Abstract:
The current invention is a closed cycle heat engine that includes a plurality of variable volume movable working chambers, each chamber having a first volume of working fluid when disposed at an isentropic expansion zone leading edge, a second volume when disposed at an isentropic expansion zone trailing edge, a third volume when disposed at an isentropic compression zone leading edge and a fourth volume of working fluid when disposed at an isentropic compression zone trailing edge. The second volume of working fluid divided by the first volume of working fluid provides a first volume ratio. The third volume of working fluid divided by the fourth volume of working fluid provides a second volume ratio. The first volume ratio equals the second volume ratio. The working fluid efficiently performs work by traversing a cycle consisting of an isothermal expansion, an isentropic expansion, an isothermal compression, and an isentropic compression.

Description:
FIELD OF INVENTION 
     This invention relates to a closed cycle rotary heat engine with confined working fluid. More particularly, the invention relates to a closed cycle heat engine having a ratio of volumes of working chambers positioned when disposed at an isentropic expansion zone trailing edge and at an isentropic expansion zone leading edge set equal to a ratio of volumes of working chambers when disposed at an isentropic compression zone leading edge and at an isentropic compression zone trailing edge. 
     BACKGROUND OF INVENTION 
     Heat engines are well known for their ability to convert heat energy to usable work. Heat engines such as steam engines, steam and gas turbines, diesel engines, and Stirling engines can provide power for transportation, machinery, or producing electricity, to name a few. 
     Rotary heat engines have a rotating hub of dynamic chambers, containing a working fluid, that are coupled to work-transfer elements to deliver mechanical work-output. They operate in a cyclical manner. Heat is added to the confined working fluid during a portion of the cycle and heat is rejected from the working fluid during another portion of the cycle. Heat causes expansion of the working fluid as work is performed. A portion of the work is used to compress the working fluid as heat is rejected. The work performed by the working fluid during expansion minus the work used to compress the working fluid during compression is the net work available to overcome friction and deliver mechanical work-output. 
     Because heat engines cannot convert all the input energy to useful work, some of the heat is not available for mechanical work, where the percentage of thermal energy that is converted to mechanical work defines the thermal efficiency of the heat engine. The theoretical upper limit of efficiency of a heat engine cycle is that of the Carnot Cycle. Practical heat engines such as the Rankine, Brayton, or Stirling engines operate on less efficient cycles. Typically, the highest thermal efficiency is achieved when the input (heat zone) temperature is as high as possible and the output (cold zone) temperature is as low as possible. 
     The Carnot cycle has long been considered the ideal heat engine cycle. It has been the goal of many heat engine designers. However, to attain Carnot cycle efficiency would be meaningless, since no power would be developed. Attempts have been made to improve the efficiency of heat engines. But, maximum power of a heat engine occurs at efficiencies considerably below Carnot cycle efficiency. Carnot cycle efficiency is only a limit of efficiency, not necessarily an ideal goal. Of course, it is desirable to balance desired power, efficiency, and cost. 
     There are many, many heat engine designs. There are internal combustion engines, external combustion engines, piston engines, turbine engines, rotary engines and many others. The instant invention is a closed cycle rotary heat engine. 
     The following patents appear to have relevancy to the instant invention:
     1. U.S. Pat. No. 3,169,375, Rotary Engines or Pumps, by Velthuis, Feb. 16, 1965   2. U.S. Pat. No. 3,698,184, Low Pollution Heat Engine, by Barrett, Oct. 17, 1972   3. U.S. Pat. No. 3,867,815, Heat Engine, by Barrett, Feb. 25, 1975   4. U.S. Pat. No. 4,089,174, Method and Apparatus for Converting Radiant Solar Energy into Mechanical Energy, by Posnansky, May 16, 1978   5. U.S. Pat. No. 4,357,800, Rotary Heat Engine, by Hecker, Nov. 9, 1982   6. U.S. Pat. No. 4,502,284, Method and Engine for the Obtainment of Quasi-isothermal Transformation in Gas Compression and Expansion, by Chrisoghilos, Mar. 5, 1985   7. U.S. Pat. No. 4,621,497, Heat Engine, by McInnes, Nov. 11, 1986   8. U.S. Pat. No. 5,325,671, Rotary Heat Engine, by Boehling, Jul. 5, 1994   

