 
     The resultant force on the piston  110  in each MCEU  100  comes from the differential pressures in the opposite end sections of the MCEU pushing on the end area of the piston. The cooling power of each MCEU  100 , the rejected heat rate, and the net work rate required to move the piston in each polytropic compression step of the MCEU cycle are a function of several design variables such as the mean MCEU operating temperature, temperature span, mean loading pressure of He working gas, diameter and wall thickness of the tube, the pressure ratio and corresponding polytropic temperature changes, etc. For example, in a system configured for liquefying natural gas, the polytropic exponent k changes from ˜1.04 at 290 K to ˜1.1 at 110 K (He alone has a value of 1.66). The inventors&#39; calculations indicate excellent promise for fabrication of small-diameter, tubular, inexpensive MCEUs driven either electromagnetically, at lower temperatures, or pneumatically, at higher temperatures, such as may enable very efficient active gas regenerative refrigerators (AGRRs) and active gas regenerative liquefiers (AGRLs) to be built. 
     The cylindrical hermetic MCEU  100  illustrated in  FIGS. 1A and 1B  includes many basic elements, according to an embodiment. The detailed MCEU cycle analysis presented below allows calculation of heat flows, work flows, pressures, temperatures, material property changes as a function of temperature, and forces for a wide range of design variables. The further description that follows gives a detailed explanation of the MCEU cycle and work input mechanisms to drive the piston as it simultaneously compresses and expands the working gas. 
     To better explain the non-obviousness and usefulness of the MCEU, an analysis is provided, of a regenerative refrigeration cycle when an array of MCEUs is combined, in accordance with an embodiment of an active gas regenerative refrigerator (AGRR). The working gas cycle in each end section of an MCEU tube consists of four steps; i) a polytropic compression by moving the piston to the right with no transverse heat transfer fluid (HTF) flow of the AGRR; ii) an isochoric (constant volume) step with cold-to-hot flow of HTF around the MCEUs with thermal energy transfer from the MCEUs to the HTF, thereby decreasing the temperature and pressure of the He working gas in hermetic MCEU tubes as the HTF is heated; iii) a polytropic expansion of the working gas in the MECUs by moving the piston to the left with no HTF flow; and iv) an isochoric step with hot-to-cold flow of HTF that causes the temperature and pressure of the He working gas in the MCEU tubes to increase as the HTF is cooled. It is important to note that the working gas in the other end section of the MCEU tube simultaneously executes exactly the opposite cycle. 
     The performance of the thermodynamic cycle executed by the working gas at each end of the MCEU tube is calculated for an ideal gas at constant temperature near room temperature, and then with real gas properties in a MCEU with realistic design specifications for an AGRR operating from near room temperature to cryogenic temperatures applicable for numerous applications. 
     For the thermodynamic analysis the variables are defined as follows:
         T w —tube wall temperature   T g —working gas temperature   m g —mass of working gas in both ends of the tube   μ g —molar mass of gas   m w —tube wall mass   n—number of moles of working gas   c v , c p —molar heat capacities of the working gas   c w —heat capacity of tube material per unit mass   R—universal gas constant, R=8.314 J/(mol K)       

     Consider a control volume around one thermally active end section of the MCEU including the working gas hermetically contained inside a thin-walled tubular shell. Apply energy conservation to the ideal working gas during the cycle and the shell and assume adiabatic processes, i.e., dQ=0 for control volume which can be expressed as:
 
 m   w   c   w   dT   w   =−dU   g   −pdV  
 
     Assume instantaneous heat transfer from the working gas to the shell wall associated with a very small Biot number which means:
 
 dT   w   =dT   g   =dT  
 
     The derivation of relationships between p, T and V are: 
     
       
         
           
             
               
                 dU 
                 g 
               
               = 
               
                 
                   nc 
                   V 
                 
                 ⁢ 
                 dT 
               
             
             , 
             
               
 
             
             ⁢ 
             
               n 
               = 
               
                 
                   m 
                   g 
                 
                 
                   μ 
                   g 
                 
               
             
           
         
       
       
         
           
             
               
                 m 
                 w 
               
               ⁢ 
               
                 c 
                 w 
               
               ⁢ 
               dT 
             
             = 
             
               
                 
                   
                     - 
                     
                       nc 
                       V 
                     
                   
                   ⁢ 
                   dT 
                 
                 - 
                 
                   
                     pdV 
                     ⁢ 
                     
                       
 
                     
                     ( 
                     
                       
                         
                           m 
                           w 
                         
                         ⁢ 
                         
                           c 
                           w 
                         
                       
                       + 
                       
                         nc 
                         V 
                       
                     
                     ) 
                   
                   ⁢ 
                   dT 
                 
               
               = 
               
                 - 
                 pdV 
               
             
           
         
       
     
     Given the ideal gas equation of state is:
 
 pV=nRT  
 
− pdV=−nRdT+Vdp  
 
     After substituting for dT into the first-law equation we have: 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       m 
                       w 
                     
                     ⁢ 
                     
                       c 
                       w 
                     
                   
                   + 
                   
                     nc 
                     V 
                   
                   + 
                   nR 
                 
                 ) 
               
               ⁢ 
               pdV 
             
             = 
             
               
                 - 
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                   
                   ) 
                 
