Patent Application
20230108485
The present invention relates to heat exchangers and more particularly to a shell and tube heat exchanger that may be used in a thermoacoustic or Stirling machine, or other application.
Known methods of construction of heat exchangers include plating a thin one-piece layer of metal on top of a sacrificial mandrel containing multiple holes using electroplating (also known as electroforming or electro-deposition) or by using electroless plating (also known as electrode-less deposition). Some relevant U.S. Pat. Nos. are 6,892,802; 5,317,805; 5,199,487; and 4,807,342—but the basic idea goes back to the year 1911 as evident in U.S. Pat. No. 997,610.
Following are the six main loss mechanisms of heat exchangers that may lead to lower efficiency of thermoacoustic or Stirling machines: (1) Insufficient net heat transfer between a face of the regenerator and its nearest heat exchanger; (2) Oscillatory heat exchange between the working fluid and the heat exchanger to the extent that it is not isothermal; (3) Flow losses due to the oscillatory motion of the working fluid through the heat exchanger; (4) Joining loss due to an abrupt change from near adiabatic conditions in the working fluid away from the thermal core to non-adiabatic thermal conditions at the heat exchanger; (5) Pressure drop loss of the secondary heat transfer fluid flowing through the heat exchanger; and (6) Poor heat transfer between the secondary heat transfer fluid and the heat exchanger. It is desirable to minimize these losses in the heat exchangers.
Some thermoacoustic and Stirling type machines may suffer from an additional loss mechanism known as Gedeon streaming, i.e. a steady flow of working fluid around the toroidal topology of such machines [see G. W. Swift, Thermoacoustics: A unifying perspective for some engines and refrigerators, pp. 177-183, Acoustical Society of America, Melville, N.Y., 2002; D. Gedeon, “DC gas flows in Stirling and pulse tube cryocoolers,” in R. G. Ross, ed., Cryocoolers 9, pp. 385-392, Plenum, N.Y., 1997; and U.S. Pat. No. 6,032,464]. It is desirable to minimize the loss due to Gedeon streaming. The working fluid in thermoacoustic or Stirling machines is often pressurized. It is desirable that the heat exchangers be able to resist high external pressure. The power density of thermoacoustic and Stirling machines is often limited by the amount of heat that can be efficiently transferred through its heat exchangers. It is desirable to increase the effectiveness of heat transfer in thermoacoustic and Stirling heat exchangers and also to fabricate the heat exchangers inexpensively. Thus, there is a need to overcome the limitations of the existing heat exchangers and provide a better solution.
The thermal core of a thermoacoustic or Stirling machine (or some other similar machines, such as pulse-tube or Vuilleumier types) usually consists of a planar regenerator with two planar heat exchangers placed closely adjacent to each face of the regenerator. The working fluid, which may be helium gas, air or other gas, is free to move in the axial, or “acoustic,” direction perpendicular to the generally planar faces of the regenerator and heat exchangers through the pores of the regenerator and passages in the heat exchanger. The working fluid goes through a cycle of compression, translation along the acoustic direction, expansion, and translation back in the acoustic direction. This cycle is known as the Stirling cycle and is equivalent to the fluid motion in a traveling wave of sound.
The hydraulic radius of the regenerator, a measure of its pore size, is usually made smaller than or equal to the so-called thermal penetration depth, δκ=√{square root over (2κ/(ρcpω))}, where κ is the thermal conductivity, ρ is the density, cp is the heat capacity of the working fluid at constant pressure, and ω is the angular frequency of the cycle. The thermal penetration depth is about the distance that heat can diffuse through the working fluid in a fraction of the acoustic period. It can be quite small, on the order of 100 microns.
Oscillatory, bi-directional, and zero-average heat may be exchanged between the working fluid and the regenerator during the acoustic cycle. The small pore size of the regenerator may be used as it may allow the above-discussed heat to be transferred almost isothermally. No entropy is produced in the limit of isothermal heat transfer, which leads to higher efficiency in the thermoacoustic or Stirling machine. A thermoacoustic stack, which is a variation of the regenerator, may be used in standing-wave type thermoacoustic machines with the pore size equal to or slightly larger than the thermal penetration depth. A thermoacoustic stack may be considered functionally equivalent to a regenerator.