     Except for Patent 7, they describe attempts to increase efficiency and power by circulating the working fluid external from the working chambers for heating and cooling. This, however, dilutes ideal isothermal expansion and isothermal compression, during the heating and cooling stages. Patents 6 and 8 more nearly provide ideal expansion and compression, since they minimize the heating and cooling areas being open to more than one working chamber at a time. 
     However, a second loss of efficiency for all of the Patents 1 through 8 occurs because heat is conducted from the hot areas to the cold areas by paths other than through the working fluid. Such a path would be through the housing. 
     A third loss of efficiency for all of the Patents 1 through 8, is the lack of defined dimensional parameters to assure proper temperature, pressure, and volume relationships of the working fluid. 
     What is needed is a heat engine that optimizes heat engine power and/or efficiency by having proper parametric relationships of temperature, pressure, and volume, as well as minimizing loss of efficiency by preventing heat loss by maximizing the amount of heat transfer from the heating areas to the cooling areas through the working fluid, and minimizing heat transfer through other conduction paths. 
     SUMMARY OF THE INVENTION 
     The current invention overcomes the teachings of the prior art by providing a closed cycle heat engine that includes a plurality of variable volume movable working chambers, each having a first volume of working fluid when disposed at an isentropic expansion zone leading edge, a second volume of working fluid when disposed at an isentropic expansion zone trailing edge, a third volume of working fluid when disposed at an isentropic compression zone leading edge, and a fourth volume of working fluid when disposed at an isentropic compression zone trailing edge. The second working fluid volume divided by the first working fluid volume provides a first volume ratio. The third working fluid volume divided by the fourth working fluid volume provides a second volume ratio. The first volume ratio is equal to the second volume ratio. The working fluid efficiently performs work by traversing a cycle consisting of an isothermal expansion, an isentropic expansion, an isothermal compression, and an isentropic compression. 
     According to one embodiment of the invention, the closed cycle heat engine includes a housing, with end closures, having a cylindrical shape with an inner surface and an outer surface. The current embodiment further includes a thermal layer that abuts the inner surface of the housing and is concentric with it. The inner surface of the thermal layer has a cylindrical-quadrant heat input span having a first temperature, a cylindrical-quadrant isentropic expansion span, a cylindrical-quadrant heat output span having a second temperature, and a cylindrical-quadrant isentropic compression span, where the first temperature is larger than the second temperature and both the temperatures are predetermined. Further included is a plurality of variable volume movable working chambers held by the housing and interfacing the thermal layer. Additionally, included is a work delivery transmission, where the working chambers convey work to the transmission and the transmission delivers the work outside the housing. According to the current embodiment, a working fluid is confined within the working chambers, where the working fluid receives heat from the heat input span and rejects heat to the heat output span, and a temperature drop in the isentropic expansion span is equal to a temperature rise in the isentropic compression span, where the cylindrical-quadrant spans of the thermal layer are disposed such that the previously mentioned first volume ratio and second volume ratio are equal and ensures a temperature range of the working fluid is less than a temperature difference between the heat input temperature and the heat output temperature and a specified power and efficiency is attained. Temperature differentials are required for heat to flow during heat input and heat output. 
     In one aspect of the current invention, the working chambers are a wedge shape having working chamber walls that include an outer surface of a vane hub, the thermal layer, planar surfaces of rectangular vanes slidingly fitted in the vane hub, and end closures. Here, the vane hub is eccentric to the thermal layer. 
     In another aspect of the invention, the working chambers have a cylindrical shape with working chamber walls that include a cylinder wall, a front surface of a moveable cylindrical piston disposed in the cylinder chamber and the thermal surface, where the piston is pivotably connected to a first end of a piston rod and a second end of the piston rod is disposed to pivot about an axis of a bearing post, where the bearing post is positioned eccentric to the thermal surface. 
     According to a third aspect of the invention, the working chambers have a cylindrical shape with working chamber walls that include a cylinder wall, a front surface of a cylindrical piston and the thermal layer, where a first piston is rigidly connected to a first end of a piston rod and a second end of the piston rod is rigidly connected a second piston, and where the piston rod has a bearing slot at the center of the rod for receiving a bearing post, where the bearing post is eccentric to the thermal surface. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The objectives and advantages of the present invention will be understood by reading the following detailed description in conjunction with the drawings, in which: 
         FIG. 1  shows a vane rotary heat engine according to the current invention. 
         FIGS. 2   a - 2   d  show temperature-entropy diagrams of the rotary heat engine cycle according to the current invention. 
         FIGS. 3   a - 3   b  show piston-based working chamber embodiments according to the current invention. 
         FIG. 4  shows a graph of temperature versus relative work. 
     
