Patent Publication Number: US-2020300512-A1

Title: Refrigerating method and apparatus

Description:
CLAIM OF PRIORITY 
     The present patent application is a non-provisional of, and claims the benefit of priority of U.S. Provisional Patent Application No. 62/858,986 filed on Jun. 8, 2019, U.S. Provisional Patent Application No. 62/872,258 filed on Jul. 10, 2019, each of which is hereby incorporated by reference herein in its entirety. 
    
    
     FIELD 
     The present invention relates to apparatuses and methods for refrigeration, heating, power generation, and propulsion. 
     BACKGROUND 
     Heat typically flows from a hot thermal reservoir to a cold thermal reservoir when these two thermal reservoirs are in thermal contact with each other. This heat can be transferred via conduction, for instance. 
     A conventional heat pump requires mechanical work to be done in order to transfer heat from a cold reservoir to a hot reservoir. For example, a conventional refrigerator consumes electricity in order to remove heat from the cold interior and deliver heat to the warm exterior, such as the room in which the refrigerator is located. 
     A conventional heat engine performs mechanical work by absorbing heat from a hot reservoir and transferring heat to a cold reservoir. For example, in a marine steam engine, the working material absorbs heat from a hot reservoir in the boiler, and subsequently performs mechanical work, e.g. on a steam turbine, whereupon the steam transfers heat to a cold reservoir, e.g. the ocean, in the condenser. 
     It would be desirable to employ devices which can directly convert thermal energy into useful mechanical work. 
     SUMMARY 
     By subjecting a volume or a bulk of a working material to a body force per unit mass, such as gravity, inertial forces, electric forces, or magnetic forces, the perceived specific heat capacity of the volume of the working material can be increased or decreased as desired. The artificial modification of the perceived macroscopic specific heat capacity of a material can be employed in a thermodynamic cycle to convert thermal energy directly into useful mechanical work, and vice versa. The entropy of a working fluid can be increased and decreased as desired. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a cross-sectional view one exemplary embodiment of the invention during an exemplary nominal operating condition.  FIG. 1A  also contains schematic plots of pressure versus position along the Y-axis at corresponding points along the X-axis within channel  722  of exemplary embodiment  720 . This embodiment can be described as a ramjet. 
         FIG. 1B  is a cross-sectional view of the embodiment shown in  FIG. 1A  when viewed along the negative X-direction. 
         FIG. 1C  is a cross-sectional view of the embodiment shown in  FIG. 1A  when viewed along the negative X-direction. 
         FIG. 2  is a cross-sectional view one exemplary embodiment of the invention during an exemplary nominal operating condition.  FIG. 2  also contains schematic plots of pressure versus position along the Y-axis at corresponding points along the X-axis within channel  792  of exemplary embodiment  790 . This embodiment can also be described as a ramjet. 
         FIGS. 3A-B  each show a cross-sectional view of an exemplary embodiment of the invention during an exemplary nominal operating condition. This embodiment can be described as a subsonic and supersonic ramjet. 
         FIGS. 4A-C  schematically show a cross-sectional view of exemplary embodiments of individual rotor discs and associated rotor blades of exemplary embodiments of the invention, such as the embodiments shown in  FIGS. 3A-B ,  FIGS. 5A-B , and  FIGS. 6A-B . 
         FIGS. 5A-B  each show a cross-sectional view of an exemplary embodiment of the invention during an exemplary nominal operating condition. This embodiment can be described as a subsonic and supersonic ramjet. 
         FIGS. 6A-B  each show a cross-sectional view of an exemplary embodiment of the invention during an exemplary nominal operating condition. This embodiment can be described as a turbojet engine, or the core of a turbofan engine, for example. 
         FIGS. 7A-J  schematically show cross-sectional views of embodiments of the invention at different points in time during an exemplary nominal operating condition. This embodiment can be considered to comprise a rotating radial engine, or a rotary engine, in which the working material does work against the piston. 
         FIG. 8  shows a plot of pressure versus specific volume for the working material in a subset of embodiments of the invention for an example method of operation, such as the example method of operation shown in  FIGS. 7A-J , or  FIGS. 9A-C . 
         FIG. 9A  shows a cross-sectional view of an exemplary embodiment of the invention employing the principles described in the context of  FIGS. 7A-J  and  FIG. 8 . This embodiment can be considered to comprise a rotating radial engine, or a rotary engine, as well as two centrifugal compressors being driven by the main drive shaft. 
         FIGS. 9B-C  show cross-sectional views of components of a coaxial differential shown in  FIG. 9A . 
         FIGS. 10A-K  schematically show cross-sectional views of embodiments of the invention at different points in time during an exemplary nominal operating condition. This embodiment can be considered to comprise a rotating radial engine, or a rotary engine, in which the piston does work on the working material. 
         FIG. 11  shows a plot of pressure versus specific volume for the working material in a subset of embodiments of the invention for an example method of operation. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1A  is a cross-sectional view of some embodiments of the invention. The exemplary embodiment  720  shown is cylindrically symmetric about an axis parallel to the X-axis and coincident with the center of exemplary embodiment  720 . Outside surface  737  is therefore the shape of a tapered cylinder. In other embodiments, outside surface  737  can be elliptical, rectangular, square, for instance. 
     Exemplary embodiment  720  comprises a channel  722  with inside surface  739  located between a first opening  723  and a second opening  728 , where the channel comprises a first contraction  724 , a first expansion  725 , a second contraction  726 , and a second expansion  727 . The cross-sectional geometry of channel  722  is circular when viewed along the X-direction. Note that the terms “contraction” and “expansion” refer to the magnitude of the radius of the axially symmetric channel. Note that the channel radius or geometry can change in a different manner as a function of position along the X-axis, or be configured differently, for other embodiments, or other operating conditions. For example, in other embodiments, the cross-sectional geometry of channel  722  can be annular or ring-shaped. In other embodiments the cross-sectional geometry of channel  722  or outside surface  737  can be square or rectangular. In other embodiments, the cross-sectional geometry of channel  722  or outside surface  737  can be polygonal, such as pentagonal, hexagonal. In some embodiments, the cross-sectional geometry of channel  722  can change from square to circular in the positive X-direction, for example. 
     Bulk material  721  can comprise a metal such as aluminium or titanium. Bulk material  721  can also comprise ceramics. In some embodiments, bulk material  721  comprises composites, such as carbon fiber or fiberglass. Bulk material  721  can also comprise electrical insulators such as glass. 
     Note that the apparatus contained within inside surface  739  and outside surface  737  does not have to be a solid material, but can contain empty or open spaces in order to not unnecessarily increase the mass or cost of exemplary embodiment  720 . 
     In  FIG. 1 , exemplary embodiment  720  moves with constant velocity magnitude and direction relative to a working material in the free stream. The velocity direction of the upstream working material relative to exemplary embodiment  720  is aligned with the X-axis on average, i.e. directed from the left of the page to the right of the page. For clarity of description, the velocity magnitude and direction of the upstream working material relative to exemplary embodiment  720  is assumed to be constant in space and time. In other modes of operation, the upstream relative velocity magnitude and direction need not be constant in space or time. For example, the upstream relative velocity magnitude can increase or decrease as a function of time. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. A working material can also be a liquid such as water. Note that water is compressible, although it is often treated as incompressible. In the embodiment shown in the figures, the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability are explained herein. 
     The working material upstream of exemplary embodiment  720 , such as at station  729 , is moving faster relative to exemplary embodiment  720  than the speed of sound in the working material in the configuration shown in  FIG. 1A . Both the first contraction  724  and the first expansion  725  of channel  722  are configured to decelerate and compress the working material flowing through channel  722  in the positive X-direction relative to exemplary embodiment  720 . The first throat  730  is defined to be the portion of channel  722  with the smallest cross-sectional area of channel  722  between first contraction  724  and first expansion  725  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  720  at the first throat  730  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  729 , the average relative speed is larger than the speed of sound, and further downstream, such as at station  731 , the average relative speed is smaller than the speed of sound within the working material in this embodiment. In some embodiments, there can be a shock wave located between the first throat  730  and station  731 . In other words, the relative flow speed of the working material downstream of the first throat  730  can be faster than the speed of sound within the working material, where the relative flow speed is reduced to a speed slower than the speed of sound throughout the shock wave. The compression of working material between stations  729  and  731  can be described as a substantially adiabatic compression in this embodiment, where the compression is adiabatic in the sense that no heat is exchanged between the working material in channel  722  and the outside environment in this idealized scenario. As explained below, the adiabatic compression between station  729  and  731  is not isentropic, even in the absence of a shock wave between station  730  and  731 . The specific entropy of the working material is reduced between stations  729  and  733  for the depicted operating condition. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  729  and  733  compared to the nominal specific heat capacity of the gas. 
     In other embodiments, the compression between stations  729  and  731  can comprise heat transfer from or to the working material. In other embodiments, this compression can at least in part be carried out by an axial compressor, such as an axial compressor found in conventional jet engines. In other embodiments, this compression can at least in part be carried out by a centrifugal compressor, for instance. In some such embodiments, the working material upstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. 
     Both the second contraction  726  and the second expansion  727  of channel  722  are configured to expand the working material flowing through channel  722  in the positive X-direction. The second throat  732  is defined to be the portion of channel  722  with the smallest cross-sectional area of channel  722  between second contraction  726  and second expansion  727  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  720  at the second throat  732  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  731 , the average relative speed is smaller than the speed of sound, and downstream, such as at station  733 , the average relative speed is larger than the speed of sound within the working material in this embodiment. The expansion of working material between stations  731  and  733  can be described as a substantially adiabatic expansion in this embodiment, where the expansion is adiabatic in the sense that no heat is exchanged between the working material in channel  722  and the outside environment in this idealized scenario. As explained below, the adiabatic expansion between station  731  and  733  is not isentropic. 
     In other embodiments, the expansion can comprise heat transfer from or to the working material. In other embodiments, this expansion can at least in part be carried out by an axial turbine, such as an axial turbine found in conventional jet engines. In other embodiments, this expansion can at least in part be carried out by a centrifugal turbine, for instance. In some such embodiments, the working material downstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. 
     Dashed lines  735  and  736  indicate stagnation streamlines which are incident on the leading edge or originate at the trailing edge of exemplary embodiment  720 . Streamlines  735  and  736  are therefore part of a streamsurface, or streamtube, which separate working material flowing around exemplary embodiment  720  from working material flowing through channel  722  of exemplary embodiment  720 . In this embodiment, the steamtube is circular in cross-section when viewed along the X-direction. The flow direction of the working material relative to exemplary embodiment  720  is indicated by arrow  769 . 
     A first body force per unit mass generating apparatus, or a first “BFGA”,  740  is located adjacent to channel  722 . First BFGA  740  is configured to be able to apply at least one body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this body force can be regulated in this embodiment. The first BFGA  740  comprises a first charge collection  741  and a second charge collection  742 . In the configuration shown, first charge collection  741  is negatively charged, and second charge collection  742  is positively charged. In other embodiments, the polarity of the charge in the charge collections can be reversed, i.e. a first charge collection  741  is positively charged, and a second charge collection  742  is negatively charged. The cross-section of first charge collection  741  is annular or ring-shaped when viewed along the X-direction. First charge collection  741  encloses channel  722 . First charge collection  741  is electrically insulated from the working material in channel  722  by an electrical insulator such as glass, ceramic, or plastic, in this embodiment. In other embodiments, the first charge collection  741  need not be electrically insulated from the working material in channel  722 . 
     Second charge collection  742  is circular in cross-section when viewed along the X-direction. Second charge collection  742  is electrically insulated from the working material in channel  722  by an electrical insulator such as glass, ceramic, or plastic, in this embodiment. In other embodiments, the second charge collection  742  need not be electrically insulated from the working material in channel  722 . Second charge collection  742  is located within an elongated cylindrical body at the center of channel  722  in this embodiment, and at least in part configured to reduce any drag forces acting on the second charge collection  742  due to the motion of the working material around the second charge collection  742 . Second charge collection  742  is structurally supported by two support beams, such as support beam  743 . The support beams are configured to rigidly connect the second charge collection  742  to the inside wall surface  739  of exemplary embodiment  720 . The streamwise geometry of the support beams is streamlined in order to reduce and drag forces acting on the support beams due to the motion or flow of the working material around the support beams. In other embodiments, there can be only one support beam supporting second charge collection  742 . In other embodiments, there can be a plurality of support beams supporting second charge collection  742 , such as three or four support beams. 
     A third BFGA  751  is configured in a similar manner as first BFGA  740 , and will therefore not be described in the same detail as first BFGA  740 . Third BFGA  751  comprises a first charge collection  752  configured in a similar manner as first charge collection  741  of first BFGA  740 . Third BFGA  751  comprises a second charge collection  753  configured in a similar manner as second charge collection  742 . Second charge collection  753  is structurally supported by two support beams. 
     The first BFGA  740  and third BFGA  751  are configured to generate a body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  722 , where the body force comprises a non-zero component in the YZ-plane and directed towards the center of channel  722 , i.e. towards the X-axis. The action of the body force per unit mass reduces the pressure on at least a portion of interior surface  739  throughout the first contraction  724  or the second contraction  726 , thereby reducing the retarding force, or drag force, acting on the exemplary embodiment  720  in the positive X-direction. This is due to the surface normal of the interior surface  739  having a component in the negative X-direction throughout the first contraction  724  or the second contraction  726 . An artificial reduction in pressure on surfaces with a surface normal which has a non-zero component in the negative X-direction can be employed to artificially reduce the retarding force, or drag force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The direction of the body force per unit mass acting on objects of the working material within channel  722  is indicated by bold arrows, such as bold arrow  772  or bold arrow  773  in  FIG. 1A . In some embodiments, the component of the body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the body force acting on the working material within the first contraction  724  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material is carried out by the first BFGA  740 . In another example, the body force acting on the working material within the second contraction  726  can comprise a component in the positive X-direction. In this case, at least a portion of the expansion and acceleration of the working material is carried out by the third BFGA  751 . 
     Due to the action of the body force per unit mass within the first contraction  724  and the second contraction  726 , the pressure within the working material within the first contraction  724  and the second contraction  726  decreases in a radially outwards direction, as indicated by line  762  and line  766  in the plot of pressure versus position along the Y-axis at the corresponding point along the X-axis, i.e. at a point in the first contraction  724  and in the second contraction  726 . A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. The change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic expansion along the radial direction in highly simplified, idealized models, for example. 
     In the first contraction  724  and the second contraction  726  the first BFGA  740  and third BFGA  751  are configured to electrically polarize atoms or molecules within the working material. The atoms or molecules can be polarized by an application of an external electric field, for instance. The first BFGA  740  and third BFGA  751  are also configured to exert a force on these polarized molecules, where the force arises from a spatial or temporal gradient in the electric field, or a spatially or temporally non-uniform electric field strength. For instance, the magnitude of the electric field strength produced by the first BFGA  740  or third BFGA  751  can increase in a radially decreasing direction, i.e. in a direction perpendicular to the X-axis and directed towards the X-axis. For instance, the positive or negative radially outward component of the electric field can decrease, i.e. become less positive or more negative, in the radially increasing direction in the case in which the electric polarization axis of the objects within the working material comprises a non-zero component the positive radial direction. For instance, the positive or negative radial component of the electric field can increase, i.e. become less negative or more positive, in the radially increasing direction in the case in which the electric polarization axis of the objects within the working material comprises a non-zero component the negative radial direction. Note that the polarization axis of a polarized molecule typically features a large component in the direction of the local electric field. This can result in a body force per unit mass acting in the negative radial direction, i.e. towards the X-axis, as indicated by the bold arrows in  FIG. 1A  in the first contraction  724  and the second contraction  726 . 
     A second BFGA  746  is located adjacent to channel  722 . Second BFGA  746  is configured to be able to apply at least one body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this body force can be regulated in this embodiment. The second BFGA  746  comprises several insulated collections of charge, such as charge collection  747 , or charge collection  748 . The longitudinal axis of each elongated collection of charge is aligned in the streamwise direction. Individual, electrically insulated collections of charge are arranged adjacent to each other in an annular or circumferential fashion around channel  722 , as shown in  FIG. 1C  and  FIG. 1A . Adjacent collections of charge, such as collection of charge  747  and collection of charge  750 , or collection of charge  748  and collection of charge  749 , are oppositely charged. 
     In other embodiments, the individual collections of charge need not be longitudinal in a streamwise direction, but can be annular in shape around channel  722 . In such embodiments the individual charge collections can be arranged adjacent to each other in a streamwise direction. In yet other embodiments, the individual charge collections can be finite in their extent along the streamwise direction and along the circumferential direction. Adjacent collections of charge can be arranged adjacent to each other in both a streamwise direction and a circumferential or annular direction around channel  722 . As before, immediately adjacent collections of charge can comprise charge of opposite polarity. The individual collections of charge of second BFGA  746  are electrically insulated from the working material in channel  722  by an electrical insulator such as glass, ceramic, or plastic, in this embodiment. In other embodiments, the individual collections of charge need not be electrically insulated from the working material in channel  722 . In other embodiments, adjacent charge collections need not be oppositely charged, but can be of the same charge. 
     A fourth BFGA  755  is configured in a similar manner as second BFGA  746 , and will therefore not be described in the same detail as second BFGA  746 . Fourth BFGA  755  comprises a charge collection  757  configured in a similar manner as charge collection  748  of second BFGA  746 . Third BFGA several longitudinal charge collections with a longitudinal axis oriented in a streamwise direction and arranged adjacent to each other in a circumferential or annular fashion around channel  722 . Adjacent collections of charge are oppositely charged. 
     The second BFGA  746  and fourth BFGA  755  are configured to generate a body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  722 , where the body force comprises a non-zero component in the YZ-plane and directed away from the center of channel  722 , i.e. away from the X-axis or in the radially outwards direction. The action of the body force per unit mass increases the pressure on at least a portion of interior surface  739  throughout the first expansion  725  or the second expansion  727  of channel  722 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  720  in the negative X-direction. This is due to the surface normal of the interior surface  739  having a component in the positive X-direction throughout the first expansion  725  or the second expansion  727  of channel  722 . An artificial increase in pressure on surfaces with a surface normal which has a non-zero component in the positive X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The direction of the body force per unit mass acting on objects of the working material within channel  722  is indicated by bold arrows, such as bold arrow  772  or bold arrow  773  in  FIG. 1A . In some embodiments, the component of the body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the body force acting on the working material within the first expansion  725  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material is carried out by the second BFGA  746 . In another example, the body force acting on the working material within the second expansion  727  can comprise a component in the positive X-direction. In this case, at least a portion of the expansion and acceleration of the working material is carried out by the fourth BFGA  727 . 
     Due to the action of the body force per unit mass within the first expansion  725  or the second expansion  727  of channel  722 , the pressure within the working material within the first expansion  725  or the second expansion  727  of channel  722  increases in a radially outwards direction, as indicated by line  764  and line  768  in the plot of pressure versus position along the Y-axis at the corresponding point along the X-axis, i.e. at a point in the first expansion  725  and in the second expansion  727 . A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. The change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. 
     In the first expansion  725  or the second expansion  727  of channel  722  the second BFGA  746  and fourth BFGA  755  are configured to electrically polarize atoms or molecules within the working material. The atoms or molecules can be polarized by an application of an external electric field, for instance. The second BFGA  746  and fourth BFGA  755  are also configured to exert a force on these polarized molecules, where the force can arise from a spatial or temporal gradient in the electric field, or a spatially or temporally non-uniform electric field strength. For instance, the magnitude of the electric field strength produced by the second BFGA  746  and fourth BFGA  755  can increase in a radially increasing direction, i.e. in a direction perpendicular to the X-axis and directed away from the X-axis, in a radially outwards direction. For instance, the positive or negative radially outward component of the electric field can decrease, i.e. become less positive or more negative, in the radially increasing direction in the case in which the electric polarization axis of the objects within the working material comprises a non-zero component the negative radial direction. For instance, the positive or negative radial component of the electric field can increase, i.e. become less negative or more positive, in the radially increasing direction in the case in which the electric polarization axis of the objects within the working material comprises a non-zero component the positive radial direction. Note that the polarization axis of a polarized molecule typically features a large component in the direction of the local electric field. This can result in a body force per unit mass acting in the positive radial direction, i.e. away from the X-axis, as indicated by the bold arrows in  FIG. 1A  in the first expansion  725  or the second expansion  727  of channel  722 . 
     In the embodiment shown in  FIG. 1A , the amount of charge in a charge collection in a first BFGA  740 , a second BFGA  746 , a third BFGA  751 , or a fourth BFGA  755  can be regulated by charging or discharging, or reducing the charge in a charge collection. In such embodiments, the charge collections can comprise electrical conductors which are able to facilitate the accumulation of charge, or the reduction in the amount of charge contained within the conductor. In some instances in time the amount of charge in a charge collection can be configured to be zero in some of such embodiments. The charging process can comprise the application of a voltage difference across charge collections of opposite polarity, such as first charge collection  741  and second charge collection  742  of first BFGA  740 , or charge collection  747  and charge collection  750 . This voltage difference can be supplied by a battery, a capacitor, an inductor, or an electric generator, for example. Oppositely charged charge collections are electrically insulated from each other as well as from portions of bulk material  721 . Electrical conductors, such as insulated copper wires, can electrically connect a charge collection to the voltage source. These electrical conductors are not shown. In between a charge collection and the channel  722  the bulk material  721  is an electrical insulator. In effect, charge collections which are oppositely charged can be considered to be the opposite plates of a capacitor, with the dielectric in between these plates comprising the working material as well as the relevant portion of bulk material  721  between the charge collections. In the embodiment shown, the charge collections are configured in a manner in which the majority of electric field lines pass through the working material within channel  722 . To that end, an individual charge collection can comprise several insulated conductors. These conductors can be wires, for instance, and can be arranged perpendicular to the streamwise direction, or arranged parallel to a radial direction. This can serve to prevent or diminish any undesirable redistribution of charge within a charge collection. 
     In other embodiments, the amount of charge contained within a charge collection is constant in time. In such embodiments, a charge collection can comprise electrons, ions or other charged particle embedded within an electrical insulator. In some such embodiments, a separate voltage source for regulating the amount of charge in a charge collection is not required. 
       FIG. 1B  is a cross-sectional view of the embodiment shown in  FIG. 1A  when viewed along the negative X-direction.  FIG. 1B  shows the support beams of second collection of charge  742  of first BFGA  740 , such as support beam  738  and support beam  743 . The electric field lines, such as electric field line  760 , schematically indicate the direction and strength of the electric field within channel  722  at the first contraction  724 . The electric field lines are directed from the positive charge within the second collection of charge  742  towards the annular first collection of charge  741 , i.e. in a positive radially outward direction. The electric field outside of the first collection of charge  741  is not shown for clarity. Due to the annular geometry of the first collection of charge  741  and the longitudinal geometry and location of the second collection of charge  742  along the center of channel  722 , the radially outward component of the electric field decreases in a radially outward direction. Objects within the working material in channel  722  can be electrically polarized by the electric field. The polarization can be proportional to the local electric field for a subset of objects within a working material, for example, as can be the case for a working material comprising monatomic molecules, for instance. The positive radially outward component of the electric field thus decreases, i.e. become less positive, in the radially increasing direction in the case in which the electric polarization axis of the objects within the working material comprises a non-zero component the positive radial direction. This can result in a body force per unit mass acting on objects within the working material, where the body force per unit mass is directed in the radially inwards direction, towards, the X-axis, towards the region of increased electric field strength, away from interior surface  739 , and towards the second collection of charge  742 . The second collection of charge is structurally supported and electrically insulated from the working material in channel  722  by bulk material  744 . The interface between bulk material  744  and the working material is described by surface  745 . 
       FIG. 1C  is a cross-sectional view of the embodiment shown in  FIG. 1A  when viewed along the negative X-direction. As shown, several collections of charge, such as collections of charge  750 ,  747 ,  749 , or  748 , are arranged in circumferential fashion around channel  722 . The collections of charge are configured to increase the electric field strength along the radially outward direction within channel  722 . This electric field strength can electrically polarize molecules within the working material, and generate a body force per unit mass acting on the polarized molecules in the radially outward direction, away from the X-axis, towards the region of increased electric field strength, towards interior surface  739 , and towards the collections of charge or the concentrations of charge. 
       FIG. 2  is a cross-sectional view of some embodiments of the invention. The exemplary embodiment  790  shown is cylindrically symmetric about an axis parallel to the X-axis and coincident with the center of exemplary embodiment  790 . Outside surface  813  is therefore the shape of a tapered cylinder. In other embodiments, outside surface  813  can be elliptical, rectangular, square, for instance. 
     Exemplary embodiment  790  comprises a channel  792  with inside surface  815  located between a first opening  793  and a second opening  801 , where the channel comprises a first contraction  794 , a first expansion  795 , a spin-up segment  796 , a second expansion  797 , a spin-down segment  798 , a second contraction  799 , and a third expansion  800 . The cross-sectional geometry of channel  792  is circular when viewed along the X-direction. Note that the terms “contraction” and “expansion” refer to the magnitude of the radius of the axially symmetric channel. Note that the channel radius or geometry can change in a different manner as a function of position along the X-axis, or be configured differently, for other embodiments, or other operating conditions. For example, in other embodiments, the cross-sectional geometry of channel  792  can be annular or ring-shaped. In other embodiments the cross-sectional geometry of channel  792  or outside surface  813  can be square or rectangular. In other embodiments, the cross-sectional geometry of at least a portion of channel  792  or outside surface  813  can be polygonal, such as pentagonal, hexagonal. In some embodiments, the cross-sectional geometry of channel  792  can change from square to circular, or vice versa, in the positive X-direction, for example. 
