Patent Publication Number: US-11649991-B2

Title: Double-ended thermoacoustic heat exchanger

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The application is a continuation-in-part of U.S. patent application Ser. No. 17/544,855, filed on Dec. 7, 2021, entitled THERMOACOUSTIC 3D PRINTED STACK AND HEAT EXCHANGER, which claims the benefit of priority from and is a non-provisional of U.S. Provisional Patent Application No. 63/144,275, filed on Feb. 1, 2021, entitled THERMOACOUSTIC 3D PRINTED STACKS AND HEAT EXCHANGERS, the entire disclosures of which are incorporated by reference. 
    
    
     STATEMENT OF GOVERNMENT INTEREST 
     The present invention was made by employees of the United States Department of Homeland Security in the performance of their official duties. The U.S. Government has certain rights in this invention. 
    
    
     FIELD 
     The discussion below relates generally to thermoacoustic refrigeration systems and assemblies and, more particularly, to double-ended symmetric standing wave thermoacoustic refrigeration systems and assemblies including thermoacoustic stacks and heat exchangers which may have configurations designed for additive manufacturing with low thermal conductivity and relatively high heat capacity. 
     BACKGROUND 
     Thermoacoustic heat pumps and refrigerators use acoustic energy to force heat transfer from lower-temperature sources to higher-temperature sinks by employing a porous medium to maintain a continuous temperature gradient. In thermoacoustic refrigerators, environmentally friendly gases are used as the thermodynamic working fluid. In addition to providing a clean technology, thermoacoustic technology has other interesting advantages, including a system with no moving parts, a simple structure, highly scalable components and requiring less equipment. Therefore, it has a longer estimated operating life than conventional refrigerators and requires less manufacturing and maintenance costs. 
     A thermoacoustic refrigerator of the standing wave variety typically includes a quarter-wavelength resonator (an open-closed tube) driven by an acoustic energy, typically from a loudspeaker or a thermoacoustic heat engine being run by a separate thermal energy source (solar or process waste heat is common). An important part of the thermoacoustic refrigerator is the stack, which has a large number of closely spaced adjacent surfaces aligned parallel to the length of the resonator tube. The stack may be constructed by winding a roll of 35-mm photographic film in an example. Lengths of a nylon fishing line may be used to separate adjacent layers of the spirally wound film stack so that air could oscillate longitudinally between the layers along the length of the stack parallel to the length of the resonator tube. The thermoacoustic effect (i.e., the formation of a longitudinal temperature gradient) is created along the porous structure of the stack. 
     SUMMARY 
     Embodiments of the present invention are directed to apparatuses and methods for providing thermoacoustic heating and cooling systems. A thermoacoustic stack permits the formation and maintenance of a thermal temperature gradient in the way of a sustained acoustic disturbance, permitting the forced movement of thermal energy from a region of lower temperature to a region of higher temperature. Specific embodiments provide thermoacoustic stacks of unique geometry and construction specifically designed for additive manufacturing techniques using polymer materials with a relatively low thermal conductivity and a relatively high heat capacity. To enhance the rate at which heat is “pumped” through the stack by the acoustic work energy, heat exchangers have been designed and fabricated to extract thermal energy from the warm end of the stack to be discharged to a heat sink and to transfer thermal energy from a refrigerated space to the cooler end of the stack. The warmer end of the stack must be at higher temperature than the environment to which heat is being rejected and the cooler end of the stack must be at lower temperature than the refrigerated space in order for the refrigerative effect to be realized. 
     A double-ended thermoacoustic refrigeration assembly has two opposed mechanical oscillators such as acoustic drivers with a symmetric stack sandwich in the center of the resonating tube. The stack sandwich has two hot side outboard heat exchangers and a cold heat exchanger in the middle of the stack sandwich. By having two adjustable mechanical oscillators the pressure and displacement waveform profiles can be more finely manipulated by adjusting the relative phase displacement and/or frequency of the two oscillators. This potentially allows the device to be more compact, as it is no longer limited to the geometrically determined resonant frequency of the tube. 
     An aspect is directed to a thermoacoustic refrigeration assembly comprising: a resonating tube having a first end and a second end; a first mechanical oscillator at the first end; and a second mechanical oscillator at the second end. A first thermoacoustic stack is disposed along a length of the resonating tube through which gas travels. The first thermoacoustic stack has a first outboard side heat exchanger disposed on a first outboard side facing away from the first mechanical oscillator and a first inboard side heat exchanger disposed on a first inboard side facing toward the first mechanical oscillator. The first inboard side heat exchanger is disposed between the first outboard side heat exchanger and the first mechanical oscillator. A second thermoacoustic stack is disposed along the length of the resonating tube and between the first thermoacoustic stack and the second mechanical oscillator. The second thermoacoustic stack has a second outboard side heat exchanger disposed on a second outboard side facing away from the second mechanical oscillator and a second inboard side heat exchanger disposed on a second inboard side facing toward the second mechanical oscillator. The second inboard side heat exchanger is disposed between the second outboard side heat exchanger and the second mechanical oscillator. 
     Another aspect is directed to a thermoacoustic refrigeration assembly comprising: a resonating tube having a first end and a second end; a first mechanical oscillator at the first end; and a second mechanical oscillator at the second end. A thermoacoustic stack sandwich is disposed along a length of the resonating tube through which gas travels. The stack sandwich includes a first outboard heat exchanger on a first side of the stack sandwich facing the first mechanical oscillator, a second outboard heat exchanger on a second side of the stack sandwich facing the second mechanical oscillator, and a center heat exchanger disposed between the first outboard heat exchanger and the second outboard heat exchanger. 
     Yet another aspect is directed to a thermoacoustic method for a resonating tube having a first end and a second end and a thermoacoustic stack sandwich disposed along a length of the resonating tube through which gas travels, the stack sandwich including a first outboard heat exchanger on a first side of the stack sandwich facing the first end, a second outboard heat exchanger on a second side of the stack sandwich facing the second end, and a center heat exchanger disposed between the first outboard heat exchanger and the second outboard heat exchanger. The method comprises driving first compressible gas parcels from the first end of the resonating tube toward the second end through the stack sandwich and driving second compressible gas parcels from the second end of the resonating tube toward the first end through the stack sandwich. 
     Other features and aspects of various embodiments will become apparent to those of ordinary skill in the art from the following detailed description which discloses, in conjunction with the accompanying drawings, examples that explain features in accordance with embodiments. This summary is not intended to identify key or essential features, nor is it intended to limit the scope of the invention, which is defined solely by the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The attached drawings disclose the embodiments. 
         FIG.  1    schematically illustrates a thermoacoustic refrigeration apparatus which includes a thermoacoustic stack with heat exchangers wherein compressible gas parcels oscillate longitudinally through the thermoacoustic stack due to an acoustic standing wave. 
         FIG.  2 A  is an oblique front elevational view of a spiral-channel stack according to an embodiment,  FIG.  2 B  is a left end view thereof, and  FIG.  2 C  is a right end view thereof. 
         FIG.  3 A  is an oblique front elevational view of a spiral-channel stack according to another embodiment,  FIG.  3 B  is a left end view thereof,  FIG.  3 C  is an oblique rear elevational view thereof, and  FIG.  3 D  is a right end view thereof. 