    
    
     DRAWINGS 
     Reference Numerals 
     
         
           100  vane rotary heat engine 
           102  hot element 
           104  working chamber 
           106  heat input port 
           108  cylindrical-quadrant isentropic expansion span (b to c) 
           110  cold element 
           112  cold input port 
           114  cylindrical-quadrant isentropic compression span (d to a) 
           116  isentropic expansion span leading edge (hot element  102  trailing edge) 
           118  isentropic expansion span trailing edge (cold element  110  leading edge) 
           120  isentropic compression span trailing edge (hot element  102  leading edge) 
           122  isentropic compression span leading edge (cold element  110  trailing edge) 
           124  rotating hub 
           126  central axis 
           128  cylindrical housing with end closures 
           130  thermally insulating liner 
           132  work delivery transmission 
           134  cylindrical-quadrant heat input span (a to b) 
           136  cylindrical-quadrant heat output span (c to d) 
           138  heat exchange cavity (in cold element  110 ) 
           140  heat exchange cavity (in hot element  102 ) 
           200  rotary heat engine cycle temperature—entropy diagrams 
           202  isothermal expansion process 
           204  isentropic expansion process 
           206  isothermal compression process 
           208  isentropic compression process 
           300  piston based working chamber 
           302  piston mechanism 
           304  pivotable independent connecting rods 
           306  eccentric post 
           308  piston chamber 
           310  rotating hub 
           312  work delivery transmission 
           320  connecting rods 
           322  centrally positioned slot 
           324  eccentric post 
           326  piston 
       