     Bulk material  791  can comprise a metal such as aluminium, steel, or titanium. Bulk material  791  can also comprise ceramics. In some embodiments, bulk material  791  comprises composites, such as carbon fiber or fiberglass. Bulk material  791  can also comprise electrical insulators such as glass. 
     In some embodiments, the apparatus contained within inside surface  815  and outside surface  813  does not have to be a solid material, but can contain empty or open spaces, as is common practice in conventional ramjet or jet engine construction. This can serve to avoid an unnecessarily large mass or cost of exemplary embodiment  790 , for instance. 
     In  FIG. 2 , exemplary embodiment  790  moves with constant velocity magnitude and direction relative to a working material in the free stream. The velocity direction of the upstream working material relative to exemplary embodiment  790  is aligned with the X-axis on average, i.e. directed from the left of the page to the right of the page. For clarity of description, the velocity magnitude and direction of the upstream working material relative to exemplary embodiment  790  is assumed to be constant in space and time. In other modes of operation, the upstream relative velocity magnitude and direction need not be constant in space or time. For example, the upstream relative velocity magnitude can increase or decrease as a function of time. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. A working material can also be a liquid such as water. In the embodiment shown in the figures, the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability are explained herein. 
     The working material upstream of exemplary embodiment  790 , such as at station  802 , is moving faster relative to exemplary embodiment  790  than the speed of sound in the working material in the configuration shown in  FIG. 2 . The first contraction  794 , the second expansion  795 , and the third expansion  797  of channel  792  are configured to decelerate and compress the working material flowing through channel  792  in the positive X-direction relative to exemplary embodiment  790 . The first throat  803  is defined to be the portion of channel  792  with the smallest cross-sectional area of channel  792  between first contraction  794  and second expansion  795  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  790  at the first throat  803  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  802 , the average relative speed is larger than the speed of sound, and further downstream, such as at station  807 , the average relative speed is smaller than the speed of sound within the working material in this embodiment. In some embodiments, there can be a shock wave located between the first throat  803  and station  805 . In other words, the relative flow speed of the working material downstream of the first throat  803  can be faster than the speed of sound within the working material, where the relative flow speed is reduced to a speed slower than the speed of sound throughout the shock wave. During nominal operations a shock wave can be located within the first expansion  795 . This can prevent or reduce the probability of an unscheduled engine unstart due to turbulence or variations in the free stream flow velocity of the working material. The compression of the working material between stations  802  and  807  can be described as a substantially adiabatic compression in this embodiment, where the compression is adiabatic in the sense that no heat is exchanged between the working material in channel  792  and the outside environment in this idealized scenario. As explained below, the adiabatic compression between station  802  and  807  is not isentropic, even in the absence of a shock wave between station  803  and  807 . The specific entropy of the working material is reduced between stations  803  and  807  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  803  and  807  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. 
     In other embodiments, the compression between stations  802  and  807  can comprise heat transfer from or to the working material. For instance, in some embodiments, fuel can be added to the working material and combusted at or before station  807 , similar to a conventional ramjet. In some embodiments, there can be heat transfer from the working material to the bulk material  791  due to temperature differences. In other embodiments, the compression between stations  802  and  803  can at least in part be carried out by an axial compressor, such as an axial compressor found in conventional jet engines. In other embodiments, the compression between stations  802  and  803  can at least in part be carried out by a centrifugal compressor, for instance. In some embodiments, the working material upstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. In other words, the free stream flow can be subsonic in some embodiments for some modes of operation. 
     Both the second contraction  799  and the third expansion  800  of channel  792  are configured to expand and accelerate the working material flowing through channel  792  in the positive X-direction. The second throat  808  is defined to be the portion of channel  792  with the smallest cross-sectional area of channel  792  between second contraction  799  and third expansion  800  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  790  at the second throat  808  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  807 , the average relative speed is smaller than the speed of sound, and downstream, such as at station  809 , the average relative speed is larger than the speed of sound within the working material in this embodiment. The expansion of the working material between stations  807  and  809  can be described as a substantially adiabatic expansion in this embodiment, where the expansion is adiabatic in the sense that no heat is exchanged between the working material in channel  792  and the outside environment in this idealized scenario. In the embodiment shown in  FIG. 2 , the adiabatic expansion between station  807  and  809  can also be described as a substantially isentropic expansion. 
     In other embodiments, the expansion between station  807  and  809  can comprise heat transfer from or to the working material. In other embodiments, this expansion can at least in part be carried out by an axial turbine, such as an axial turbine found in conventional jet engines. In other embodiments, this expansion can at least in part be carried out by a centrifugal turbine, for instance. In some such embodiments, the working material downstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. 
     Dashed lines  811  and  812  indicate stagnation streamlines which are incident on the leading edge or originate at the trailing edge of exemplary embodiment  790 . Streamlines  811  and  812  are therefore part of a streamsurface, or streamtube, which separate working material flowing around exemplary embodiment  790  from working material flowing through channel  792  of exemplary embodiment  790 . In this embodiment, the steamtube is circular in cross-section when viewed along the X-direction. The flow direction of the working material relative to exemplary embodiment  790  is indicated by arrow  841 . 
     A first body force per unit mass generating apparatus, or a first “BFGA”,  816  is located within channel  792 . First BFGA  816  is configured to be able to apply an effective body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this effective body force can be regulated in this embodiment. The first BFGA  816  comprises a rotating drum  817  which rotates relative to bulk material  791  about axis  822 . The drum  817  comprises a bulk material which is annular in cross-section when viewed along the X-axis and which encloses channel  792 . The drum  817  is axially symmetric about axis  822 , and can thus be considered to be in the shape of a tapered cylinder, or a cylinder of variable radius along the longitudinal length of the cylinder. The drum  817  comprises a first opening  818  and a second opening  819  through which the working material can flow into and out of the volume enclosed by the annular drum  817 . The rotating drum can be structurally supported by bulk material  791  or the remainder of exemplary embodiment  790  via bearings, such as ball or roller bearings, fluid bearings, or magnetic bearings, for example. 
     The first BFGA  816  comprises a spin-up segment  796  which is configured to induce or increase the rate of rotation in the bulk flow of the working material in channel  792  about axis  822 . The spin-up segment  796  comprises at least one rotor disc, such as rotor disc  826 . In  FIG. 2  there are five rotor discs, although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the rotor disc is at least in part structurally supported by drum  817 . In other embodiments, the rotor blades can be at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The axis of the central shaft or support disc can be coincident with axis  822 , and the radius of the outer surface of the central shaft or the support disc can be smaller than the radius of channel  792  at the location of the central shaft or support disc. At least a portion of the working material can be configured to flow around the central shaft or support disc through channel  792 . In some embodiments, at least a portion of the working material can be configured to flow through the central shaft or through the support disc. 
     The rate of rotation of the bulk flow of the working material through channel  792  about axis  822  can be configured to be very large, or substantially increased, at station  805  compared to station  803  due to the action of the spin-up segment  796 . 
     The rotor blades in a rotor disc can be configured in a similar manner as the rotor blades or baffles in a conventional centrifugal compressor. Note that, apart from the deflection of fluid flow in the radially outwards direction by the rotor blades of the rotor discs of the spin-up segment  796  and by the effective centrifugal forces, the axial flow direction of the working material is maintained throughout the spin-up segment  796 . This is in contrast to conventional centrifugal compressors, in which the bulk flow of the working material is typically twice deflected through ninety degrees, at the inlet and outlet of a centrifugal compressor, such as a centrifugal compressor found in a conventional turboprop engine. The spin-up segment  796  can thus be considered to be an axial flow centrifugal compressor. 
     The rotor blades in a rotor disc in the spin-up segment  796  can also be configured in a similar manner as the rotor blades or baffles in a conventional axial compressor. In some such configurations, an absence of stator discs or stator blades in the spin-up segment  796  can facilitate the increase of the rate of rotation or swirl of the bulk flow of the working material about axis  822  throughout the spin-up segment  796 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-up segment  796 , such as between rotor disc  826  and the rotor disc immediately downstream of rotor disc  826 , can be employed to enhance the increase of the rate of rotation or swirl of the bulk flow of the working material about axis  822  throughout the spin-up segment  796 . In a subset of embodiments, the first expansion  795  of channel  792  can be employed to reduce the maximum local relative flow velocity of the working material relative to the rotor blades of the rotor disc of spin-up segment  796  to subsonic speeds during nominal operations. This can reduce the wave drag associated with the formation of shock waves at the rotor discs of spin-up segment  796 . 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about axis  822  in the spin-up segment  796  can be provided by a separate electrical motor, for example. The electrical motor can be configured to rotate drum  817 , and thereby rotate, and supply mechanical power to, the rotor discs of spin-up segment  826 . The electrical power supplied to the electrical motor can be provided by a battery, or by an electrical generator which is driven by a separate turbine, such as the turbine in an auxiliary power unit. For instance, an electrical motor can be employed to power the first BFGA  798  and increase the rate of rotation of drum  817  and the associated rotor discs of the spin-up segment  796  during the starting of the engine  790 , i.e. the increase of the net thrust of the exemplary embodiment  790  from a value which is zero or less than zero, i.e. directed in the positive X-direction, to a value which is above zero, i.e. directed in the negative X-direction. 
     The working material flowing through second expansion  797  comprises an axial flow component as well as a rotational or swirl component due to the rotation about axis  822  imparted to the working material by the spin-up segment  796 . In order to maintain the rate of rotation of the bulk flow of the working material about axis  822 , second expansion  797  can comprise baffles arranged in a streamwise direction, i.e. along the X-direction. The baffles can be rigidly connected to drum  817 , and therefore rotate about axis  822 . The baffles can be configured to prohibit, or restrict or reduce, the circumferential motion of the working material about the X-axis relative to the drum  817  or relative to the baffles. In this scenario, since the drum  817  and the baffles are rotating, the angular rate of rotation of the bulk flow of the working material in the second expansion  797  is substantially equal to the angular rate of rotation of the drum  817  and the baffles about axis  822 . Thus the baffles can be employed to control and regulate the rate of rotation of the working material flowing through second expansion  797 . 
     The first BFGA  816  comprises a spin-down segment  798  which is configured to decrease the rate of rotation in the bulk flow of the working material in channel  792  about axis  822 . The spin-down segment  798  comprises at least one rotor disc, such as rotor disc  832 . The rotor discs in the spin-down segment  798  can be configured in a similar manner as the rotor discs in the spin-up segment  796 . In  FIG. 2  there are three rotor discs in spin-down segment  798 , although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the rotor disc is at least in part structurally supported by drum  817 . In other embodiments, the rotor blades can be at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines, and as described in the context of the spin-up segment  796 . 
     The rate of rotation of the bulk flow of the working material through channel  792  about axis  822  can be configured to be negligible, or substantially reduced, at station  807  compared to station  806  or station  805  due to the action of the spin-down segment  798 . 
     The rotor discs of spin-down segment  798  can be configured in a similar manner as the rotor blades or baffles in a conventional centrifugal turbine. As described in the context of the spin-up segment  796 , the spin-down segment  798  can be considered to be an axial flow centrifugal turbine. 
     The rotor discs of spin-down segment  798  can also be configured in a similar manner as the rotor blades or baffles in a conventional axial turbine. In some such configurations, an absence of stator discs or stator blades in the spin-down segment  798  can facilitate the decrease of the rate of rotation or swirl of the bulk flow of the working material about axis  822  throughout the spin-down segment  798 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-down segment  798 , such as between rotor disc  832  and the rotor disc immediately upstream of rotor disc  832 , can be employed to enhance the decrease of the rate of rotation or swirl of the bulk flow of the working material about axis  822  throughout the spin-down segment  798 . 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about axis  822  in the spin-up segment  796  can be provided by the mechanical power extracted from the working material during the decrease in the rate of rotation of the working material about axis  822  in the spin-down segment  798 . This decrease in the rate of rotation of the working material in spin-down segment  798  can be generate a torque which acts on the rotor discs of spin-down segment  798  about axis  822 , and which can be mechanically transferred to drum  817 , and to the rotor discs of spin-up segment  796 . In other embodiments, the rotor discs in spin-down segment  8798  can be configured to drive an electrical generator. At least a portion of the energy recovered by the electrical generator can be employed to deliver electrical power to an electrical motor configured to drive the rotor discs of spin-up segment  796 . 
     In the embodiment shown in  FIG. 790 , the first BFGA  816  is comprises a single spool connecting the rotor discs of the spin-up segment  796  with the rotor discs of the spin-down segment  798  via a single shaft  817 . In other embodiments, the first BFGA can comprise two spools, three spools, or a larger number of spools. For example, a first drive shaft can connect the rotor disc  826  of spin-up segment  796  to the rotor disc  832  of spin-down segment  798 , thereby forming a first spool. A second drive shaft can connect the remaining rotor discs of spin-up segment  796  to the remaining rotor discs of spin-down segment  798 , thereby forming a second spool. The first drive shaft can be configured to pass through the center of the second drive shaft. Such multi-spool architectures are common in conventional turbofan engines, for example. 
     The first BFGA  816  is configured to generate an effective body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  792 , where the effective body force comprises a non-zero component in the YZ-plane and directed away from the center of channel  792 , i.e. away from the X-axis or in the radially outwards direction. The effective body force per unit mass acting on the working material arises from the rotation of the working material in an inertial frame, about axis  822 , within the second expansion  797  of channel  722 . Due to the lack of a centripetal body force per unit mass acting on the objects of a working material in the negative radial direction, there is an effective or perceived centrifugal body force per unit mass acting in the positive radial direction on objects in the working material, as indicated by the bold arrows, such as bold arrow  843 . In the steady state, the effective centrifugal force is balanced by the interior surface  815  of drum  817  of BFGA  816 , and an increase in the pressure and density of the working material in the radially increasing direction, i.e. in the direction in the YZ-plane and away from axis  822 , or away from the X-axis. 
     The action of the effective body force per unit mass increases the pressure on at least a portion of interior surface  815  throughout the second expansion  797  of channel  792 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  790  in the negative X-direction. This is due to the surface normal of the interior surface  815  having a component in the positive X-direction throughout the second expansion  797  of channel  792 . An artificial increase in pressure on surfaces with a surface normal which has a non-zero component in the positive X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The direction of the effective body force per unit mass acting on objects of the working material within channel  792  is indicated by bold arrows, such as bold arrow  843  in  FIG. 2 . In some embodiments, the component of the effective body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the effective body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the effective body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the effective body force acting on the working material within the second expansion  797  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material is carried out by the first BFGA  816 . 
     Due to the action of the effective body force per unit mass within the second expansion  797  of channel  792 , the pressure within the working material within the second expansion  797  of channel  792  increases in a radially outwards direction, as indicated by line  836  in the plot of pressure versus position along the Y-axis at the corresponding point along the X-axis, i.e. at a point in the second expansion  797 . A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. The change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. Upstream of the spin-up segment  796  and downstream of the spin-down segment  798  the rate of rotation of the working material is negligible in the simplified embodiment shown in  FIG. 2 . The radial variation of the pressure at these locations is approximately uniform, as indicated by lines  834 ,  838 , and  840  in the plots of pressure versus position along the Y-axis at the corresponding point along the X-axis, i.e. at a point along the X-axis in the first contraction  794 , the second contraction  799 , and the third expansion  800 , respectively. 
     A wide variety of body force generating apparatuses, or combinations thereof, can be employed in embodiments of the invention. For example, in other embodiments, the pressure at a point along the X-axis in the first contraction  794 , the second contraction  799 , or the third expansion  800  need not be substantially uniform in the radial direction, but can vary in the radial direction, as exemplified by the embodiment shown in  FIG. 1A . For example, a second BFGA configured in a similar manner as the first BFGA  816  can be located within at least a portion of the third expansion  800  and configured to generate a body force on the working material directed in the radially outward direction, as is the case in the first BFGA  816 . In another example, other types of BFGA, such as the type of BFGA described in  FIG. 1A  can be used in place of, or concurrently with, the first BFGA  816  in the exemplary embodiment shown in  FIG. 2 , or throughout portions of channel  792  in which no dedicated BFGA is being employed, such as the first contraction  794 , the second contraction  799 , or the third expansion  800 . As discussed in the context of  FIG. 1A , a BFGA can be employed to generate a body force on the working material in the first contraction  794  or the second contraction  799 , where the body force can comprise a component in a radially inward direction, away from interior surface  815 . As discussed in the context of  FIG. 1A  and  FIG. 2 , a BFGA can be employed to generate a body force on the working material in the first expansion  795 , the second expansion  797 , or the third expansion  800 , where the body force can comprise a component in a radially outward direction, towards interior surface  815 . 
     In another example, several embodiments, such as embodiment  790 , or embodiment  720 , can be connected in series. For example, an embodiment of the invention can comprise a first and a second embodiment  790  of the type shown in  FIG. 2  connected in series, such that the station  808  of the first embodiment is coincident with station  803  of the second embodiment. Due to the cooling of the working material throughout successive embodiments, and the unchanged maximum cross-sectional area of channel  722  or channel  792 , the amount of thrust produced by two embodiments connected in series can exceed the thrust produced by two identical and equivalent embodiments connected in parallel, i.e. operated independently of each other. 
     The exemplary embodiment  720  and the exemplary embodiment  790 , as well as other embodiments operated in accordance with the invention, can be employed to reduce the specific entropy of the working material interacting with the embodiments. This can be employed to convert thermal energy of the working material directly into useful work. For instance, embodiment  720  can generate a net thrust force in the negative X-direction. The power associated with the generation of this force can be provided by the thermal energy of the working material flowing through channel  722 . Thus, the working material at station  733  is at a lower temperature than the working material at station  729 , and the relative velocity of the working material at station  733  is larger than the relative velocity of the working material at station  729  relative to embodiment  720 . The acceleration of the working material and the cooling of the working material is a consequence of the work done by the working material on embodiment  720 . Embodiment  790  can interact with the working material flowing through channel  792  in similar fashion. 
       FIGS. 3A-B  show a cross-sectional view of one embodiment of the invention. This embodiment can be described as a subsonic and supersonic ramjet. The embodiment shown in  FIGS. 3A-B  is configured in a similar manner as the embodiment shown in  FIG. 2 , and will therefore not be described in the same detail. 
     The exemplary embodiment  100  shown is substantially cylindrically symmetric about an axis parallel to the long axis of the embodiment, denoted the X-axis, and coincident with the center of exemplary embodiment  100 . The X-axis is parallel to, and coincident with, a line connecting the tip  105  of the spike  104  with the tip  108  of the exit fairing  109 , and directed to the right of the page. Outside surface  129  is therefore the shape of a tapered cylinder. In other embodiments, outside surface  129  can be elliptical, rectangular, or square, for instance. 
     Exemplary embodiment  100  comprises a channel  102  with inside surface of drum  133  located between a first opening  103  and a second opening  117 , where the channel comprises a first contraction  110 , a first expansion  111 , a spin-up segment  112 , a second expansion  113 , a spin-down segment  114 , a second contraction  115 , and a third expansion  116 . The cross-sectional geometry of channel  102  is circular when viewed along the X-direction. Note that the terms “contraction” and “expansion” refer to the magnitude of the maximum radius of the axially symmetric channel. 
     Note that the channel radius or geometry can change in a different manner as a function of position along the X-axis, or be configured differently, for other embodiments, or other operating conditions. For example, in other embodiments, the cross-sectional geometry of channel  102  can be annular or ring-shaped. In other embodiments the cross-sectional geometry of channel  102  or outside surface  129  can be square or rectangular. In other embodiments, the cross-sectional geometry of at least a portion of channel  102  or outside surface  129  can be polygonal, such as pentagonal, hexagonal. In some embodiments, the cross-sectional geometry of channel  102  can change from square to circular, or vice versa, in the positive X-direction, for example. For instance, the channel can be rectangular at the first opening  103 , annular at the opening  134  to the spin-up segment  112  and station  123 , and rectangular again at station  124  and exit  117 . The rectangular opening and exit can facilitate easier modification of the channel cross-sectional area, since the cross-sectional area of a rectangular channel can be modified by simple ramps, as exemplified by the rectangular variable area inlet of the Concorde engines or the F-14 Tomcat engines. By contrast, the cross-sectional area of a circular channel can be modified by the mechanically more complex circular or polygonal variable area nozzles, as exemplified by the exhaust nozzles of the Concorde engines. The modification of the cross-sectional area of the channel is particularly relevant for flight at variable subsonic and supersonic speeds. Engines with different channel geometries, as well as different methods for modifying the cross-sectional area of a channel, are within the scope of the invention. 
     In the embodiment shown in  FIGS. 3A-B , the inlet is configured to be able to modify the smallest cross-sectional area of the channel  102 . This is accomplished by a translating spike  104  which is able to move along the positive and negative X-direction along support shaft  107  inside slot  106 . In  FIG. 3A  the spike is shown in an extended position, which facilitates an increased cross-sectional area at the inlet of channel  102 . The spike can be in this position at low subsonic speeds, high subsonic speeds, and low supersonic speeds, for example. In  FIG. 3B  the spike is shown in a retracted position, which facilitates a reduced cross-sectional area at the inlet of channel  102  and at throat  119 . The spike can be in such a position at low and high supersonic speeds, for example. The mode of operation of the spike and the purpose of the spike is similar to the spike at the inlet of the engines of the Lockheed SR-71 Blackbird. 
     In the embodiment shown in  FIGS. 3A-B , the exhaust nozzle  150  is configured to be able to modify the smallest cross-sectional area of the channel  102 . This is facilitated by variable cross-sectional area nozzles, such as primary nozzle  151 , secondary nozzle  153 , and fairing nozzle  155 . In  FIG. 3B , the primary nozzle  151  is configured to reduce the cross-sectional area of the channel  102 , while secondary nozzle  153  is configured to increase the cross-sectional area of the channel  102 . The exhaust nozzle  150  can be in such a position at high subsonic speeds, as well as low and high supersonic speeds, for example. The elements of secondary nozzle  153  are rotably coupled to the tip of primary nozzle  151 . The individual primary nozzle  151 , secondary nozzle  153 , and fairing nozzle  155  are configured with overlapping leaves or panels, as is the case for exhaust nozzles of conventional supersonic turbojet engines, such as the engines on the Eurofighter Typhoon. In other embodiments, the variable geometry exhaust nozzle of engine  100  can be configured in a similar manner as the variable geometry exhaust nozzle of conventional supersonic turbojet engines, such as the ejector type exhaust nozzle of the SR-71. In  FIG. 3B , the exhaust nozzle is configured to accelerate the flow in channel  102  to supersonic speeds. In  FIG. 3A , the exhaust nozzle  150  is shown in a stored configuration. The primary nozzle  151  and the secondary nozzle  153  are folded in a radially outward direction. The exhaust nozzle  150  can be in such a position at low and high subsonic speeds, or low supersonic speeds, for example. 
     Bulk material  101  can comprise a metal such as aluminium, steel, or titanium. Bulk material  101  can also comprise ceramics. In some embodiments, bulk material  101  comprises composites, such as carbon fiber or fiberglass. 
     In some embodiments, the apparatus contained within inside surface of drum  133  and outside surface  129  does not have to be a solid material, but can contain empty or open spaces, as is common practice in conventional ramjet or jet engine construction. This can serve to avoid an unnecessarily large mass or cost of exemplary embodiment  100 , for instance. 
     In  FIGS. 3A-B , exemplary embodiment  100  moves with constant velocity magnitude and direction relative to a working material in the free stream. The velocity direction of the upstream working material relative to exemplary embodiment  100  is aligned with the X-axis on average, i.e. directed from the left of the page to the right of the page. For clarity of description, the velocity magnitude and direction of the upstream working material relative to exemplary embodiment  100  is assumed to be constant in space and time. In other modes of operation, the upstream relative velocity magnitude and direction need not be constant in space or time. For example, the upstream relative velocity magnitude can increase or decrease as a function of time. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. A working material can also be a liquid such as water. In the embodiment shown in the figures, the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability are explained herein. 
     The working material upstream of exemplary embodiment  100 , such as at station  118 , is moving faster relative to exemplary embodiment  100  than the speed of sound in the working material in the configuration shown in  FIG. 3B . The first contraction  110 , the second expansion  111 , and the third expansion  113  of channel  102  are configured to decelerate and compress the working material flowing through channel  102  in the positive X-direction relative to exemplary embodiment  100 . The first throat  119  is defined to be the portion of channel  102  with the smallest cross-sectional area of channel  102  between first contraction  110  and second expansion  111  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  100  at the first throat  119  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  118 , the average relative speed is larger than the speed of sound, and further downstream, such as at station  123 , the average relative speed is smaller than the speed of sound within the working material in this embodiment. In some embodiments, there can be a shock wave located between the first throat  119  and station  120 . In other words, the relative flow speed of the working material downstream of the first throat  119  can be faster than the speed of sound within the working material, where the relative flow speed is reduced to a speed slower than the speed of sound throughout the shock wave. During nominal operations a shock wave can be located within the first expansion  111 . This can prevent or reduce the probability of an unscheduled engine unstart due to turbulence or variations in the free stream flow velocity of the working material. The compression of the working material between stations  118  and  123  can be described as a substantially adiabatic compression in this embodiment, where the compression is adiabatic in the sense that no heat is exchanged between the working material in channel  102  and the outside environment in this idealized scenario. The adiabatic compression between station  121  and  122  is not isentropic, even in the absence of a shock wave between stations  121  and  122 . The specific entropy of the working material is reduced between stations  121  and  122  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  120  and  123  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. 