The purpose of the heat exchangers is to bring net, steady, average, and non-oscillatory heat to or from the faces of the regenerator. This may allow the thermoacoustic or Stirling machine to perform functions, such as refrigeration or the production of work, in the form of sound, from heat. Net heat can be brought into or away from the regenerator via secondary heat transfer fluids—such as water-glycol mixture, alcohol, brine, oil, external air, combustion product, or other fluid. The secondary heat transfer fluid may be placed into close proximity to the working fluid within the heat exchangers.
A monocoque plating-over-mandrel technique allows inexpensive construction of a shell and tube heat exchanger for a thermoacoustic and Stirling machine with tens of thousands of millimeter scale tubes, without having to cut, handle, and join so many small parts. Small tubes lead to higher efficiencies in the thermoacoustic and Stirling machines because of the small size of the thermal penetration depth. Other fabrication methods may also be advantageously employed. Additive manufacturing (also known as 3-D printing), for example, provides another method for monocoque fabrication of a shell-and-tube heat exchanger with tens of thousands of tubes without handling individual small parts.
It is desirable to have large net heat transfer between the regenerator and the heat exchanger. One may achieve this and an almost isothermal oscillatory heat exchange by using a heat exchanger having the working-fluid features that are very small and which may scale with the thermal penetration depth. The monocoque plating-on-mandrel technique or other technique may allow an economical and practical way of accomplishing this in a shell-and-tube geometry. Instead of cutting, arranging, and joining tens of thousands of very small tubes for each heat exchanger, casting mandrels in a mold with many tiny pins that become holes in the mandrel allows the relatively high one-time cost of making the mold to be spread over many mandrels and/or heat exchangers.
Casting the mandrel in a mold with non-cylindrical pin shapes additionally allows for tube shapes that are better suited to thermoacoustic and Stirling machines than the cylindrical tubes of the prior art. Additional advantages, objects, and features of the invention will be set forth in part in the description that follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principle of the invention. In the drawings:
Another mesh/screen (not shown in
There may be several improvements associated with embodiments of this design. Not all improvements discussed herein apply to all embodiments and the herein discussed improvements should not be considered as limitations on any embodiments. First, the tubes of the monocoque tube bundle may be tapered at both ends of each tube portion with each tube having mouths at each end that are wider in cross section than the cross section of the tube waist region between the two mouths in the manner of an hourglass or Venturi tube. The cross section of the tubes of the monocoque bundle varies between the respective two mouths and the region that has the narrowest cross section is defined as the tube waist region. In some embodiments, the tube waist region lies near the middle of the tube, while in other embodiments, the tube waist region lies toward either of the two mouths of the respective tube. In some embodiments, the tube waist region is disposed at a distance of about 10% to 40% of the tube length from the first mouth end or the second mouth end. In other embodiments, the tube waist region is disposed at a distance of about 7% to 92% of the tube length from the first mouth end or the second mouth end. The adjacent mouth ends may be smoothly blended together such that the smoothly blended surface between adjacent mouth ends may have a continuous curved surface. In some embodiments, the blended surface may have sections of circles, parabolas, ellipses or other smooth curves. In other embodiments, the blended surface is curved and without abrupt steps or transitions. In some other embodiments, the blended surface may have a sharp change of slope near the boundaries of adjacent mouth ends. In other embodiments there may be a small raised boss near the boundaries of the adjacent mouth ends. These embodiment may have the following advantages: (1) The near elimination of the flat tube-sheet portion of the monocoque of
Another improvement is that the narrowest part of the tube waist region is closer to the regenerator facing face of the heat exchanger than the face away from the regenerator. This embodiment may also have an advantage of having a smaller angle taper of the tubes on the side away from the regenerator to minimize the flow separation of working fluid from the tube walls and thus minimizing jetting into the open duct space outside the thermal core. This decreases or eliminates the jetting pressure drop losses and facilitates the flow in the open duct outside the thermal core to be more laminar, thus giving the working fluid a second chance to exchange heat with the heat exchanger as it reenters the heat exchanger on the other half of the acoustic cycle.