               
               ⁢ 
               Vdp 
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                     + 
                     nR 
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 dV 
                 V 
               
             
             = 
             
               - 
               
                 dp 
                 p 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                     + 
                     nR 
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 ln 
                 ⁡ 
                 
                   ( 
                   V 
                   ) 
                 
               
             
             = 
             
               
                 - 
                 
                   ln 
                   ⁡ 
                   
                     ( 
                     p 
                     ) 
                   
                 
               
               + 
               
                 ln 
                 ⁡ 
                 
                   ( 
                   const 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               pV 
               ⁢ 
               
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                     + 
                     nR 
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                   
                   ) 
                 
               
             
             = 
             const 
           
         
       
       
         
           
             γ 
             = 
             
               
                 c 
                 p 
               
               
                 c 
                 V 
               
             
           
         
       
       
         
           
             
               
                 c 
                 p 
               
               - 
               
                 c 
                 V 
               
             
             = 
             
               R 
               = 
               
                 8.3144 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   J 
                   
                     mol 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     K 
                   
                 
               
             
           
         
       
       
         
           
             k 
             = 
             
               
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                     + 
                     nR 
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       
                         m 
                         w 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       nc 
                       V 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       
                         
                           m 
                           w 
                         
                         n 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       c 
                       V 
                     
                     + 
                     R 
                   
                   
                     
                       
                         
                           m 
                           w 
                         
                         n 
                       
                       ⁢ 
                       
                         c 
                         w 
                       
                     
                     + 
                     
                       c 
                       V 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         
                           
                             m 
                             w 
                           
                           n 
                         
                         ⁢ 
                         
                           c 
                           w 
                         
                       
                       + 
                       
                         c 
                         p 
                       
                     
                     
                       
                         
                           
                             m 
                             w 
                           
                           n 
                         
                         ⁢ 
                         
                           c 
                           w 
                         
                       
                       + 
                       
                         c 
                         V 
                       
                     
                   
                   = 
                   
                     
                       c 
                       p 
                       a 
                     
                     
                       c 
                       V 
                       a 
                     
                   
                 
               
             
           
         
       
     
     This equation defines k as the polytropic compression or expansion exponent. In the limit of massless tube walls, it reduces to c p /c v  for the working gas as expected. 
     
       
         
           
             
               pV 
               k 
             
             = 
             const 
           
         
       
       
         
           or 
         
       
       
         
           
             
               p 
               2 
             
             = 
             
               
                 
                   p 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       V 
                       1 
                     
                     
                       V 
                       2 
                     
                   
                   ) 
                 
               
               k 
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               T 
               2 
             
             = 
             
               
                 
                   T 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       V 
                       1 
                     
                     
                       V 
                       2 
                     
                   
                   ) 
                 
               
               
                 k 
                 - 
                 1 
               
             
           
         
       
     
     The polytropic exponent, k and the compression ratios of working gas in the MCEU show the importance of the ratio of thermal mass of the He working gas and the walls of the tube (the drive piston can be selected to minimize its thermal mass), the mean pressure of the He gas in the MCEU, and the geometry of the MCEU design. This derivation also shows that an adiabatic process for the entire control volume at either end of the MCEU means a polytropic process for the working gas during the compression or expansion caused by the moveable piston. 
     The specific work per mole for the working gas in a non-flow, hermetic MCEU is: 
     
       
         
           
             
               w 
               
                 polytropic 
                 , 
                 nonflow 
               
             
             = 
             
               
                 
                   
                     - 
                     
                       RT 
                       1 
                     
                   
                   
                     k 
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         ( 
                         
                           
                             p 
                             2 
                           
                           
                             p 
                             1 
                           
                         
                         ) 
                       
                       
                         
                           k 
                           - 
                           1 
                         
                         k 
                       
                     
                     - 
                     1 
                   
                   ] 
                 
               
               = 
               
                 
                   c 
                   V 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       T 
                       1 
                     
                     - 
                     
                       T 
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The work of compression for a polytropic process is then: 
     
       
         
           
             
               w 
               
                 polytropic 
                 , 
                 nonflow 
               
             
             = 
             
               
                 
                   - 
                   
                     nRT 
                     1 
                   
                 
                 
                   k 
                   - 
                   1 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ( 
                       
                         
                           p 
                           2 
                         
                         
                           p 
                           1 
                         
                       
                       ) 
                     
                     
                       
                         k 
                         - 
                         1 
                       
                       k 
                     
                   
                   - 
                   1 
                 
                 ] 
               
             
           
         
       
     
     Define 
               r   =         V   1       V   2       &gt;   1       ,         
so the work of compression done on the working gas becomes:
 
     
       
         
           
             
               w 
               polytropic 
             
             = 
             
               
                 
                   nRT 
                   1 
                 
                 
                   k 
                   - 
                   1 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     r 
                     
                       k 
                       - 
                       1 
                     
                   
                   - 
                   1 
                 
                 ] 
               
             
           
         
       