     In other embodiments, the compression between stations  118  and  123  can comprise heat transfer from or to the working material. For instance, in some embodiments, fuel can be added to the working material and combusted at or before station  123 , similar to a conventional ramjet. In some embodiments, there can be heat transfer from the working material to the bulk material  101  due to temperature differences. In other embodiments, the compression between stations  118  and  120  can at least in part be carried out by an axial compressor, such as an axial compressor found in conventional jet engines. In other embodiments, the compression between stations  118  and  120  can at least in part be carried out by a centrifugal compressor, for instance. In some embodiments, the working material upstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. In other words, the free stream flow can be subsonic in some embodiments for some modes of operation. 
     Both the second contraction  115  and the third expansion  116  of channel  102  are configured to expand and accelerate the working material flowing through channel  102  in the positive X-direction. The second throat  124  is defined to be the portion of channel  102  with the smallest cross-sectional area of channel  102  between second contraction  115  and third expansion  116  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  100  at the second throat  124  is approximately equal to the speed of sound within the working material at that location. Upstream, such as at station  123 , the average relative speed is smaller than the speed of sound, and downstream, such as at station  125 , the average relative speed is larger than the speed of sound within the working material in this embodiment. The expansion of the working material between stations  123  and  125  can be described as a substantially adiabatic expansion in this embodiment, where the expansion is adiabatic in the sense that no heat is exchanged between the working material in channel  102  and the outside environment in this idealized scenario. In the embodiment shown in  FIGS. 3A-B , the adiabatic expansion between station  123  and  125  can also be described as a substantially isentropic expansion. 
     In other embodiments, there can be heat transfer from or to the working material between stations  120  and  125 , or stations  123  and  125 . In some embodiments, there can be heat transfer from the working material to the bulk material  101  due to temperature differences, for example. In other embodiments, heat can be deliberately added or removed from the working material. In other embodiments, the expansion between stations  123  and  125  can at least in part be carried out by an axial turbine, such as an axial turbine found in conventional jet engines. In other embodiments, this expansion can at least in part be carried out by a centrifugal turbine, for instance. In some such embodiments, the working material downstream of the embodiment can move relative to the embodiment at a speed slower than the speed of sound in the working material. In other words, the exhaust can be subsonic relative to the engine  100  in some embodiments or some modes of operation. 
     A first body force per unit mass generating apparatus, or a first “BFGA”,  132  is located within channel  102 . First BFGA  132  is configured to be able to apply an effective body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this effective body force can be regulated in this embodiment. The first BFGA  132  comprises a rotating drum  133  which rotates relative to bulk material  101  about the X-axis. The drum  133  comprises a bulk material which is annular in cross-section when viewed along the X-axis and which encloses channel  102 . The drum  133  is axially symmetric about the X-axis, and can thus be considered to be in the shape of a tapered cylinder, or a cylinder of variable radius along the longitudinal length of the cylinder. The drum  133  comprises a first opening  134  and a second opening through which the working material can flow into and out of the volume enclosed by the annular drum  133 . The rotating drum can be structurally supported by bulk material  101  or the remainder of exemplary embodiment  100  via bearings, such as ball or roller bearings, fluid bearings, or magnetic bearings, for example. 
     The first BFGA  132  comprises a spin-up segment  112  which is configured to induce or increase the rate of rotation in the bulk flow of the working material in channel  102  about the X-axis. The spin-up segment  112  comprises at least one rotor disc, such as rotor discs  142 ,  143 , and  144 . In  FIGS. 3A-B  there are three rotor discs, although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the rotor disc  144  is at least in part structurally supported by drum  133 . The rotor blades of the rotor discs  142  and  143  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  142  are supported by shaft  149 . The rotor blades of the rotor disc  143  are supported by shaft  148 . The axis of the central shaft or support disc can be coincident with the X-axis, and the radius of the outer surface of the central shaft or the support disc can be smaller than the radius of channel  102  at the location of the central shaft or support disc. At least a portion of the working material can be configured to flow around the central shaft or support disc through channel  102 . In some embodiments, at least a portion of the working material can be configured to flow through the central shaft or through the support disc. 
     The rate of rotation of the bulk flow of the working material through channel  102  about the X-axis can be configured to be very large, or substantially increased, at station  121  compared to station  119  due to the action of the spin-up segment  112 . 
     The rotor blades in a rotor disc can be configured in a similar manner as the rotor blades or baffles in a conventional centrifugal compressor. Note that, apart from the deflection of fluid flow in the radially outwards direction by the rotor blades of the rotor discs of the spin-up segment  112  and by the effective centrifugal forces, the axial flow direction of the working material is maintained throughout the spin-up segment  112 . This is in contrast to conventional centrifugal compressors, in which the bulk flow of the working material is typically deflected through ninety degrees at least once, at the inlet of a centrifugal compressor, such as a centrifugal compressor found in a conventional turboprop engine. The spin-up segment  112  can thus be considered to be an “axial flow centrifugal compressor”. The rate of rotation of the rotor blades can be modified to regulate the rate of rotation of the working material within the BFGA  132  for a given free stream flow speed and a given desired thrust. 
     The rotor blades in a rotor disc in the spin-up segment  112  can also be configured in a similar manner as the rotor blades in a conventional axial compressor. In some such configurations, an absence of stator discs or stator blades in the spin-up segment  112  compared to a conventional axial compressor can facilitate the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  112 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-up segment  112 , such as between rotor disc  142  and rotor disc  143 , can be employed to enhance the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  112 . In a subset of embodiments, the first expansion  111  of channel  102  can be employed to reduce the maximum local relative flow velocity of the working material relative to the rotor blades of the rotor disc of spin-up segment  112  to subsonic speeds during nominal operations. This can reduce the wave drag associated with the formation of shock waves at the rotor discs of spin-up segment  112 . In other embodiments, the spin-up segment  112  can consist only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to impart a rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  101  of engine  100 . In other embodiments, the stator blades are rotably coupled to the bulk material  101  of the engine  100 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  132  for a given free stream flow speed and a given desired thrust. 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  112  can be provided by a separate electrical motor, for example. The electrical motor can be configured to rotate drum  133 , and thereby rotate, and supply mechanical power to, the rotor discs of spin-up segment  142 . The electrical power supplied to the electrical motor can be provided by a battery, or by an electrical generator which is driven by a separate turbine, such as the turbine in an auxiliary power unit, or a turbine located downstream of station  123 . For instance, an electrical motor can be employed to power the first BFGA  132  and increase the rate of rotation of drum  133  and the associated rotor discs of the spin-up segment  112  during the starting of the engine  100 , i.e. the increase of the net thrust of the exemplary embodiment  100  from a value which is zero or less than zero, i.e. directed in the positive X-direction, to a value which is above zero, i.e. directed in the negative X-direction. 
     The working material flowing through second expansion  113  comprises an axial flow component as well as a rotational or swirl component due to the rotation about the X-axis imparted to the working material by the spin-up segment  112 . In order to maintain the rate of rotation of the bulk flow of the working material about the X-axis, second expansion  113  can comprise baffles arranged in a streamwise direction, i.e. along the X-direction. The baffles can be rigidly connected to drum  133 , and therefore rotate about the X-axis. The baffles can be configured to prohibit, or restrict or reduce, the circumferential motion of the working material about the X-axis relative to the drum  133  or relative to the baffles. In this scenario, since the drum  133  and the baffles are rotating, the angular rate of rotation of the bulk flow of the working material in the second expansion  113  is substantially equal to the angular rate of rotation of the drum  133  and the baffles about the X-axis. Thus the baffles can be employed to control and regulate the rate of rotation of the working material flowing through second expansion  113 . In other embodiments, there need not be any baffles between station  121  and  122 , allowing the working material to rotate substantially freely about the X-axis between stations  121  and  122 . Note that the viscous drag from drum  133  can also contribute a rate of rotation to the working material flowing through drum  133 . 
     The first BFGA  132  comprises a spin-down segment  114  which is configured to decrease the rate of rotation in the bulk flow of the working material in channel  102  about the X-axis. The spin-down segment  114  comprises at least one rotor disc, such as rotor disc  146  or rotor disc  147 . The rotor discs in the spin-down segment  114  can be configured in a similar manner as the rotor discs in the spin-up segment  112 . In  FIGS. 3A-B  there are three rotor discs in spin-down segment  114 , although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the upstream rotor disc of spin down segment  114  is at least in part structurally supported by drum  133 . The rotor blades of the rotor discs  146  and  147  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  147  are supported by shaft  149 . The rotor blades of the rotor disc  146  are supported by shaft  148 . 
     The rate of rotation of the bulk flow of the working material through channel  102  about the X-axis can be configured to be negligible, or substantially reduced, at station  123  compared to station  122  or station  121  due to the action of the spin-down segment  114 . 
     As described in the context of the spin-up segment  112 , the spin-down segment  114  can be considered to be an axial flow centrifugal turbine. 
     The rotor discs of spin-down segment  114  can also be configured in a similar manner as the rotor blades in a conventional axial turbine. In some such configurations, an absence of stator discs or stator blades in the spin-down segment  114  can facilitate the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  114 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-down segment  114 , such as between rotor disc  146  and rotor disc  147 , can be employed to enhance the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  114 . In other embodiments, the spin-down segment  114  can consist only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to reduce the rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  101  of engine  100 . In other embodiments, the stator blades are rotably coupled to the bulk material  101  of the engine  100 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  132  for a given free stream flow speed and a given desired thrust. The function of the stator blades is identical to the function of the rotor blades in the spin-down segment  114 . 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  112  can be provided by the mechanical power extracted from the working material during the decrease in the rate of rotation of the working material about the X-axis in the spin-down segment  114 . This decrease in the rate of rotation of the working material in spin-down segment  114  can be generate a torque which acts on the rotor discs of spin-down segment  114  about the X-axis, and which can be mechanically transferred to drum  133  and to the drive shafts  148  and  149 , and to the rotor discs of spin-up segment  112 . In other embodiments, the rotor discs in spin-down segment  114  can be configured to drive an electrical generator. At least a portion of the energy recovered by the electrical generator can be employed to deliver electrical power to an electrical motor configured to drive the rotor discs of spin-up segment  112 . In the case in which the spin-up segment comprises stator discs, the torque acting on the stator discs in the spin-down segment  114  can be cancelled by the torque acting on the working material in the spin-up segment  112  during nominal operations, such that there is no net torque on the engine  100 . 
     In the embodiment shown in  FIG. 100 , the first BFGA  132  is comprises a three spools connecting the rotor discs of the spin-up segment  112  with the rotor discs of the spin-down segment  114  via external drive shaft or drum  133 , and internal drive shafts  149  and  148 . In other embodiments, the first BFGA can comprise two spools, four spools, or a larger number of spools. Such multi-spool architectures are common in conventional turbofan engines, for example. 
     The first BFGA  132  is configured to generate an effective body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  102 , where the effective body force comprises a non-zero component in the YZ-plane and directed away from the center of channel  102 , i.e. away from the X-axis or in the radially outwards direction. The effective body force per unit mass acting on the working material arises from the rotation of the working material in an inertial frame, about the X-axis, within the second expansion  113  of channel  722 . Due to the lack of a centripetal body force per unit mass acting on the objects of a working material in the negative radial direction, there is an effective or perceived centrifugal body force per unit mass acting in the positive radial direction, i.e. in a radially outward direction, on objects in the working material. In the steady state, the effective centrifugal force is balanced by the interior surface of drum  133  of BFGA  132 , and an increase in the pressure and density of the working material in the radially increasing direction, i.e. in the direction in the YZ-plane and away from the X-axis, or away from the X-axis. 
     The action of the effective body force per unit mass increases the pressure on at least a portion of interior surface of drum  133  throughout the second expansion  113  of channel  102 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  100  in the negative X-direction. This is due to the surface normal of the interior surface of drum  133  having a component in the positive X-direction throughout the second expansion  113  of channel  102 . An artificial increase in pressure on surfaces with a surface normal which has a non-zero component in the positive X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The specific entropy of the working material is reduced between stations  121  and  122  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  121  and  122  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. The modification of the specific heat capacity between stations  121  and  122  is similar in principle to the modification of the specific heat capacity discussed in the context of  FIGS. 7A-J . 
     The direction of the effective body force per unit mass acting on objects of the working material within channel  102  is in the radially outward direction. In some embodiments, the component of the effective body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the effective body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the effective body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the effective body force acting on the working material within the second expansion  113  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material would be carried out by the first BFGA  132 . 
     Due to the action of the effective body force per unit mass within the second expansion  113  of channel  102 , the pressure within the working material within the second expansion  113  of channel  102  increases in a radially outwards direction at a given location along the length of the channel. A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. At a given location along the length of the channel the change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. Upstream of the spin-up segment  112  and downstream of the spin-down segment  114  the rate of rotation of the working material about the X-axis is negligible in the simplified embodiment shown in  FIGS. 3A-B . The radial variation of the pressure between stations  118  and  120 , and between stations  123  and  125  is approximately uniform. 
     A wide variety of body force generating apparatuses, or combinations thereof, can be employed in embodiments of the invention. For example, in other embodiments, the pressure at a point along the X-axis in the first contraction  110 , the second contraction  115 , or the third expansion  116  need not be substantially uniform in the radial direction, but can vary in the radial direction, as exemplified by the embodiment shown in  FIG. 1A . For example, a second BFGA configured in a similar manner as the first BFGA  132  can be located within at least a portion of the third expansion  116  and configured to generate a body force on the working material directed in the radially outward direction, as is the case in the first BFGA  132 . In another example, other types of BFGA, such as the type of BFGA described in  FIG. 1A  can be used in place of, or concurrently with, the first BFGA  132  in the exemplary embodiment shown in  FIGS. 3A-B , or throughout portions of channel  102  in which no dedicated BFGA is being employed, such as the first contraction  110 , the second contraction  115 , or the third expansion  116 . As discussed in the context of  FIG. 1A , a BFGA can be employed to generate a body force on the working material in the first contraction  110  or the second contraction  115 , where the body force can comprise a component in a radially inward direction, away from interior surface  131 . As discussed in the context of  FIG. 1A  and  FIGS. 3A-B , a BFGA can be employed to generate a body force on the working material in the first expansion  111 , the second expansion  113 , or the third expansion  116 , where the body force can comprise a component in a radially outward direction, towards interior surface  131 . 
     In another example, several embodiments, such as embodiment  100 , can be connected in series. For example, an embodiment of the invention can comprise a first and a second embodiment  100  of the type shown in  FIGS. 3A-B  connected in series, such that the station  124  of the first embodiment is coincident with station  119  of the second embodiment, or such that station  124  of the first embodiment is coincident with station  120  of the second embodiment. Due to the cooling of the working material throughout successive embodiments, and the unchanged maximum cross-sectional area of channel  102 , the amount of thrust produced by two embodiments connected in series can exceed the thrust produced by two identical and equivalent embodiments connected in parallel, i.e. operated independently of each other. 
     The exemplary embodiment  100 , as well as other embodiments operated in accordance with the invention, can be employed to reduce the specific entropy of the working material interacting with the embodiments. This can be employed to convert thermal energy of the working material directly into useful work. For instance, embodiment  100  can generate a net thrust force in the negative X-direction. The power associated with the generation of this force can be provided by the thermal energy of the working material flowing through channel  102 . Thus, the working material at station  125  is at a lower temperature than the working material at station  118 , and the relative velocity of the working material at station  125  is larger than the relative velocity of the working material at station  118  relative to embodiment  100 . The acceleration of the working material and the cooling of the working material is a consequence of the work done by the working material on embodiment  100 . 
       FIGS. 4A-C  schematically show a cross-sectional view of exemplary embodiments of individual rotor discs and associated rotor blades of exemplary embodiments of the invention, such as the embodiments shown in  FIGS. 3A-B ,  FIGS. 5A-B , and  FIGS. 6A-B . 
       FIG. 4A  schematically shows a circumferential cross-section of  5  rotor discs, namely rotor discs  180 ,  182 ,  186 ,  191 , and  193 . The radially inward direction is directed into the page, and the flow direction is directed from the left of the page to the right of the page. Each rotor disc comprises several rotor blades, such as rotor blades  181 ,  183 ,  190 ,  192 , and  194 . All rotor blades rotate in a downwards direction, i.e. from the top of the page towards the bottom of the page during nominal operations. A spin-up segment apparatus can be considered to comprise rotor discs  180  and  182 , as well as spin-up segment  187  and baffle segment  188  of rotor disc  186 . A spin-down segment apparatus can be considered to comprise rotor discs  191  and  193 , as well as spin-down segment  189  of rotor disc  186 . The spin-up segment apparatus, i.e. the first rotor disc  180 , the second rotor disc  182 , and the spin-up portion  187  of the third rotor disc  186  as well as the baffle portion  188  of the third rotor disc  186  are configured to increase the rate of rotation of the working material in an inertial frame in the direction towards the bottom of the page, or in a positive sense according to the right hand rule about the X-axis. Recall that the X-axis is coincident with the axis of rotation and directed from the left of the page to the right of the page. The spin-down segment apparatus, i.e. the spin-down portion  189  of the third rotor disc  186 , as well as the fourth rotor disc  191  and the fifth rotor disc  193  are configured to decrease the rate of rotation of the working material in an inertial frame. For some embodiments, such as the embodiments shown in  FIGS. 3A-B , the rate of rotation of the working material about the X-axis upstream of rotor disc  180  and downstream of the rotor disc  193  is negligible or zero during nominal operations. For some embodiments, or some modes of operation, the rate of rotation of the working material about the X-axis upstream of rotor disc  180  and/or downstream of the rotor disc  193  can be non-zero. 
     Rotor disc  180  can correspond to rotor disc  142  in  FIGS. 3A-B , rotor disc  404  in  FIGS. 5A-B , or rotor disc  694  in  FIGS. 6A-B , for example. Rotor disc  182  can correspond to rotor disc  143  in  FIGS. 3A-B , rotor disc  406  in  FIGS. 5A-B , or rotor disc  696  in  FIGS. 6A-B . Rotor disc  186  can correspond to rotor disc  144  in  FIGS. 3A-B , rotor disc  408  in  FIGS. 5A-B , or rotor disc  698  in  FIGS. 6A-B . Note that  FIGS. 3A-B ,  FIGS. 5A-B , and  FIGS. 6A-B  only show the spin-up segment  187  and the spin down-segment  189  of rotor disc  186  in cross-sectional view for simplicity and clarity, as indicated by spin-up segment  698  and spin-down segment  712  in  FIGS. 6A-B . As indicated in  FIGS. 3A-B ,  FIGS. 5A-B , and  FIGS. 6A-B , rotor disc  180  can be driven by a drive shaft coupled to rotor disc  193 , and rotor disc  182  can be driven by a drive shaft coupled to rotor disc  191 . The rotor blades of rotor disc  186  can be structurally supported by a drive shaft as well, where the drive shaft can be located at the center of the channel, or at the outside surface of the channel. Thus at least a portion of the mechanical power extracted by the spin-down segment apparatus can be delivered mechanically to the spin-up segment apparatus via the drive shafts. 
     The axial flow speed of the working material in the X-direction can vary along the length of the X-axis throughout the rotor discs. For example, the axial flow speed can increase or decrease in the positive X-direction. The axial flow speed of the working material in the X-direction throughout a spin-up segment apparatus or a spin-down segment apparatus can vary along the length of the X-axis. Note that the axial flow speed is a function of the cross-sectional area of a channel comprising the rotor discs. The axial flow speed of the working material in the X-direction throughout a spin-up segment apparatus or a spin-down segment apparatus can also remain substantially constant along the length of the X-axis. 
     In order to increase the rate of rotation of the working material in the spin-up segment apparatus, the rate of rotation of the third rotor disc  186  is typically larger than the rate of rotation of the second rotor disc  182 , which in turn is larger than the rate of rotation of the first rotor disc  180  during nominal operations. 
     The embodiment shown in  FIG. 4B  is configured in a similar manner as the embodiment shown in  FIG. 4A , and will therefore not be described in the same detail.  FIG. 4B  schematically shows a circumferential cross-section of  5  rotor discs, namely rotor discs  210 ,  212 ,  216 ,  221 , and  223 . The radially inward direction is directed into the page, and the flow direction is directed from the left of the page to the right of the page. Each rotor disc comprises several rotor blades, such as rotor blades  221 ,  213 ,  220 ,  222 , and  224 . All rotor blades rotate in a downwards direction, i.e. from the top of the page towards the bottom of the page during nominal operations. A spin-up segment apparatus can be considered to comprise rotor discs  210  and  212 , as well as spin-up segment  217  and baffle segment  218  of rotor disc  216 . A spin-down segment apparatus can be considered to comprise rotor discs  221  and  223 , as well as spin-down segment  219  of rotor disc  216 . The spin-up segment apparatus is configured to increase the rate of rotation of the working material in an inertial frame in the direction towards the bottom of the page, or in a positive sense according to the right hand rule about the X-axis. Recall that the X-axis is coincident with the axis of rotation and directed from the left of the page to the right of the page. The spin-down segment apparatus is configured to decrease the rate of rotation of the working material in an inertial frame. For some embodiments, such as the embodiments shown in  FIGS. 3A-B , the rate of rotation of the working material about the X-axis upstream of rotor disc  210  and downstream of the rotor disc  223  is negligible or zero during nominal operations. For some embodiments, or some modes of operation, the rate of rotation of the working material about the X-axis upstream of rotor disc  210  and/or downstream of the rotor disc  223  can be non-zero. 
     Compared to  FIG. 4A  the rotor blades of rotor discs  210  and  212  have a stronger camber. Note that the trailing edges of the rotor blades of rotor discs  210  and  212  are substantially parallel to the X-axis. This can enhance the rate of rotation of the working material downstream of rotor disc  212  and upstream of rotor disc  216  compared to the embodiment shown in  FIG. 4A  for some modes of operation, ceteris paribus. Note that the camber of the rotor blades of rotor discs  251  and  253  is configured to reduce the rate of rotation of the working material about the X-axis by a desired amount for a given rotational speed of the rotor discs. 
     The embodiment shown in  FIG. 4C  is configured in a similar manner as the embodiment shown in  FIG. 4A  and  FIG. 4B , and will therefore not be described in the same detail.  FIG. 4C  schematically shows a circumferential cross-section of  5  rotor discs, namely rotor discs  240 ,  242 ,  246 ,  251 , and  253 . The radially inward direction is directed into the page, and the flow direction is directed from the left of the page to the right of the page. Each rotor disc comprises several rotor blades, such as rotor blades  241 ,  243 ,  250 ,  252 , and  254 . All rotor blades rotate in a downwards direction, i.e. from the top of the page towards the bottom of the page during nominal operations. A spin-up segment apparatus can be considered to comprise rotor discs  240  and  242 , as well as spin-up segment  247  and baffle segment  248  of rotor disc  246 . A spin-down segment apparatus can be considered to comprise rotor discs  251  and  253 , as well as spin-down segment  249  of rotor disc  246 . The spin-up segment apparatus is configured to increase the rate of rotation of the working material in an inertial frame in the direction towards the bottom of the page, or in a positive sense according to the right hand rule about the X-axis. Recall that the X-axis is coincident with the axis of rotation and directed from the left of the page to the right of the page. The spin-down segment apparatus is configured to decrease the rate of rotation of the working material in an inertial frame. For some embodiments, such as the embodiments shown in  FIGS. 3A-B , the rate of rotation of the working material about the X-axis upstream of rotor disc  240  and downstream of the rotor disc  253  is negligible or zero during nominal operations. For some embodiments, or some modes of operation, the rate of rotation of the working material about the X-axis upstream of rotor disc  240  and downstream of the rotor disc  253  can be non-zero. 
     Compared to  FIG. 4A  and  FIG. 4B , the rotor blades of rotor discs  240  and  242  have a stronger camber. This can enhance the rate of rotation of the working material downstream of rotor disc  242  and upstream of rotor disc  246  compared to the embodiment shown in  FIG. 4A  and  FIG. 4B , ceteris paribus. Note that the camber of the rotor blades of rotor discs  251  and  253  is configured to reduce the rate of rotation of the working material about the X-axis by a desired amount for a given rotational speed of the rotor discs. 
       FIGS. 5A-B  show a cross-sectional view of one embodiment of the invention. This embodiment can be described as a subsonic and supersonic ramjet. The embodiment shown in  FIGS. 5A-B  is configured in a similar manner as the embodiment shown in in  FIG. 2  and  FIGS. 3A-B , and will therefore not be described in the same detail. 
     The exemplary embodiment  360  shown is substantially cylindrically symmetric about an axis parallel to the long axis of the embodiment, denoted the X-axis, and coincident with the center of exemplary embodiment  360 . The X-axis is parallel to, and coincident with, a line connecting the tip  365  of the spike  364  with the tip  368  of the exit fairing  369 , and directed to the right of the page. Outside surface  391  is therefore the shape of a tapered cylinder. In other embodiments, outside surface  391  can be elliptical, rectangular, or square, for instance. 
     Exemplary embodiment  360  comprises a channel  362  with inside surface of drum  395  located between a first opening  363  and a second opening  378 , where the channel comprises a first contraction  370 , a first expansion  371 , a spin-up segment  372 , a second expansion  373 , a second contraction  374 , a spin-down segment  375 , a third contraction  376 , and a third expansion  377 . The cross-sectional geometry of channel  362  is circular when viewed along the X-direction. Note that the terms “contraction” and “expansion” refer to the magnitude of the maximum radius of the axially symmetric channel. 