         FIG.  4 A  is a perspective view of a slot-channel stack according to another embodiment and  FIG.  4 B  is an end view thereof. 
         FIG.  5 A  is a perspective view of a tube-channel stack according to another embodiment and  FIG.  5 B  is an end view thereof. 
         FIG.  6 A  is a perspective view of a recessed-type external-loop fluid-enhanced heat exchanger according to an embodiment,  FIG.  6 B  is a perspective view thereof showing the external loop fluid passages, and  FIG.  6 C  is an internal end view thereof. 
         FIG.  7 A  is a perspective view of a protruding-type fluid-enhanced heat exchanger according to an embodiment,  FIG.  7 B  is a perspective view thereof showing the fluid passages, and  FIG.  7 C  is an internal end view thereof. 
         FIG.  8 A  is a perspective view of a protruding-type passive heat exchanger according to an embodiment and  FIG.  8 B  is an internal end view thereof. 
         FIG.  9 A  is a perspective view of a constructed refrigeration assembly and  FIG.  9 B  is a perspective view thereof with internal features shown in broken lines, illustrating, in the circled portion, the recessed-type fluid-enhanced heat exchanger on the right side of the stack, a spiral-type stack, and a protruding-type fluid-enhanced heat exchanger on the left side. 
         FIG.  10    is a schematic illustration of a double-ended symmetric standing wave thermoacoustic refrigerator according to an embodiment. 
         FIGS.  11 A- 11 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube as influenced simultaneously by the left acoustic driver and right acoustic driver. 
         FIGS.  12 A- 12 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube as influenced simultaneously by the left acoustic driver and right acoustic driver. 
         FIGS.  13 A- 13 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube as influenced simultaneously by the left acoustic driver and right acoustic driver. 
         FIGS.  14 A- 14 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube as influenced simultaneously by the left acoustic driver and right acoustic driver. 
         FIG.  15    is an example of a graph of actual performance from prototypes showing temperature versus time for a thermoacoustic test conducted with no forced heat rejection on the hot side and an unloaded cold side. 
         FIG.  16    is an example of a graph of actual performance from prototypes showing temperature versus time for a thermoacoustic test conducted with the hot side thermally anchored via active heat rejection and an unloaded cold side. 
         FIG.  17    is another example of a graph of actual performance from prototypes showing temperature versus time for a thermoacoustic test conducted with the hot side thermally anchored via active heat rejection and an unloaded cold side. 
     
    
    
     DETAILED DESCRIPTION 
     A number of examples or embodiments of the present invention are described and disclosed herein. The present invention provides many applicable inventive concepts that have been disclosed and can be embodied in a variety of ways. Rather, as will be appreciated by one of skill in the art, the teachings and disclosures herein can be combined or rearranged with other portions of this disclosure along with the knowledge of one of ordinary skill in the art. 
     Mechanical and Thermal Interactions in Thermoacoustic Stack 
       FIG.  1    schematically illustrates a thermoacoustic refrigeration apparatus which includes a thermoacoustic stack with heat exchangers wherein compressible gas parcels oscillate longitudinally through the thermoacoustic stack due to an acoustic standing wave. The schematic view of  FIG.  1    illustrates the basic theory of a thermoacoustic refrigeration apparatus  100  by showing mechanical and thermal interactions. A thermoacoustic stack  110  is disposed along a length of a resonating tube  120  through which gas travels (leftward and rightward in the example shown). On the right side of the resonating tube is a mechanical oscillator  130  that can be used to produce acoustic disturbance. 
     Compressible gas “parcels” oscillate longitudinally through the stack due to an acoustic standing wave. When moving toward the pressure antinode (leftward, as shown), the parcel encounters higher pressure and its temperature rises as a result of nearly adiabatic compression, allowing it to shed thermal energy to the relatively cooler local portion of the stack surface. As seen in the example of  FIG.  1   , the pressure antinode on the left side is a region of maximum compression and rarefaction. It is also referred to as a displacement node which is a region of minimum molecular motion. 
     Next, the parcel travels toward the pressure node (rightward, as shown) and experiences rarefaction, causing its temperature to drop as a result of nearly adiabatic expansion, permitting the absorption of thermal energy from the relatively warmer local stack surface. As seen in  FIG.  1   , the pressure node on the right side is a region of minimum compression and rarefaction. It is also referred to as a displacement antinode which is a region of maximum molecular motion. 
     In this way, adjacent gas parcels incrementally “pump” a net positive amount of thermal energy toward the pressure antinode (leftward). Removal of the aggregate thermal energy at the warm end (left side as shown) of the stack via a warm side heat exchanger  150  permits continued absorption of low-temperature thermal energy from a refrigerated target at the cool end (right side as shown) of the stack  110  via a cool side heat exchanger  160 . 
     Embodiments 
     The thermoacoustic stack provides narrow longitudinal channel(s) between adjacent wall surfaces oriented parallel to the propagation of directionally bound acoustic disturbances in a standing-wave resonating chamber or tube. Due to oscillatory movement of the thermoacoustic medium (e.g., compressible gas in a tested embodiment) and position-based oscillatory pressurization, the microscopically scaled constituent “parcels” of the medium simultaneously travel longitudinally toward one end of the stack and experience greater pressure (leftward travel toward the left end in  FIG.  1   ). As such, they experience polytropic compression, with a polytropic exponent nearing the value yielding adiabatic compression. This compression causes each constituent to experience a rise in temperature. 
     A main function of the stack is to create a temperature gradient by absorbing a large amount of heat. The spacing between adjacent wall surfaces of the stack is determined by the thermal penetration depth, δ k . The thermal penetration depth is the distance over which effective heat transfer may occur between the gas parcel and an adjacent surface. On the one hand, the spacing should be as small as possible so that more gas is available within this region to increase the heat interaction between the working gas and the stack surface along with an increase in the heat transfer area. On the other hand, a spacing that is very small may create pressure disturbance near the stack and restrict the oscillatory displacement of the gas through viscous influence. In an embodiment, the stack wall spacing is typically between 2&amp; and 4 δ k  (e.g., 3 δ k ). 
       FIG.  1    illustrates the theory behind the thermoacoustic apparatus while  FIGS.  9 A and  9 B  illustrate a prototyped embodiment. Because each constituent is within the thermal penetration depth region of the stack channel&#39;s walls, the higher-temperature compressed medium is able to transfer thermal energy to the stack whenever its temperature exceeds the stack material&#39;s temperature. Conversely, when the acoustic standing wave causes movement of the medium toward the other end of the stack (rightward travel toward the right end in  FIG.  1   ), the pressure decreases and the constituent parcel expands polytropically (again, nearing the adiabatic threshold); its temperature decreases accordingly. Since the parcel had just relinquished some of its thermal energy to the warmer end (left end in  FIG.  1   ) of the stack, its temperature now drops further and, in aggregate, provides a refrigerative effect. If heat exchangers are placed at either end of the stack, with the warmer (i.e., compressively heated) end of the stack shedding its heat to the ambient environment and the cooler (i.e., expanded) end connected to a refrigerated environment, the stack allows the mechanically induced acoustic standing wave to act as a refrigerator. The stack “pumps” thermal energy from the cooler end to the warmer end via mechanically induced longitudinal motion of the thermoacoustic medium within the stack. 