Furthermore, another improvement associated with this design is that a mesh/screen or similar porous planar structure (e.g. one or more sheets of woven wire mesh, perforated or chemically etched sheet, expanded metal, sintered powder, felt, open-cell foam, or additively manufactured porous structure), with pore size finer than the tube mouth hydraulic radius, is joined to the monocoque tube bundle on the face away from the regenerator. The mesh/screen or similar porous structure has minute openings or constrictions of complicated three dimensional shape through which working fluid may pass. According to the present invention, fluid may pass through the mesh/screen or porous structure and the dimension of the opening or constriction corresponds to the pore size. As discussed above, the joining of the mesh/screen or similar porous structure to the monocoque tube bundle may have the following advantages: (1) A further decrease of jetting of the working fluid as it flows away from the regenerator and exits the heat exchanger into the open duct space outside the thermal core helps the flow in the open duct outside the thermal core to be more laminar and thus gives the working fluid a second chance to exchange heat with the heat exchanger as it reenters the heat exchanger on the other half of the acoustic cycle; and (2) Additional heat exchange with the working fluid via the mesh/screen and the thermal conduction path of the mesh/screen, the means for joining of the mesh/screen to the monocoque tube bundle, and the monocoque tube bundle to the secondary heat transfer fluid.
Moreover, another improvement associated with this design is that a mesh/screen or similar porous planar structure (e.g. one or more sheets of woven wire mesh, perforated or chemically etched sheet, expanded metal, sintered powder, felt, open-cell foam, or additively manufactured porous structure), with pore size finer than the tube mouth hydraulic radius, is joined to the monocoque tube bundle on the face adjacent to the regenerator. This may have an advantage of an additional heat exchange with the working fluid via the mesh/screen and the thermal conduction path of the mesh/screen to the monocoque tube bundle. The mesh/screen may be acting as the initial portion of the regenerator and thus may put the regenerator into direct thermal contact with the secondary heat transfer fluid via the thermal conduction path of the lateral and axial thermal conductivity of the mesh/screen over each tube mouth, the thermal conduction of the means for joining of the mesh/screen to the monocoque tube bundle, and the thermal conduction of the monocoque tube bundle to the secondary heat transfer fluid.
Yet another improvement associated with this design is that porous packing (e.g. open-cell metal foam, sintered powder, felt, additively manufactured porous structure, or one or more segments of woven wire mesh, perforated or chemically etched sheet, or expanded metal), with pore size finer than the tube hydraulic radius, are joined within the tubes to the monocoque tube bundle. The porous packing may fully or partially fill the tubes. This may have the following advantages: (1) Additional heat exchange with the working fluid via the porous packing and the thermal conduction path of the porous packing, the means for joining of the porous packing to the monocoque tube bundle, and the monocoque tube bundle to the secondary heat transfer fluid; (2) Reduction, through more isothermal contact of the working fluid with the heat exchanger, of losses due to non-isothermal oscillating heat exchange; (3) A further decrease of jetting of the working fluid as it flows away from the regenerator and exits the heat exchanger into the open duct space outside the thermal core, which helps the flow in the open duct outside the thermal core to be more laminar and thus gives the working fluid a second chance to exchange heat with the heat exchanger as it reenters the heat exchanger on the other half of the acoustic cycle; and (4) If it is placed against the regenerator, the porous packing may be acting as the initial portion of the regenerator and thus may put the regenerator into direct thermal contact with the secondary heat transfer fluid via the thermal conduction path of the lateral and axial thermal conductivity of the porous packing near each tube mouth facing the regenerator, the thermal conduction of the means for joining of the porous packing to the monocoque tube bundle, and the thermal conduction of the monocoque tube bundle to the secondary heat transfer fluid.
In some embodiments, the tubes may have a non-circular cross-section if the plating thickness is sufficient for the monocoque to be able to resist the mean-pressure induced non-membrane (azimuthal bending) stress this would entail. This may have the following advantages: (1) Improved heat transfer between the working fluid and the secondary heat transfer fluid because of an increase in heat transfer area; (2) Reduction, through more isothermal contact of the working fluid with the heat exchanger, of losses due to non-isothermal oscillating heat exchange; (3) More freedom in blending the tube ends to the monocoque face; (4) More freedom of construction methods of the mold to cast the sacrificial mandrel; and (5) The possibility of more tube perimeter near the narrowest part of the tube waist region, which may be obtained in certain embodiments with fluted tubes, or low-order polygonal, star-shaped, or multi-lobed cross-sections. This may resist mean-pressure induced axial compressional stress in the tube waist region that may become the dominant stress when the ratio of tube waist dimension (such as average diameter) to tube spacing becomes small.