     Note that the channel radius or geometry can change in a different manner as a function of position along the X-axis, or be configured differently, for other embodiments, or other operating conditions, as described in the context of  FIGS. 3A-B . 
     In the embodiment shown in  FIGS. 5A-B , the inlet is configured to be able to modify the smallest cross-sectional area of the channel  362 . This is accomplished by a translating spike  364  which is able to move along the positive and negative X-direction along support shaft  367  inside slot  366 . In  FIG. 5A  the spike is shown in an extended position, which facilitates an increased cross-sectional area at the inlet of channel  362 . The spike can be in this position at low subsonic speeds, high subsonic speeds, and low supersonic speeds, for example. In  FIG. 5B  the spike is shown in a retracted position, which facilitates a reduced cross-sectional area at the inlet of channel  362  and at throat  380 . The spike can be in such a position at low and high supersonic speeds, for example. The mode of operation of the spike and the purpose of the spike is similar to the spike at the inlet of the engines of the Lockheed SR-71 Blackbird, for example. 
     In the embodiment shown in  FIGS. 5A-B , the exhaust nozzle is configured to be able to modify the smallest cross-sectional area of the channel  362 . This is facilitated by variable cross-sectional area nozzles, such as primary nozzle  421 , and annular sliding ring  424 . In  FIG. 5B , the annular sliding ring  424  is moved in a negative X-direction relative to the embodiment shown in  FIG. 5A . This reduces the cross-sectional area of the channel  362  to a local minimum at throat  386 . Downstream of the throat  386 , the cross-sectional area of the channel  362  is increased again by primary nozzle  421  and exit fairing  369 . The exhaust nozzle can be in a similar position at high subsonic speeds, as well as low and high supersonic speeds, for example. The elements of primary nozzle  421  are rotably coupled to the bulk material  361 . The primary nozzle  421  is configured with overlapping leaves or panels, as is the case for exhaust nozzles of conventional supersonic turbojet engines, such as the engines on the Eurofighter Typhoon. In other embodiments, the variable geometry exhaust nozzle of engine  360  can be configured in a similar manner as the variable geometry exhaust nozzle of conventional supersonic turbojet engines, such as the ejector type exhaust nozzle of the SR-71, or the type of nozzle shown in  FIGS. 3A-B . In  FIG. 5B , the exhaust nozzle is configured to accelerate the subsonic flow in channel  362  to supersonic speeds. In  FIG. 5A , the exhaust nozzle is shown in a stored configuration. The primary nozzle  421  are folded in a radially outward direction compared to the configuration shown in  FIG. 5A . The exhaust nozzle can be in such a position at low and high subsonic speeds, or low supersonic speeds, for example. Annular sliding ring  424  is configured to slide on rails along inside surface  393  of channel  362  in the positive and negative X-direction. Due to the sloping surface of exit fairing  369 , the cross-sectional area of throat  386  can be regulated in flight during nominal operations in this manner. 
     Bulk material  361  can comprise a metal such as aluminium, steel, or titanium. Bulk material  361  can also comprise ceramics. In some embodiments, bulk material  361  comprises composites, such as carbon fiber or fiberglass. 
     In some embodiments, the apparatus contained within inside surface of drum  395  and outside surface  391  does not have to be a solid material, but can contain empty or open spaces, as is common practice in conventional ramjet or jet engine construction. This can serve to avoid an unnecessarily large mass or cost of exemplary embodiment  360 , for instance. 
     In  FIGS. 5A-B , exemplary embodiment  360  moves with constant velocity magnitude and direction relative to a working material in the free stream. The velocity direction of the upstream working material relative to exemplary embodiment  360  is aligned with the X-axis on average, i.e. directed from the left of the page to the right of the page. For clarity of description, the velocity magnitude and direction of the upstream working material relative to exemplary embodiment  360  is assumed to be constant in space and time. In other modes of operation, the upstream relative velocity magnitude and direction need not be constant in space or time. For example, the upstream relative velocity magnitude can increase or decrease as a function of time. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. In the embodiment shown in the figures, the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability comprise compressibility, for example. Note that all materials are compressible to some extent. 
     The working material upstream of exemplary embodiment  360 , such as at station  379 , is moving faster relative to exemplary embodiment  360  than the speed of sound in the working material in the configuration shown in  FIG. 5B . The first contraction  370 , the first expansion  371 , and the second expansion  373  of channel  362  are configured to decelerate and compress the working material flowing through channel  362  in the positive X-direction relative to exemplary embodiment  360 . The first throat  380  is defined to be the portion of channel  362  with the smallest cross-sectional area of channel  362  between first contraction  370  and first expansion  371  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  360  at the first throat  380  is approximately equal to the speed of sound within working material at that location. Upstream, such as at station  379 , the average relative speed is larger than the speed of sound, and further downstream, such as at station  385 , the average relative speed is smaller than the speed of sound within the working material in this embodiment. In some embodiments, there can be a shock wave located between the first throat  380  and station  381 . In other words, the relative flow speed of the working material downstream of the first throat  380  can be faster than the speed of sound within the working material, where the relative flow speed is reduced to a speed slower than the speed of sound throughout the shock wave. During nominal operations a shock wave can be located within the first expansion  371 . This can prevent or reduce the probability of an unscheduled engine unstart due to turbulence or variations in the free stream flow velocity of the working material. The compression of the working material between stations  379  and  385  can be described as a substantially adiabatic compression in this embodiment, where the compression is adiabatic in the sense that no heat is exchanged between the working material in channel  362  and the outside environment in this idealized scenario. The specific entropy of the working material is reduced between stations  382  and  384  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  381  and  385  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. 
     Both the third contraction  376  and the third expansion  377  of channel  362  are configured to expand and accelerate the working material flowing through channel  362  in the positive X-direction. The second throat  386  is defined to be the portion of channel  362  with the smallest cross-sectional area of channel  362  between third contraction  376  and third expansion  377  when viewed along the X-direction. The average speed of the working material relative to exemplary embodiment  360  at the second throat  386  is approximately equal to the speed of sound within the working material at that location. Upstream, such as at station  385 , the average relative speed is smaller than the speed of sound, and downstream, such as at station  387 , the average relative speed is larger than the speed of sound within the working material in this embodiment. The expansion of the working material between stations  385  and  387  can be described as a substantially adiabatic expansion in this embodiment, where the expansion is adiabatic in the sense that no heat is exchanged between the working material in channel  362  and the outside environment in this idealized scenario. In the embodiment shown in  FIGS. 5A-B , the adiabatic expansion between station  385  and  387  can also be described as a substantially isentropic expansion. 
     A first body force per unit mass generating apparatus, or a first “BFGA”,  394  is located within channel  362 . First BFGA  394  is configured to be able to apply an effective body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this effective body force can be regulated in this embodiment. This can be accomplished by controlling the rate of rotation of drive shafts  412 ,  413 , and drum  395 , relative to bulk material  361  or engine  360  or an inertial frame, for example. The first BFGA  394  comprises a rotating drum  395  which rotates relative to bulk material  361  about the X-axis. The drum  395  comprises a bulk material which is annular in cross-section when viewed along the X-axis and which encloses channel  362 . The drum  395  is substantially axially symmetric about the X-axis, and can thus be considered to be in the shape of a tapered cylinder, or a cylinder of variable radius along the longitudinal length of the cylinder. The drum  395  comprises a first opening  396  and a second opening through which the working material can flow into and out of the volume enclosed by the annular drum  395 . The rotating drum can be structurally supported by bulk material  361  or the remainder of exemplary embodiment  360  via bearings, such as ball or roller bearings, fluid bearings, or magnetic bearings, for example. 
     The first BFGA  394  comprises a spin-up segment  372  which is configured to induce or increase the rate of rotation in the bulk flow of the working material in channel  362  about the X-axis. The spin-up segment  372  comprises at least one rotor disc, such as rotor discs  404 ,  406 , and  408 . In  FIGS. 5A-B  there are three rotor discs, although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the rotor disc  408  is at least in part structurally supported by drum  395 . The rotor blades of the rotor discs  404  and  406  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  404  are supported by shaft  413 . The rotor blades of the rotor disc  406  are supported by shaft  412 . The axis of the central shaft or support disc can be coincident with the X-axis, and the radius of the outer surface of the central shaft or the support disc can be smaller than the radius of channel  362  at the location of the central shaft or support disc. At least a portion of the working material can be configured to flow around the central shaft or support disc through channel  362 . In some embodiments, at least a portion of the working material can be configured to flow through the central shaft or through the support disc. 
     The rate of rotation of the bulk flow of the working material through channel  362  about the X-axis can be configured to be very large, or substantially increased, at station  382  compared to station  380  due to the action of the spin-up segment  372 . 
     The rotor blades in a rotor disc can be configured in a similar manner as the rotor blades in a conventional centrifugal compressor. Note that, apart from the deflection of fluid flow in the radially outwards direction by the rotor blades of the rotor discs of the spin-up segment  372  and by the effective centrifugal forces, the axial flow of the working material is maintained throughout the spin-up segment  372 . This is in contrast to conventional centrifugal compressors, in which the bulk flow of the working material is typically deflected through ninety degrees at least once, at the inlet of a centrifugal compressor, such as a centrifugal compressor found in a conventional turboprop engine. The spin-up segment  372  can thus be considered to be an “axial flow centrifugal compressor”. The rate of rotation of the rotor blades can be modified to regulate the rate of rotation of the working material within the BFGA  394  for a given free stream flow speed and a given desired thrust. 
     The rotor blades in a rotor disc in the spin-up segment  372  can also be configured in a similar manner as the rotor blades in a conventional axial compressor. In some such configurations, an absence of stator discs or stator blades in the spin-up segment  372  compared to a conventional axial compressor can facilitate the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  372 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-up segment  372 , such as between rotor disc  404  and rotor disc  406 , can be employed to enhance the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  372 . In a subset of embodiments, the first expansion  371  of channel  362  can be employed to reduce the maximum local relative flow velocity of the working material relative to the rotor blades of the rotor disc of spin-up segment  372  to subsonic speeds during nominal operations. This can reduce the wave drag associated with the formation of shock waves at the rotor discs of spin-up segment  372 . In other embodiments, the spin-up segment  372  can consist only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to impart a rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  361  of engine  360 . In other embodiments, the stator blades are rotably coupled to the bulk material  361  of the engine  360 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  394  for a given free stream flow speed and a given desired thrust. 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  372  can be provided by a separate electrical motor, for example. The electrical motor can be configured to rotate drum  395 , and thereby rotate, and supply mechanical power to, the rotor discs of spin-up segment  404 . The electrical power supplied to the electrical motor can be provided by a battery, or by an electrical generator which is driven by a separate turbine, such as the turbine in an auxiliary power unit, or a turbine located downstream of station  385 . For instance, an electrical motor can be employed to power the first BFGA  394  and increase the rate of rotation of drum  395  and the associated rotor discs of the spin-up segment  372  during the starting of the engine  360 , i.e. the increase of the net thrust of the exemplary embodiment  360  from a value which is zero or less than zero, i.e. directed in the positive X-direction, to a value which is above zero, i.e. directed in the negative X-direction. 
     The working material flowing through second expansion  373  and second contraction  374  comprises an axial flow component as well as a rotational or swirl component due to the rotation about the X-axis imparted to the working material by the spin-up segment  372 . In order to maintain the rate of rotation of the bulk flow of the working material about the X-axis, second expansion  373  and second contraction  374  can comprise baffles arranged in a streamwise direction, i.e. along the X-direction. The baffles can be rigidly connected to drum  395 , and therefore rotate about the X-axis. The baffles can be configured to prohibit, or restrict or reduce, the circumferential motion of the working material about the X-axis relative to the drum  395  or relative to the baffles. In this scenario, since the drum  395  and the baffles are rotating, the angular rate of rotation of the bulk flow of the working material in the second expansion  373  is substantially equal to the angular rate of rotation of the drum  395  and the baffles about the X-axis. Thus the baffles can be employed to control and regulate the rate of rotation of the working material flowing through second expansion  373 . In other embodiments, there need not be any baffles between station  382  and  384 , allowing the working material to rotate substantially freely about the X-axis between stations  382  and  384 . Note that the viscous drag from drum  395  can also contribute a rate of rotation to the working material flowing through drum  395 . 
     The first BFGA  394  comprises a spin-down segment  375  which is configured to decrease the rate of rotation in the bulk flow of the working material in channel  362  about the X-axis. The spin-down segment  375  comprises at least one rotor disc, such as rotor disc  410  or rotor disc  411 . The rotor discs in the spin-down segment  375  can be configured in a similar manner as the rotor discs in the spin-up segment  372 . In  FIGS. 5A-B  there are three rotor discs in spin-down segment  375 , although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the upstream rotor disc of spin down segment  375  is at least in part structurally supported by drum  395 . The rotor blades of the rotor discs  410  and  411  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  411  are supported by shaft  413 . The rotor blades of the rotor disc  410  are supported by shaft  412 . 
     The rate of rotation of the bulk flow of the working material through channel  362  about the X-axis can be configured to be negligible, or substantially reduced, at station  385  compared to station  384  or station  382  due to the action of the spin-down segment  375 . 
     As described in the context of the spin-up segment  372 , the spin-down segment  375  can be considered to be an axial flow centrifugal turbine. 
     The rotor discs of spin-down segment  375  can also be configured in a similar manner as the rotor blades in a conventional axial turbine. In some such configurations, an absence of stator discs or stator blades in the spin-down segment  375  can facilitate the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  375 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-down segment  375 , such as between rotor disc  410  and rotor disc  411 , can be employed to enhance the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  375 . In other embodiments, the spin-down segment  375  can consist only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to reduce the rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  361  of engine  360 . In other embodiments, the stator blades are rotably coupled to the bulk material  361  of the engine  360 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  394  for a given free stream flow speed and a given desired thrust. The function of the stator blades is identical to the function of the rotor blades in the spin-down segment  375 . 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  372  can be provided by the mechanical power extracted from the working material during the decrease in the rate of rotation of the working material about the X-axis in the spin-down segment  375 . This decrease in the rate of rotation of the working material in spin-down segment  375  can be generate a torque which acts on the rotor discs of spin-down segment  375  about the X-axis, and which can be mechanically transferred to drum  395  and to the drive shafts  412  and  413 , and to the rotor discs of spin-up segment  372 . In other embodiments, the rotor discs in spin-down segment  375  can be configured to drive an electrical generator. At least a portion of the energy recovered by the electrical generator can be employed to deliver electrical power to an electrical motor configured to drive the rotor discs of spin-up segment  372 . In the case in which the spin-up segment comprises stator discs, the torque acting on the stator discs in the spin-down segment  375  can be cancelled by the torque acting on the working material in the spin-up segment  372  during nominal operations, such that there is no net torque on the engine  360 . 
     In the embodiment shown in  FIG. 360 , the first BFGA  394  is comprises a three spools connecting the rotor discs of the spin-up segment  372  with the rotor discs of the spin-down segment  375  via external drive shaft or drum  395 , and internal drive shafts  413  and  412 . In other embodiments, the first BFGA can comprise two spools, four spools, or a larger number of spools. Such multi-spool architectures are common in conventional turbofan engines, for example. 
     The first BFGA  394  is configured to generate an effective body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  362 , where the effective body force comprises a non-zero component in the YZ-plane and directed away from the center of channel  362 , i.e. away from the X-axis or in the radially outwards direction. The effective body force per unit mass acting on the working material arises from the rotation of the working material in an inertial frame, about the X-axis, within the second expansion  373  of channel  722 . Due to the lack of a centripetal body force per unit mass acting on the objects of a working material in the negative radial direction, there is an effective or perceived centrifugal body force per unit mass acting in the positive radial direction, i.e. in a radially outward direction, on objects in the working material. In the steady state, the effective centrifugal force is balanced by the interior surface of drum  395  of BFGA  394 , and an increase in the pressure and density of the working material in the radially increasing direction, i.e. in the direction in the YZ-plane and away from the X-axis, or away from the X-axis. 
     The action of the effective body force per unit mass increases the pressure on at least a portion of interior surface of drum  395  throughout the second expansion  373  of channel  362 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  360  in the negative X-direction. This is due to the surface normal of the interior surface of drum  395  having a component in the positive X-direction throughout the second expansion  373  of channel  362 . An artificial increase in pressure on surfaces with a surface normal which has a non-zero component in the positive X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The specific entropy of the working material is reduced between stations  382  and  383  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  382  and  383  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. The modification of the specific heat capacity between stations  382  and  383  is similar in principle to the modification of the specific heat capacity discussed in the context of  FIGS. 7A-J . 
     The direction of the effective body force per unit mass acting on objects of the working material within channel  362  is in the radially outward direction. In some embodiments, the component of the effective body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the effective body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the effective body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the effective body force acting on the working material within the second expansion  373  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material would be carried out by the first BFGA  394 . 
     Due to the action of the effective body force per unit mass within the second expansion  373  of channel  362 , the pressure within the working material within the second expansion  373  of channel  362  increases in a radially outwards direction at a given location along the length of the channel. A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. At a given location along the length of the channel the change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. Upstream of the spin-up segment  372  and downstream of the spin-down segment  375  the rate of rotation of the working material about the X-axis is negligible in the simplified embodiment shown in  FIGS. 5A-B . The radial variation of the pressure between stations  379  and  381 , and between stations  385  and  387  is approximately uniform. 
     The action of the effective body force per unit mass decreases the pressure on at least a portion of interior surface of wall  414  throughout the second contraction  374  of channel  362 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  360  in the negative X-direction. This is due to the surface normal of the interior surface of wall  414  having a component in the negative X-direction throughout the second expansion  373  of channel  362 . An artificial decrease in pressure on surfaces with a surface normal which has a non-zero component in the negative X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on engine  360  due to the larger pressure of the working material acting on the other surfaces, such as the surface of exit fairing  369  in third expansion  377 , as well as the interior surface of first expansion  371  and second expansion  373 . The specific entropy of the working material is reduced between stations  383  and  384  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  383  and  384  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. The modification of the specific heat capacity between stations  383  and  384  is similar in principle to the modification of the specific heat capacity discussed in the context of  FIGS. 10A-K . 
     Due to the action of the effective body force per unit mass within the second contraction  374  of channel  362 , the pressure within the working material within the second contraction  374  of channel  362  increases in a radially outwards direction at a given location along the length of the channel. A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. At a given location along the length of the channel the change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. 
     A wide variety of body force generating apparatuses, or combinations thereof, can be employed in embodiments of the invention. For example, in other embodiments, the pressure at a point along the X-axis in the first contraction  370 , the third contraction  376 , or the third expansion  377  need not be substantially uniform in the radial direction, but can vary in the radial direction, as exemplified by the embodiment shown in  FIG. 1A  and discussed in the context of  FIGS. 3A-B . For example, a second BFGA configured in a similar manner as the first BFGA  394  can be located within at least a portion of the third expansion  377  and configured to generate a body force on the working material directed in the radially outward direction, as is the case in the first BFGA  394 . In another example, other types of BFGA, such as the type of BFGA described in  FIG. 1A  can be used in place of, or concurrently with, the first BFGA  394  in the exemplary embodiment shown in  FIGS. 5A-B , or throughout portions of channel  362  in which no dedicated BFGA is being employed, such as the first contraction  370 , the third contraction  376 , or the third expansion  377 . As discussed in the context of  FIG. 1A , a BFGA can be employed to generate a body force on the working material in the first contraction  370  or the third contraction  376 , where the body force can comprise a component in a radially inward direction, away from interior surface  393 . As discussed in the context of  FIG. 1A  and  FIGS. 5A-B , a BFGA can be employed to generate a body force on the working material in the first expansion  371 , the second expansion  373 , or the third expansion  377 , where the body force can comprise a component in a radially outward direction, towards interior surface  393 . 
     In another example, several embodiments, such as embodiment  360 , can be connected in series. For example, an embodiment of the invention can comprise a first and a second embodiment  360  of the type shown in  FIGS. 5A-B  connected in series, such that the station  386  of the first embodiment is coincident with station  380  of the second embodiment, or such that station  386  of the first embodiment is equivalent to station  381  of the second embodiment. An annular duct can direct the working material from the exit of a spin-down segment of a first BFGA to the entrance of a spin-up segment of a second BFGA. Due to the cooling of the working material throughout successive embodiments, and the unchanged maximum cross-sectional area of channel  362 , the amount of thrust produced by two embodiments connected in series can exceed the thrust produced by two identical and equivalent embodiments connected in parallel, i.e. operated independently of each other. 
     The exemplary embodiment  360 , as well as other embodiments operated in accordance with the invention, can be employed to reduce the specific entropy of the working material interacting with the embodiments. This can be employed to convert thermal energy of the working material directly into useful work. For instance, embodiment  360  can generate a net thrust force in the negative X-direction. The power associated with the generation of this force can be provided by the thermal energy of the working material flowing through channel  362 . Thus, the working material at station  387  is at a lower temperature than the working material at station  379 , and the relative velocity of the working material at station  387  is larger than the relative velocity of the working material at station  379  relative to embodiment  360 . The acceleration of the working material and the cooling of the working material is a consequence of the work done by the working material on embodiment  360 . 
       FIGS. 6A-B  show a cross-sectional view of one embodiment of the invention. This embodiment can be described as a subsonic and supersonic ramjet. The embodiment shown in  FIGS. 6A-B  is configured in a similar manner as the embodiment shown in  FIG. 2 ,  FIGS. 3A-B , and  FIGS. 6A-B , and will therefore not be described in the same detail. 
     The exemplary embodiment  630  shown is substantially cylindrically symmetric about an axis parallel to the long axis of the embodiment, denoted the X-axis, and coincident with the center of exemplary embodiment  630 . The X-axis is parallel to, and coincident with, a line connecting the tip  719  of the hub  634  with the tip  635  of the exit fairing  636 , and directed to the right of the page. Outside surface  661  is therefore the shape of a tapered cylinder. In other embodiments, outside surface  661  can be elliptical, rectangular, or square, for instance. 
     Exemplary embodiment  630  comprises a channel  632  with inside surface  659  located between a first opening  633  and a second opening  644 , where the channel comprises a first contraction  637 , a spin-up segment  638 , a first expansion  639 , a second contraction  640 , a spin-down segment  641 , and a second expansion  642 . The cross-sectional geometry of channel  632  is circular or annular when viewed along the X-direction. Note that the terms “contraction” and “expansion” refer to the magnitude of the maximum radius of the axially symmetric channel. 
     Note that the channel radius or geometry can change in a different manner as a function of position along the X-axis, or be configured differently, for other embodiments, or other operating conditions, as described in the context of  FIGS. 3A-B . 
     Embodiment  630  comprises a compressor with a first compressor spool  661  and a second compressor spool  669 . The first spool  661  comprises  3  stages, such as a first stage consisting of a first rotor disc  663  and a first stator disc  664 . The first compressor spool  661  is driven by drive shaft  668 . The second compressor spool  669  comprises  6  stages, such as fourth stage consisting of a first rotor disc  674  and a first stator disc  675 . The second compressor spool  669  is driven by drive shaft  668 . 
     Embodiment  630  comprises a turbine  677  with a first turbine spool  678  and a second turbine spool  681 . The first spool  678  comprises  1  stage consisting of a first rotor disc  679  and a first stator disc  680 . The first turbine spool  678  drives drive shaft  676 . The second turbine spool  681  comprises  1  stage, consisting of a first rotor disc  682  and a first stator disc  683 . The second turbine spool  681  drives drive shaft  668 . 
     In the embodiment shown in  FIGS. 6A-B , the inlet is configured to be able to modify the smallest cross-sectional area of the channel  632 . This is accomplished by a sliding annular ring  714  which is able to move along the positive and negative X-direction along shaft  676 , into and out of slot  718 . In  FIG. 6A  the annular ring  714  is shown in an extended position, which facilitates a reduced cross-sectional area in channel  632 . The annular ring  714  can be in this position at engine start, and at low subsonic speeds, for example. In  FIG. 6B  the annular ring  714  is shown in a retracted position, which facilitates an increased cross-sectional area at the inlet of channel  632  and at throat  647 . The annular ring  714  can be in such a position at high subsonic speeds, for example. The annular ring  714  can be configured to regulate the mass flow rate of air through the first BFGA  684  by choking the channel  632  for some modes of operation. In other words, the local speed of the working material at the smallest cross-sectional area of the channel  632  at the annular ring  714  relative to the engine  630  can be approximately equal to the speed of sound during some modes of operation. In some such modes of operation, the local speed of the working material downstream of the smallest cross-sectional area of the channel  632  at the annular ring  714  relative to the engine  630  can be larger than the local speed of sound until a shock wave decelerates the flow to subsonic speeds again before inlet  686 . The regulation of the mass flow rate of the working material flowing through channel  632  by the constriction of channel  632  by annular ring  714  can be employed to regulate the thrust and the performance of first BFGA  684  for some modes of operation. In other embodiments, there need not be a variable geometry inlet to first BFGA  684 , i.e. there need not be a device such as sliding ring  714 . 
     Bulk material  631  can comprise a metal such as aluminium, steel, or titanium. Bulk material  631  can also comprise ceramics. In some embodiments, bulk material  631  comprises composites, such as carbon fiber or fiberglass. 
     In some embodiments, the apparatus contained within inside surface of drum  685  and outside surface  661  does not have to be a solid material, but can contain empty or open spaces, as is common practice in conventional ramjet or jet engine construction. This can serve to avoid an unnecessarily large mass or cost of exemplary embodiment  630 , for instance. 