     
     If no HTF flows in the regenerator of the AGRR, the temperature T 2  of the helium working gas in the MCEUs does not change after polytropic compression so the working gas upon polytropic expansion returns exactly to T 1 . This is exactly what is expected in an ideal working gas with instantaneous heat transfer, no friction or leakage in the drive piston, no thermal conduction along shell walls, and perfect insulation between the working gas and the drive piston. 
     Now consider what happens when HTF flows over/around the MCEUs in the respective regenerator arrays to change T 2  to T 3  before the polytropic expansion step occurs. 
                 p   2     =         p   1     ⁡     (       V   1       V   2       )       k       ,     
     ⁢       T   2     =         T   1     ⁡     (       V   1       V   2       )         k   -   1         ,     
     ⁢       p   3     =       p   2     ⁢       T   3       T   2           ,         
Choose
 
               T   3     =           T   1     +     T   2       2     =         T   1     ⁡     (     1   +       (       V   1       V   2       )       k   -   1         )       2             
because the temperature approach between the HTF and the MCEU shell at that position in the regenerator of the AGRR decreases from a maximum of T 2 −T 1  to ˜0 during the optimum flow period of the HTF (this average value of T 3  assumes linear temperature change which is a reasonable choice).
 
               p   3     =       ⁢             p   1     ⁡     (       V   1       V   2       )       k     ⁢       T   3       T   2         =       ⁢             p   1     ⁡     (       V   1       V   2       )       k     ⁢       (     1   +       (       V   1       V   2       )       k   -   1         )     2     ⁢       (       V   1       V   2       )       1   -   k         =       ⁢       p   1     ⁢       (         V   1       V   2       +       (       V   1       V   2       )     k       )     2                         p   4     =       ⁢           p   3     ⁡     (       V   3       V   4       )       k     =       ⁢         p   1     ⁢       (         V   1       V   2       +       (       V   1       V   2       )     k       )     2     ⁢       (       V   1       V   2       )       -   k         =       ⁢       p   1     ⁢           (       V   1       V   2       )       1   -   k       +   1     2                 
From isochoric cooling/heating:
 
     
       
         
           
             
               T 
               4 
             
             = 
             
               
                 
                   T 
                   1 
                 
                 ⁢ 
                 
                   
                     p 
                     4 
                   
                   
                     p 
                     1 
                   
                 
               
               = 
               
                 
                   T 
                   1 
                 
                 ⁢ 
                 
                   
                     
                       
                         ( 
                         
                           
                             V 
                             1 
                           
                           
                             V 
                             2 
                           
                         
                         ) 
                       
                       
                         1 
                         - 
                         k 
                       
                     
                     + 
                     1 
                   
                   2 
                 
               
             
           
         
       
     
     Two MCEU cycles, as illustrated in  FIGS. 2A and 2B  below, are simultaneously executed 180° out of phase by the same mass of working gas at each dual regenerator section at opposite end sections of the tube. The working gas changes in pressure and temperature as the piston in the MCEU is driven to one end or the other end of the MCEU tube. The diagrams described below illustrate the idealized cycle for the working gas in each end of the MCEU, as follows (mass transfer through leaky seals on drive piston neglected): 
     Calculating the temperature after polytropic expansion as a check: 
     
       
         
           
             
               T 
               4 
             
             = 
               
             ⁢ 
             
               
                 
                   
                     T 
                     3 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         V 
                         3 
                       
                       
                         V 
                         4 
                       
                     
                     ) 
                   
                 
                 
                   k 
                   - 
                   1 
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     
                       
                         T 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 
                                   V 
                                   1 
                                 
                                 
                                   V 
                                   2 
                                 
                               
                               ) 
                             
                             
                               k 
                               - 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           V 
                           2 
                         
                         
                           V 
                           1 
                         
                       
                       ) 
                     
                     
                       k 
                       - 
                       1 
                     
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     1 
                   
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               V 
                               1 
                             
                             
                               V 
                               2 
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           k 
                         
                       
                       + 
                       1 
                     
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Looks 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     O 
                     . 
                     K 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 T 
                 1 
               
               - 
               
                 T 
                 4 
               
             
             = 
               
             ⁢ 
             
               
                 
                   T 
                   1 
                 
                 - 
                 
                   
                     T 
                     1 
                   
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               V 
                               1 
                             
                             
                               V 
                               2 
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           k 
                         
                       
                       + 
                       1 
                     
                     2 
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     T 
                     1 
                   
                   ⁢ 
                   
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             
                               V 
                               1 
                             
                             
                               V 
                               2 
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           k 
                         
                       
                     
                     2 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     1 
                   
                   ⁢ 
                   
                     
                       1 
                       - 
                       
                         r 
                         
                           1 
                           - 
                           k 
                         
                       
                     
                     2 
                   
                 
               
             
           
         
       
     
     The resultant work input needed for a complete cycle of the working gas (ideal gas) in a thermally active end section of the MCEU is given by the difference between work of compression from T 1  and the work from expansion from T 3 , a slightly lower temperature: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 W 
                 polytropic 
               
             
             = 
             
               
                 
                   W 
                   
                     1 
                     - 
                     2 
                   
                 
                 - 
                 
                   W 
                   
                     3 
                     - 
                     4 
                   
                 
               
               = 
               
                 
                   
                     
                       n 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         RT 
                         1 
                       
                     
                     
                       k 
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         r 
                         
                           k 
                           - 
                           1 
                         
                       
                       - 
                       1 
                     
                     ] 
                   