An embodiment with the non-circular tube cross-section improvement is shown in
The improved design may minimize the above mentioned six loss mechanisms of a heat exchanger due to the following reasons: (1) Net heat transfer between a regenerator and its adjoining heat exchanger may be maximized by having many small diameter tubes, by thermal conduction of the mesh/screens that are joined to the heat exchanger on either face and/or by thermal conduction of the porous packing that is joined to the interior of the heat exchanger tubes, by having tubes of non-circular cross section, and/or by the promotion of laminar flow in the open duct opposite the thermal core via the gradual tapering of the tubes on the side away from the regenerator and by the mesh/screen on the face of the heat exchanger away from the regenerator and/or by porous packing within the heat exchanger tubes; (2) Non-isothermal oscillatory heat flow between the working fluid and the heat exchanger may be minimized by having the tube hydraulic radius being about or smaller than the thermal penetration depth, wherein the hydraulic radius is half of the tube's local radius for circular cross-section tubes, or the local tube cross sectional area divided by the local tube perimeter for tubes of non-circular cross-section; and/or by including porous packing within the tubes possessing pores of hydraulic radius that are on the order of or smaller than the thermal penetration depth wherein the pore hydraulic radius is the working fluid volume within the pores divided by the surface area of the pores in contact with the working fluid; (3) Flow losses due to motion of the working fluid may be minimized by the streamlined shape of the doubly tapered tubes, by the minimization of jetting in the open space outside the thermal core, and by not making the tubes any longer than necessary to achieve highly effective heat transfer to the working fluid; (4) Joining loss may be proportional to pm−1, where pm is the mean pressure of the working fluid—and this loss may be lowered by increasing the mean pressure, which the present invention facilitates by reducing or eliminating the structurally weak flat tube-sheet of the prior art; (5) Pressure drop loss due to the flow of the secondary heat transfer fluid may be minimized by the wide flow path between the narrow waists of the tubes; and (6) Heat transfer between the secondary heat transfer fluid and the heat exchanger may be improved over other heat exchanger geometries by the relatively high Nusselt number of flow over a bank of tubes. In some non-limiting embodiments, the ratio of the pore hydraulic radius to the thermal penetration depth may be about 2% to 20%.
In some thermoacoustic and Stirling machine topologies, those with a toroidal path for the acoustic power that allows for the free flow of working fluid mass, it may be advantageous to apply an asymmetric drag force between working fluid acoustical flow that occurs with and against the direction of the acoustic power. The asymmetric drag force may cause a second-order time-averaged working fluid pressure difference that reduces or eliminates time-averaged mass flow (the so-called Gedeon streaming) that may decrease the efficiency of such machines. The subject heat exchangers may straightforwardly supply this second-order time-averaged pressure difference with some structural differences between the heat exchanger on one side of the regenerator and the other. For example, more jetting of the flow in one duct than the other may be induced by placing a mesh between the heat exchanger and the duct on one heat exchanger but not on the other.