     In  FIGS. 6A-B , exemplary embodiment  630  moves with constant velocity magnitude and direction relative to a working material in the free stream  645 . The velocity direction of the upstream working material relative to exemplary embodiment  630  is aligned with the X-axis on average, i.e. directed from the left of the page to the right of the page. For clarity of description, the velocity magnitude and direction of the upstream working material relative to exemplary embodiment  630  is assumed to be constant in space and time. In other modes of operation, the upstream relative velocity magnitude and direction need not be constant in space or time. For example, the upstream relative velocity magnitude can increase or decrease as a function of time. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. In the embodiment shown in the figures, the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability comprise compressibility, for example. Note that all materials are compressible to some extent. 
     The working material upstream of exemplary embodiment  630 , such as at station  645 , is moving slower relative to exemplary embodiment  630  than the speed of sound in the working material in the configuration shown in  FIG. 6B . Note that, in some embodiments, the engine  630  can move supersonically through the atmosphere, where the flow is decelerated by a suitably configured inlet, such as the inlet with translating inlet spike shown in  FIGS. 5A-B . Typically, the local free stream flow at station  645  is subsonic, such that the local free stream flow at the rotor tips of the rotor discs, such as rotor disc  663 , is below Mach  1 . In other embodiments, or other modes of operation, the tip speed can be above Mach  1 . The compression of the working material between stations  645  and  647  can be described as a substantially adiabatic compression in this embodiment, where the compression is adiabatic in the sense that no heat is exchanged between the working material in channel  632  and the outside environment in this idealized scenario. The specific entropy of the working material is reduced between stations  648  and  650  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  648  and  650  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. p The average speed of the working material relative to exemplary embodiment  630  in the free stream, i.e. at station  645 , is approximately equal to the speed of sound within the working material at that location. Upstream, such as at station  651 , the average relative speed is smaller than the speed of sound, and downstream, such as at station  653 , the average relative speed is larger than the speed of sound within the working material in this embodiment. The expansion of the working material between stations  651  and  653  can be described as a substantially adiabatic expansion in this embodiment, where the expansion is adiabatic in the sense that no heat is exchanged between the working material in channel  632  and the outside environment in this idealized scenario. In the embodiment shown in  FIGS. 6A-B , the adiabatic expansion between station  651  and  653  can also be described as a substantially isentropic expansion. 
     A first body force per unit mass generating apparatus, or a first “BFGA”,  684  is located within channel  632 . First BFGA  684  is configured to be able to apply an effective body force per unit mass on objects, e.g. atoms or molecules, of the working material. The magnitude of this effective body force can be regulated in this embodiment. This can be accomplished by controlling the rate of rotation of drive shafts  702 ,  676 , and drum  685 , relative to bulk material  631  or engine  630  or an inertial frame, for example. The first BFGA  684  comprises a rotating drum  685  which rotates relative to bulk material  631  about the X-axis. The drum  685  comprises a bulk material which is annular in cross-section when viewed along the X-axis and which encloses channel  632 . The drum  685  is substantially axially symmetric about the X-axis, and can thus be considered to be in the shape of a tapered cylinder, or a cylinder of variable radius along the longitudinal length of the cylinder. The drum  685  comprises a first opening  686  and a second opening through which the working material can flow into and out of the volume enclosed by the annular drum  685 . The rotating drum can be structurally supported by bulk material  631  or the remainder of exemplary embodiment  630  via bearings, such as ball or roller bearings, fluid bearings, or magnetic bearings, for example. 
     The first BFGA  684  comprises a spin-up segment  638  which is configured to induce or increase the rate of rotation in the bulk flow of the working material in channel  632  about the X-axis. The spin-up segment  638  comprises at least one rotor disc, such as rotor discs,  696 , and  698 . In  FIGS. 6A-B  there are three rotor discs in the spin-up segment, although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the rotor disc  698  is at least in part structurally supported by drum  685 . The rotor blades of the rotor discs  694  and  696  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  694  are supported by shaft  676 . The rotor blades of the rotor disc  696  are supported by shaft  702 . The axis of the central shaft or support disc can be coincident with the X-axis, and the radius of the outer surface of the central shaft or the support disc can be smaller than the radius of channel  632  at the location of the central shaft or support disc. At least a portion of the working material can be configured to flow around the central shaft or support disc through channel  632 . In some embodiments, at least a portion of the working material can be configured to flow through the central shaft or through the support disc. 
     The rate of rotation of the bulk flow of the working material through channel  632  about the X-axis can be configured to be very large, or substantially increased, at station  648  compared to station  647  due to the action of the spin-up segment  638 . 
     The rotor blades in a rotor disc can be configured in a similar manner as the rotor blades in a conventional centrifugal compressor. Note that, apart from the deflection of fluid flow in the radially outwards direction by the rotor blades of the rotor discs of the spin-up segment  638  and by the effective centrifugal forces, the axial flow of the working material is maintained throughout the spin-up segment  638 . This is in contrast to conventional centrifugal compressors, in which the bulk flow of the working material is typically deflected through ninety degrees at least once, at the inlet of a centrifugal compressor, such as a centrifugal compressor found in a conventional turboprop engine. The spin-up segment  638  can thus be considered to be an “axial flow centrifugal compressor”. The rate of rotation of the rotor blades can be modified to regulate the rate of rotation of the working material within the BFGA  684  for a given free stream flow speed and a given desired thrust. 
     The rotor blades in a rotor disc in the spin-up segment  638  can also be configured in a similar manner as the rotor blades in a conventional axial compressor. In some such configurations, an absence of stator discs or stator blades in the spin-up segment  638  compared to a conventional axial compressor can facilitate the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  638 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-up segment  638 , such as between rotor disc  694  and rotor disc  696 , can be employed to enhance the increase of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-up segment  638 . In other embodiments, the spin-up segment  638  can consist of only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to impart a rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  631  of engine  630 . In other embodiments, the stator blades are rotably coupled to the bulk material  631  of the engine  630 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  684  for a given free stream flow speed and a given desired thrust. 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  638  can be provided by a separate electrical motor, for example. The electrical motor can be configured to rotate drum  685 , and thereby rotate, and supply mechanical power to, the rotor discs of spin-up segment  694 . The electrical power supplied to the electrical motor can be provided by a battery, or by an electrical generator which is driven by a separate turbine, such as the turbine in an auxiliary power unit, or a turbine located downstream of station  651 . For instance, an electrical motor can be employed to power the first BFGA  684  and increase the rate of rotation of drum  685  and the associated rotor discs of the spin-up segment  638  during the starting of the engine  630 , i.e. the increase of the net thrust of the exemplary embodiment  630  from a value which is zero or less than zero, i.e. directed in the positive X-direction, to a value which is above zero, i.e. directed in the negative X-direction. 
     The working material flowing through first expansion  639  and second contraction  640  comprises an axial flow component as well as a rotational or swirl component due to the rotation about the X-axis imparted to the working material by the spin-up segment  638 . In order to maintain the rate of rotation of the bulk flow of the working material about the X-axis, first expansion  639  and second contraction  640  can comprise baffles arranged in a streamwise direction, i.e. along the X-direction. The baffles can be rigidly connected to drum  685 , and therefore rotate about the X-axis. The baffles can be configured to prohibit, or restrict or reduce, the circumferential motion of the working material about the X-axis relative to the drum  685  or relative to the baffles. In this scenario, since the drum  685  and the baffles are rotating, the angular rate of rotation of the bulk flow of the working material in the first expansion  639  is substantially equal to the angular rate of rotation of the drum  685  and the baffles about the X-axis. Thus the baffles can be employed to control and regulate the rate of rotation of the working material flowing through first expansion  639 . In other embodiments, there need not be any baffles between station  648  and  650 , allowing the working material to rotate substantially freely about the X-axis between stations  648  and  650 . Note that the viscous drag from drum  685  can also contribute a rate of rotation to the working material flowing through drum  685 . 
     The first BFGA  684  comprises a spin-down segment  641  which is configured to decrease the rate of rotation in the bulk flow of the working material in channel  632  about the X-axis. The spin-down segment  641  comprises at least one rotor disc, such as rotor disc  700  or rotor disc  701 . The rotor discs in the spin-down segment  641  can be configured in a similar manner as the rotor discs in the spin-up segment  638 . In  FIGS. 6A-B  there are three rotor discs in spin-down segment  641 , although other embodiments can have one rotor disc, or a plurality of rotor discs, or any suitable number of rotor discs. Each rotor blade of the upstream rotor disc of spin down segment  641  is at least in part structurally supported by drum  685 . The rotor blades of the rotor discs  700  and  701  are at least in part structurally supported by a central shaft or a support disc, as is the case in conventional turbofan engines. The rotor blades of the rotor disc  700  are supported by shaft  702 . The rotor blades of the rotor disc  701  are supported by shaft  676 . Note that the drive shaft  676  being driven by rotor disc  701  is also driven by rotor disc  679 . In other embodiments, this need not be the case, i.e. the drive shaft driven by rotor disc  701  and driving rotor disc  694  can be uncoupled from another drive shaft driven by rotor disc  679  and driving second compressor spool  669 . 
     The rate of rotation of the bulk flow of the working material through channel  632  about the X-axis can be configured to be negligible, or substantially reduced, at station  651  compared to station  650  or station  648  due to the action of the spin-down segment  641 . 
     As described in the context of the spin-up segment  638 , the spin-down segment  641  can be considered to be an axial flow centrifugal turbine. 
     The rotor discs of spin-down segment  641  can also be configured in a similar manner as the rotor blades in a conventional axial turbine. In some such configurations, an absence of stator discs or stator blades in the spin-down segment  641  can facilitate the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  641 . In other such configurations, stator discs or stator blades between adjacent rotor discs in the spin-down segment  641 , such as between rotor disc  700  and rotor disc  701 , can be employed to enhance the decrease of the rate of rotation or swirl of the bulk flow of the working material about the X-axis throughout the spin-down segment  641 . In other embodiments, the spin-down segment  641  can consist only one stator disc, or only a plurality of stator discs. In such embodiments, the individual blades or guide vanes of the stator disc are configured to reduce the rate of rotation on the flow about the X-axis. In some embodiments, the stator blades are rigidly connected to the bulk material  631  of engine  630 . In other embodiments, the stator blades are rotably coupled to the bulk material  631  of the engine  630 , where the axis of rotation can be substantially parallel to the radial direction. This allows the angle of attack of the individual stator blades to be adjusted during nominal operations. The angle of attack of the stator blades can be employed to regulate the rate of rotation of the working material within the BFGA  684  for a given free stream flow speed and a given desired thrust. The function of the stator blades is identical to the function of the rotor blades in the spin-down segment  641 . 
     At least a portion of the mechanical power required for the increase in the rate of rotation of the working material about the X-axis in the spin-up segment  638  can be provided by the mechanical power extracted from the working material during the decrease in the rate of rotation of the working material about the X-axis in the spin-down segment  641 . This decrease in the rate of rotation of the working material in spin-down segment  641  can generate a torque which acts on the rotor discs of spin-down segment  641  about the X-axis, and which can be mechanically transferred to drum  685  and to the drive shafts  702  and  676 , and to the rotor discs of spin-up segment  638 . In other embodiments, the rotor discs in spin-down segment  641  can be configured to drive an electrical generator. At least a portion of the energy recovered by the electrical generator can be employed to deliver electrical power to an electrical motor configured to drive the rotor discs of spin-up segment  638 . In the case in which the spin-up segment comprises stator discs, the torque acting on the stator discs in the spin-down segment  641  can be cancelled by the torque acting on the working material in the spin-up segment  638  during nominal operations, such that there is no net torque on the engine  630 . 
     In the embodiment shown in  FIG. 630 , the first BFGA  684  is comprises a three spools connecting the rotor discs of the spin-up segment  638  with the rotor discs of the spin-down segment  641  via external drive shaft or drum  685 , and internal drive shafts  676  and  702 . In other embodiments, the first BFGA can comprise two spools, four spools, or a larger number of spools. Such multi-spool architectures are common in conventional turbofan engines, for example. 
     The first BFGA  684  is configured to generate an effective body force per unit mass which acts on objects, such as atoms or molecules, in the working material, such as air, within channel  632 , where the effective body force comprises a non-zero component in the YZ-plane and directed away from the center of channel  632 , i.e. away from the X-axis or in the radially outwards direction. The effective body force per unit mass acting on the working material arises from the rotation of the working material in an inertial frame, about the X-axis, within the first expansion  639  of channel  722 . Due to the lack of a centripetal body force per unit mass acting on the objects of a working material in the negative radial direction, there is an effective or perceived centrifugal body force per unit mass acting in the positive radial direction, i.e. in a radially outward direction, on objects in the working material. In the steady state, the effective centrifugal force is balanced by the interior surface of drum  685  of BFGA  684 , and an increase in the pressure and density of the working material in the radially increasing direction, i.e. in the direction in the YZ-plane and away from the X-axis, or away from the X-axis. 
     The action of the effective body force per unit mass increases the pressure on at least a portion of interior surface of drum  685  throughout the first expansion  639  of channel  632 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  630  in the negative X-direction. This is due to the surface normal of the interior surface of drum  685  having a component in the positive X-direction throughout the first expansion  639  of channel  632 . An artificial increase in pressure on surfaces with a surface normal which has a non-zero component in the positive X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on these surfaces due to the pressure of the working material acting on these surfaces. The specific entropy of the working material is reduced between stations  648  and  649  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  648  and  649  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. The modification of the specific heat capacity between stations  648  and  649  is similar in principle to the modification of the specific heat capacity discussed in the context of  FIGS. 7A-J . 
     The direction of the effective body force per unit mass acting on objects of the working material within channel  632  is in the radially outward direction. In some embodiments, the component of the effective body force per unit mass along the X-direction is negligible, resulting in no direct contribution to thrust or drag by the BFGA acting on the working material. In other embodiments, the component of the effective body force per unit mass long the positive or negative X-direction can be non-zero. In such embodiments, the effective body force per unit mass can be employed to decelerate or compress the working material, or to accelerate or expand the working material. For example, the effective body force acting on the working material within the first expansion  639  can comprise a component in the negative X-direction. In this case, at least a portion of the compression and deceleration of the working material would be carried out by the first BFGA  684 . 
     Due to the action of the effective body force per unit mass within the first expansion  639  of channel  632 , the pressure within the working material within the first expansion  639  of channel  632  increases in a radially outwards direction at a given location along the length of the channel. A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. At a given location along the length of the channel the change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. Upstream of the spin-up segment  638  and downstream of the spin-down segment  641  the rate of rotation of the working material about the X-axis is negligible in the simplified embodiment shown in  FIGS. 6A-B . The radial variation of the pressure between stations  645  and  647 , and between stations  651  and  653  is approximately uniform. 
     The action of the effective body force per unit mass decreases the pressure on at least a portion of interior surface of wall  704  throughout the second contraction  640  of channel  632 , thereby increasing the propulsive force, or thrust force, acting on the exemplary embodiment  630  in the negative X-direction. This is due to the surface normal of the interior surface of wall  704  having a component in the negative X-direction throughout the first expansion  639  of channel  632 . An artificial decrease in pressure on surfaces with a surface normal which has a non-zero component in the negative X-direction can be employed to artificially increase the propulsive force, or thrust force, acting on engine  630  due to the larger pressure of the working material acting on the other surfaces, such as the surface of exit fairing  636  in second expansion  642 , as well as the interior surface of first expansion  639 . The specific entropy of the working material is reduced between stations  649  and  650  for the depicted operating condition, while the stagnation pressure is increased. As explained below, this is due to a modification of the perceived specific heat capacity of the gas between stations  649  and  650  compared to the nominal specific heat capacity of the gas due to the rotation of the gas about the X-axis and the associated centrifugal body forces acting on the gas. The modification of the specific heat capacity between stations  649  and  650  is similar in principle to the modification of the specific heat capacity discussed in the context of  FIGS. 10A-K . 
     Due to the action of the effective body force per unit mass within the second contraction  640  of channel  632 , the pressure within the working material within the second contraction  640  of channel  632  increases in a radially outwards direction at a given location along the length of the channel. A radial direction is a direction which is perpendicular to the X-axis and directed away from the X-axis. At a given location along the length of the channel the change in pressure, density, or temperature can be modelled as a conventional, isentropic and adiabatic compression along the radial direction in highly simplified, idealized models, for example. 
     A wide variety of body force generating apparatuses, or combinations thereof, can be employed in embodiments of the invention. For example, in other embodiments, the pressure at a point along the X-axis in the first contraction  637 , or the second expansion  642  need not be substantially uniform in the radial direction, but can vary in the radial direction, as exemplified by the embodiment shown in  FIG. 1A  and discussed in the context of  FIGS. 3A-B . For example, a second BFGA configured in a similar manner as the first BFGA  684  can be located within at least a portion of the second expansion  642  and configured to generate a body force on the working material directed in the radially outward direction, as is the case in the first BFGA  684 . In another example, other types of BFGA, such as the type of BFGA described in  FIG. 1A  can be used in place of, or concurrently with, the first BFGA  684  in the exemplary embodiment shown in  FIGS. 6A-B , or throughout portions of channel  632  in which no dedicated BFGA is being employed, such as the first contraction  637 , or the second expansion  642 . As discussed in the context of  FIG. 1A , a BFGA can be employed to generate a body force on the working material in the first contraction  637 , where the body force can comprise a component in a radially inward direction, away from interior surface  659 . As discussed in the context of  FIG. 1A  and  FIGS. 6A-B , a BFGA can be employed to generate a body force on the working material in the first expansion  639 , or the second expansion  642 , where the body force can comprise a component in a radially outward direction, towards interior surface  659 . 
     In another example, several embodiments, such as embodiment  630 , can be connected in series. For example, an embodiment of the invention can comprise a first and a second embodiment  630  of the type shown in  FIGS. 6A-B  connected in series, such that the station  651  of the first embodiment is coincident with station  647  of the second embodiment. An annular duct can direct the working material from the exit of a spin-down segment of a first BFGA to the entrance of a spin-up segment of a second BFGA. Due to the cooling of the working material throughout successive embodiments, and the unchanged maximum cross-sectional area of channel  632 , the amount of thrust produced by two embodiments connected in series can exceed the thrust produced by two identical and equivalent embodiments connected in parallel, i.e. operated independently of each other. 
     The exemplary embodiment  630 , as well as other embodiments operated in accordance with the invention, can be employed to reduce the specific entropy of the working material interacting with the embodiments. This can be employed to convert thermal energy of the working material directly into useful work. For instance, embodiment  630  can generate a net thrust force in the negative X-direction. The power associated with the generation of this force can be provided by the thermal energy of the working material flowing through channel  632 . Thus, the working material at station  653  is at a lower temperature than the working material at station  645 , and the relative velocity of the working material at station  653  is larger than the relative velocity of the working material at station  645  relative to embodiment  630 . The acceleration of the working material and the cooling of the working material is a consequence of the work done by the working material on embodiment  630 . 
       FIGS. 7A-J  schematically show cross-sectional views of embodiments of the invention at different points in time during an exemplary nominal operating condition. 
     Exemplary embodiment  860  comprises a first work exchange apparatus  873  comprising a first chamber  880  and a second work exchange apparatus  895  comprising a second chamber  903 . The working material within the first chamber is subject to a body force per unit mass provided by a body force generating apparatus during nominal operations. A wide variety of body force generating apparatuses can be used. In embodiment  860 , the body force per unit mass is inertial in nature. First chamber  880  is configured to rotate about axis  872 , thereby experiencing an effective centrifugal acceleration, as described in the context of  FIG. 2 . An axis coincident with and parallel to axis  872  and directed from the left of the page to the right of the page is denoted the “X-axis”. An axis perpendicular to the X-axis and in the plane of the page and directed from the bottom of the page to the top of the page is denoted the “Y-axis”. A radial direction is a direction perpendicular to the X-axis, lying in the YZ-plane, and directed away from the X-axis. Due to the rotation of the first chamber  880 , an effective body force per unit mass is acting on the working material within first chamber  880  in the positive radial direction. In the steady state, this results in a temperature gradient within the working material in chamber  880 , where the temperature increases in an increasing, or outward, or positive, radial direction. The pressure and density also increases in a positive radial direction in this scenario. The first and second work exchange apparatuses comprise reciprocating pistons. The second work exchange apparatus can comprise an axial compressor, a centrifugal compressor, or a different type of compressor in other embodiments, for example. 
     In order to enhance the change in temperature throughout first chamber  880  in the positive radial direction, the walls of chamber  880 , such as the walls  864 , can comprise thermally insulating material. The insulating material can comprise polystyrene, ceramics, or fiberglass, and can encompass chamber  880 . This can minimize or reduce the flow of heat from the regions of large temperature within chamber  880  to regions of low temperature within chamber  880  through the walls of chamber  880 . This can increase the magnitude of the temperature difference or the magnitude of the spatial temperature gradient within chamber  880 . 
     The first work exchange apparatus  873  is contained within a rotating apparatus  864  which is configured to rotate about axis  872  relative to apparatus  861 . Rotating apparatus  864  is supported by ball bearings, such as ball bearing  867  or ball bearing  866 . The bulk material  865  of rotating apparatus  864  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. The bulk material  863  of apparatus  861  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. A drive flange  868  can allow external apparatuses, such as electric generators, propellers, or drive shafts to be mechanically coupled to the rotating apparatus  864 . The volume  869  between the rotating apparatus  864  and apparatus  861  is evacuated, i.e. forms a vacuum, in the depicted embodiment. In other embodiments, the volume  869  can comprise a low pressure gas or a fluid specially configured or selected to reduce the viscous drag associated with the relative motion of the rotating apparatus  864  relative to apparatus  861 . 
     The rotating apparatus  864  can comprise several work exchange apparatuses of the same type as the first work exchange apparatus  873 . These work exchange apparatuses can be arranged adjacent to each other in circumferential fashion about axis  872 . The work exchange apparatuses within rotating apparatus  864 , such as first work exchange apparatus  873 , can be considered to be the cylinders of a rotary engine, i.e. a radial engine rotating about a central axis, or axis  872 . For instance, rotating apparatus  864  can comprise six or seven work exchange apparatuses of the same type and general construction as the first work exchange apparatus  873  arranged in circumferential fashion in the YZ-plane about axis  872 . In other embodiments, rotating apparatus  864  can comprise one such work exchange apparatus, where the centrifugal loads are balanced by a counterweight. In other embodiments, rotating apparatus  864  can comprise eight or nine such work exchange apparatuses. In other embodiments, rotating apparatus  864  can comprise a plurality of such work exchange apparatuses. In  FIG. 7G  and  FIG. 7H , the connecting rod  892  and the piston shaft  893  of another first work exchange apparatus are shown, where the other first work exchange apparatus is part of the rotating apparatus  864  and configured in a similar manner as the first work exchange apparatus  873 . Another second work exchange apparatus configured in a similar manner as the second work exchange apparatus  895  can be employed to compress the working material from the other first work exchange apparatus, in a similar manner in which the second work exchange apparatus  895  is employed to compress the working material from the first work exchange apparatus. In other embodiments, a single second work exchange apparatus can be employed to compress the working material of more than one first work exchange apparatus of rotating apparatus  864 . 
     Some embodiments can comprise more than one rotating apparatus of the same type as rotating apparatus  864 . In some such embodiments, the rotating apparatuses can be configured to rotate in opposite directions. This can mitigate or at least partially cancel any gyroscopic effects associated with the rotation of the masses associated with the rotating apparatuses and the change in the orientation of the associated axes of rotation of the rotating masses in an inertial space. In a subset of such embodiments, the axis of rotation of a first rotating apparatus is parallel to and coincident with an axis of rotation of a second rotating apparatus. 
     The bulk material  905  of second work exchange apparatus  895  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. A working material can also be a liquid such as water. In the embodiment shown in  FIGS. 7A-J , the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability are explained herein. 
     Upstream of first opening  870  the working material can be compressed by an upstream compressor. This can increase the power output of embodiment  860  during nominal operations. The upstream compressor can be a centrifugal or axial flow compressor, or a reciprocating engine, for example. The upstream compressor can also be configured in a similar manner as a turbocharger or a supercharger in a conventional internal combustion engine, or a compressor in a conventional turbojet engine. The upstream compressor can also be referred to as a third work exchange apparatus. Downstream of the upstream compressor and upstream of first opening  870 , heat can also be removed from the working material in a heat exchanger. The heat exchanger can be configured in a similar manner as an intercooler, for example. Embodiments in which an expander, such as an axial or centrifugal turbine, is located upstream of first opening  870  are also within the scope of the invention. The upstream expander can also be referred to as a third work exchange apparatus. Embodiments in which a heat exchanger downstream of the third work exchange apparatus and upstream of the first opening  870  is configured to deliver heat to the working material are also within the scope of the invention. In some embodiments, the second work exchange apparatus  895  and the aforementioned third work exchange apparatus can be the same. In other words, the second work exchange apparats  895  can also be employed to expand or compress the working material prior to entering chamber  880 . 
     In the embodiment shown in  FIGS. 7A-J , and throughout one thermodynamic cycle during nominal operation, the volume  882  beyond the piston  883  is maintained at a vacuum. In other embodiments, the volume  882  need not comprise a vacuum, but can comprise a gas of suitable pressure. The average pressure of the gas in volume  882  can be selected and maintained by a pressure regulating apparatus in a manner in which the structural loads on the components of the reciprocating apparatus are reduced, where the reciprocating apparatus comprises the piston  883 , the piston shaft  884 , connecting rod  888 , connecting plate  891 , or crankshaft  885 , for example. The volume located on the opposite side of piston  907  of the second work exchange apparatus  895  compared to chamber  903  can be configured in a similar manner. The volume can comprise a vacuum, or comprise a gas, where the average pressure of the gas throughout one thermodynamic cycle during nominal operations can be selected and regulated to optimize the performance, or maximize the length of the lifecycle, or minimize the total cost of maintenance, of the second work exchange apparatus  895 . 