                 
                 - 
                 
                   
                     nRT 
                     
                       k 
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         r 
                         
                           k 
                           - 
                           1 
                         
                       
                       - 
                       1 
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 W 
                 polytropic 
               
             
             = 
             
               
                 
                   W 
                   
                     1 
                     - 
                     2 
                   
                 
                 - 
                 
                   W 
                   
                     3 
                     - 
                     4 
                   
                 
               
               = 
               
                 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       R 
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             1 
                           
                           - 
                           
                             T 
                             4 
                           
                         
                         ) 
                       
                     
                   
                   
                     k 
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       r 
                       
                         k 
                         - 
                         1 
                       
                     
                     - 
                     1 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 W 
                 polytropic 
               
             
             = 
             
               
                 
                   W 
                   
                     1 
                     - 
                     2 
                   
                 
                 - 
                 
                   W 
                   
                     4 
                     - 
                     3 
                   
                 
               
               = 
               
                 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       RT 
                       1 
                     
                   
                   
                     k 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     ⌊ 
                     
                       
                         r 
                         
                           k 
                           - 
                           1 
                         
                       
                       + 
                       
                         r 
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       2 
                     
                     ⌋ 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             x 
             = 
               
             ⁢ 
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     W 
                     polytropic 
                   
                 
                 
                   W 
                   
                     4 
                     - 
                     3 
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     T 
                     1 
                   
                   ⁢ 
                   
                     
                       ⌊ 
                       
                         
                           r 
                           
                             k 
                             - 
                             1 
                           
                         
                         + 
                         
                           r 
                           
                             1 
                             - 
                             k 
                           
                         
                         - 
                         2 
                       
                       ⌋ 
                     
                     2 
                   
                   ⁢ 
                   
                     1 
                     
                       
                         T 
                         1 
                       
                       ⁢ 
                       
                         
                           
                             [ 
                             
                               
                                 r 
                                 
                                   1 
                                   - 
                                   k 
                                 
                               
                               + 
                               1 
                             
                             ] 
                           
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               r 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                     
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     ⌊ 
                     
                       
                         r 
                         
                           k 
                           - 
                           1 
                         
                       
                       + 
                       
                         r 
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       2 
                     
                     ⌋ 
                   
                   
                     
                       r 
                       
                         k 
                         - 
                         1 
                       
                     
                     - 
                     
                       r 
                       
                         1 
                         - 
                         k 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             x 
             = 
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     W 
                     polytropic 
                   
                 
                 
                   W 
                   
                     4 
                     - 
                     3 
                   
                 
               
               = 
               
                 
                   
                     r 
                     
                       k 
                       - 
                       1 
                     
                   
                   + 
                   
                     r 
                     
                       1 
                       - 
                       k 
                     
                   
                   - 
                   2 
                 
                 
                   
                     r 
                     
                       k 
                       - 
                       1 
                     
                   
                   - 
                   
                     r 
                     
                       1 
                       - 
                       k 
                     
                   
                 
               
             
           
         
       
     
       FIG. 3  illustrates the relative work input in a complete cycle for the working gas in one end section of a MCEU, according to an embodiment. The curves shown in  FIG. 3  indicate that to make an effective MCEU cycle, the design choices must achieve k of ˜1.05 to ˜1.10 with a piston geometry that gives a compression ratio of ˜2. Such values can be obtained with MCEU tube dimensions of 0.125″ o.d. with a wall thickness of 0.003″ with overall length of 8″ and thermally active sections 2″ long with 5.0 MPa (˜750 psia) mean pressure with a piston sized to give a compression ratio of ˜1.2 to ˜2.0 (see  FIG. 1 ). If k˜1 (the isothermal limit), x is close to zero no matter what the compression ratio is, i.e., there is no work recovered because no work is input and there is no cooling. This limit is approached only for very large thermal mass of the MCEU shell, very little working gas in the MECU, and/or a small compression ratio. These regions of design space are easy to avoid in fabricating an effective MCEU. 
     Similarly, the heat and entropy flows for the working gas in the thermally active end sections of the MCEU can be calculated.  FIG. 4  illustrates the entropy-temperature diagram for the cycle of the working gas in the thermally active end sections of a MCEU, according to an embodiment. 
     In  FIG. 4 , the path between points  1  and  2  of the entropy-temperature diagram represents a polytropic compression of a working gas (with heat flow from the working gas to a metal shell); the path between points  2  and  3  of the entropy-temperature diagram represents isochoric cooling of the working gas from a separate heat transfer fluid; the path between points  3  and  4  of the entropy-temperature diagram represents polytropic expansion of the working gas (with heat flow from the metal shell to the working gas); and the path between points  4  and  1  of the entropy-temperature diagram represents isochoric heating of the working gas from a separate heat transfer fluid. 
     