     In one embodiment, the stack is manufactured by 3D printing additively with PLA (polylactic acid) filament. 3D printing provides an effective technique for creating the flow path shape of the stack. Polylactic acid can easily be fabricated into the desired geometries via extruded additive manufacturing techniques, has a relatively high heat capacity, and has a relatively low thermal conductivity to discourage the longitudinal migration of thermal energy within the stack itself, which maintains the desired thermal gradient. When a tighter tolerance is desired, stereolithographic techniques using resin may be ideal, as it also possesses low thermal conductivity. Maintenance of the longitudinal thermal gradient can be further enhanced via microscopic conduction disruption effects inherent of very thin geometries. 
                 General   ⁢         polytropic   ⁢         process   :           Pv   n       =         Constant   ⟹     P   1       ⁢     v   1   n       =       P   2     ⁢     v   2   n           ⁢   
       n   adiabatic     =     γ   =         c   p       c   v       =         1.005         kJ     kg   ·   K           0.718         kJ     kg   ·   K           =     1.4           (     for   ⁢         air   ⁢         at   ⁢         room   ⁢         temperature     )               ⁢   
       Adiabatic   ⁢         process   :           P   1     ⁢     v   1   1.4       =       P   2     ⁢     v   2   1.4               
where P is pressure and v is volume of the thermoacoustic system.
 
     A thermoacoustic system may comprise the thermoacoustic stack, a first heat exchanger connected with the outer wall at the first end, and a second heat exchanger connected with the outer wall at the second end. At least one of the first heat exchanger and the second heat exchanger may have a perforated core region including a plurality of parallel longitudinal openings and may be of a recessed construction to form a female socket into which a resonating tube recesses to make a male-to-female connection. At least one of the first heat exchanger and the second heat exchanger may have a channel to receive a fluid passing therethrough to absorb or discharge heat. 
     The following describes examples of three different channel configurations: 1) single-channel bound by a spiral wall (e.g., spiral stack geometry), 2) multi-channel “slots” bound by linear walls (e.g., parallel stack geometry), and 3) multi-channel square tubes arranged in a grid (e.g., pin array stack geometry). 
       FIG.  2 A  is an oblique front elevational view of a spiral-channel stack according to an embodiment,  FIG.  2 B  is a left end view thereof, and  FIG.  2 C  is a right end view thereof. The spiral-channel stack  200  includes a spiral wall  210  disposed inside an outer wall  220 . The outer wall  220  has a length extending between a first end and a second end. In the embodiment shown, the outer wall  220  is a circular cylindrical wall, but other geometrical shapes may be used in other embodiments (e.g., elliptical or polygonal). 
     An internal wall structure is disposed inside the outer wall and may extend between the first end and the second end. The internal wall structure includes a plurality of spaced adjacent wall surfaces extending along the length of the outer wall between the first end and the second end to provide open flow passages between the spaced adjacent wall surfaces. The open flow passages may extend between the first end and the second end. In general, the internal wall structure may include a plurality of adjacent wall surfaces that are substantially parallel to the length of the outer wall (e.g., parallel ±5° or parallel ±1°). In embodiments, the inner wall structure may have one or more through passages between interior walls allowing flow between inner wall surfaces. 
     In this embodiment, the internal wall structure includes a spiral wall. The spiral wall  210  has an outer edge connected to an interior surface of the outer wall  220  and an inner edge spiraling circumferentially outward to the outer edge. The inner edge may be disposed at a center of the cross-sectional opening provided by the outer wall  220 . The spiral wall  210  may extend parallel to the length of the outer wall  220  between the first end and the second end to provide a spiral-shaped open flow passage between adjacent wall surfaces of the spiral wall. 
     The spiral wall  210  has closely spaced adjacent surfaces aligned parallel to the length of the resonator tube  120 . The stack may be constructed by 3D printing additively with PLA resin. 3D printing provides an effective technique for creating the flow path shape of the stack. PLA can be fabricated easily into the desired geometries via extruded additive manufacturing techniques, has a relatively high heat capacity, and has a relatively low thermal conductivity to discourage the longitudinal migration of thermal energy within the stack itself, which maintains the desired thermal gradient. When a tighter tolerance is desired, stereolithographic techniques using resin may be employed to provide a relatively low thermal conductivity. 
     3D printing generates closely spaced surfaces of the spiral wall  210  having a uniform spacing (e.g., a deviation of less than ±20%, or less than ±10%, or less than ±5%). The spiral stack layers are a few thermal penetration depths apart (e.g., 2 to 4); in some cases, 4 thermal penetration depths may be the optimum layer separation. In one example, the thermal penetration depth is the square root of ((thermal conductivity)/(pi*standing wave frequency*density*isobaric specific heat per unit mass)). If the stack layers are too far apart, the gas cannot effectively transfer heat to and from the stack walls. If the stack layers are too close together, viscous effects hamper the motion of the gas particles. 
     To maintain the structural integrity or stability of the spiral wall  210  and the uniform stack spacing, one or more cross bars or cross members are provided at or near the two ends of the spiral-channel stack  200 .  FIGS.  2 A to  2 C  show a first cross member  230  at the first end of the stack  200  and a second cross member  240  at the second end of the stack  200  opposite from the first end. The first cross member  230  extends across the spiral wall  210  of the stack  200  at the first end between the outer wall  220  and is connected to the spiral wall  210  at the first end. The second cross member  240  extends across the spiral wall  210  of the stack  200  at the second end between the outer wall  220  and is connected to the spiral wall  210  at the second end. The cross members may be formed integrally with the outer wall  220  and the spiral wall  210  by 3D printing or the like. 
     In the embodiment shown, the first cross member  230  and the second cross member  240  are circumferentially spaced from one another by an angle. The angle may be about 45 to 135 degrees, or about 60 to 120 degrees, or about 75 to 105 degrees, or about 90 degrees (e.g., 90°±5%). 
       FIG.  3 A  is an oblique front elevational view of a spiral-channel stack according to another embodiment,  FIG.  3 B  is a left end view thereof,  FIG.  3 C  is an oblique rear elevational view thereof, and  FIG.  3 D  is a right end view thereof. The spiral-channel stack  300  includes a spiral wall  310  disposed inside an outer wall  320 . The spiral-channel stack  300  is similar to the spiral-channel stack  200  of  FIGS.  2 A- 2 C . 
     The spiral-channel stack  300  has two cross members  330 ,  340  at a first end of the stack to stabilize the spiral wall structure and no cross members at the second end of the stack. The cross members  330 ,  340  each extend across the spiral wall  310  of the stack  300  at the first end between the outer wall  320  and are connected to the spiral wall  310  at the first end. The cross members may be formed integrally with the outer wall  320  and the spiral wall  310  by 3D printing or the like. The first cross member  330  and the second cross member  340  are circumferentially spaced from one another by a circumferential angle. The angle may be about 45 to 135 degrees, or about 60 to 120 degrees, or about 75 to 105 degrees, or about 90 degrees (e.g., 90°±5%). 