     In the embodiment shown in  FIGS. 7A-J , the crankshaft  885  does rotate in an inertial frame during nominal operations. In some such embodiments, crankshaft  885  can rotate about axis  872 . In such embodiments, crankshaft  885  can comprise a counterweight, such as counterweight  886 . In some such embodiments, the rate of rotation of crankshaft  885  about axis  872  can be at a different angular frequency compared to the rate of rotation of rotating apparatus  864 . This allows the transfer of mechanical power from rotating apparatus  864  to the crankshaft, or vice versa. In other embodiments, or other methods of operation, or other operating conditions, crankshaft  885  need not rotate in an inertial frame. 
     A connecting plate  891  is rotably coupled to crank  887  of crankshaft  885 , where the axis of relative rotation is parallel to axis  872 . A connecting rod, such as connecting rod  891 , is rotably coupled to connecting plate  891  via a connecting pin  890 , where the axis of relative rotation is parallel to axis  872 . Connecting rod  891  is also rotably coupled to piston shaft  884  via connecting pin  889  in the crankcase  894 . 
     The nominal operation of the exemplary embodiment  860  for a nominal operating condition throughout one thermodynamic cycle can be described as follows. Throughout this nominal operating condition, the rate of rotation of the first work exchange apparatus  873 , and in particular of chamber  880 , is constant in time and greater than zero. 
     As shown in  FIG. 7A ,  FIG. 7B , and  FIG. 7C , at the beginning of the thermodynamic cycle, the first valve  878  of the first work exchange apparatus  873  is opened, and the piston  883  is moved in the radially outward direction, while the second valve  879  remains closed. This increases the volume of first chamber  880  and draws working material through the annular pipe at first opening  870  of the rotating apparatus  864 , through the inlet pipe  874  with pipe wall  875 , and through the open first valve  878  into first chamber  880 . The motion of piston  883  in the radially outward direction is indicated by the bold arrow in chamber  880  in  FIG. 7B . Due to the body force acting on the working material in chamber  880 , there is an increase in temperature, pressure, and density of the working material along the radially outward direction throughout chamber  880 . 
     Between the configurations shown in  FIG. 7C  and  FIG. 7D , the first valve  878  of the first work exchange apparatus  873  is closed while the second valve  879  remains closed. At this point, the piston  883  has increased the volume of chamber  880  to slightly less than half the total available volume of chamber  880  in this embodiment, and this example method of operation. The ratio of the volume of chamber  880  at the point at which the first valve  878  is closed at this stage in the thermodynamic cycle to the total available volume of chamber  880  is denoted the “initial volume fraction”. In other embodiments, or other example methods of operation, the initial volume fraction can be less than the magnitude of the volume fraction shown in  FIG. 7D . In other embodiments, or other example methods of operation, the initial volume fraction can be larger than the magnitude of the volume fraction shown in  FIG. 7D . In other embodiments, or other example methods of operation, the initial volume fraction can be less than or equal to 0.5. In other embodiments, or other example methods of operation, the initial volume fraction can be larger than 0.5. The optimal initial volume fraction, or the most suitable initial volume fraction, for a given embodiment of the invention, for a given operating condition, and for a given application, can be readily determined using theoretical or empirical methods known in the art. 
     As shown in  FIG. 7E ,  FIG. 7F , and  FIG. 7G , the working material within chamber  880  is subsequently expanded as the volume within chamber  880  is increased further while the first valve  878  and the second valve  879  remain closed. Throughout this expansion the working material does work on piston  883  in this embodiment. In a simplified model this expansion can be described as an adiabatic expansion in the sense that no heat is exchanged between the working material and the environment, such as bulk material  865 . Due to the effective body force per unit mass acting on the working material within chamber  880  in the negative Y-direction, or, in this case, in the radially outward direction, this expansion is associated with a reduction in the specific entropy of the working material within chamber  880 . [ 000203 ] Between the configurations shown in  FIG. 7G  and  FIG. 7H , the second valve  879  of the first work exchange apparatus  873 , and the first valve  901  of the second work exchange apparatus  895  are opened, while the first valve  878  of the first work exchange apparatus  873 , and the second valve  902  of the second work exchange apparatus  895  remain closed. 
     As shown in  FIG. 7I , and  FIG. 7J , the piston  883  is subsequently moved in the radially inward direction, and the piston  907  and piston shaft  908  is moved in the negative Y-direction, while the first valve  878  and the second valve  902  remain closed. This decreases the volume of first chamber  880  and increases the volume of the second chamber  903  and pushes the working material out of first chamber  880  through the open second valve  879 , through outlet pipe  876  with pipe wall  877 , through the annular pipe at second opening  871  of the rotating apparatus  864 , through the inlet pipe  896  with pipe wall  897  of the second work exchange apparatus  895 , through the open first valve  901  and into second chamber  903  of the second work exchange apparatus  895 . The motion of piston  883  in the radially inward direction and the motion of piston  907  in the negative Y-direction is indicated by the bold arrow in chamber  880  and the bold arrow in chamber  903  in  FIG. 7I . 
     Between the configurations shown in  FIG. 7J  and  FIG. 7A , the first valve  901  of the second work exchange apparatus  895  and the second valve  879  of the first work exchange apparatus  873  are subsequently closed. The first valve  878  of the first work exchange apparatus  873  is subsequently opened, as shown in  FIG. 7A . 
     As shown in  FIGS. 7A-E , the working material within chamber  903  is subsequently compressed as the volume within chamber  903  is decreased while the first valve  901  and the second valve  902  remain closed. Throughout this compression piston  907  does work on the working material in chamber  903  in this embodiment. In a simplified model this compression can be described as an adiabatic and isentropic compression. 
     As shown in  FIGS. 7A-E , the working material within chamber  903  is subsequently compressed as the volume within chamber  903  is decreased while the first valve  901  and the second valve  902  remain closed. Throughout this compression piston  907  does work on the working material in chamber  903  in this embodiment. In a simplified model this compression can be described as an adiabatic and isentropic compression. Recall that no body force per unit mass is acting on the working material within chamber  903  of second work exchange apparatus  895  in this example. In other embodiments, a body force per unit mass can act on the working material in chamber  903 , where the component of the body force can be in the positive Y-direction. In other embodiments, a body force per unit mass can act on the working material in chamber  903 , where the component of the body force can be in the negative Y-direction, where the magnitude of the component of the body force per unit mass in the negative Y-direction in chamber  903  is smaller than the component of the body force per unit mass in the negative Y-direction, or the radially outwards direction, in chamber  880 . Throughout this compression, the pressure, temperature and density of the working material in chamber  903  increases. 
     Once the pressure of the working material in chamber  903  has reached the value of the ambient pressure, or the pressure beyond third opening  900 , the second valve  902  can be opened, which occurs between the configurations shown in  FIG. 7E  and  FIG. 7F . 
     As shown in  FIGS. 7F-G  the piston  907  of second work exchange apparatus  895  is subsequently moved further into the positive Y-direction, reducing the volume of chamber  903  and expelling the working material through the open second valve  902 , through the outlet pipe  898  with pipe wall  899 , and out of third opening  900 . 
     Following the expulsion out of the third opening  900  the temperature of the working material is lower than the temperature of the working material at the beginning of the thermodynamic cycle. The temperature of the working material can be subsequently increased to the temperature of the working material at the beginning of the thermodynamic cycle. When the working material is expelled into a large reservoir of working material, such as air expelled into the atmosphere, the temperature increase occurs at substantially constant pressure. This completes the thermodynamic cycle described in  FIGS. 7A-J . In other embodiments the working material can remain in a closed cycle as opposed to an open cycle. In such embodiments, the increase in temperature can occur isobarically, isochorically, or polytropically, for example. 
     As used herein, the term “interaction cycle” describes the properties of the working material throughout its interaction with exemplary embodiment  860 . The interaction cycle is equivalent to the aforementioned closed thermodynamic cycle with the exception of the isobaric heating of the working material after having exited through the third opening  900 . An exemplary interaction cycle can comprise: the drawing or pulling of working material into a first chamber  880 ; the subjecting of the working material within the first chamber  880  to a body force per unit mass, where the body force per unit mass comprises a non-zero component in a first direction, e.g. in the negative Y-direction; the expansion of the working material within the first chamber  880 , where the expansion comprises a non-zero component in the first direction, e.g. in the negative Y-direction; the expulsion of the working material from the first chamber  880  and the drawing or pulling of the working material into a second chamber  903 , where the component of the body force per unit mass is negligible in magnitude along a second direction; the compression of the working material within the second chamber  903 , where the compression comprises a non-zero component in the second direction, e.g. in the positive Y-direction; and the expulsion of the working material from the second chamber  903 . For instance, the interaction cycle described in  FIGS. 7A-J  is approximately described by this exemplary interaction cycle. An interaction cycle can be described as an open thermodynamic cycle, or an incomplete thermodynamic cycle. Due to the reduction of the specific entropy of the working material in chamber  880  during the expansion of the working material, the working material experiences a reduction in temperature throughout an interaction cycle in which the pressure of the working material at the beginning and end of the interaction cycle is identical. Throughout such an interaction cycle the working material need not absorb heat from the environment, or deliver heat to the environment. In this case, the interaction cycle can be described as a substantially adiabatic interaction cycle. Throughout such an interaction cycle, the working material can do a net amount of work on its environment, e.g. on piston  883  of first work exchange apparatus  873  and piston  907  of second work exchange apparatus  895 . According to the first law of thermodynamics, and in an idealized, frictionless scenario, the amount of work done by the working material on its environment throughout a complete, or closed, thermodynamic cycle is equal to the amount of heat absorbed by the working material throughout the cycle. Thus, embodiments of the invention can be employed to convert thermal energy, or heat, contained within the working material, or provided by an external heat source, directly into useful energy, or mechanical work. In some embodiments the mechanical work can be converted into other forms of useful energy, such as electrical energy, or gravitational potential energy. 
     In some embodiments, the interaction cycle also comprises a compression or expansion of the working material upstream of the first opening  870 , as described previously. In some embodiments, the second chamber  903  comprises a body force per unit mass directed in a third direction, e.g. in the negative Y-direction, where the component of the body force per unit mass is smaller than the magnitude of an equivalent component of the body force per unit mass in the first chamber  880  in the first direction, e.g. in the negative Y-direction, and where the compression of the working material in the second chamber  903  comprises a component in the negative third direction, i.e. in the positive Y-direction. In some embodiments, the second chamber  903  comprises a body force per unit mass directed in a fourth direction, e.g. in the positive Y-direction, and where the compression of the working material in the second chamber  903  comprises a component in the fourth direction, e.g. in the positive Y-direction. 
     Since the working material experiences a reduction in temperature throughout the aforementioned interaction cycle, embodiments of the invention can also be employed in applications requiring refrigeration of a thermal reservoir. For example, a closed thermodynamic cycle can be formed by a heat exchanger, where the heat exchanger is configured to allow the working material to flow through the heat exchanger located between the third opening  900  and the first opening  870 . The heat exchanger can be configured to isobarically deliver heat to the working material, for example. The heat exchanger can be configured to remove heat from the interior of a refrigerator, or a room which is to be cooled. The useful mechanical work generated by apparatus  860  can be converted into electrical energy by an electric generator. The electrical energy can be delivered to a national electricity grid, or converted into thermal energy in a different thermal reservoir, such as the atmosphere or outer space, for example. The conversion into thermal energy can comprise Joule heating, or the emission of electromagnetic waves, or photons. In the latter case, the frequencies of the photons can be configured to correspond to the frequencies for which the atmosphere has a low coefficient of absorptivity, such that a large portion of the photons are able to travel through the atmosphere into outer space. Such methods are well known in the field of radiative cooling. 
     In some embodiments, or some example methods of operation, the working material can be returned to the first chamber  880  of the first work exchange apparatus  873  after having been compressed in the second chamber  903  of the second work exchange apparatus  895 . In this manner the working material can be subjected to several consecutive interaction cycles before being expelled through third opening  900 . In other words, several interaction cycles can be connected in series, i.e. arranged sequentially in time. As described in the context of  FIG. 2 , the cooling of the working material throughout successive interaction cycles, and the unchanged maximum volume of first chamber  880 , can result in the amount of work done by the working material throughout at least two consecutive interaction cycles increasing for a subset of embodiments and operating conditions. 
     In other embodiments, an exemplary interaction cycle can comprise: the subjecting of the working material within the first chamber to a body force per unit mass, where the body force per unit mass comprises a non-zero component in a first direction; the compression of the working material within the first chamber, where the compression comprises a non-zero component in the first direction. A similar scenario is also described in  FIG. 11 . For example, the first valve  878  and the second valve  879  can be located at the radially outward location of chamber  882  instead of the radially inward location of chamber  880  shown in  FIGS. 7A-J . In other words the first valve  878  and the second valve  879  can be located on the opposing side of piston  883 , at the radially outward facing side of chamber  882 . Throughout an interaction cycle, the working material can be drawn into chamber  882  through the open first valve. In this case the piston  883  can be drawn back to the innermost radially inward position, such as the position shown in  FIG. 7A , before the first valve is closed. As before, the working material in chamber  882  is subject to a body force per unit mass acting in the radially outwards direction, resulting in a decrease in temperature, pressure, and density of the working material along the radially inward direction throughout chamber  882 . Following the closure of the first valve the piston  883  can do work on the working material and compress the working material in chamber  882 , while the first and second valve to chamber  882  remain closed. Due to the body force acting on the working material throughout this compression, and due to the compression being performed by piston  883  in a radially outward direction, the compression is associated with a reduction in the specific entropy of the working material. Following the compression, the piston  883  can be located at almost the outermost radial position within chamber  882 . Subsequently the second valve of chamber  882  can be opened, and the working material can be expelled from chamber  882  and into a second work exchange apparatus such as second work exchange apparatus  895 . Following the expulsion of the working material, the piston  883  can be located at the outermost radial position within chamber  882  once more, and the second valve can be closed, and the first valve can be opened in anticipation of the next pull or draw of working material into chamber  882 . In the second work exchange apparatus the working material from chamber  883  can be expanded. The expansion can be described as an adiabatic and isentropic expansion in a simplified model. Once the pressure of the working material in the second work exchange apparatus has decreased to a level approximately equivalent to the pressure of the working material in an adjacent thermal reservoir, the working material can be expelled into the adjacent thermal reservoir. The working material can be air, and the adjacent thermal reservoir can be the atmosphere, for example. This completes this interaction cycle. As described in the context of the interaction cycle depicted in  FIGS. 7A-J , a wide variety of alternative configurations, alternative methods of operation, and alternative utilizations or applications of such an interaction cycle are within the scope of the invention. For instance, the interaction cycle can be part of a closed thermodynamic cycle. The working material can also be compressed or expanded prior to entering the interaction cycle, or after exiting the interaction cycle. Several such interaction cycles can also be arranged in series or sequentially in time, for example. 
     In some embodiments, the first work exchange apparatus  873  can comprise four valves, two for chamber  880 , and two for chamber  882 . In such embodiments, both chambers, i.e. both chamber  880  and chamber  882 , can be employed concurrently to compress and expand the working materials located within both chambers. In other words, the piston  883  can simultaneously interact with working material on both the radially inward side, as described in the context of  FIGS. 7A-J , and the working material on the radially outwards side, as described in the preceding paragraph. This can increase the power output of an embodiment of the invention by simultaneously or concurrently utilizing chamber  880  and chamber  882  to reduce the specific entropy of the working material on both sides of piston  883 . Similarly, in some embodiments, the second work exchange apparatus  895  can comprise four valves, two for chamber  903 , and two for the opposite chamber  906 , i.e. the chamber on the opposite side of piston  907 , on the side of the piston located in the negative Y-direction. In some such embodiments, both chamber  903  and the opposite chamber  906  can be employed to compress the working material from chamber  880 , or expand the working material from chamber  882 . In other such embodiments, both chamber  903  can be employed to compress the working material from chamber  880  or expel the working material through the third opening  900 , while the opposite chamber  906  is employed to draw or pull working material from chamber  882 , or from another chamber of another first work exchange apparatus. 
     In some embodiments, or some methods of operation, the second work exchange apparatus  895  can be employed to expand the working material from chamber  880  instead of compressing the working material from chamber  880 . In some embodiments, or some methods of operation, a second work exchange apparatus, such as second work exchange apparatus  895 , need not be required. In such embodiments, the pressure of the working material at the second opening  871  can already be substantially equal to the ambient pressure, or the pressure in the thermal reservoir into which the working material is to be expelled. 
     In some embodiments, the second work exchange apparatus  895  can be part of an inline reciprocating engine. In some embodiments, the second work exchange apparatus  895  can be part of a radial engine. In other embodiments, the second work exchange apparatus  895  can comprise an axial or centrifugal compressor, an axial or centrifugal turbine, or a converging diverging duct, for example. 
       FIG. 8  shows a plot  930  of pressure  932  versus specific volume  931  for the working material in a subset of embodiments of the invention for an example method of operation, or an example thermodynamic cycle, such as the example method of operation shown in  FIGS. 7A-J . 
     Prior to interacting with an embodiment of the invention, the state of the working material is described by station  933  in this example thermodynamic cycle. Station  933  can describe the thermodynamic properties of the working material in a first thermal reservoir, for example. The first thermal reservoir can be the atmosphere of the earth, for example. Between station  933  and station  934 , the working material is compressed  942  adiabatically and isentropically in this example. Following the compression  942 , the working material is pulled or drawn into a first chamber, in which the working material is subject to a body force per unit mass. As a result, a spatial variation  943  in pressure, temperature, and density of the working material is established within the first chamber. The thermodynamic properties  934  of the working material in the first thermal reservoir provide a boundary condition for the spatial variation  943  of the thermodynamic properties of the working material within the first chamber. The thermodynamic properties of the working material at the opposing side of the first chamber, such as at the side of first chamber facing piston  883 , are described by station  936 . In other words, station  936  describes the thermodynamic state of the working material in the first chamber as perceived by piston  883 . 
     Following the pulling or drawing of working material into a first chamber by piston  883 , the valves of first chamber are closed. Subsequently, the working material within the first chamber is expanded by the retraction of piston  883  and an increase in the volume within the first chamber. The resulting change in the thermodynamic properties of the working material within the first chamber as perceived by piston  883 , i.e. at the location of piston  883 , is described by line  944 . The change in the thermodynamic properties of the working material within the first chamber as perceived by the opposite side of the first chamber, e.g. the side facing the valves to the first chamber, such as first chamber  880 , is described by dashed line  945 . Following the completion of the expansion of the working material in the first chamber, the spatial variation of the thermodynamic properties of the working material throughout the first chamber is described by line  946 . Station  937  describes the thermodynamic properties of the working material within the first chamber as perceived by piston  883 , i.e. at the location of piston  883  at this point in the thermodynamic cycle. Station  938  describes the thermodynamic properties of the working material within the first chamber as perceived by the opposing side of the first chamber, e.g. the side facing the valves to the first chamber, such as first chamber  880 , at this point in the thermodynamic cycle. 
     Following the completion of the expansion of the working material in the first chamber, the working material can be expelled out of the first chamber and pulled or drawn into a second chamber. In this simplified example, there is no body force per unit mass acting on the working material in the second chamber. The thermodynamic properties of the working material at the location of the valves, i.e. at station  938 , provides a boundary condition for the thermodynamic properties of the working material within the second chamber. The thermodynamic properties of the working material within the entirety of the second chamber are therefore described by the thermodynamic properties at station  938  at this stage in the thermodynamic cycle. The valves between the second chamber and the first chamber are closed following the expulsion of the working material from the first chamber. The working material in the second chamber can subsequently be compressed  947  adiabatically and isentropically this simplified model. Note that a portion of line  946  between stations  937  and  938  overlaps with line  947  between stations  938  and  940 . Following the adiabatic compression  947 , the thermodynamic state of the working material is described by station  940 . Following the adiabatic compression  947 , the working material can be expelled from the second chamber through a valve into a second reservoir. In some embodiments, the second reservoir and the first reservoir are identical, or one and the same. Within the second reservoir, the working material can be heated isobarically  948  and return to station  933 , thus completing the thermodynamic cycle. Throughout this thermodynamic cycle the working material absorbs heat from the environment, and does a net amount of work on the environment. 
     At least a portion of the mechanical work done on the working material in the second chamber can be provided by the mechanical work done by the working material during the expansion of the working material in the first chamber. For example, the rotating apparatus  864  or the crankshaft  885  can be employed to deliver mechanical power to the second work exchange apparatus  895  in the embodiment shown in  FIGS. 7A-J . 
       FIG. 9A  shows a cross-sectional view of an exemplary embodiment of the invention employing the principles described in the context of  FIGS. 7A-J  and  FIG. 8 . This embodiment can be considered to comprise a rotating radial engine, or a rotary engine, as well as two centrifugal compressors being driven by the main drive shaft. The embodiment shown in  FIG. 9A  is configured in a similar manner as the embodiment shown in  FIGS. 7A-J , and will therefore not be described in the same detail. 
     The apparatus  1442  comprises a first centrifugal compressor  1514  with an impeller  1516  with an inlet  1515  and exit  1518  located within casing  1522 . The first centrifugal compressor  1514  is configured in a similar manner as conventional centrifugal compressors found in turbochargers and superchargers in conventional automobile and aircraft engines. The first centrifugal compressor  1514  comprises a diffuser  1524  and a volute  1525  and exit pipe  1526 . During nominal operations the first centrifugal compressor  1514  compresses the working material after entering through inlet  1515  and before exiting into volute  1525  and pipe  1526 . Following the compression in the first centrifugal compressor  1514 , the working material enters the rotary engine  1444 . 
     Note that the first centrifugal compressor  1514  and the compression of the working material therein is not an essential part of embodiments of the invention. The purpose of the first centrifugal compressor  1514  is to increase the density of the working material at the inlet to rotary engine  1444 , and thereby increase the mass flow rate of the working material through rotary engine  1444 . This can increase the power output of rotary engine  1444 . The purpose of the first centrifugal compressor  1514  is therefore similar to the purpose of a conventional turbocharger or supercharger on a conventional piston engine, such as a conventional automobile or aircraft engine. In some embodiments, the working material can also pass through an intercooler between the exit of supercharger  1514  and the inlet to rotary engine  1444 . The intercooler can cool down the working material by facilitating a heat exchange with the ambient working material, such as air in the atmosphere, for example. The intercooler can alternatively or concurrently cool down the working material by facilitating a heat exchange with the working material in the exhaust of engine  1442 , i.e. the working material exiting the second centrifugal compressor  1501  through pipe  1513 , for example. Such methods are well known in the field of turbocharging or supercharging reciprocating engines. 
     The apparatus  1442  comprises a rotary engine  1444 , which is shown in cross-sectional view in  FIG. 9A . The engine is located inside a casing  1443  which contains a volume of reduced pressure, or a vacuum, in order to reduce the viscous drag of rotary engine  1444  rotating within casing  1443  about axis  1448  during nominal operations. The rotary engine  1444  comprises a plurality of cylinders, pistons, or work exchange apparatuses, such as work exchange apparatus  1450  and work exchange apparatus  1459 . The rotary engine  1444  can comprise two, eight or twelve work exchange apparatuses, for example. In other embodiments, the rotary engine  1444  can comprise any number of work exchange apparatuses, such as three, seven, ten, or eleven work exchange apparatuses or cylinders. Each work exchange apparatus is configured similarly. Work exchange apparatus  1450  comprises a piston  1451  and piston shaft  1452 . Work exchange apparatus  1459  comprises a piston  1470  and piston shaft  1471  rotably coupled to piston rod  1472 , which in turn is rotably coupled to connecting plate  1457 , which in turn is rotably coupled to crank  1455 . Valves  1464  and  1465  allow working material to enter and exit chamber  1466 . Counterweight  1454  balances the mass of the crank  1455  of crankshaft  1453 . 
     Working material can flow through pipe  1526  through several circular channels, such as channel  1446  through exit shaft  1449  of inside casing  1444 , and into the cylinders via the valves. Similarly, the working material can flow through several circular channels, such as channel  1447  through exit shaft  1449  of inside casing  1444 , out of the cylinders and into centrifugal compressor  1501 . 
     The apparatus  1442  comprises a second centrifugal compressor  1501  with an impeller  1503  with an inlet  1502  and exit  1505  and back plate  1508  located within a casing. The second centrifugal compressor  1501  is configured in a similar manner as conventional centrifugal compressors found in turbochargers and superchargers in conventional automobile and aircraft engines. The second centrifugal compressor  1501  comprises a diffuser  1511  and a volute  1512  and exhaust pipe  1513 . During nominal operations the second centrifugal compressor  1501  compresses the working material after entering through inlet  1502  and before exiting into volute  1512  and pipe  1513 . Following the compression in the second centrifugal compressor  1501 , the working material is exhausted into the ambient reservoir, such as the atmosphere of earth, at a colder temperature than at inlet  1515  to engine  1442  during nominal operations. 