    
     DETAILED DESCRIPTION 
     Although the following detailed description contains many specifics for the purposes of illustration, anyone of ordinary skill in the art will readily appreciate that many variations and alterations to the following exemplary details are within the scope of the invention. Accordingly, the following preferred embodiment of the invention is set forth without any loss of generality to, and without imposing limitations upon, the claimed invention. 
     An efficient closed cycle rotary heat engine is described that uses a known constant-temperature heat input and a known constant-temperature heat output for providing work output. Referring to the figures,  FIG. 1  shows an exemplary vane rotary heat engine  100  according to the current invention. The closed cycle rotary heat engine  100  has a thermal cycle that includes a cylindrical-quadrant heat input span  134 , shown spanning from position (a) to position (b), where the working fluid in a working chamber  104  undergoes an isothermal expansion as heat is provided by a hot element  102  with a known constant temperature heat source (not shown) through at least one heat input port  106 . Here it is understood that a plurality of heat input ports  106  to the hot element  102  is within the scope of the invention. Further shown is a cylindrical-quadrant isentropic expansion span  108  spanning from position (b) to position (c), where the working fluid in the working chamber  104  undergoes isentropic expansion without additional energy provided to the working fluid within the working chamber  104 . Additionally, a cylindrical-quadrant heat output span  136  is shown spanning from position (c) to position (d), where heat is removed from the working fluid in the working chamber  104  by a cold element  110  with a known constant temperature cold source (not shown) via at least one cold input port  112 . Here it is understood that a plurality of cold input ports  112  to the cold element  110  is within the scope of the invention. Further shown is a cylindrical-quadrant isentropic compression span  114  spanning from position (d) to position (a), where isentropic compression of the working fluid in the working chamber  104  continues without any additional energy removed. As described,  FIG. 1  defines four processes: isothermal expansion, isentropic expansion, isothermal compression and isentropic compression. Working fluid is confined within variable volume movable working chambers  104  of the system for acting on a work delivery transmission  132 . The working fluid receives heat from the hot element  102  and rejects heat to the cold element  110 , and the temperature drop in the isentropic expansion is equal to the temperature rise in the isentropic compression. 
     In the current invention, efficiency is achieved by setting the absolute value of the ratio of the volume of the working chamber  104  when positioned at the isentropic expansion zone trailing edge  118  to the volume of the working chamber positioned at the isentropic expansion zone leading edge  116  equal to the absolute value of the ratio of the volume of the working chamber  104  positioned at the isentropic compression zone leading edge  122  to the volume of the working chamber positioned at the isentropic compression zone trailing edge  120 . Providing a known constant hot element  102  temperature and a known constant cold element  110  temperature enables the arc-spans across the isentropic zones to be determined and the chamber volume ratios may be made equal for optimizing engine efficiency. 
     Some known constant heat input sources include geothermal, nuclear and fossil fuels, where some known constant cooling output sources include large bodies of water and radiators coupled to large heat sinks, to name a few. 
     Further shown in  FIG. 1 , the variable volume working chambers  104  are coupled to a rotating hub  124  affixed to a work delivery transmission  132 , eccentric to a central axis  126  by a value (E). The working chambers  104  contain a confined, pressurized working fluid or gas such as helium, nitrogen, air or other gas having relatively high thermal conductivity. 
     In the closed-cycle system  100  of the current invention, the working fluid temperature is determined from the known values of the hot element  102  temperature and the cold element  110  temperature. Specifically, it is desirable to determine the working fluid temperature when the net heat is maximum, where the net heat of the system is the difference of the heat added H A =t h (S 2 −S 1 ) and the heat rejected H R =t 1 (S 2 −S 1 ) such that the net heat is H N =(t h −t 1 )(S 2 −S 1 ). Here, t h  is the working fluid high temperature, t 1  is the working fluid low temperature, S 1  is the entropy across the isentropic compression zone  114  beginning at the trailing edge  122  of the cold element  110  and ending at the leading edge  120  of the hot element  102 , S 2  is the entropy across the isentropic expansion zone  108  beginning at the trailing edge  116  of the hot element  102  and ending at the leading edge  118  of the cold element  110 . 
     From this, the system efficiency is equal to the ratio of the net heat divided by the heat added: 
               e   =           H   A     -     H   R         H   A       =         (       t   h     -     t   l       )     ⁢     (       S   2     -     S   1       )           t   h     ⁡     (       S   2     -     S   1       )             ,         
which simplifies to
 
             e   =           t   h     -     t   l         t   h       .           
The heat added and heat rejected can be expressed using thermodynamic principles that show the change in heat in a material is equal to the specific heat of the material multiplied by the mass, and the change in temperature e.g. ΔQ=c i mΔT. This can be expressed using the previously defined terms: H A =a(T H −t h ) and H R =b(t 1 −T L ). The coefficients (a) and (b) relate to the heat transfer between the working fluid and the hot element  102  and cold element  110  (T H  and T L , respectively) where the working fluid has a known mass and the hot element  102  and cold element  110  have specific heat transfer properties and surface areas.
 
     The efficiency can now be expressed as 
             e   =           a   ⁡     (       T   H     -     t   h       )       -     b   ⁡     (       t   l     -     T   L       )           a   ⁡     (       T   H     -     t   h       )         .           
The right side of that equation can be set equal to the right side of the previous equation
 
             e   =         t   h     -     t   l         t   h             
so that the temperatures of the hot working fluid and cold working fluid can be expressed in terms of each other, that is
 
                 t   h     =             at   l     ⁢     T   H             (     a   +   b     )     ⁢     t   l       -     bT   L         ⁢           ⁢   and   ⁢           ⁢     t   l       =         bt   h     ⁢     T   L             (     a   +   b     )     ⁢     t   h       -     aT   H             ,         
respectively.
 