Additionally or alternatively, the shape of the tube tapers between the two heat exchangers may differ. For example, the tube shape of at least one of the heat exchangers may be adjusted so that the heat exchanger provides hydrodynamic mass flux suppression (acting as a so-called jet-pump) as taught in U.S. Pat. No. 6,032,464. To do so, the waist position and/or taper amount or shape may be changed. For example, the position of the waist of the tubes on one exchanger or the other may be moved from near the regenerator to further away from the regenerator, perhaps even most of the way towards the duct, in order to provide mass flux suppression, if needed. Such an embodiment is shown in the cutaway detail of
As mentioned above, not all of the improvements discussed herein need to be incorporated to have a workable monocoque shell and tube heat exchanger according to the present invention. The cross-section detail of
Workable heat exchangers may be made with a wide range of dimensions. As a specific example, a series of computational fluid dynamics calculations suggests excellent heat transfer with the heat exchanger represented in
The optimal dimensions for the heat exchanger, when adapted for use in thermoacoustic, Stirling, or similar machines, depend on a multitude of factors of the thermoacoustic, Stirling, or similar machine in which it is incorporated. To give some sense for the range of desirable dimensions, six example optimizations are described here, the results of which are shown in
In the analytic optimizations that follow, the nearest neighbor center-to-center spacing D of the tubes in an assumed hexagonal lattice is stepped over a range of discrete values. At each D, the tube length L and the secondary heat transfer fluid mass flow rate M are varied until the minimum in the total lost work WLostTOTAL, described below, is found for that D. The tube geometry scales with D and L. To make progress, some assumptions about the tube geometry are made based on experience with flow separation from the tube walls based on the previous computational fluid dynamics calculations. The tubes are assumed to have an hourglass-like shape of two truncated cones that come together at a waist with an OD that is 60% of the tube spacing D at a position that is 15% of the tube length L from the regenerator. The ODs of the open mouths of the tubes at each end of the tube are assumed to be 95% of the tube spacing D so that adjacent tubes almost touch. The tube wall thickness t is 3.75% of the tube spacing D. When mesh/screens are added to the face of the monocoque shell, the optimization drives the tube length L to be on the order of the tube spacing D for large D so that the tubes can be rather squat and the tube walls have a large angle from their central axis. This angle is taken into account in the wall thickness t, which is calculated normal to the wall rather than normal to the tube central axis, to determine the tube ID along the length of the tube, which in turn affects slightly the Reynolds number and losses at the mouths and within the tubes. In some embodiments, the tapering angle is determined by the tube spacing D and tube length L.
To interpolate between the discrete stepped values of D, parabolic fits are made to the three points in WLostTOTAL, L, and M nearest the minimum in WLostTOTAL vs. D. The optimal value of D that gives the lowest total lost work is found from the WLostTOTAL vs. D fit, and that value of D is then used to find the optimal values of L and M from their parabolic fits.
Various thermoacoustic-Stirling machines may be used for inputs on the working fluid side of the optimizations, the choice of which affects the resulting optimal heat exchanger geometry. The inputs used are: (1) the external ambient bath temperature T0, the external non-ambient (hot or cold) bath temperature T$, the heat into the machine (or out of the machine if negative) from the ambient bath 0in, and the heat into the machine from the non-ambient bath
$in, from which the machine's work, First Law efficiency and Second Law efficiency can be derived; (2) whether the heat exchanger under consideration is the ambient or the non-ambient heat exchanger, which affects how lost work is calculated and whether the heat leak from the external loop to the ambient contributes to the lost work tally; (3) the working fluid thermodynamic properties and machine operating frequency, which determine the thermal and viscous penetration depths and are important for the thermoacoustic losses and beneficial heat transfer; (4) thermodynamic properties of the secondary heat transfer fluid; (5) the acoustic pressure amplitude, acoustic volumetric velocity amplitude, and the heat exchanger frontal width and the length along the secondary heat transfer fluid flow direction (for simplicity a rectangular exchanger presented to the working fluid is assumed), which determines the machine power density and affects the relative weighting of losses between the beneficial heat transfer thermal resistance and losses from secondary heat transfer fluid flow friction.
For accurate results, the secondary heat transfer fluid external loop plumbing, the external fan coil unit (FCU) that makes contact to the external bath connected to the heat exchanger under consideration, the number of heat exchangers under consideration connected to the external loop plumbing and FCU, and the loop pump efficiency (taken here to be 75%) also need to be defined because this affects the lost work associated with the secondary heat transfer fluid mass flow rate M. For example, an external loop and FCU with high flow friction will drive the optimization to have a lower M, which in turn allows for the heat exchanger under consideration to have a tighter geometry on the secondary heat transfer fluid side than it would otherwise have. This will be shown in one example below to have a relatively small effect, which is nevertheless taken into account in the optimization.