       
         
           
             dS 
             = 
             
               
                 
                   
                     C 
                     V 
                   
                   T 
                 
                 ⁢ 
                 dT 
               
               + 
               
                 
                   
                     ( 
                     
                       
                         ∂ 
                         P 
                       
                       
                         ∂ 
                         T 
                       
                     
                     ) 
                   
                   V 
                 
                 ⁢ 
                 dV 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   or 
                 
               
             
           
         
       
       
         
           
             dS 
             = 
             
               
                 
                   
                     C 
                     p 
                   
                   T 
                 
                 ⁢ 
                 dT 
               
               - 
               
                 
                   
                     ( 
                     
                       
                         ∂ 
                         V 
                       
                       
                         ∂ 
                         T 
                       
                     
                     ) 
                   
                   p 
                 
                 ⁢ 
                 dp 
               
             
           
         
       
     
     For an ideal gas, the change in entropy is: 
     
       
         
           
             
               
                 S 
                 i 
               
               - 
               
                 S 
                 f 
               
             
             = 
             
               
                 
                   ∫ 
                   i 
                   f 
                 
                 ⁢ 
                 
                   d 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   S 
                 
               
               = 
               
                 
                   
                     nc 
                     V 
                   
                   ⁢ 
                   
                     ln 
                     ⁡ 
                     
                       ( 
                       
                         
                           T 
                           f 
                         
                         
                           T 
                           i 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   n 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   R 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ln 
                     ⁡ 
                     
                       ( 
                       
                         
                           V 
                           f 
                         
                         
                           V 
                           i 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   or 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   S 
                   i 
                 
                 - 
                 
                   S 
                   f 
                 
               
               = 
               
                 
                   
                     ∫ 
                     i 
                     f 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     S 
                   
                 
                 = 
                 
                   
                     
                       nc 
                       p 
                     
                     ⁢ 
                     
                       ln 
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             f 
                           
                           
                             T 
                             i 
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     R 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ln 
                       ⁡ 
                       
                         ( 
                         
                           
                             p 
                             f 
                           
                           
                             p 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     Let&#39;s define 
     
       
         
           
             
               Q 
               if 
             
             = 
             
               
                 
                   Q 
                   f 
                 
                 - 
                 
                   Q 
                   i 
                 
               
               = 
               
                 
                   ∫ 
                   i 
                   f 
                 
                 ⁢ 
                 
                   T 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   d 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   S 
                 
               
             
           
         
       
     
     For the isochoric processes in the working gas (dV=0):
 
 Q   23   =nc   V ( T   3   −T   2 )&lt;0,  Q   41   =nc   V ( T   1   −T   4 )&gt;0
 
     These equations show that heat (thermal energy) flows out of the selected control volume of the working gas in one end section of a MCEU in the hot-to-cold flow ( 2  to  3 ) of heat transfer fluid through an AGRR comprised of an array of MCEUs and heat flows into the control volume of the working gas in the cold-to-hot flow ( 4  to  1 ) of the HTF in the same AGRR. 
     For the polytropic processes in the working gas: 
     
       
         
           
             
               
                 
                   
                     Q 
                     12 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         nc 
                         V 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             2 
                           
                           - 
                           
                             T 
                             1 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∫ 
                         1 
                         2 
                       
                       ⁢ 
                       
                         TnR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             V 
                           
                           V 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         nc 
                         V 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             2 
                           
                           - 
                           
                             T 
                             1 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∫ 
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               1 
                             
                             ⁢ 
                             
                               V 
                               1 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                           
                             V 
                             
                               k 
                               - 
                               1 
                             
                           
                         
                         ⁢ 
                         nR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             V 
                           
                           V 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               Q 
               12 
             
             = 
               
             ⁢ 
             
               
                 
                   
                     nc 
                     V 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         T 
                         2 
                       
                       - 
                       
                         T 
                         1 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     nRT 
                     1 
                   
                   ⁢ 
                   
                     V 
                     1 
                     
                       k 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       1 
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         V 
                       
                       
                         V 
                         k 
                       
                     
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     nc 
                     V 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         T 
                         2 
                       
                       - 
                       
                         T 
                         1 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     nRT 
                     1 
                   
                   ⁢ 
                   
                     V 
                     1 
                     
                       k 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     1 
                     
                       1 
                       - 
                       k 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         V 
                         2 
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       
                         V 
                         1 
                         
                           1 
                           - 
                           k 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               Q 
               12 
             
             = 
               
             ⁢ 
             
               
                 
                   
                     nc 
                     V 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         T 
                         2 
                       
                       - 
                       
                         T 
                         1 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     nRT 
                     1 
                   
                   ⁢ 
                   
                     1 
                     
                       1 
                       - 
                       k 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           ( 
                           
                             
                               V 
                               2 
                             
                             
                               V 
                               1 
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       1 
                     
                     ) 
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     
                       nc 
                       V 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           T 
                           2 
                         
                         - 
                         
                           T 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       nRT 
                       1 
                     
                     ⁢ 
                     
                       1 
                       
                         1 
                         - 
                         k 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           r 
                           
                             k 
                             - 
                             1 
                           
                         
                         - 
                         1 
                       
                       ) 
                     
                   
                 
                 &lt; 
                 0 
               
             
           
         
       
       
         
           
             
               Q 
               34 
             
             = 
               
             ⁢ 
             
               
                 
                   
                     nc 
                     V 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         T 
                         4 
                       
                       - 
                       
                         T 
                         3 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     nRT 
                     3 
                   
                   ⁢ 
                   
                     1 
                     
                       1 
                       - 
                       k 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           ( 
                           
                             
                               V 
                               4 
                             
                             
                               V 
                               3 
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       1 
                     
                     ) 
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     nc 
                     V 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         T 
                         4 
                       
                       - 
                       
                         T 
                         3 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     nRT 
                     3 
                   