     The net heat is expressed in a useful form, where H N =H A −H R =a(T H −t h )−b(t 1 −T L ), and substituting for t 1  provides the expression 
     
       
         
           
             
               H 
               N 
             
             = 
             
               
                 aT 
                 H 
               
               - 
               
                 at 
                 h 
               
               - 
               
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     t 
                     h 
                   
                   ⁢ 
                   
                     T 
                     L 
                   
                 
                 
                   
                     
                       ( 
                       
                         a 
                         + 
                         b 
                       
                       ) 
                     
                     ⁢ 
                     
                       t 
                       h 
                     
                   
                   - 
                   
                     aT 
                     H 
                   
                 
               
               + 
               
                 
                   bT 
                   L 
                 
                 . 
               
             
           
         
       
     
     To determine the maximum net heat, the derivative is set to zero, that is 
                   ⅆ     H   N         ⅆ     t   h         =   0     ,         
or
 
                 ⅆ     H   N         ⅆ     t   h         =       a   +           (         (     a   +   b     )     ⁢     t   h       -     aT   H       )     ⁢     (       b   2     ⁢     T   L       )       -       (       b   2     ⁢     t   h     ⁢     T   L       )     ⁢     (     a   +   b     )             (         (     a   +   b     )     ⁢     t   h       -     aT   H       )     2         =             a   ⁡     (         (     a   +   b     )     ⁢     t   h       -     aT   H       )       2     +       (         (     a   +   b     )     ⁢     t   h       -     aT   H       )     ⁢     (       b   2     ⁢     T   L       )       -       (       b   2     ⁢     t   h     ⁢     T   L       )     ⁢     (     a   +   b     )             (         (     a   +   b     )     ⁢     t   h       -     aT   H       )     2       =   0.             
Expressing this as a quadratic equation:
 
                 t   h   2     -       (       2   ⁢     aT   H         (     a   +   b     )       )     ⁢     t   h       +     (           a   2     ⁢     T   H   2       -       b   2     ⁢     T   H     ⁢     T   L             (     a   +   b     )     2       )       =   0.         
Solving for the working fluid temperature t h  when the net heat H N  is maximum gives
 
                 t   h     =         aT   H     +     b   ⁢         T   H     ⁢     T   L               a   +   b         ,         
and
 
               t   h     =           aT   H     -     b   ⁢         T   H     ⁢     T   L               a   +   b       .           
t h  must be greater than the value where t h =t 1 . Previously, an equation was shown where t 1  was expressed in terms of t h . So, substituting t h  for t 1  in that equation gives:
 
               t   h     =           bt   h     ⁢     T   L             (     a   +   b     )     ⁢     t   h       -     aT   H         .           
Solving the equation for t h  results in:
 
                 t   h     =         aT   H     +     bT   L         (     a   +   b     )         ,         
the value where t h =t 1 . Since t h  must be greater than t 1 , the equation
 
               t   h     =         aT   H     +     b   ⁢         T   H     ⁢     T   L               a   +   b             
is the only root that qualifies. The equation for the maximum net heat is derived by substituting the right side of the equation for t h  in the equation for the net heat, giving:
 
               H   N   Max     =           abT   H     -     2   ⁢   ab   ⁢         T   H     ⁢     T   L           +     abT   L         (     a   +   b     )       .           
The relative work, W R , is provided as
 
               W   R     =       H   N       H   N   Max             
or
 
               W   R     =           aT   H     -     at   h     -         b   2     ⁢     t   h     ⁢     T   L             (     a   +   b     )     ⁢     t   h       -     aT   H         +     bT   L             abT   H     -     2   ⁢   ab   ⁢         T   H     ⁢     T   L           +     abT   L         (     a   +   b     )         .           
Solving for t h  gives the following quadratic equation:
 
( a   2 +2 ab+b   2 ) t   h   2  
 
−(2 a   2   T   H   +abT   H   +abT   L +2 abT   H   +b   2   T   H   +b   2   T   L   −abW   R   T   H  
 
+2 abW   R √{square root over ( T   H   T   L )}−abW R   T   L   −b   2   W   R   T   H +2 b   2   W   R √{square root over ( T   H   T   L )}−b 2   W   R   T   L ) t   h  
 
+( a   2   T   H   2   +abT   H   T   L   +abT   H   2   +b   2   T   H   T   L   −abW   R   T   H   2 +2 abW   R   T   H √{square root over ( T   H   T   L )}−abW R   T   H   T   L )=0
 
     As previously determined, 
               t   h     =         aT   H     +     b   ⁢         T   H     ⁢     T   L               a   +   b             
provides the temperature t h  when the net heat H N  is maximum, thus
 
                 H   N       H   N   Max       =   1.         
Because H N  is equivalent to the net work, H N   Max  is equivalent to the maximum net work, W N   Max , where the relative net work W R  is also equal to one at that point.
 