The beneficial heat transfer between the tubes of the heat exchanger under consideration and the working fluid is calculated with a quasi-static approximation, where heat transfer is integrated in time over the half of the cycle that the working fluid is flowing away from the regenerator, through the heat exchanger, and into the duct, and integrated in space along the length of the tube, using a standard static Nusselt number correlation for straight tubes [see F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, N.Y.,
Tube thermo-viscous losses—the dissipated power WTubeThermal caused by oscillatory heat transfer accompanied by undesirable temperature swings in the working fluid within the tubes, and the viscous dissipation WTubeViscous caused by the oscillating working fluid motion along the tube walls—are calculated by integrating the local inverse thermal resistance and the local viscous resistance, respectively, of the five-element model of Swift [see G. W. Swift, Thermoacoustics: A unifying perspective for some engines and refrigerators, Acoustical Society of America, Melville, N.Y., eqs. 4.78 and 4.74, pp. 92-94, 2002] over the length of the tubes.
Beneficial net heat transfer from the mesh/screens, if they are used, is calculated using the stacked square-weave screen matrix Colburn J-Factor correlation found in
The mesh/screens also present a conductive thermal resistance. To calculate the total time-dependent convective plus conductive thermal resistance, the mesh/screen is treated like a circular fin, with an arbitrary assumed constant temperature difference imposed between the working fluid at infinity and the temperature of the screen-fin at its (assumed) circular perimeter where the screen makes contact to the tube mouth. For simplicity, the heat capacity of the mesh/screen is ignored, the assumption being that a time integral described below near the end of the calculation is sufficient to capture the intermediate averaging effect of the mesh/screen heat capacity. The analytic solution of the mesh/screen temperature minus the working fluid temperature at infinity—the temperature difference that drives heat out of the mesh/screen and into the working fluid—is the zero-order modified Bessel function of the first kind, where the characteristic decay number of the radial mesh/screen temperature drop is the square root of the heat transfer coefficient times the wet area of the mesh/screen, divided by the total mesh/screen frontal area, the mesh/screen length in the acoustic direction (its thickness), and the effective volumetric thermal conductivity of the mesh/screen metal and its pores. The total time-dependent conductive plus convective thermal resistance is then taken to be the arbitrary assumed imposed temperature difference divided by the total heat found by integrating the mesh/screen to working fluid at infinity temperature difference over the area of the mesh/screen, multiplied by the mesh/screen convective heat transfer coefficient.
As a diagnostic tool, the convective thermal resistance is subtracted from the total convective plus conductive thermal resistance, and what is left over is considered to be a weighted effective mesh/screen conduction thermal resistance; although this distinction of the total resistance into convective and conductive components is not important to finding the lost work due to the total thermal resistance and the optimization of the heat exchanger geometry. A time-dependent fin efficiency for the mesh/screen is calculated as the ratio of the convective screen thermal resistance to the total convective plus conductive screen thermal resistance.
In some of the optimizations, a sandwich assembly of a coarse screen between two or more fine screens is considered, the coarse screen being used to increase the lateral thermal conduction of the sandwich assembly and give mechanical support to the fine screens. In this case, the fine screens are calculated as described above over a circular area equal to the square unit cell area of a coarse square-weave screen single pore. The fin efficiency of the fine screen calculated in this way is used to enhance the thermal contact of the coarse screen to the working fluid, and the conductivity of all the screens in parallel is used to calculate the conduction loss of the sandwich assembly as a whole. In this fine screen fin efficiency calculation the fine screen thermal resistance is increased by 30% in an attempt to correct for the likely spotty contact between the fine and coarse screens. In the two optimizations with the large low-density engine described below (
The dissipated power WScreenThermal from non-isothermal oscillatory heat exchange between the working fluid and the mesh/screens, if they are used, is calculated using the lumped thermal loss resistor model of Swift [see G. W. Swift, Thermoacoustics: A unifying perspective for some engines and refrigerators, Acoustical Society of America, Melville, N.Y., eq. 4.78, p. 94, 2002], which connects the loss due to undesirable working fluid temperature oscillations to the mesh/screen frontal area and the hydraulic radius of the screen. Flow loss WScreenViscous through the mesh/screens is calculated with the time integral of the product of the sinusoidal volumetric velocity through the screens with the time dependent square-mesh single screen pressure drop due to oscillating flow found through the correlation of Wakeland and Keolian [see R. S. Wakeland and R. M. Keolian, Measurements of Resistance of Individual Square-Mesh Screens to Oscillating Flow at Low and Intermediate Reynolds Numbers, J. Fluids Eng., 125, pp. 851-862, 2003].