     When the first cross member ( 230 ,  330 ) and/or the second cross member ( 240 ,  340 ) are connected to the first end or the second end of the stack, they cover a portion of an area of an outer wall opening surrounded by the outer wall ( 220 ,  320 ) at the first end or the second end. The covered portion is generally kept to a minimum. For example, the covered portion but the cross members may be less than about 5% or less than about 1% of the area of the outer wall opening. 
       FIG.  4 A  is a perspective view of a slot-channel stack according to another embodiment and  FIG.  4 B  is an end view thereof. The slot-channel stack  400  includes a plurality of parallel walls  410  disposed inside an outer wall  420 . The parallel walls  410  are substantially uniformly spaced (e.g., uniform spacing ±10% deviation). In the embodiment shown, the outer wall  420  is a circular cylindrical wall, but other geometrical shapes may be used in other embodiments. The parallel walls  410  have closely spaced adjacent surfaces aligned parallel to the length of the resonator tube  120 . The stack may be constructed by 3D printing additively with PLA resin. The slot-channel stack layers (i.e., parallel walls  410 ) are a few thermal penetration depths apart (e.g., 2 to 4). 
     The parallel walls  410  typically are structurally more stable than the spiral wall  310  of  FIGS.  2 A- 3 C . If additional structural stability is desired, a cross member similar to those shown in  FIGS.  2 A- 3 C  may be added. In one example, a cross member may be provided at or near one end of the slot-channel stack  400  and extend between the outer wall  420  perpendicularly to the parallel walls  410 . The cross member is connected to the parallel walls  410 . In another example, another cross member may be provided at or near an opposite end of the slot-channel stack  400  and extend between the outer wall  420  perpendicularly to the parallel walls  410  at the opposite end. The cross member(s) may be formed integrally with the outer wall  420  and the parallel walls  410  by 3D printing or the like. 
       FIG.  5 A  is a perspective view of a tube-channel stack according to another embodiment and  FIG.  5 B  is an end view thereof. The tube-channel stack  500  includes a plurality of uniformly spaced parallel and transverse walls  510  disposed inside an outer wall  520 . In the embodiment shown, the outer wall  520  is a circular cylindrical wall, but other geometrical shapes may be used in other embodiments. The parallel and transverse walls  510  have closely spaced adjacent surfaces aligned parallel to the length of the resonator tube  120 . The stack may be constructed by 3D printing additively with PLA resin. The tube-channel stack layers are a few thermal penetration depths apart (e.g., 2 to 4). The plurality of parallel walls are substantially uniformly spaced (e.g., uniform spacing ±10% deviation) and the plurality of transverse walls are substantially uniformly spaced (e.g., uniform spacing ±10% deviation) and substantially perpendicular to the parallel walls (e.g., 90°±10°). 
     The parallel and transverse walls  510  are relatively more stable structurally than the spiral wall  310  of  FIGS.  2 A- 3 C  and the parallel walls  410  of  FIGS.  4 A- 4 C . If additional structural stability is desired, a cross member similar to those shown in  FIGS.  2 A- 3 C  may be added. In one example, a first cross member may be provided at or near a first end of the slot-channel stack  500  and extend between the outer wall  520  at an angle of about 45° to the parallel and transverse walls  510 . The first cross member is connected to the parallel and transverse walls  510 . In another example, an additional second cross member may be provided at or near a second end of the tube-channel stack  500  opposite from the first end and extend between the outer wall  520  at an angle of about 45° to the parallel and transverse walls  510  at the second end. The second cross member may be parallel to or disposed at a circumferential angle relative to the first cross member. In one example, the first and second cross members may be circumferentially spaced by about 90° in a manner similar to those shown in  FIGS.  2 A- 2 C . The cross member(s) may be formed integrally with the outer wall  520  and the parallel and transverse walls  510  by 3D printing or the like. 
       FIG.  6 A  is a perspective view of a recessed-type external-loop fluid-enhanced heat exchanger  600  according to an embodiment,  FIG.  6 B  is a perspective view thereof showing the external loop fluid passages, and  FIG.  6 C  is an internal end view thereof. 
     This heat exchanger  600  may be constructed of a material with a large thermal conductivity (e.g., greater than about 80 W/m-K or greater than about 150 W/m-K) and a low heat capacity (e.g., lower than about 1200 J/kg-K or lower than about 900 J/kg-K). An example is aluminum alloy which was used in a prototype due to its ease of machinability and machining equipment available to the inventors. Copper would be another viable material and silver would be the best. The heat exchanger  600  has a perforated core region  610  including a plurality of parallel longitudinal openings  612  through which acoustic waves may propagate and a U-shaped channel  620  through which a thermal fluid may flow to absorb or discharge heat. The perforated core region  610  is of a recessed construction to form a female portion or socket into which the resonating tube recesses to make a male-to-female connection. When placed on the warm end of the stack, the heat exchanger solid material directly absorbs thermal energy from the warm end of the stack and passes it to the fluid, which advects it away via bulk fluid motion precipitated by a pump through the U-shaped channel  620 . Similarly, at the cold end, a thermal fluid sheds thermal energy to the heat exchanger solid material, which is then absorbed by the thermoacoustic medium only to be “pumped” up the stack to the warmer end. 
       FIG.  7 A  is a perspective view of a protruding-type fluid-enhanced heat exchanger according to an embodiment,  FIG.  7 B  is a perspective view thereof showing the fluid passages, and  FIG.  7 C  is an internal end view thereof. 
     This heat exchanger  700  may be constructed of a material with a large thermal conductivity and a low heat capacity, similar to the heat exchanger  600 . An example is aluminum alloy. The heat exchanger  700  has a protruding perforated core region  710  including parallel longitudinal openings  712  through which acoustic waves may propagate and parallel channels  720  penetrating the core through which a thermal fluid may flow to absorb or discharge heat. The channels  720  displace areas where perforations could be, somewhat limiting the free area through which acoustic energy may propagate. When placed on the warm end of the stack, the heat exchanger solid material directly absorbs thermal energy from the warm end of the stack and passes it to the fluid, which advects it away via bulk fluid motion precipitated by a pump. Similarly, at the cold end, a thermal fluid sheds thermal energy to the heat exchanger solid material, which is then absorbed by the thermoacoustic medium only to be “pumped” up the stack to the warmer end. The design of the heat exchanger  700  is such that it provides a protruding male plug that recesses into the resonating tube. 
       FIG.  8 A  is a perspective view of a protruding-type passive heat exchanger according to an embodiment and  FIG.  8 B  is an internal end view thereof. This heat exchanger  800  may be constructed of a material with a large thermal conductivity and a low heat capacity, similar to the heat exchanger  600 . An example is aluminum alloy. The heat exchanger  800  has a protruding perforated core region  810  including a plurality of parallel longitudinal openings through which acoustic waves may propagate and an external surface  820  through which heat may be transferred by convection to or from the immersed environment. When placed on the warm end of the stack, the heat exchanger solid material directly absorbs thermal energy from the warm end of the stack and passes it to the outermost surface  820 , which communicates via free or forced convection with the immersed environment. Similarly, at the cold end, a relatively cooler immersed environment sheds thermal energy to the heat exchanger&#39;s outer surface  820 , which is then absorbed by the thermoacoustic medium only to be “pumped” up the stack to the warmer end. The design of the heat exchanger  800  is such that it provides a male plug that recesses into the resonating tube. 