                   ⁢ 
                   
                     1 
                     
                       1 
                       - 
                       k 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         r 
                         
                           1 
                           - 
                           k 
                         
                       
                       - 
                       1 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               Let 
               ⁢ 
               
                   
               
               ⁢ 
               
                 Q 
                 12341 
               
             
             = 
             
               
                 Q 
                 12 
               
               + 
               
                 Q 
                 23 
               
               + 
               
                 Q 
                 34 
               
               + 
               
                 Q 
                 41 
               
             
           
         
       
     
     All the nc V  terms cancel each other and: 
               Q   12341     =         nRT   1     ⁢     1     1   -   k       ⁢     (       r     k   -   1       -   1     )       +     nR   ⁢         T   1     ⁡     (     1   +     r     k   -   1         )       2     ⁢     1     1   -   k       ⁢     (       r     1   -   k       -   1     )                       Q   12341     =       ⁢           nRT   1       1   -   k       [         2   ⁢     r     k   -   1         -   2   +       (     1   +     r     k   -   1         )     ⁢     (       r     1   -   k       -   1     )         2     ]     =       ⁢         nRT   1       1   -   k       [         r     k   -   1       +     r     1   -   k       -   2     2     ]             
This result shows that Q 12341 =−ΔW polytropic , as it should be.
 