     During nominal operations, i.e. during power production by engine  1442 , there is a difference in the rotational speeds of crankshaft  1453  and exit shaft  1449  of inside casing  1444 . Note that exit shaft  1449  is rigidly attached to the crankcase and the inside casing  1444 . The crankshaft  1453  can rotate faster or slower than exit shaft  1449  in an inertial frame. In order to reduce the centrifugal loads and stresses, it is preferable for the crankshaft  1453  to rotate more slowly than exit shaft  1449  in an inertial frame. The differential rotation between the exit shaft  1449  and the crankshaft  1453  in a rotating frame can be converted into a net rotation in an inertial frame by a differential gear. In the embodiment shown in  FIG. 9A , a coaxial differential gear  1475  is employed. The differential gear  1475  comprises two planetary gears. 
     The first planetary gear  1476  comprises a sun gear  1477  which is rigidly coupled to the exit shaft  1449 . The first planetary gear  1476  also comprises several planet gears, such as planet gear  1478  rotably coupled to support shaft  1479 , which in turn is rigidly mounted on support plate  1481 , which does not rotate in an inertial frame during nominal operations. The first planetary gear  1476  also comprises other planet gears, such as planet gears  1482  and  1483 . The first planetary gear  1476  also comprises a ring gear  1484  which can rotate freely relative to the engine  1442 , and is also the ring gear for the second planetary gear  1485 . 
     The second planetary gear  1485  comprises a sun gear  1486  which is rigidly coupled to the crankshaft  1453 . The second planetary gear  1485  also comprises several planet gears, such as planet gear  1487  rotably coupled to a support shaft, which in turn is rigidly mounted on carrier gear  1489 , which rotates freely during nominal operations and is rigidly coupled to drive shaft  1490 . The second planetary gear  1485  also comprises other planet gears, such as planet gears  1491  and  1492 . The second planetary gear  1485  also comprises a ring gear  1484 , as mentioned. 
     The radius of the sun gear of the first planetary gear  1476  is denoted “R 1 ”. The radius of the planetary gear of the first planetary gear  1476  is denoted “RP 1 ”. The radius of the ring gear of the first planetary gear  1476  is denoted “R 1 B”. The radius of the sun gear of the second planetary gear  1485  is denoted “R 2 ”. The radius of the planetary gear of the second planetary gear  1485  is denoted “RP 2 ”. The radius of the ring gear of the second planetary gear  1485  is denoted “R 2 B”. For a typical coaxial differential the ratio of R 1  to RP 1  is equal to the ratio of R 2  to RP 2 . For a typical coaxial differential the ratio of R 1  to R 1 B is equal to the ratio of R 2  to R 2 B. In other words, the first planetary gear  1476  is geometrically similar to the second planetary gear  1485 . In this case, the rate of rotation of the drive shaft  1490  is proportional to the difference in the rates of rotation of the crankshaft  1453  and the exit shaft  1449 . The constant of proportionality is a function of the radius R 1  and RP 1 , for example. In this manner, the drive shaft  1490  can be powered by the difference in the rotational speeds of two rotating drive shafts. 
     The first centrifugal compressor  1514  and the second centrifugal compressor  1501  are driven by drive shaft  1490  via a gear train. For example, the carrier gear  1489  drives a first gear  1496  mounted on drive shaft  1494 . A second gear  1497  is mounted on the drive shaft driven by carrier gear  1489 . The second gear  1497  drives a third gear  1498 , which drives the impeller  1503  of the second centrifugal compressor  1501 . Similarly, impeller  1516  of first centrifugal compressor  1514  is driven by third gear  1500 , which is driven by second gear  1499 , which in turn is coupled to a drive shaft driven by carrier gear  1489 . 
     In other embodiments, the gear train coupling the drive shaft  1490  to the impellers of first centrifugal compressor  1514  and the second centrifugal compressor  1501  can be configured differently. For example, the gear train can comprise more gears, clutches, gearboxes or transmissions, and other such mechanical devices. 
       FIG. 9B  shows the first planetary gear  1476  in cross-sectional view.  FIG. 9C  shows the second planetary gear  1485  in cross-sectional view. 
       FIGS. 10A-K  schematically show cross-sectional views of embodiments of the invention at different points in time during an exemplary nominal operating condition. This embodiment can be considered to comprise a rotating radial engine, or a rotary engine, in which the piston does work on the working material. 
     Exemplary embodiment  1570  comprises a first work exchange apparatus  1587  comprising a first chamber  1594  and a second work exchange apparatus  1626  comprising a second chamber  1633 . The working material within the first chamber is subject to a body force per unit mass provided by a body force generating apparatus during nominal operations. A wide variety of body force generating apparatuses can be used. In embodiment  1570 , the body force per unit mass is inertial in nature. First chamber  1594  is configured to rotate about axis  1585 , thereby experiencing an effective centrifugal acceleration, as described in the context of  FIG. 2 . An axis coincident with and parallel to axis  1585  and directed from the left of the page to the right of the page is denoted the “X-axis”. An axis perpendicular to the X-axis and in the plane of the page and directed from the bottom of the page to the top of the page is denoted the “Y-axis”. A radial direction is a direction perpendicular to the X-axis, lying in the YZ-plane, and directed away from the X-axis. Due to the rotation of the first chamber  1594 , an effective body force per unit mass is acting on the working material within first chamber  1594  in the positive radial direction. In the steady state, this results in a temperature gradient within the working material in chamber  1594 , where the temperature increases in an increasing, or outward, or positive, radial direction. The pressure and density also increases in a positive radial direction in this scenario. The first and second work exchange apparatuses comprise reciprocating pistons in this simplified example. The second work exchange apparatus can comprise an axial turbine, a centrifugal turbine, or a different type of expander in other embodiments, for example. 
     In order to enhance the change in temperature throughout first chamber  1594  in the positive radial direction, the walls of chamber  1594 , such as the walls  1574 , can comprise thermally insulating material. The insulating material can comprise polystyrene, ceramics, or fiberglass, and can encompass chamber  1594 . This can minimize or reduce the flow of heat from the regions of large temperature within chamber  1594  to regions of low temperature within chamber  1594  through the walls of chamber  1594 . This can increase the magnitude of the temperature difference or the magnitude of the spatial temperature gradient within chamber  1594 . 
     The first work exchange apparatus  1587  is contained within a rotating apparatus  1574  which is configured to rotate about axis  1585  relative to apparatus  1571 . Rotating apparatus  1574  is supported by ball bearings, such as ball bearing  1576 . The bulk material of rotating apparatus  1574  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. The bulk material  1573  of apparatus  1571  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. The differential rotation between exit shaft  1586  and crankshaft  1604  can be employed to drive a differential, which in turn can drive external apparatuses, such as electric generators, propellers, or drive shafts to be mechanically coupled to engine  1570 . The volume  1579  between the rotating apparatus  1574  and apparatus  1571  is evacuated, i.e. forms a vacuum, in the depicted embodiment. In other embodiments, the volume  1579  can comprise a low pressure gas or a fluid specially configured or selected to reduce the viscous drag associated with the relative motion of the rotating apparatus  1574  relative to casing apparatus  1571 . 
     The rotating apparatus  1574  can comprise several work exchange apparatuses of the same type as the first work exchange apparatus  1587 . These work exchange apparatuses can be arranged adjacent to each other in circumferential fashion about axis  1585 . The work exchange apparatuses within rotating apparatus  1574 , such as first work exchange apparatus  1587 , can be considered to be the cylinders of a rotary engine, i.e. a radial engine rotating about a central axis, or axis  1585 . For instance, rotating apparatus  1574  can comprise six or seven work exchange apparatuses of the same type and general construction as the first work exchange apparatus  1587  arranged in circumferential fashion in the YZ-plane about axis  1585 . In other embodiments, rotating apparatus  1574  can comprise one such work exchange apparatus, where the centrifugal loads are balanced by a counterweight. In other embodiments, rotating apparatus  1574  can comprise eight or nine such work exchange apparatuses. In other embodiments, rotating apparatus  1574  can comprise a plurality of such work exchange apparatuses. In  FIG. 10A ,  FIG. 10E ,  FIG. 1OF  and  FIG. 10K , the connecting rod  1623  of another first work exchange apparatus are shown, where the other first work exchange apparatus is part of the rotating apparatus  1574  and configured in a similar manner as the first work exchange apparatus  1587 . Another second work exchange apparatus configured in a similar manner as the second work exchange apparatus  1626  can be employed to expand the working material from the other first work exchange apparatus, in a similar manner in which the second work exchange apparatus  1626  is employed to expand the working material from the first work exchange apparatus. In other embodiments, a single second work exchange apparatus can be employed to expand the working material of more than one first work exchange apparatus of rotating apparatus  1574 . 
     Some embodiments can comprise more than one rotating apparatus of the same type as rotating apparatus  1574 . In some such embodiments, the rotating apparatuses can be configured to rotate in opposite directions. This can mitigate or at least partially cancel any gyroscopic effects associated with the rotation of the masses associated with the rotating apparatuses and the change in the orientation of the associated axes of rotation of the rotating masses in an inertial space. In a subset of such embodiments, the axis of rotation of a first rotating apparatus is parallel to and coincident with an axis of rotation of a second rotating apparatus. 
     The bulk material of second work exchange apparatus  1626  can comprise metal such as aluminium, steel, or titanium, or a composite such as carbon fiber or fiberglass, or a ceramic. 
     A working material can be a gas, such as air, helium, or nitrogen, for example. A working material can also be a liquid such as water. In the embodiment shown in  FIGS. 10A-K , the working material is treated as an ideal gas for simplicity. In other embodiments, the working material can be any suitable material, where the conditions for suitability are explained herein. 
     The working material flows from a first opening into inlet pipe  1588 , which directs the working material into the cylinders, such as cylinder  1587 . Upstream of the first opening the working material can be compressed by an upstream compressor. This can increase the power output of embodiment  1570  during nominal operations. The upstream compressor can be a centrifugal or axial flow compressor, or a reciprocating engine, for example. The upstream compressor can also be configured in a similar manner as a turbocharger or a supercharger in a conventional internal combustion engine, or a compressor in a conventional turbojet engine. The upstream compressor can also be referred to as a third work exchange apparatus. Downstream of the upstream compressor and upstream of the first opening, heat can also be removed from the working material in a heat exchanger. The heat exchanger can be configured in a similar manner as an intercooler, for example. Embodiments in which an expander, such as an axial or centrifugal turbine, is located upstream of the first opening are also within the scope of the invention. The upstream expander can also be referred to as a third work exchange apparatus. Embodiments in which a heat exchanger downstream of the third work exchange apparatus and upstream of the first opening is configured to deliver heat to the working material are also within the scope of the invention. In some embodiments, the second work exchange apparatus  1626  and the aforementioned third work exchange apparatus can be the same. In other words, the second work exchange apparats  1626  can also be employed to expand or compress the working material prior to entering chamber  1594 . 
     In the embodiment shown in  FIGS. 10A-K , and throughout one thermodynamic cycle during nominal operation, the crankcase  1609 , or volume  1609  behind the piston  1598 , is maintained at a desired pressure. The average pressure of the gas in volume  1609  can be selected and maintained by a pressure regulating apparatus in a manner in which the structural loads on the components of the reciprocating apparatus are reduced, where the reciprocating apparatus comprises the piston  1598 , connecting rod  1600 , connecting plate  1608 , or crankshaft  1585 , for example. The volume located on the opposite side of piston  1637  of the second work exchange apparatus  1626  compared to chamber  1633  can be configured in a similar manner. The volume can comprise a vacuum, or comprise a gas, where the average pressure of the gas throughout one thermodynamic cycle during nominal operations can be selected and regulated to optimize the performance, or maximize the length of the lifecycle, or minimize the total cost of maintenance, of the second work exchange apparatus  1626 . In other embodiments, the volume  1609  can comprise a vacuum. 
     In the embodiment shown in  FIGS. 10A-K , the crankshaft  1585  rotates in an inertial frame during nominal operations. In some such embodiments, crankshaft  1585  can rotate about axis  1585 . In such embodiments, crankshaft  1585  can comprise a counterweight, such as counterweight  886 . In some such embodiments, the rate of rotation of crankshaft  1585  about axis  1585  can be at a different angular frequency compared to the rate of rotation of rotating apparatus  1574  in an inertial frame. This allows the transfer of mechanical power from rotating apparatus  1574  to the crankshaft, or vice versa. In other embodiments, or other methods of operation, or other operating conditions, crankshaft  1585  need not rotate in an inertial frame. 
     A connecting plate  1608  is rotably coupled to crank  1606  of crankshaft  1585 , where the axis of relative rotation is parallel to axis  1585 . A connecting rod, such as connecting rod  1608 , is rotably coupled to connecting plate  1608  via a connecting pin, where the axis of relative rotation is parallel to axis  1585 . Connecting rod  1608  is also rotably coupled to the piston via connecting pin  1602  in the crankcase  1609 . 
     The nominal operation of the exemplary embodiment  1570  for a nominal operating condition throughout one thermodynamic cycle can be described as follows. Throughout this nominal operating condition, the rate of rotation of the first work exchange apparatus  1587 , and in particular of chamber  1594 , is constant in time and greater than zero. 
     As shown in  FIGS. 10A-E , at the beginning of the thermodynamic cycle, the first valve  1592  of the first work exchange apparatus  1587  is open, and the piston  1598  is moved in the radially inward direction, while the second valve  1593  remains closed. This increases the volume of first chamber  1594  and draws working material through the inlet pipe  1588  at the first opening of the rotating apparatus  1574 , through the open first valve  1592  into first chamber  1594 . The motion of piston  1598  in the radially inward direction is indicated by the bold arrow in chamber  1594 . Due to the body force acting on the working material in chamber  1594 , there is an increase in temperature, pressure, and density of the working material along the radially outward direction throughout chamber  1594 . 
     Between the configurations shown in  FIG. 10E  and  FIG. 10F , the first valve  1592  of the first work exchange apparatus  1587  is closed while the second valve  1593  remains closed. At this point, the piston  1598  has increased the volume of chamber  1594  in this embodiment, and this example method of operation. 
     As shown in  FIG. 10G , and  FIG. 10H , the working material within chamber  1594  is subsequently compressed as the volume within chamber  1594  is decreased while the first valve  1592  and the second valve  1593  remain closed. Throughout this compression the piston  1598  does work on the working material in this embodiment. In a simplified model this compression can be described as an adiabatic compression in the sense that no heat is exchanged between the working material and the environment, such as the bulk material of inside casing  1574 . Due to the effective body force per unit mass acting on the working material within chamber  1594  in the negative Y-direction, or, in this case, in the radially outward direction, this compression is associated with a reduction in the specific entropy of the working material within chamber  1594 . 
     Between the configurations shown in  FIG. 10H  and  FIG. 10I , the second valve  1593  of the first work exchange apparatus  1587 , and the first valve  1631  of the second work exchange apparatus  1626  are opened, while the first valve  1592  of the first work exchange apparatus  1587 , and the second valve  1632  of the second work exchange apparatus  1626  remain closed. 
     As shown in  FIG. 10J , and  FIG. 10K , the piston  1598  is subsequently moved in the radially outward direction, and the piston  1637  and piston shaft  1638  is moved in the negative Y-direction, while the first valve  1592  and the second valve  1632  remain closed. This decreases the volume of first chamber  1594  and increases the volume of the second chamber  1633  and pushes the working material out of first chamber  1594  through the open second valve  1593 , through outlet pipe  1590 , through the annular pipe at a second opening, through circular channels, such as channel  1582  and channel  1583 , through exit shaft  1586  of the rotating apparatus  1574 , through the exit pipe  1584  into inlet pipe  1627  of the second work exchange apparatus  1626 , through the open first valve  1631  and into second chamber  1633  of the second work exchange apparatus  1626 . The motion of piston  1598  in the radially outward direction and the motion of piston  1637  in the negative Y-direction is indicated by the bold arrow in chamber  1594  and the bold arrow in chamber  1633  in  FIG. 10K . 
     Between the configurations shown in  FIG. 10K  and  FIG. 10A , the first valve  1631  of the second work exchange apparatus  1626  and the second valve  1593  of the first work exchange apparatus  1587  are subsequently closed. The first valve  1592  of the first work exchange apparatus  1587  is subsequently or concurrently opened, as shown in  FIG. 10A . 
     As shown in  FIGS. 10A-E , the working material within chamber  1633  is subsequently compressed as the volume within chamber  1633  is decreased while the first valve  1631  and the second valve  1632  remain closed. Throughout this compression piston  1637  does work on the working material in chamber  1633  in this embodiment. In a simplified model this compression can be described as an adiabatic and isentropic compression. 
     As shown in  FIGS. 10A-C , the working material within chamber  1633  is subsequently expanded as the volume within chamber  1633  is increased while the first valve  1631  and the second valve  1632  remain closed. Throughout this expansion the working material in chamber  1633  does work on piston  1637  in this embodiment. In a simplified model this compression can be described as an adiabatic and isentropic compression. Recall that no body force per unit mass is acting on the working material within chamber  1633  of second work exchange apparatus  1626  in this example. In other embodiments, a body force per unit mass can act on the working material in chamber  1633 , where the component of the body force can be in the negative Y-direction. In other embodiments, a body force per unit mass can act on the working material in chamber  1633 , where the component of the body force can be in the positive Y-direction, where the magnitude of the component of the body force per unit mass in the positive Y-direction in chamber  1633  is smaller than the component of the body force per unit mass in the negative Y-direction, or the radially outwards direction, in chamber  1594 . Throughout this expansion, the pressure, temperature and density of the working material in chamber  1633  decreases. 
     Once the pressure of the working material in chamber  1633  has reached the value of the ambient pressure, or the pressure beyond third opening of exit pipe  1629 , the second valve  1632  can be opened, which occurs between the configurations shown in  FIG. 10C  and  FIG. 10D . 
     As shown in  FIGS. 10E-H  the piston  1637  of second work exchange apparatus  1626  is subsequently moved further into the positive Y-direction, reducing the volume of chamber  1633  and expelling the working material through the open second valve  1632 , through the outlet pipe  1629 , and out of the third opening. 
     Following the expulsion out of the third opening  1629  the temperature of the working material is lower than the temperature of the working material at the beginning of the thermodynamic cycle. The temperature of the working material can be subsequently increased to the temperature of the working material at the beginning of the thermodynamic cycle by absorbing heat. When the working material is expelled into a large reservoir of working material, such as air expelled into the atmosphere, the temperature increase occurs at substantially constant pressure during the absorption of energy from the atmosphere. This completes the thermodynamic cycle described in  FIGS. 10A-K . In other embodiments the working material can remain in a closed cycle as opposed to an open cycle. In such embodiments, the increase in temperature can occur isobarically, isochorically, or polytropically, for example. The increase in temperature can be facilitated by a heat exchanger thermally coupled to another thermal reservoir. 
     As used herein, the term “interaction cycle” describes the properties of the working material throughout its interaction with exemplary embodiment  1570 . The interaction cycle is equivalent to the aforementioned closed thermodynamic cycle with the exception of the isobaric heating of the working material after having exited through the third opening  1629 . An exemplary interaction cycle can comprise: the drawing or pulling of working material into a first chamber  1594 ; the subjecting of the working material within the first chamber  1594  to a body force per unit mass, where the body force per unit mass comprises a non-zero component in a first direction, e.g. in the negative Y-direction; the compression of the working material within the first chamber  1594 , where the compression comprises a non-zero component in the first direction, e.g. in the negative Y-direction; the expulsion of the working material from the first chamber  1594  and the drawing or pulling of the working material into a second chamber  1633 , where the component of the body force per unit mass is negligible in magnitude along a second direction; the expansion of the working material within the second chamber  1633 , where the expansion comprises a non-zero component in the second direction, e.g. in the negative Y-direction; and the expulsion of the working material from the second chamber  1633 . For instance, the interaction cycle described in  FIGS. 10A-K  is approximately described by this exemplary interaction cycle. An interaction cycle can be described as an open thermodynamic cycle, or an incomplete thermodynamic cycle. Due to the reduction of the specific entropy of the working material in chamber  1594  during the compression of the working material, the working material experiences a reduction in temperature throughout an entire interaction cycle in which the pressure of the working material at the beginning and end of the interaction cycle is identical. Throughout such an interaction cycle the working material need not absorb heat from the environment, or deliver heat to the environment. In this case, the interaction cycle can be described as a substantially adiabatic interaction cycle. Throughout such an interaction cycle, the working material can do a net amount of work on its environment, e.g. on piston  1637  of first work exchange apparatus  1587  and piston  1637  of second work exchange apparatus  1626 . According to the first law of thermodynamics, and in an idealized, frictionless scenario, the amount of work done by the working material on its environment throughout a complete, or closed, thermodynamic cycle is equal to the amount of heat absorbed by the working material throughout the cycle. Thus, embodiments of the invention can be employed to convert thermal energy, or heat, contained within the working material, or provided by an external heat source, directly into useful energy, or mechanical work. In some embodiments the mechanical work can be converted into other forms of useful energy, such as electrical energy, or gravitational potential energy. In the presence of friction, a fraction of the thermal energy extracted from the working material is converted back into thermal energy or heat. 
     In some embodiments, the interaction cycle also comprises a compression or expansion of the working material upstream of the first opening, as described previously. In some embodiments, the second chamber  1633  comprises a body force per unit mass directed in a third direction, e.g. in the positive Y-direction, where the component of the body force per unit mass is smaller than the magnitude of an equivalent component of the body force per unit mass in the first chamber  1594  in the first direction, e.g. in the negative Y-direction, and where the expansion of the working material in the second chamber  1633  comprises a component in the negative third direction, i.e. in the negative Y-direction. In some embodiments, the second chamber  1633  comprises a body force per unit mass directed in a fourth direction, e.g. in the negative Y-direction, and where the compression of the working material in the second chamber  1633  comprises a component in the fourth direction, e.g. in the negative Y-direction. 
     Since the working material experiences a reduction in temperature throughout the aforementioned interaction cycle, embodiments of the invention can also be employed in applications requiring refrigeration of a thermal reservoir. For example, a closed thermodynamic cycle can be formed by a heat exchanger, where the heat exchanger is configured to allow the working material to flow through the heat exchanger located between the third opening  1629  and the first opening upstream of pipe  1588 . The heat exchanger can be configured to isobarically deliver heat to the working material, for example. The heat exchanger can be configured to remove heat from the interior of a refrigerator, or a room which is to be cooled. The useful mechanical work generated by apparatus  1570  can be converted into electrical energy by an electric generator. The electrical energy can be delivered to a national electricity grid, or converted into thermal energy in a different thermal reservoir, such as the atmosphere or outer space, for example. The conversion into thermal energy can comprise Joule heating, or the emission of electromagnetic waves, or photons, for example. In the latter case, the frequencies of the photons can be configured to correspond to the frequencies for which the atmosphere has a low coefficient of absorptivity, such that a large portion of the photons are able to travel through the atmosphere into outer space. Such methods are well known in the field of radiative cooling. 
       FIG. 11  shows a plot  970  of pressure  972  versus specific volume  971  for the working material in a subset of embodiments of the invention for an example method of operation. 
     Prior to interacting with an embodiment of the invention, the state of the working material is described by station  973  in this example thermodynamic cycle. Station  973  can describe the thermodynamic properties of the working material in a first thermal reservoir, for example. The first thermal reservoir can be the atmosphere of the earth, and the working material can be air, for example. Station  976  immediately follows station  973 . Between station  973  and station  976 , the working material is compressed  982  adiabatically and isentropically in this example. Following the compression  982 , the working material is pulled or drawn into a first chamber, in which the working material is subject to a body force per unit mass. As a result, a spatial variation  983  in pressure, temperature, and density of the working material is established within the first chamber. Note that a portion of line  982  between stations  973  and  976  overlaps with line  983  between stations  976  and  974 . The thermodynamic properties  976  of the working material in the first thermal reservoir provide a boundary condition for the spatial variation  983  of the thermodynamic properties of the working material within the first chamber. The thermodynamic properties of the working material at the opposing side of the first chamber, such as at the side of first chamber, such as chamber  882 , facing piston  883 , are described by station  974 . Note that the reference designators refer to an embodiment adapted from the embodiment shown in  FIGS. 7A-J  for the purposes of the thermodynamic cycle shown in  FIG. 11 . The principles of one such exemplary adaptation are discussed in the context of  FIGS. 7A-J . In other words, station  974  describes the thermodynamic state of the working material in the first chamber  882  as perceived by piston  883  of the adapted embodiment. 
     Following the pulling or drawing of working material into a first chamber by piston  883 , the valves of first chamber are closed. Subsequently, the working material within the first chamber is compressed by the extension of piston  883  and a decrease in the volume within the first chamber. The resulting change in the thermodynamic properties of the working material within the first chamber  882  as perceived by piston  883 , i.e. at the location of piston  883 , is described by line  985 . The change in the thermodynamic properties of the working material within the first chamber as perceived by the opposite side of the first chamber, e.g. the side facing the valves to the first chamber, such as first chamber  882 , is described by dashed line  984 . Following the completion of the compression of the working material in the first chamber, the spatial variation of the thermodynamic properties of the working material throughout the first chamber is described by line  986 . Station  978  describes the thermodynamic properties of the working material within the first chamber as perceived by piston  883 , i.e. at the location of piston  883  at this point in the thermodynamic cycle. Station  977  describes the thermodynamic properties of the working material within the first chamber as perceived by the opposing side of the first chamber, e.g. the side facing the valves to the first chamber, such as first chamber  882 , at this point in the thermodynamic cycle. 