       FIG.  9 A  is a perspective view of a constructed refrigerative assembly  900  in which a thermoacoustic stack  910  is disposed along a length of a resonating tube  920  through which gas travels (leftward and rightward in the example shown). The stack  910  is a spiral-type stack in a prototype. On the right side of the resonating tube  920  is a mechanical oscillator  930  such as a loudspeaker that can be used to produce a standing wave proportional to the length of the resonating tube. The stack  910  is connected to a left-side heat exchanger at a left end or first end and is connected to a right-side heat exchanger at a right end or second end. A left portion of the resonating tube  920  is connected to the left-side heat exchanger. A right portion of the resonating tube  920  is connected to the right-side heat exchanger. 
       FIG.  9 B  is a perspective view of the refrigeration or cooling assembly  900  with internal features shown in broken lines, illustrating, in the circled portion, the recessed-type fluid-enhanced heat exchanger  600  on the right side of the stack  910  as the right-side heat exchanger and a protruding-type fluid-enhanced heat exchanger  700  on the left side of the stack  910  as the left-side heat exchanger. Other embodiments may include a different stack, different heat exchangers, and/or a different mechanical oscillator. 
     Double-Ended Symmetric Standing Wave Thermoacoustic Refrigerator 
       FIG.  10    is a schematic illustration of a double-ended symmetric standing wave thermoacoustic refrigerator  1000  according to an embodiment. 
     A right thermoacoustic stack  1010 R is disposed along a length of a resonating tube  1020  through which gas travels (leftward and rightward as shown). On the right side of the resonating tube  1020  is a right mechanical oscillator  1030 R (e.g., a right acoustic driver) that can be used to produce acoustic disturbance. A left thermoacoustic stack  1010 L is disposed to the left of the right thermoacoustic stack  1010 R. On the left side of the resonating tube  1020  is a left mechanical oscillator  1030 L (e.g., a left acoustic driver) that can be used to produce acoustic disturbance. Compressible gas “parcels” oscillate longitudinally through the stacks  1010 L &amp;  1010 R due to acoustic standing waves generated by the mechanical oscillators  1030 L &amp;  1030 R, which are coupled to a mechanical oscillator controller or acoustic driver controller  1070 . 
     For the right mechanical oscillator  1030 R (first mechanical oscillator) and corresponding right stack  1010 R (first stack), when a right parcel (first parcel) moves toward the pressure antinode leftward, the right parcel encounters higher pressure and its temperature rises as a result of nearly adiabatic compression, allowing it to shed thermal energy to the relatively cooler local portion of the stack surface on the right side. The pressure antinode on the left side is a region of maximum compression and rarefaction, referred to as a left displacement node which is a region of minimum molecular motion. When the right parcel travels toward the pressure node (rightward) and experiences rarefaction, it causes its temperature to drop as a result of nearly adiabatic expansion, permitting the absorption of thermal energy from the relatively warmer local stack surface on the left side. The pressure node on the right side is a region of minimum compression and rarefaction, as a right displacement antinode which is a region of maximum molecular motion. In this way, adjacent gas parcels incrementally “pump” a net positive amount of thermal energy toward the pressure antinode (leftward). Removal of the aggregate thermal energy at the warm end (left side as shown) of the right stack  1010 R via a warm side heat exchanger  1050  permits continued absorption of low-temperature thermal energy from a refrigerated target at the cool end (right side as shown) of the right stack  1010 R via a cool side heat exchanger on the right  1060 R. 
     For the left mechanical oscillator  1030 L (second mechanical oscillator) and corresponding left stack  1010 L (second stack), when a left parcel (second parcel) moves toward the pressure antinode rightward, the left parcel encounters higher pressure and its temperature rises as a result of nearly adiabatic compression, allowing it to shed thermal energy to the relatively cooler local portion of the stack surface on the left side. The pressure antinode on the right side is a region of maximum compression and rarefaction, referred to as a right displacement node which is a region of minimum molecular motion. When the left parcel travels toward the pressure node (leftward) and experiences rarefaction, it causes its temperature to drop as a result of nearly adiabatic expansion, permitting the absorption of thermal energy from the relatively warmer local stack surface on the right side. The pressure node on the left side is a region of minimum compression and rarefaction, as a left displacement antinode which is a region of maximum molecular motion. In this way, adjacent gas parcels incrementally “pump” a net positive amount of thermal energy toward the pressure antinode (rightward). Removal of the aggregate thermal energy at the warm end (right side as shown) of the left stack  1010 L via the warm side heat exchanger  1050  permits continued absorption of low-temperature thermal energy from a refrigerated target at the cool end (left side as shown) of the left stack  1010 L via a cool side heat exchanger on the left  1060 L. 
     The left stack  1010 L and the right stack  1010 R may combine to form a symmetric stack sandwich at the center of the resonating tube  1020 . The stack sandwich includes a left (first) side outboard heat exchanger  1060 R facing the left (first) mechanical oscillator  1030 L and a right (second) side outboard heat exchanger  1060 L facing the right (second) mechanical oscillator  1030 R. It is symmetrical in geometry and dimension with respect to the center heat exchanger  1050 . 
     Having two opposed acoustic drivers  1030 L &amp;  1030 R with a symmetric stack sandwich  1010 L &amp;  1010 R in the center of the resonating tube was conceptually explored via simple simulators and schematics. The basic thermoacoustic principle remains the same, but by having two acoustic drivers the pressure and displacement waveform profiles can be more finely manipulated by adjusting the relative phase displacement and/or frequency of the two adjustable acoustic drivers  1030 L &amp;  1030 R using the acoustic driver controller  1070 . This potentially allows the device  1000  to be more compact, as it is no longer limited to the geometrically determined resonant frequency of the tube  1020 . Another advantage is the hot side outboard heat exchangers  1060 L &amp;  1060 R in the stack sandwich face the acoustic drivers  1030 L &amp;  1030 R; when the cold side heat exchanger does this, as in the simple resonating device shown in  FIG.  1   , a secondary heating effect occurs which partially negates the cooling effect. By sequestering the cold side heat exchanger  1050  in the center of the stack sandwich as the cold center heat exchanger  1050 , it is shielded from this secondary heating effect and is therefore more effective at cooling the refrigerated space. 
     In one embodiment, the drivers  1030 L &amp;  1030 R can be driven in-phase (i.e., clapping mode) with the respective left driver diaphragm and right driver diaphragm moving in opposite directions with respect to one another. In the clapping mode, the total separation distance between the left and right diaphragms widens and narrows by an amount equal to twice the driven amplitude. Conversely, in another embodiment, the drivers  1030 L &amp;  1030 R can be driven out-of-phase (i.e., sloshing mode) with the respective left diaphragm and right diaphragm moving in the same direction with respect to one another. This reverses the roles of the heat exchangers on either the left side ( 1050  &amp;  1060 L) or the right side ( 1050  &amp;  1060 R). In the sloshing mode, the total separation distance between the driver diaphragms remains constant since their respective motions are identical. 