     The variables a, b, T H , T L  and W R  must be known to determine t h . Assuming that a, b, T H  and T L  are known, values for W R  can be chosen from 0 to 1. 
     Referring again to the drawings,  FIG. 1  shows the vane rotary heat engine  100  including a housing  128  of cylindrical shape with a concentric thermal layer abutting its inner surface. The thermal layer includes a thermally insulating liner  130  with an embedded hot element  102  and an embedded cold element  110 . The inside surface of the thermal layer provides a cylindrical-quadrant heat input span  134 , a cylindrical-quadrant isentropic expansion span  108 , a cylindrical-quadrant heat output span  136 , and a cylindrical quadrant isentropic compression span  114 . 
     The outer surface of the thermally insulating liner abuts the inner surface of the housing  128 . The inner surface of the thermally insulating liner  130  provides the cylindrical-quadrant isentropic expansion span  108  with an arc-length, set for a predetermined temperature drop of the working fluid, that spans from the isentropic expansion span leading edge  116  to the isentropic expansion span trailing edge  118 . The thermally insulating layer  130  further provides the cylindrical-quadrant isentropic compression span  114  that extends concentrically with an arc-length, set for a predetermined temperature rise of the working fluid, spanning from the isentropic compression span leading edge  122  to the isentropic compression span trailing edge  120 , where the absolute value of the temperature drop across the cylindrical-quadrant isentropic expansion span  108  is equal to the absolute value of the temperature rise across the cylindrical-quadrant isentropic compression span  114 . The thermally insulating liner  130  is made from material having properties low in thermal conductivity, such as plastic, ceramic or glass and can be formed or machined to required mechanical tolerances. The insulating liner  130  isolates the hot element  102  and the cold element  110  from each other and from the cylindrical housing  128  confining the heat flow from the thermally conductive hot element  102  to the working fluid and from the working fluid to the thermally conductive cold element  110 , providing higher efficiency. It is desirable that all parts of the heat engine, except for the hot element  102  and the cold element  110 , have low thermal conductivity for maximum efficiency. 
     The thermally conductive hot element  102  is of cylindrical-quadrant shape and is positioned between the isentropic zones  108 / 114  having a hot element  102  leading edge  120  and a hot element  102  trailing edge  116  with at least one hot element  102  heat input port  106  extending there through. The outer surface of the hot element  102  abuts an inner surface of the thermally insulating liner  130  and an inner surface of the hot element  102  providing an isothermal cylindrical-quadrant heat input span  134  substantially flush with the cylindrical-quadrant isentropic spans  108 / 114 . According to one embodiment, the hot element  102  can further have a plurality of heat exchange cavities  140  (only one is shown) extending radially into the inner surface of the hot element  102 . 
     A thermally conductive cold element  110  has a cylindrical-quadrant shape positioned between the isentropic spans  108 / 114  having a cold element  110  leading edge  118  and a cold element  110  trailing edge  122  with at least one cold input port  112  extending there through. The outer surface of the cold element  110  abuts the inner surface of the thermally insulating liner  130  and the inner surface of the cold element  110  providing an isothermal compression span substantially flush with the isentropic spans  108 / 114 . According to one embodiment, the cold element  110  further has a plurality of heat exchange cavities  138  (only one is shown) extending radially into the inner surface of the cold element  110 . 
     The heat exchange cavities  138  and  140  enhance heat flow from the hot element  102  to the working fluid and from the working fluid to the cold element  110 . As an example, if one half of the surface area is provided with holes having a depth equal to four times their diameter, the heat transfer area becomes approximately nine times as great, a considerable increase in that case. It should be noted that the heat exchange cavities  138  and  140  should not intersect the heat input ports  106  and the cold input ports  112 , since the working fluid must remain confined. It is important that the heat exchange cavities  138  and  140  not be open to more than one working chamber  104  at a time. 