     Following the completion of the compression of the working material in the first chamber, the working material can be expelled out of the first chamber and pulled or drawn into a second chamber. In this simplified example, there is no body force per unit mass acting on the working material in the second chamber. The thermodynamic properties of the working material at the location of the valves, i.e. at station  977 , provides a boundary condition for the thermodynamic properties of the working material within the second chamber. The thermodynamic properties of the working material within the entirety of the second chamber are therefore described by the thermodynamic properties at station  977  at this stage in the thermodynamic cycle. The valves between the second chamber and the first chamber are closed following the expulsion of the working material from the first chamber. The working material in the second chamber can subsequently be expanded  987  adiabatically and isentropically this simplified model. Note that a portion of line  987  between stations  977  and  980  is coincident with line  986  between stations  977  and  978 . Following the adiabatic expansion  987 , the thermodynamic state of the working material is described by station  980 . Following the adiabatic expansion  987 , the working material can be expelled from the second chamber through a valve into a second reservoir. In some embodiments, the second reservoir and the first reservoir are identical, or one and the same. Within the second reservoir, the working material can be heated isobarically  988  and return to station  973 , thus completing the thermodynamic cycle. Throughout this thermodynamic cycle the working material absorbs heat from the environment, and does a net amount of work on the environment. 
     At least a portion of the mechanical work done on the working material in the first chamber can be provided by the mechanical work done by the working material during the expansion of the working material in the second chamber. For example, the rotating apparatus  864  or the crankshaft  885  can be employed to deliver mechanical power to the first work exchange apparatus  873  in an embodiment adapted from the embodiment shown in  FIGS. 7A-J  for the purposes of the thermodynamic cycle shown in  FIG. 11 . 
     In other embodiments, other types of body force generating apparatuses can be employed to modify the component of the body force per unit mass acting on objects or elements within a working material, as explained below. In general, a first work exchange apparatus can be configured to establish a spatial temperature gradient within the working material. A second work exchange apparatus can be employed to modify the local pressure and temperature of the working material, resulting the generation of acoustic waves, pressure waves, or shock waves within the working material. When these waves travel through the spatial gradient in the temperature, the local or global specific entropy of a working material can be reduced. For instance, the waves can travel through the temperature gradient in a direction such that thermal energy is transferred from a region of low temperature to a region of high temperature within the working material. Note that the first and second work exchange apparatus can be identical, i.e. at least a portion of the first work exchange apparatus can be employed to perform the operation, function, or task of the second work exchange apparatus 
     For example, a first work exchange apparatus can comprise a body force generating apparatus which can apply a body force per unit mass to objects within a working material, and thus generate a spatial temperature gradient within a working material. A second work exchange apparatus can be employed to allow the working material at the large temperature side of the temperature gradient within the working material to do work on the work exchange apparatus. In the process of doing work on the second work exchange apparatus, the local working material at the large temperature side of the temperature gradient expands, which is associated with an instantaneous reduction in pressure, temperature, and density of the working material at the large temperature side of the temperature gradient. The local and instantaneous reduction in pressure and density results in pressure waves, or expansion waves, or acoustic waves, or phonons travelling from the large temperature side of the temperature gradient through the temperature gradient to the low temperature side of the temperature gradient at the speed of sound. This expansion wave is associated with a cooling or a reduction in temperature of the working material, as well as a reduction in pressure and a reduction in density throughout the temperature gradient and on the low temperature side of the temperature gradient. Thus, the working material on the low temperature side of the temperature gradient, as well as the working material within the temperature gradient, experiences a reduction in temperature. Effectively, a portion of the energy consumed by the working material at the large temperature side of the temperature gradient while doing work on the second work exchange apparatus is replenished by, or provided by, the portion of the working material at the low temperature side of the temperature gradient and the working material within the temperature gradient. In this process, thermal energy is transferred from the region of low temperature in the working material to a region of large temperature in the working material. This process can lead to a reduction in the specific entropy of the working material. The scenario described in this example is also exemplified by  FIG. 8 ,  FIGS. 7A-J , the first expansion  725  and the second expansion  727  in  FIG. 1A , as well as the second expansion  797  in  FIG. 2 . Since the instantaneous temperature change at the large temperature side of the temperature gradient is also associated with a pressure change, the temperature change is transmitted through the spatial temperature gradient via a pressure wave at the speed of sound. In some embodiments, this allows the transfer of thermal energy through a spatial temperature gradient at much larger rates than thermal conduction, for example. 
     In another example, a first work exchange apparatus can comprise a body force generating apparatus which can apply a body force per unit mass to objects within a working material, and thus generate a spatial temperature gradient within a working material. A second work exchange apparatus can be employed to do work on the working material at the low temperature side of the temperature gradient within the working material. In the process of work being done on the working material, the local working material at the low temperature side of the temperature gradient is compressed, which is associated with an instantaneous increase in pressure, temperature, and density of the working material at the low temperature side of the temperature gradient. The local and instantaneous increase in pressure and density results in pressure waves, compression waves, or acoustic waves, or phonons travelling from the low temperature side of the temperature gradient through the temperature gradient to the high temperature side of the temperature gradient at the speed of sound. This compression wave is associated with a heating or an increase in temperature of the working material, as well as an increase in pressure and an increase in density throughout the temperature gradient and on the large temperature side of the temperature gradient. Thus, the working material on the large temperature side of the temperature gradient, as well as the working material within the temperature gradient, experiences an increase in temperature. Effectively, a portion of the energy delivered to the working material at the low temperature side of the temperature gradient in the process of work being done by the second work exchange apparatus on the working material is delivered to the portion of the working material at the large temperature side of the temperature gradient and the working material within the temperature gradient. In this process, thermal energy is transferred from the region of low temperature in the working material to a region of large temperature in the working material. This process can lead to a reduction in the specific entropy of the working material. The scenario described in this example is also exemplified by  FIG. 11 , and the first contraction  724  and the second contraction  726  in  FIG. 1A . Since the instantaneous temperature change at the low temperature side of the temperature gradient is also associated with a pressure change, the temperature change is transmitted through the spatial temperature gradient via a pressure wave at the speed of sound. In some embodiments, this allows the transfer of thermal energy through a spatial temperature gradient at much larger rates than thermal conduction, for example. 
     A work exchange apparatus can be configured to do work on a working material, or allow a working material to do work on a work exchange apparatus. A work exchange apparatus can comprise another BFGA, or the same BFGA that is being used to induce a spatial temperature gradient within a working material. A work exchange apparatus can also comprise a converging duct, a converging diverging duct, or a diverging duct. A work exchange apparatus can also comprise an axial or centrifugal compressor. A work exchange apparatus can also comprise a propeller or a thrust generating apparatus. A work exchange apparatus can also comprise a reciprocating piston. 
     Note that the specific entropy of a working material can also be increased when the thermodynamic cycle, and the associated thermodynamic apparatuses, are operated in reverse. In this manner mechanical work can be converted into thermal energy. Such embodiments of the invention can be employed in heating applications, for example. 
     There are numerous ways in which such body forces per unit mass can be generated. 
     One type of such a body force per unit mass is the gravitational acceleration acting on a thermal medium. To that end a first chamber can be subjected to a gravitational field, resulting in a gravitational body force per unit mass acting on the elements of a working material in the first chamber. A piston can be employed to compress the working material in the first chamber in the direction of the gravitational acceleration, e.g. “from above”, or to expand the working material in the first chamber in the direction of the gravitational acceleration, e.g. “from below”. In this manner the working material in the first chamber can be compressed or expanded in a manner in which the specific entropy of the working material is reduced, as described herein. A second chamber can also be located in the gravitational field. A piston can be employed to compress or expand the working material in a direction perpendicular to the direction of the gravitational body force per unit mass acting on the working material in the second chamber. In other words, in an adapted embodiment of the embodiment shown in  FIGS. 7A-J , the long axis of the first chamber  880  can be parallel to the local acceleration due to gravity, and the piston  883  can move in a direction parallel to the acceleration due to gravity. In this adapted embodiment, the second work exchange apparatus  895  can be rotated by ninety degrees about an axis in the XZ-plane, e.g. about an axis out of the page. In this manner the long axis of the second chamber  903  is oriented perpendicularly to the acceleration due to gravity, and the piston  907  can move in a direction perpendicular to the acceleration due to gravity. In this manner the working material in the second chamber can be compressed or expanded in substantially isentropically and adiabatically, as described herein. 
     A body force can also arise from the existence of a potential field gradient. One such example is the force which arises from the gradient of an electric potential. For example, the elements of a thermal medium can be configured to be electrically charged. In the context of a thermal medium, the term “elements” refers to the constituent parts of the thermal medium, such as sub-molecular particles, molecules, or a distinct or specified collection of molecules, for example. In the case of a gas, the molecules could be positively or negatively ionized, for instance. The thermal medium may also comprise a collection of mobile electrons. Note that this collection may be contained in a solid, such as a conductor, or it may be described as a gas. By applying an electric field within a reservoir, body forces per unit mass can be generated on the electrically charged elements of the thermal medium inside the reservoir. 
     For other embodiments it may be impossible or inconvenient to use, procure, or create a thermal medium with mobile electrical charges. In this case, elements of the thermal medium may be polarized by applying an electric field, or these elements may already have an intrinsic polarization, as in the case of polar molecules, such as dihydrogen monoxide. When placed in an electric field gradient, these polarized elements can experience a body force. Note that the magnitude of said force depends on the orientation of the polarization axis relative to the electric field, amongst other parameters. Thus an electric field can be configured to generate body forces per unit mass on the polar elements in the thermal medium in a reservoir, as well as polarize elements in the thermal medium, if necessary. The electric field can be applied in a myriad of ways known in the art. 
     Magnetism can also be employed to generate body forces. The thermal medium may comprise diamagnetic, paramagnetic, or ferromagnetic elements. When magnetized, the individual elements in the thermal medium may form magnetic dipoles, or these elements may already have an intrinsic magnetic dipole, such as an electron. When these magnetic dipoles are placed in a magnetic field with a non-zero curl or gradient, they can experience a body force. Note that the magnitude of the body force is a function of the orientation of the magnetic dipole relative to the local magnetic field, amongst other parameters. Thus an external magnetic field can be configured to generate body forces per unit mass on the magnetized elements in the thermal medium in a reservoir, as well as magnetize the elements in the thermal medium, if necessary. The magnetic field can be generated by permanent magnets, ferromagnets, other at least instantaneously magnetized elements, or by an electrical current flowing through an electromagnet, amongst other methods known in the art. 
     The body forces per unit mass may also arise from inertial effects. For instance, a reservoir may be subject to an acceleration in an inertial frame. This results an acceleration of the thermal medium relative to the reservoir. When accelerating a reservoir at a constant rate of acceleration in an inertial frame in a direction vertically upwards towards the top of the page in  FIG. 1 , the thermal medium inside the reservoir will experience an acceleration relative to the reservoir, where the acceleration is directed vertically downwards towards the bottom of the page. Inertial forces can be generated by linear acceleration, i.e. motion of the reservoir along a straight line in the inertial frame. Inertial forces can also be generated by angular acceleration, i.e. motion of the reservoir along a curved path. In general, inertial forces can be generated by any accelerating motion in an inertial frame. The embodiments shown in  FIG. 2  and  FIGS. 7A-J  employ radial acceleration of a reservoir or a chamber. Note that the centripetal acceleration varies linearly with radius in this embodiment. If a substantially uniform body force per unit mass of thermal medium is desired, the depicted apparatus can be located at a larger radius, where the radial dimension of the chamber is only a fraction of said radius. For instance, the radius can be increased by placing the horizontal axis of rotation further upwards towards the top of the page in  FIGS. 7A-J . In some embodiments, the direction vector of the axis of rotation can lie anywhere in a plane perpendicular to the plane of the page and intersecting the plane of the page horizontally. Other embodiments can have different locations and orientations of the axis of rotation, as well as different rotational velocities. These parameters can also vary in time. Embodiments employing other types of forces or combinations thereof are within the spirit and scope of the invention. 
     Unless specified or clear from context, the term “or” is equivalent to “and/or” throughout this paper. 
     The embodiments and methods described in this paper are only meant to exemplify and illustrate the principles of the invention. This invention can be carried out in several different ways and is not limited to the examples, embodiments, arrangements, configurations, or methods of operation described in this paper or depicted in the drawings. This also applies to cases where just one embodiments is described or depicted. Those skilled in the art will be able to devise numerous alternative examples, embodiments, arrangements, configurations, or methods of operation, that, while not shown or described herein, embody the principles of the invention and thus are within its spirit and scope. 
     ASPECTS OF THE INVENTION 
     The invention is further defined by the following aspects. 
     Aspect 1. A fluid interaction apparatus, wherein the fluid interaction apparatus comprises: a working material; a work exchange apparatus, wherein the work exchange apparatus comprises an active surface against which the working material can do work, or with which the work exchange apparatus can do work on the working material; a body force generating apparatus, wherein the direction of the body force applied to the working material by the body force generating apparatus comprises a non-zero component in the positive or negative direction of the external or outward surface normal of the active surface of the work exchange apparatus. 
     Aspect 2. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises a converging duct, and wherein the active surface is the interior wetted surface of the duct, and wherein the active external surface normal has a non-zero component directed against or upstream of the free stream flow direction, and where the component of the body force acting on at least a portion of the volume of fluid entering the converging duct has a non-zero component in the positive or same direction of the active external surface normal 
     Aspect 3. The fluid interaction apparatus of aspect 2, wherein the component of the body force acting on at least a portion of the volume of fluid entering the converging duct has a substantial component in the radially inward direction of the duct, perpendicular to the local free stream flow during level cruise 
     Aspect 4. The fluid interaction apparatus of aspect 2, wherein the body force is configured to reduce the perceived pressure on the exterior active surface of the duct, such that thermal energy can be extracted from the working material and converted into useful mechanical or electrical work at a later time or space, such as in a subsequent and downstream work exchange apparatus, such as a conventional diverging duct. 
     Aspect 5. The fluid interaction apparatus of aspect 2, wherein the duct can be circular, elliptical, polygonal, rectangular, or square 
     Aspect 6. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises a converging duct, and wherein the active surface is the interior wetted surface of the duct, and wherein the active external surface normal has a non-zero component directed against or upstream of the streamwise flow direction, and where the component of the body force acting on at least a portion of the volume of fluid entering the converging duct has a non-zero component in the negative or opposite direction of the active external surface normal 
     Aspect 7. The fluid interaction apparatus of aspect 6, wherein the component of the body force acting on at least a portion of the volume of fluid entering the converging duct has a substantial component in the radially outward direction of the duct, perpendicular to the local free stream flow during level cruise 
     Aspect 8. The fluid interaction apparatus of aspect 6, wherein the body force is configured to increase the perceived pressure on the exterior active surface of the duct, such that thermal energy can be delivered to, or applied on, the working material and by the application of mechanical work onto the working material by the active surface of the duct. 
     Aspect 9. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises a diverging duct, and wherein the active surface is the interior wetted surface of the duct, and wherein the active external surface normal has a non-zero component directed streamwise, or downstream of the free stream flow direction, and where the component of the body force acting on at least a portion of the volume of fluid entering the diverging duct has a non-zero component in the negative or opposite direction of the active external surface normal 
     Aspect 10. The fluid interaction apparatus of aspect 9, wherein the component of the body force acting on at least a portion of the volume of fluid entering the diverging duct has a substantial component in the radially outward direction of the duct, perpendicular to the local free stream flow during level cruise 
     Aspect 11. The fluid interaction apparatus of aspect 9, wherein the body force is configured to increase the perceived pressure on the exterior active surface of the duct, such that thermal energy can be extracted from the working material and converted into useful mechanical work, such as thrust or electricity. 
     Aspect 12. The fluid interaction apparatus of aspect 9, wherein the duct can be circular, elliptical, polygonal, rectangular, or square 
     Aspect 13. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises a diverging duct, and wherein the active surface is the interior wetted surface of the duct, and wherein the active external surface normal has a non-zero component directed streamwise, or downstream of the free stream flow direction, and where the component of the body force acting on at least a portion of the volume of fluid entering the diverging duct has a non-zero component in the positive or same direction of the active external surface normal 
     Aspect 14. The fluid interaction apparatus of aspect 6, wherein the component of the body force acting on at least a portion of the volume of fluid entering the diverging duct has a substantial component in the radially inward direction of the duct, perpendicular to the local free stream flow during level cruise 
     Aspect 15. The fluid interaction apparatus of aspect 13, wherein the body force is configured to reduce the perceived pressure on the exterior active surface of the duct, such that thermal energy can be delivered to, or applied on, the working material and by the application of mechanical work onto the working material by the active surface of the duct. 
     Aspect 16. The fluid interaction apparatus of aspect 1, wherein the working fluid is compressible, such as air, nitrogen, helium 
     Aspect 17. The fluid interaction apparatus of aspect 1, wherein the local free stream fluid flow is supersonic or faster than compression or expansion waves within the fluid 
     Aspect 18. The fluid interaction apparatus of aspect 1, wherein the local free stream fluid flow is subsonic. 
     Aspect 19. The fluid interaction apparatus of aspect 1, wherein the working fluid is substantially incompressible, such as water. 
     Aspect 20. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises turbomachinery, such as an axial or centrifugal compressor, where the active surface can comprise the propeller or rotor blades. 
     Aspect 21. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises propeller blades, or rotor discs, or turbomachinery of any kind, where the active surface can comprise the propeller or rotor blades. 
     Aspect 22. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises reciprocating pistons, and where the active surface is the wetted surface of the piston head which is in contact with the working fluid within any adjacent chambers 
     Aspect 23. The fluid interaction apparatus of aspect 1, wherein the work exchange apparatus comprises a separately arranged, specially configured additional body force generating apparatus configured to do work on the fluid or allow the fluid to do work against it. 
     Aspect 24. The fluid interaction apparatus of aspect 1, wherein the component of the body force is substantially perpendicular to the local free stream flow which interacts with the fluid interaction apparatus, such as a duct or otherwise conventional jet engine. 
     Aspect 25. The fluid interaction apparatus of aspect 1, wherein the body force per unit mass generating apparatus gravitational in nature. 
     Aspect 26. The fluid interaction apparatus of aspect 1, wherein the body force per unit mass generating apparatus inertial in nature. 
     Aspect 27. The fluid interaction apparatus of aspect 26, wherein the body force per unit mass generating apparatus is configured to rotate a volume or bulk of a working fluid in order to provide a perceived inertial body force per mass to the molecules in the working fluid. 
     Aspect 28. The fluid interaction apparatus of aspect 26, wherein the body force per unit mass generating apparatus is configured to accelerate in inertial space a volume or bulk of a working fluid in order to provide a perceived inertial body force per mass to the molecules in the working fluid. 
     Aspect 29. The fluid interaction apparatus of aspect 1, wherein the body force per unit mass generating apparatus electrical in nature. 
     Aspect 30. The fluid interaction apparatus of aspect 29, wherein the body force per unit mass generating apparatus comprises an electrical field generating apparatus, and wherein the working material comprises mobile electrical charges 
     Aspect 31. The fluid interaction apparatus of aspect 29, wherein the body force per unit mass generating apparatus comprises an electrical field generating apparatus, and wherein the working material comprises molecules or objects which carry a permanent or induced electrical polarization 
     Aspect 32. The fluid interaction apparatus of aspect 1, wherein the body force per unit mass generating apparatus magnetic in nature. 
     Aspect 33. The fluid interaction apparatus of aspect 32, wherein the body force per unit mass generating apparatus comprises a magnetic field generating apparatus, and wherein the working material comprises molecules or objects which carry a permanent or induced magnetic dipole or multipole. 
     Aspect 34. The fluid interaction apparatus of aspect 1, wherein the body force per unit mass generating apparatus mechanical in nature. 
     Aspect 35. The fluid interaction apparatus of aspect 34, wherein the body force per unit mass generating apparatus comprises annular, but not necessarily circular, airfoils or ducts configured to induce a pressure gradient substantially perpendicularly to the flow direction in a manner similar to a conventional body force generating apparatus. 
     Aspect 36. The fluid interaction apparatus of any one of aspects 1-35 wherein the thrust produced by such an apparatus is employed to propel and aircraft, such as commercial airliners or transport, watercraft, such as cruise ships or container ships, or land vehicles, such as a car, truck, motorcycle, bike. 
     Aspect 37. A system comprising two or more apparatuses of any one of aspects 1 to 35. 
     Aspect 38. A system comprising two or more apparatuses of any one of aspects 1 to 35, where at least two are connected in series, with the outlet of a first fluid interaction apparatus is at the same time the inlet of a second fluid interaction apparatus. 
     Aspect 39. A system comprising two or more apparatuses of any one of aspects 1 to 35, where at least two are connected in series, with the outlet of a first fluid interaction apparatus is at the same time the inlet of a second fluid interaction apparatus. 
     Aspect 40. The system of aspect 39, wherein a converging duct can be arranged upstream of a diverging duct 
     Aspect 41. The system of aspect 39, wherein a converging duct can be arranged adjacent to a diverging duct 
     Aspect 42. A system comprising at least two systems of claim 41, wherein the fluid flow between any two such systems can comprise supersonic or subsonic flow velocities 
     Aspect 43. A system comprising at least two systems of claim 41, wherein the fluid flow between any two such systems can comprise supersonic flow velocities 
     Aspect 44. A system comprising at least two systems of claim 41, wherein the fluid flow between any two such systems can comprise subsonic flow velocities 
     Aspect 45. The fluid interaction apparatus of aspect 1, wherein the apparatus also comprises a working chamber, apparatuses such as valves configured for drawing and expelling fluid from the chamber, and in which work can be done on a working material by a piston, and in which the working material can do work on the piston; wherein the work exchange apparatus comprises reciprocating pistons, where the active surface is the wetted surface of the piston head which is in contact with the working fluid within any adjacent chamber, and wherein at least a portion of the working material within the working camber can be subjected to the body force per unit mass of at least one body force generating apparatus, wherein the body force per unit mass has a non-zero component in the positive or negative surface normal of the piston, or the positive or negative instantaneous stroke direction of the piston in the chamber. 
     Aspect 46. The fluid interaction apparatus of aspect 45, in which the component of the body force per unit mass acting on the working fluid has a non-zero component in the direction of the active piston head, in the opposite direction of the inward normal of the active piston head, and wherein the active piston head can be retracted from the chamber and increase the volume of the fluid inside the chamber in order to allow the working material to do work on the piston head and cool down and experience a reduction in entropy. 
     Aspect 47. The fluid interaction apparatus of aspect 45, in which the component of the body force per unit mass acting on the working fluid has a non-zero component in the opposite direction of the active piston head, in the same direction of the inward normal of the active piston head, and wherein the active piston head can be inserted into the chamber and decrease the volume of the fluid inside the chamber in order to allow the piston to do work on the fluid and heat the fluid while also reducing the entropy of the fluid. 
     Aspect 48. The fluid interaction apparatus of aspect 45, in which the component of the body force per unit mass acting on the working fluid has a non-zero component in the direction of the active piston head, in the opposite direction of the inward normal of the active piston head, and wherein the active piston head can be inserted into the chamber and decrease the volume of the fluid inside the chamber in order to allow the piston to do work on the fluid and heat the fluid while also increasing the entropy of the fluid. 
     Aspect 49. The fluid interaction apparatus of aspect 45, in which the component of the body force per unit mass acting on the working fluid has a non-zero component in the opposite direction of the active piston head, in the same direction of the inward normal of the active piston head, and wherein the active piston head can be retracted from the chamber and increase the volume of the fluid inside the chamber in order to allow the working material to do work on the piston head and cool down and experience an increase in entropy of the fluid. 
     Aspect 50. The fluid interaction apparatus of aspect 45, wherein a the fluid interaction apparatus also comprises a compressor, such as a centrifugal compressor, axial compressor, or turbocharger, or supercharger, or a reciprocating piston compressor, upstream of the inlet valves of the working chamber, in order to increase the nominal operating pressure and mass flow rate through the working chamber 
     Aspect 51. The fluid interaction apparatus of aspect 45, wherein a the fluid interaction apparatus also comprises an expander, such as a centrifugal turbine, axial turbine, or a reciprocating piston engine, downstream of the outlet valves of the working chamber, in order to recuperate or recover any excess work performed by the piston in the working chamber 
     Aspect 52. A method of interacting with a fluid, the method comprising: providing at least one fluid interaction apparatus of any one of aspects 1 to 35, providing and employing a body force generating apparatus to artificially facilitate a reduced pressure on an active surface of a fluid interaction apparatus with an outward surface normal with non-zero component in the local upstream direction, and contributing to a net thrust and a cooling of the working material as a result 
     Aspect 53. A method of interacting with a fluid, the method comprising: providing at least one fluid interaction apparatus of any one of aspects 1 to 35, providing and employing a body force generating apparatus to artificially facilitate an increased pressure on an active surface of a fluid interaction apparatus with an outward surface normal with non-zero component in the local downstream direction, and contributing to a net thrust and a cooling of the working material as a result 
     Aspect 54. A method of interacting with a fluid, the method comprising: providing at least one fluid interaction apparatus of any one of aspects 1 to 35, providing and employing a body force generating apparatus to artificially facilitate an increase in pressure on an active surface of a fluid interaction apparatus with an outward surface normal with non-zero component in the local upstream direction, and contributing to a drag force and a heating of the working material as a result 
     Aspect 55. A method of interacting with a fluid, the method comprising: providing at least one fluid interaction apparatus of any one of aspects 1 to 35, providing and employing a body force generating apparatus to artificially facilitate an reduced pressure on an active surface of a fluid interaction apparatus with an outward surface normal with non-zero component in the local downstream direction, and contributing to a drag force and a heating of the working material as a result