     Visualization of Double-Ended Symmetric Standing Wave Thermoacoustic Refrigerator Using Simple Simulator for a Resonator Length of ½ Wavelength in Clapping Mode 
     In this clapping mode setup, the acoustic drivers  1030 L &amp;  1030 R are being run in-phase with each other (both move inward/outward simultaneously) at a frequency corresponding to double the natural resonant frequency of the tube  1020 . In this arrangement, the pressure antinodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . The pressure nodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion antinodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion nodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . 
       FIGS.  11 A- 11 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube  1020  as influenced simultaneously by the left acoustic driver  1030 L and right acoustic driver  1030 R. The 8 diagrams illustrate graphs from cycle time zero advancing with increments of ⅛ cycle to ⅞ cycle. They are developed using the principle of superposition of waves without modeling wave reflections. The diagrams provide insight into the combined acoustic profile of the overall resonating tube in way of the stack sandwich  1010 L &amp;  1010 R. 
     In  FIG.  11 A , starting at cycle time  0 , the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are at their extreme or maximum inboard locations and temporarily motionless. The overall pressure throughout the tube is neutral (no gradient) and the gas is quiescent (no bulk motion). The pressure gradient induced by the left acoustic driver&#39;s previous cycle is being perfectly cancelled out by the pressure gradient induced by the right acoustic driver&#39;s precious cycle. There is no instantaneous motion of the gas (i.e., it is temporarily stationary). 
     In  FIG.  11 B , at cycle time of ⅛ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving away from each other and accelerating. The pressure is increasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the left and accelerating. At the right quadrant, the gas is flowing to the right and accelerating. As such, all gas is currently center-avoiding. 
     In  FIG.  11 C , at cycle time of ¼ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated to their neutral positions and are moving away from each other (i.e., translating outboard) at maximum velocity. The pressure (i.e., compression) is maximum at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the left at maximum velocity. At the right quadrant, the gas is flowing to the right at maximum velocity. In other words, maximum velocity molecular displacement is occurring at both ends and its direction is outboard. As such, all gas is currently center-avoiding. The gas is motionless at the center. In this way, the adiabatically compressed high temperature gas at center is shedding heat to the hot center heat exchanger  1050  (i.e., some of its thermal energy is extracted for ultimate rejection to the surroundings) and as the gas displaces outward from center, it expands and cools through the stacks  1010 L &amp;  1010 R. By the time it has reached the cold heat exchangers  1060 L &amp;  1060 R at either end of the stack sandwich, it has expanded and cooled enough to accept heat from the refrigerated space. As such, when the device  1000  is run with zero phase difference (i.e., simultaneously converging/diverging diaphragms), the outboard heat exchangers  1060 L &amp;  1060 R are colder and the center one 1050 is the hotter one, which is the opposite configuration as when the driver diaphragms run 180° out-of-phase (see  FIG.  12 E  below). 
     In  FIG.  11 D , at cycle time of ⅜ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving away from each other and decelerating. The pressure is decreasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the left and decelerating. At the right quadrant, the gas is flowing to the right and decelerating. All gas is still center-avoiding. 
     In  FIG.  11 E , at cycle time of ½ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated as far outboard as they can, at their extreme outboard locations. The pressure throughout the tube  1020  is neutral (no gradient) and the gas is quiescent (no bulk motion). There is no instantaneous motion of the gas (i.e., it is temporarily stationary). The tube  1020  has no pressure gradient and no translation. This is similar to the state at cycle time  0 , with the exception of the diaphragms&#39; present position (extreme outboard in  FIG.  11 E  as opposed to extreme inboard in  FIG.  11 A ). Again, the respective pressure profiles of the drivers  1030 L &amp;  1030 R cancel each other out due to symmetry. 
     In  FIG.  11 F , at cycle time of ⅝ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving toward each other and accelerating. The pressure is decreasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right and accelerating. At the right quadrant, the gas is flowing to the left and accelerating. As such, all gas is currently center-seeking. 
     In  FIG.  11 G , at cycle time of % cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated to their neutral positions and are moving toward each other (i.e., translating inboard) at maximum velocity. The pressure is minimum at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right at maximum velocity. At the right quadrant, the gas is flowing to the left at maximum velocity. In other words, maximum velocity molecular displacement is occurring at both ends and its direction is inboard. As such, all gas is currently center-seeking. The gas is motionless at the center. The state at % cycle in  FIG.  11 G  is similar to the state at ¼ cycle in  FIG.  11 C , but now the diaphragms are at their neutral position translating at maximum velocity back toward the center (inboard). The gas is under maximum rarefaction at the center of the tube  1020  where the gas is temporarily motionless. Maximum gas velocity occurs at either end of the tube  1020  and is center-seeking to fill the void. 
     In  FIG.  11 H , at cycle time of ⅞ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving toward each other and decelerating. The pressure is increasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right and decelerating. At the right quadrant, the gas is flowing to the left and decelerating. All gas is still center-seeking. The cycle now repeats back to the 0 cycle state in  FIG.  11 A . 
     Visualization of Double-Ended Symmetric Standing Wave Thermoacoustic Refrigerator Using Simple Simulator for a Resonator Length of ½ Wavelength in Sloshing Mode 
     In this sloshing mode setup, the acoustic drivers  1030 L &amp;  1030 R are being run 180° out-of-phase with each other (both move left simultaneously or right simultaneously) at a frequency corresponding to double the natural resonant frequency of the tube. In this arrangement, the pressure antinodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . The pressure nodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion antinodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion nodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . 
       FIGS.  12 A- 12 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube  1020  as influenced simultaneously by the left acoustic driver  1030 L and right acoustic driver  1030 R. The 8 diagrams illustrate graphs from cycle time zero advancing with increments of ⅛ cycle to ⅞ cycle. They provide insight into the combined acoustic profile of the overall resonating tube in way of the stack sandwich  1010 L &amp;  1010 R. 
     In  FIG.  12 A , starting at cycle time  0 , the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are at their rightmost locations and temporarily motionless at their neutral positions. The pressure is maximum in the leftmost quadrant and minimum in the rightmost quadrant. The gas is traversing at maximum velocity to the right at the center of the assembly  1000 . The gas is under maximum compression at the tube&#39;s far left and maximum rarefaction at its far right and is motionless at both drivers  1030 L &amp;  1030 R while translating rightward at maximum velocity at center. In this way, the high temperature gas on the left is translating rightward through the left-side hot heat exchanger  1060 L (from which some of its thermal energy is extracted for ultimate rejection to the surroundings) and continues to be adiabatically expand (and correspondingly drop in temperature) as it approaches the cold heat exchanger  1050  (where it is able to absorb thermal energy from the refrigerated space). 
     In  FIG.  12 B , at cycle time of ⅛ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling leftward and accelerating. The pressure is decreasing in the leftmost quadrant and increasing in the rightmost quadrant. The gas is traversing to the right at the center of the assembly  1000  and decelerating. 
     In  FIG.  12 C , at cycle time of ¼ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  130 R are at their neutral locations and traveling leftward at maximum velocity. The pressure and gas motion are temporarily neutral with no pressure gradient and no bulk motion throughout the tube  1020 . 