     The inventors have prepared detailed design calculations, according to an embodiment, for a new MCEU with He working gas at up to 5.0 MPa mean pressure at 290 K using ⅛″ diameter Al alloy seamless tubing of type 2024-T6 with 0.003″ wall thickness with pistons ranging in diameter from ⅞ to ⅜ of the i.d. of the MCEU tube. With typical MCEU dimensions listed above, using real gas properties for helium working gas at starting pressure of 5.0 MPa at 290 K, and the temperature-dependent heat capacity of 2024-T6 Al alloy tube material, the calculated P-T cycle for an achievable MCEU piston design with He working gas at about 100 K is shown in  FIG. 5 . This module could be one of three AGRRs in an efficient AGRL for liquid natural gas (LNG). 
       FIG. 6  illustrates details of a piston structure of a MCEU  600 , according to an embodiment, with one or more sets  602  of opposing permanent magnets  604 , with a magnetic coupler  606 , to create a stronger transverse flux, compared to a single permanent magnet. In one embodiment of the invention, illustrated in  FIG. 6 , two small-diameter cylindrical high-field Nd 2 Fe 14 B permanent magnets  604 , which together form one set  602 , are inserted as opposing each other into a cylindrical drive piston assembly  610  within a Rulon sleeve seal (not shown) in the center section  612  of the MCEU  600 . The N-S poles of the permanent magnets  604  are aligned as S-N-N-S. This embodiment includes an iron flux coupler  606  to help concentrate the magnetic flux of the of the radial magnetic field B R  created by the opposing permanent magnets  604 . Two or more sets  602  of such opposing permanent magnets  604  are envisioned to increase the Lorenz force applicable on the drive piston. 
       FIG. 6  also shows a drive mechanism, according to an embodiment. As an example, a thin annular coil  614  with several layers of good electrical conducting or superconducting wire such as AWG 20-30, is assembled surrounding the center section of a hermetic MCEU tube  616  with the piston, seals, and working gas in it (the complete piston and seals are not shown in detail in  FIG. 6 , but are shown and described elsewhere). The magnetic field from the energized coil  614  couples tightly to the concentrated magnetic flux from all sets  602  of opposing Nd 2 Fe 14 B magnets  604  within the piston assembly  610 . As the d.c. power supply to each MCEU drive coil  614  charges with appropriate polarity during different steps within the MCEU cycle, the current in the coil creates a Lorenz force on the permanent magnets  604  to thereby move the drive piston inside the MCEU  600  in either axial direction. The Lorenz force in this electromagnetic drive can be adjusted in strength by adjusting the length of the center section  612  of the MCEU  600  relative to the thermally active sections of the MCEU to keep the Joule heating from the drive coils to a small parasitic heat load compared to the cooling power of the MCEU  600  (or vice-versa). 
     In  FIG. 7  an embodiment of the invention illustrates another drive mechanism for a MCEU  710 . In this second embodiment of the invention two or more sets of two small-diameter cylindrical high-field Nd 2 Fe 14 B permanent magnets  742  are inserted as opposing each other into a cylindrical drive piston  718  assembly within a Rulon sleeve seal  726  in the center section of the MCEU. A cylindrical soft iron or other high magnetic permeability material  738  is mounted in the seal section  726  of the MCEU  710  to augment coupling of the magnetic flux of the two permanent magnet arrays. Outside the Al tube  714  another cylindrical annular Nd 2 Fe 14 B permanent magnet array  734  is mounted inside a close-fitting, low-friction hermetic tube  730  such that gas at either end of this surrounding tube can change pressure to move the annular magnet array back and forth. The magnetic flux from the opposing permanent magnets in this shell couples tightly to the flux of similar sets of Nd 2 Fe 14 B magnets  742  inside the central MCEU piston. This outer magnet array  734  in its close fitting housing  730  is pneumatically driven, and drives in turn the central piston inside the MCEU  710 , back and forth to alternatively compress or expand its working He gas  722 . One or more cylindrical, thin annular Nd 2 Fe 14 B permanent magnets  734  are assembled inside a close-fitting, low-friction hermetic tube  730  surrounding the center section of the hermetic MCEU tube  710  with the piston  718 , seals  726 , and working gas  722  in it. The magnetic flux from annular permanent magnet array  734  couples tightly to the concentrated magnetic flux from all sets of opposing Nd 2 Fe 14 B magnets  742  within the piston assembly  718 . When the outer annular magnet array  734  in its close fitting housing  730  is pneumatically moved back and forth over the center section of the MCEU  710 , it will thereby move the drive piston  718  inside the MCEU  710 . The pneumatic drive in each MCEU  710  is fed by a separate pressurized gas supply (not shown) into either end of the thin hermetic shell  730  around the MCEU  710 . This gas is supplied by gas via a small tube  746  from a common feed gas source with adjustable pressures as necessary to move the annular magnet back and forth. Correspondingly, the gas on the other end of the annular shell  730  around the center section of the MECU  710  will be returned to a common lower pressure vessel from which the suction port of the gas pump will be fed to return higher pressure gas to the supply tank. Two-way valves on the manifolds out of the higher pressure vessel and into the lower pressure vessel of the pneumatic gas drive subsystem (not shown) allow properly-timed connections required to execute MCEU cycles via this pneumatically driven subsystem for an entire array of MCEUs (not shown). 
       FIGS. 8A and 8B  are schematic diagrams of an AGRR system  800  showing the system during respective isochoric steps of a refrigeration cycle, according to an embodiment. The AGRR system  800  includes an array  802  of MCEUs  804 , each having a cylinder  805  and a double-ended drive piston  806  positioned within the cylinder and configured to be driven back and forth to alternately compress and expand equal masses of working gas in respective ends of the MCEU  804 . Each MCEU  804  further includes a seal  807  positioned between the inside of the cylinder  805  and the drive piston  806 . The seal  807  is configured to permit axial movement of the drive piston  806  within the cylinder  805  while preventing movement of the working gas between the ends of the MCEUs  804 . 
     The drive pistons  806  can be driven by any appropriate mechanism, such as, for example, either of the mechanisms described above with reference to  FIGS. 6 and 7 . 
     First ends  808  of each of the MCEUs  804  are positioned within a first heat transfer chamber  810 , while second ends  812  of each of the MCEUs are positioned within a second heat transfer chamber  814 . The first heat transfer chamber  810  includes first and second fluid ports  816 ,  818  and the second heat transfer chamber  814  includes third and fourth fluid ports  820 ,  822 . A thermal load  824  is in fluid communication with the first and third fluid ports  816 ,  820 , while a heat sink  826  is in fluid communication with the second and fourth fluid ports  818 ,  822 . A reversible fluid pump  828  is configured to drive a heat transfer fluid (HTF) through a heat transfer circuit formed by the first and second heat transfer chambers  812 ,  814 , the thermal load  824 , and the heat sink  826 . 
     In operation, during a first operating step, the drive pistons  806  are driven to a first position, defined by an extreme of travel in a first direction, as shown in  FIG. 8A , radially compressing the working gas in the first ends  808  of the MCEUs  804  into first annular gaps  830  between radial surfaces of the drive pistons  806  and inner radial surfaces of the first ends  808 , while expanding the working gas in the second ends  812 . This causes the temperature of the working fluid in the first ends  808  to rise, and the temperature of the working fluid in the second ends  812  to drop. During this step, the pump  828  is not in operation. 