       FIG. 2(   a ) through  FIG. 2(   d ) show rotary heat engine cycle diagrams  200  according to the current invention. Shown are rectangles of the four thermodynamic processes plotted on a temperature-entropy diagram, where the cycle progresses in the clockwise direction. The ordinate is temperature (T) and the abscissa is entropy (S), where the abscissa is shown in broken lines to illustrate that the absolute values of the entropy are unknown and only differences in entropy can be determined (T H ) is the temperature of the hot element  102 , and (T L ) is the temperature of the cold element  110 . As shown, the rotary heat engine cycle has the four processes: isothermal expansion  202  from point (a) to point (b), isentropic expansion  204  from point (b) to point (c), isothermal compression  206  from point (c) to point (d) and isentropic compression  208  from point (d) to point (a) to complete the cycle. In the isothermal expansion  202 , work is performed on the working chamber  104  by the expanding working fluid as heat is added at temperature (T H ) to the working fluid. Here, the working fluid expands while maintaining constant high temperature (t h ). This expansion of the working fluid is converted into mechanical work as an eccentric rotating hub  124  (see  FIG. 1) , for example, rotates to turn a work delivery transmission  132  extending from inside to outside of the closed cycle heat engine. 
     During isentropic expansion  204 , work is further performed on the working chamber  104  by the expanding working fluid as the hub  124  moves the working chamber  104  across the isentropic expansion zone  204  from point (b) to point (c). Here, work is exchanged for a temperature reduction in the working fluid to a low temperature (t 1 ) from point (b) to point (c). 
     In the isothermal compression  206  from point (c) to point (d), the working fluid is compressed and heat is removed to the cold element  110  at temperature (T L ) while maintaining the working fluid temperature (t 1 ). 
     In the isentropic compression  208 , work is required in exchange for heating the working fluid to temperature (t h ) as the rotating hub  124  moves the working chamber  104  across the isentropic compression zone from point (d) to point (a) to complete the cycle. 
     The ratio of the change in chamber volumes across the isentropic zones  204 / 208  are made equal to ensure that the absolute value of the temperature drop from point (b) to point (c) is equal to the absolute value of the temperature rise from point (d) to point (a). The linear and angular dimensions, eccentricities and extents of the various components are adjusted to provide the required volume ratios that optimize the system. 
     The difference in the work performed and the work required is the net work available to overcome friction and to power external devices of the system. Further, the net work correlates to the difference between the heat added and the heat removed by the hot element  102  and cold element  110 , respectively. In  FIG. 2(   b ), the crosshatch area below the isothermal expansion  202  represents the heat added to the system. In  FIG. 2(   c ), the crosshatch area below the isothermal compression  206  represents the heat removed from the system. In  FIG. 2(   d ), the net heat is the difference between the heat added and the heat removed, represented by the area enclosed within the full-cycle rectangle. 
     The heat energy added (H A ) is the product of the working fluid high temperature (t h ) and the change in entropy from point (a) to point (b). Similarly, the heat energy removed (H R ) is the product of the working fluid low temperature (t 1 ) and the change in entropy from point (c) to point (d). The net heat energy (H N ) is the heat energy added less the heat energy rejected. The efficiency (e) of the current invention is the ratio of net heat energy (H N ) to the heat energy added to the system (H A ). The current invention provides an optimized rotary heat engine efficiency when the net heat energy (H N ) is a known value. 
       FIG. 3(   a ) and  FIG. 3(   b ) show a piston-based working chamber  300  embodiment of the current invention. Shown in  FIG. 3(   a ) are piston mechanisms  302  having pivotable independent connecting rods  304  that are contained within piston chambers  308  of the rotating hub  310 , where the connecting rods  304  are pivotably connected to the piston  302 . The connecting rods  304  are rotatably connected to an eccentric post  306  projecting from one end closure of the cylindrical housing  128  and eccentrically positioned relative to the center axis of the cylindrical-quadrant heat input span  134 , the cylindrical-quadrant isentropic expansion span  108 , the cylindrical-quadrant heat output span  136 , and the cylindrical-quadrant isentropic compression span  114 , as discussed above. The work delivery transmission  312 , attached to the rotating hub, projects through the other end closure. 