     In  FIG.  12 D , at cycle time of ⅜ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling leftward and decelerating. The pressure is decreasing in the leftmost quadrant and increasing in the rightmost quadrant. The gas is traversing to the left at the center of the assembly  1000  and accelerating. 
     In  FIG.  12 E , at cycle time of ½ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  130 R have translated to their leftmost positions and are now temporarily motionless. Maximum rarefaction is occurring at far left and maximum compression is occurring at far right. Maximum velocity molecular displacement is occurring at the tube&#39;s center position and its direction is to the left; the gas is motionless at both drivers  1030 L &amp;  1030 R. As compared to the 0 cycle state shown in  FIG.  12 A , the same principle of gas motion/temperature/heat exchange is occurring, but now the left side hot heat exchanger  1060 L is the primary absorber of the gas&#39;s heat (again, to be rejected to the ambient environment) instead of the right side. The cold heat exchanger  1050  in the center retains its job of transferring low-temperature heat to the lower-temperature acoustically expanded gas. 
     In  FIG.  12 F , at cycle time of ⅝ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling rightward and accelerating. The pressure is increasing in the leftmost quadrant and decreasing in the rightmost quadrant. The gas is traversing to the left at the center of the assembly  1000  and decelerating. 
     In  FIG.  12 G , at cycle time of % cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are back in their neutral positions but are now traversing rightward at maximum velocity. The pressure and gas motion are temporarily neutral throughout the tube  1020  with no pressure gradient and no translation. This is similar to the ¼ cycle state in  FIG.  12 C , with the exception of the diaphragms&#39; velocity direction (rightward in  FIG.  12 G  as opposed to leftward in  FIG.  12 C ). 
     In  FIG.  12 H , at cycle time of ⅞ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling rightward and decelerating. The pressure is increasing in the leftmost quadrant and decreasing in the rightmost quadrant. The gas is traversing to the right at the center of the assembly  1000  and accelerating. The cycle now repeats back to the 0 cycle state in  FIG.  12 A . 
     Visualization of Double-Ended Symmetric Standing Wave Thermoacoustic Refrigerator Using Simple Simulator for a Resonator Length of 1 Wavelength in Clapping Mode 
     In this clapping mode setup, the acoustic drivers  1030 L &amp;  1030 R are being run in-phase with each other (both move inward/outward simultaneously) at a frequency corresponding to the natural resonant frequency of the tube  1020 . In this arrangement, the pressure antinodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . The pressure nodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion antinodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion nodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . 
       FIGS.  13 A- 13 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube  1020  as influenced simultaneously by the left acoustic driver  1030 L and right acoustic driver  1030 R. The 8 diagrams illustrate graphs from cycle time zero advancing with increments of ⅛ cycle to ⅞ cycle. They provide insight into the combined acoustic profile of the overall resonating tube in way of the stack sandwich  1010 L &amp;  1010 R. 
     In  FIG.  13 A , starting at cycle time  0 , the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are at their extreme or maximum inboard locations and temporarily motionless. The pressure is maximum in regions adjacent to each acoustic driver  1030 L,  1030 R. The pressure is minimum at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right at maximum velocity. At the right quadrant, the gas is flowing to the left at maximum velocity. All gas is currently center-seeking. In this way, the high temperature gas on the left is translating rightward through the left-side hot heat exchanger  1060 L (from which some of its thermal energy is extracted for ultimate rejection to the surroundings) and continues to be adiabatically expand (and correspondingly drop in temperature) as it approaches the cold heat exchanger  1050  (where it is able to absorb thermal energy from the refrigerated space). 
     In  FIG.  13 B , at cycle time of ⅛ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving away from each other and accelerating. The pressure is decreasing in the regions adjacent to each acoustic driver  1030 L,  1030 R. The pressure is increasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right at a decelerating rate. At the right quadrant, the gas is flowing to the left at a decelerating rate. As such, all gas is still center-seeking but is slowing down. 
     In  FIG.  13 C , at cycle time of ¼ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated to their neutral positions and are moving away from each other (i.e., translating outboard) at maximum velocity. The pressure is neutral throughout the tube  1020 . The pressure gradient induced by the left acoustic driver&#39;s previous cycle is being perfectly cancelled out by the pressure gradient induced by the right acoustic driver&#39;s precious cycle. There is no instantaneous motion of the gas (i.e., it is temporarily stationary). 
     In  FIG.  13 D , at cycle time of ⅜ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving away from each other and decelerating. The pressure is increasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the left and accelerating. At the right quadrant, the gas is flowing to the right and accelerating. All gas is now center-avoiding and accelerating. 
     In  FIG.  13 E , at cycle time of ½ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated as far outboard as they can, at their extreme outboard locations. The pressure is minimum in the regions adjacent to each acoustic driver  1030 L,  1030 R. At the left quadrant, the gas is flowing to the left at maximum velocity. At the right quadrant, the gas is flowing to the right at maximum velocity. All gas is center-avoiding. In this way, the outboard heat exchangers  1060 L &amp;  1060 R are colder and the center heat exchanger  1050  is the hotter one. The outboard heat exchangers  1060 L &amp;  1060 R will encounter the gas after it has had a chance to cool down and will therefore be cooler than the center heat exchanger  1050 . 
     In  FIG.  13 F , at cycle time of ⅝ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving toward each other and accelerating. The pressure is increasing in the regions adjacent to each acoustic driver  1030 L,  1030 R. The pressure is decreasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the left and decelerating. At the right quadrant, the gas is flowing to the right and decelerating. As such, all gas is still center-avoiding but decelerating. 
     In  FIG.  13 G , at cycle time of % cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R have translated to their neutral positions and are moving toward each other (i.e., translating inboard) at maximum velocity. The pressure is neutral throughout and there is no instantaneous motion of the gas (i.e., it is temporarily stationary). The tube  1020  has no pressure gradient and no translation. This is similar to the state at cycle time of ¼ cycle, with the exception of the diaphragms&#39; velocity direction (moving toward each other at maximum velocity in  FIG.  13 G  as opposed to moving away from each other at maximum velocity in  FIG.  13 C ). Again, the respective pressure profiles of the drivers  1030 L &amp;  1030 R cancel each other out due to symmetry. 
     In  FIG.  13 H , at cycle time of ⅞ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are moving toward each other and decelerating. The pressure is increasing in the regions adjacent to each acoustic driver  1030 L,  1030 R. The pressure is decreasing at the center of the assembly  1000 . At the left quadrant, the gas is flowing to the right and accelerating. At the right quadrant, the gas is flowing to the left and accelerating. All gas is currently center-seeking and accelerating. The cycle now repeats back to the 0 cycle state in  FIG.  13 A . 
     Visualization of Double-Ended Symmetric Standing Wave Thermoacoustic Refrigerator Using Simple Simulator for a Resonator Length of 1 Wavelength in Sloshing Mode 
     In this sloshing mode setup, the acoustic drivers  1030 L &amp;  1030 R are being run 180° out-of-phase with each other (both move left simultaneously or right simultaneously) at a frequency corresponding to the natural resonant frequency of the tube. In this arrangement, the pressure antinodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . The pressure nodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion antinodes include those at one-quarter of the resonator length inboard of each acoustic driver  1030 L,  1030 R. The motion nodes include one immediately inboard of each acoustic driver  1030 L,  1030 R and one at the center of the assembly  1000 . 