     During a second step, the pump  828  operates to drive the HTF in a first direction D 1  through the fluid circuit, as shown in  FIG. 8A , so that fluid heated by the thermal load  824  is carried into the first heat transfer chamber  810 , where it is heated as it flows across the outsides of the first ends  808  of the MCEUs  804 , while cooling the working fluid within the first ends. HTF from the first heat transfer chamber  810  is carried to the heat sink  826 , where the heated fluid is cooled by contact with the heat sink. From the heat sink  826 , the cooled HTF is carried into the second heat transfer chamber  814 , where it is cooled as it flows across the outsides of the second ends  812  of the MCEUs  804 , while warming the working fluid within the second ends. Lastly, cooled HTF is carried from the second heat transfer chamber  814  to the thermal load  824 , where it efficiently chills the thermal load, being heated itself in return. 
     During a third operational step, the flow of fluid is shut down, and the drive pistons  806  are driven to a second position defined by an extreme of travel in a second direction, opposite the first direction, as shown in  FIG. 8B , radially compressing the working gas in the second ends  812  of the MCEUs  804  into second annular gaps  832  between the radial surfaces of the drive pistons  806  and the inner radial surfaces of the second ends  812 , while expanding the working gas in the first ends  808 . This causes the temperature of the working fluid in the second ends  812  to rise, and the temperature of the working fluid in the first ends  808  to drop. 
     Finally, during a fourth step, the pump  828  operates to drive the HTF in a second direction D 2  through the fluid circuit, as shown in  FIG. 8B . Accordingly, HTF is driven from the heat sink  824  to the second heat transfer chamber  814 , from the second heat transfer chamber to the heat sink  826 , from the heat sink to the first heat transfer chamber  810 , and from the first heat transfer chamber to the thermal load. The HTF cools the thermal load  824  while being heated in exchange, cools the second ends  812  of the MCEUs  804  while being heated in exchange, transfers heat to the heat sink  826 , which is configured to remove the heat to a remote location, while being cooled thereby, warms the first ends  808  while being cooled, and back to the thermal load. 
     The four-step process outlined above is repeated continuously during operation of the device. 
     The term thermally active section is used here to refer to the outer surface of the portion of a cylinder that is in direct contact, on its inner surface, with a working fluid. Because the MCEUs  804  are configured to form the first and second annular gaps  830 ,  832 , the working fluid remains in contact with the inner surfaces of the first and second ends  808 ,  812  along a length of the respective cylinders  805  that remains constant throughout the operational cycle. Accordingly, the surface area of the active sections of each of the first and second ends  808 ,  812  of the MCEUs  804  also remains unchanged throughout the cycle, even as the respective drive pistons  806  move reciprocally within the cylinders  805 . This means that the ability of the heat transfer fluid outside the MCEUs  804  to exchange heat with the working fluid inside the MCEUs is not affected by the position of the pistons  806 . 
     This is in contrast to devices in which a piston seal sweeps an inner face of a cylinder as the piston moves, compressing a working fluid into an end of the cylinder. In such a device, the active section is defined by the distance between the piston seal and the end of the cylinder, such that as the piston moves back and forth within the cylinder, the surface area of the active section continually changes, reaching a minimum when the working fluid is at maximum compression. Thus, the heat exchange capacity of the cylinder is at a minimum when the temperature difference across the cylinder wall is at a maximum, which can significantly reduce the heat transfer efficiency of the associated system. 
     In the embodiment of  FIGS. 8A and 8B , the end surfaces of the cylinders  805  lying transverse to the cylinder axes are positioned against the walls of the first and second heat transfer chambers  810 ,  814  such that they are not exposed to the HTF as it flows through the chambers. According to another embodiment, the first and second ends  808 ,  812  of each of the MCEUs  804  are positioned within the first and second heat transfer chambers  810 ,  814 , respectively, and the HTF flows over and in contact with the transverse end surfaces, such that the active sections of each MCEU  804  are increased by the area of the transverse end surfaces as well. In this embodiment, the array  802  is configured such that when the drive pistons  806  of the MCEUs  804  are in either of their first or second positions, a gap remains between transverse ends of the pistons and the transverse ends of the respective cylinders  805 . Accordingly, working fluid remains in contact with the transverse ends of the cylinders  805  throughout the operational cycle. 
     The array  802  of MCEUs  804  is represented in  FIGS. 8A and 8B  by a small number of MCEUs in a single row. It will be understood that in practice, the number of MCEUs in the array can number in the hundreds, or more, and can be arranged in any appropriate configuration, including rows and columns, hexagonal grids, etc. 
     In the embodiment illustrated in  FIGS. 8A and 8B , the AGRR system  800  is configured for use with a gaseous HTF. According to other embodiments, liquid heat transfer fluids may also be employed. It is important to avoid heat transfer fluids that might freeze during operation, which reduces the number of suitable fluids, especially liquids, particularly when the system is to be operated at cryogenic temperatures. Hydrogen and helium are among the fluids that can be employed in most cryogenic applications. According to a preferred embodiment, He gas, at a pressure of around 500 psia, is employed as the heat transfer fluid. 
     Although in most embodiments, a gaseous HTF is maintained at an elevated pressure of several hundred psia, in some embodiments in which the HTF is not pressurized, ambient air may be used as the HTF, in which case the heat sink can be omitted, so that the air is drawn directly into one or the other heat transfer chamber, then vented back to the atmosphere after exiting the other chamber, or even after passing through the thermal load. 
     The abstract of the present disclosure is provided as a brief outline of some of the principles of the invention according to one embodiment, and is not intended as a complete or definitive description of any embodiment thereof, nor should it be relied upon to define terms used in the specification or claims. The abstract does not limit the scope of the claims. Elements of the various embodiments described above can be combined, and further modifications can be made, to provide further embodiments without deviating from the spirit and scope of the invention. All of the patents and non-patent publications referred to in this specification and/or listed in the Application Data Sheet are incorporated herein by reference, in their entirety. Aspects of the embodiments can be modified, if necessary to employ concepts of the various patents and publications to provide yet further embodiments. 
     While various aspects and embodiments have been disclosed herein, other aspects and embodiments are contemplated. The various aspects and embodiments disclosed herein are for purposes of illustration and are not intended to be limiting, with the true scope and spirit being indicated by the following claims.
An active gas regenerative refrigerator includes a plurality of compressor-expander units, each having a hermetic cylinder with a drive piston configured to be driven reciprocally therein, and a quantity of working fluid in each end of the cylinder. A piston seal in a central portion of the cylinder prevents passage of the working fluid between ends of the cylinder. Movement of the piston to a first extreme results in radial compression of one of the quantities of working fluid in a cylindrical gap formed between one end of the piston and an inner surface of the cylinder, while the other quantity is expanded in the opposite end of the cylinder. The piston includes a plurality of magnets arranged in pairs, with magnets of each pair positioned with like-poles facing each other. A piston drive is configured to couple with transverse magnetic flux regions formed by the magnets.