     In another embodiment,  FIG. 3(   b ) shows another piston-based working chamber  300 , where the rods  320  are non-pivotable having a centrally positioned slot  322  where an eccentric post  324  is disposed in the slot  322 . Opposing pistons  326  are connected at each end of the rod  320 . As the heat is exchanged, the pistons  326  operate on the slots  322  of the rods  320  that move about the eccentric post  324  to provide work for output through the work delivery transmission  312 . 
       FIG. 4  is a graph plotted using results from the included equations. The value of b is assumed to be 1.5 times the value of a. Values of T H  and T L  are assumed to be 1500 degrees Rankine and 500 degrees Rankine, respectively. Values of t h  and t 1  are plotted to form the curves. 
     The graph shows the value of t h  to be 1120 degrees Rankine and the value of t 1  to be 646 degrees Rankine when the net work is maximum. With those values, it is seen that the efficiency is equal to 42 percent. 
     Efficiency can be increased by increasing the value of t h , with a corresponding decrease in the value of t 1 . For example, when t h  is assigned the value of 1437 degrees Rankine, the corresponding value of t 1  is 515 degrees Rankine. The efficiency with those values is seen to be 64 percent. However, the power will be less, since the work relative is seen to be 25 percent of the maximum. 
     It is seen that all values of t h  must be less than T H  in order for heat to flow, and all values of t 1  must be greater than T L  in order for heat to flow. Of course, t 1  must be less than t h . They become equal with the chosen parameters at a temperature of 900 degrees Rankine. At that point, of course, the efficiency is zero. 
     Efficiency is maximum when t h =T H  and t 1 =T L . However, although the efficiency equals the Carnot cycle efficiency of 67 percent, the Net Work is zero. Although it seems contradictory, it should be understood that the efficiency is a limit. No heat engine can operate at that efficiency with the chosen parameters. So it is with the Carnot cycle. No engine can operate with the efficiency defined by the Carnot cycle. 
     Other graphs similar to  FIG. 4  can be developed by varying the parameters a, b, T H , and T L . 
     Assuming that it is desired to operate the engine at maximum power with the above parameters, th equals 1120 degrees Rankine and t1 equals 646 degrees Rankine. Using air as the working fluid and the thermodynamic equation (t2/t1=(V1/V2) k-1 ), the volume ratio can be determined. For air, the specific heat ratio, k, equals 1.40. The equation can be rewritten as (V1/V2=(t2/t1) 1/(k-1) ). Letting t2=1120 and t1=646, the volume ratio (V c /V b )=(V d /V a )=(1120/646) 2.5 =3.958. The various dimensional parameters would need to be manipulated to give that volume ratio. 
     It should be noted that all parts, except the hot element  102  and the cold element  110 , of the engine should, desirably, have low thermal conductivity so that maximum heat is transferred from the hot element  102  to the working fluid and from the working fluid to the cold element  110  in order to maximize the thermal efficiency. Also, power can be varied by increasing or decreasing the amount of working fluid within the engine, thereby increasing or decreasing the pressure and heat transfer to and from the working fluid. The means for increasing or decreasing the amount of the working fluid is not shown, since there are many ways of accomplishing that. 
     The present invention has now been described in accordance with several exemplary embodiments, which are intended to be illustrative in all aspects, rather than restrictive. Thus, the present invention is capable of many variations in detailed implementation, which may be derived from the description contained herein by a person of ordinary skill in the art. For example in reverse mode, by manipulating the various parameters, the invention is a refrigerator engine for removing heat from a body. Heat is absorbed by the working fluid from the cool zone and rejected to the heat zone. 
     All such variations are considered to be within the scope and spirit of the present invention as defined by the following claims and their legal equivalents.