       FIGS.  14 A- 14 H  show 8 diagrams depicting the instantaneous pressure distribution and longitudinal motion of gas flow in the resonating tube  1020  as influenced simultaneously by the left acoustic driver  1030 L and right acoustic driver  1030 R. The 8 diagrams illustrate graphs from cycle time zero advancing with increments of ⅛ cycle to ⅞ cycle. They provide insight into the combined acoustic profile of the overall resonating tube in way of the stack sandwich  1010 L &amp;  1010 R. 
     In  FIG.  14 A , starting at cycle time  0 , the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are at their rightmost locations and temporarily motionless at their neutral positions. The pressure is neutral throughout and there is no instantaneous motion of the gas (i.e., it is temporarily stationary). 
     In  FIG.  14 B , at cycle time of ⅛ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling leftward and accelerating. The pressure is increasing in the leftmost quadrant and decreasing in the rightmost quadrant. The gas is moving leftward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving rightward in the center of the assembly  1000  and accelerating. 
     In  FIG.  14 C , at cycle time of ¼ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  130 R are at their neutral locations and traveling leftward at maximum velocity. The pressure is maximum in the left quadrant and minimum in the right quadrant. The gas is moving leftward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving rightward at the center of the assembly  1000  at maximum velocity. The center heat exchanger  1050  is receiving expanded gas coming from the left and is therefore at a cooler temperature than the left heat exchanger  1060 L, which will be warmer. The right heat exchanger  1060 R may also be getting cooled by gas flowing from the left, but it will be heated later. In this way, the center heat exchanger  1050  is always cooled by the gas motion while the outboard heat exchangers  1060 L and  1060 R are either heated or slightly cooled, with the heated stage being the primary goal of this running configuration. 
     In  FIG.  14 D , at cycle time of ⅜ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling leftward and decelerating. The pressure is decreasing in the left quadrant and increasing in the right quadrant. The gas is moving leftward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving rightward at the center of the assembly  1000  and decelerating. 
     In  FIG.  14 E , at cycle time of ½ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  130 R have translated to their leftmost positions and are now temporarily motionless. The pressure is neutral throughout and there is no instantaneous motion of the gas (i.e., it is temporarily stationary). This is similar to the 0 cycle state in  FIG.  14 A , with the exception of the diaphragms&#39; present position (leftmost in  FIG.  14 E  as opposed to rightmost in  FIG.  14 A ). 
     In  FIG.  14 F , at cycle time of ⅝ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling rightward and accelerating. The pressure is decreasing in the left quadrant and increasing in the right quadrant. The gas is moving rightward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving leftward at the center of the assembly  1000  and accelerating. 
     In  FIG.  14 G , at cycle time of % cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are back in their neutral positions but are now traversing rightward at maximum velocity. The pressure is minimum in the left quadrant and maximum in the right quadrant. The gas is moving rightward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving leftward at the center of the assembly  1000  at maximum velocity. The center heat exchanger  1050  is receiving expanded gas coming from the right and is therefore at a cooler temperature than the right heat exchanger  1060 R, which will be warmer. The right heat exchanger  1060 R may also be getting cooled by gas flowing from the right, but it will be heated later. In this way, the center heat exchanger  1050  is always cooled by the gas motion while the outboard heat exchangers  1060 L and  1060 R are either heated or slightly cooled, with the heated stage being the primary goal of this running configuration. 
     In  FIG.  14 H , at cycle time of ⅞ cycle, the diaphragms of the acoustic drivers  1030 L &amp;  1030 R are traveling rightward and decelerating. The pressure is increasing in the left quadrant and decreasing in the right quadrant. The gas is moving rightward in the region adjacent to each acoustic driver  1030 L,  1030 R and moving leftward at the center of the assembly  1000  and decelerating. The cycle now repeats back to the 0 cycle state in  FIG.  14 A . 
     Performance of Prototypes of Refrigeration Assembly 
       FIG.  15    is an example of a graph of actual performance from prototypes of the refrigeration assembly of  FIGS.  9 A- 9 B  showing temperature versus time for a thermoacoustic test conducted with no forced heat rejection on the hot side (e.g.,  150  in  FIG.  1   ) and an unloaded cold side (e.g.,  160  in  FIG.  1   ) of the stack (e.g.,  110  in  FIG.  1   ). The temperature profile of the hot side of the stack rises over time. The temperature profile of the cold side of the stack drops over time. They are somewhat mirror images of one another in shape but not in magnitude. 
       FIG.  16    is an example of a graph of actual performance from prototypes of the refrigerative assembly of  FIGS.  9 A- 9 B  showing temperature versus time for a thermoacoustic test conducted with the hot side (e.g.,  150  in  FIG.  1   ) thermally anchored via active heat rejection and an unloaded cold side (e.g.,  160  in  FIG.  1   ). For this prototype, the temperature profile of the thermally anchored hot side remains relatively constant and slightly below the ambient temperature. The temperature profile of the unloaded cold side drops over time at a decreasing rate until it levels off to a relatively constant low temperature. 
       FIG.  17    is another example of a graph of actual performance from prototypes of the refrigerative assembly of  FIGS.  9 A- 9 B  showing temperature versus time for a thermoacoustic test conducted with the hot side (e.g.,  150  in  FIG.  1   ) thermally anchored via active heat rejection and an unloaded cold side (e.g.,  160  in  FIG.  1   ). For this other prototype, the temperature profile of the thermally anchored hot side remains relatively constant and quickly merges with the ambient temperature. The temperature profile of the unloaded cold side drops over time rather sharply at a decreasing rate and quickly levels off to a relatively constant low temperature. 
     The inventive concepts taught by way of the examples discussed above are amenable to modification, rearrangement, and embodiment in several ways. For example, this invention may be applicable in other systems having different geometries, sizes, or arrangements of components. Accordingly, although the present disclosure has been described with reference to specific embodiments and examples, persons skilled in the art will recognize that changes may be made in form and detail without departing from the spirit and scope of the disclosure. 
     An interpretation under 35 U.S.C. § 112(f) is desired only where this description and/or the claims use specific terminology historically recognized to invoke the benefit of interpretation, such as “means,” and the structure corresponding to a recited function, to include the equivalents thereof, as permitted to the fullest extent of the law and this written description, may include the disclosure, the accompanying claims, and the drawings, as they would be understood by one of skill in the art. 
     To the extent the subject matter has been described in language specific to structural features or methodological steps, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or steps described. Rather, the specific features and steps are disclosed as example forms of implementing the claimed subject matter. To the extent headings are used, they are provided for the convenience of the reader and are not be taken as limiting or restricting the systems, techniques, approaches, methods, or devices to those appearing in any section. Rather, the teachings and disclosures herein can be combined or rearranged with other portions of this disclosure and the knowledge of one of ordinary skill in the art. It is intended that this disclosure encompass and include such variation. 
     The indication of any elements or steps as “optional” does not indicate that all other or any other elements or steps are mandatory. The claims define the invention and form part of the specification. Limitations from the written description are not to be read into the claims.