An approximation is used for calculating the pressure drop of the secondary heat transfer fluid flowing over the tubes. A standard correlation for flow over a hexagonal tube bank of staggered, straight-wall tubes [see F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, N.Y.,
The same secondary heat transfer fluid velocity distribution is used to calculate Reynolds number, Nusselt number, and heat transfer coefficient as a function of position along the length of the tube. The heat transfer over the tube OD is then integrated along the tube to find the thermal resistance between the secondary heat transfer fluid and the tube wall OD.
The electrical power needed to pump secondary heat transfer fluid flow in the external plumbing loop WSHTFDragLoopElec is calculated with the aid of the Churchill formula for the Darcy friction factor, which is applicable to the laminar, transitional, and turbulent regimes [see S. W. Churchill “Friction factor equation spans all fluid-flow regimes” Chem. Eng. 84 (24) pp. 91-92 (1977)]. Multiple heat exchangers under consideration may be connected to a single loop, so their flows are added in calculating the Reynolds number, and the resulting power loss is divided among the exchangers, taking pump efficiency into account. The same is true for summing flows for the FCU attached to the loop in order to get the electrical pump power dissipated by FCU flow WSHTFDragFCUElec. Here, an appropriate commercially available FCU is selected, and the manufacturer supplied pressure drop at a certain flow rate is scaled by velocity squared for calculating flow loss (for simplicity, turbulent flow is assumed). Additionally, the manufacturer supplied temperature drop at a heat rate is used to calculate a (linear) FCU thermal resistance to the external bath.
If the heat exchanger under consideration is connected to the non-ambient temperature bath, the power WLoopHeatLeak dissipated due to the heat leak between the ambient and the secondary heat transfer fluid within the external plumbing loop, covered by an assumed amount of insulation, is calculated. This is done to help optimize the external loop pipe diameter to minimize the total lost work, which in turn influences the optimization of the secondary heat transfer fluid mass flow rate M and thus the dimensions of the tubes of the heat exchanger under consideration.
The thermal resistances described above for the heat exchanger under consideration are combined into a total time-dependent heat exchanger thermal resistance R(ψ), calculated as the sum of the time independent thermal resistance of the secondary heat transfer fluid to the monocoque shell, and the parallel combination of the time-dependent thermal resistance of the tubes to the working fluid and, if mesh/screens are used, the time-dependent convective plus conductive thermal resistance of mesh/screens on either heat exchanger face to the working fluid, where ψ is the time phase of the time-dependent working fluid volumetric velocity U(ψ)=U0cos(ψ). For simplicity, the thermal resistance of the monocoque tube bundle shell itself has been neglected, justified by experience gained from the earlier computational fluid dynamics computations that included the shell conductivity. A time-dependent working fluid “convective heat,” Q(ψ)=πQHXcos(ψ), during the outward half of the stroke when −π/2<ψ<π/2, and Q(ψ)=0 during the inward half of the stoke when ψ is outside this range, is assumed to flow from the secondary heat transfer fluid, through the heat exchanger under consideration, and into the working fluid as the working fluid flows out of the regenerator, where HX, which is equal to either
0in or
$in, is the time averaged heat flowing into the heat exchanger under consideration from the secondary heat transfer fluid. After these long calculations, the centrally important time averaged temperature drop between the working fluid and the secondary heat transfer fluid is then calculated as an average temperature drop
The temperature drop ΔTFCU between the secondary heat transfer fluid external fan coil unit outlet temperature and the relevant external bath temperature TBath—where TBath is fixed by one of the inputs of the optimization to be either the external ambient bath temperature T0 or the external non-ambient hot or cold bath temperature T$—is determined by the heat into the heat exchanger under consideration HX, the number of heat exchangers connected to the FCU, and the input thermal resistance for the assumed external FCU. The secondary heat transfer fluid temperature drop ΔTSHTF, averaged across the transverse extent of the heat exchanger under consideration, is taken to be ΔTSHTF=QHX/(2Mcp), where cp is the secondary heat transfer fluid heat capacity. From TBath, ΔTFCU, ΔTSHTF, and ΔTHX, the external fan coil unit secondary heat transfer fluid outlet temperature TFCU=TBath+ΔTFCU, the average secondary heat transfer fluid temperature in the loop TSHTF=TFCU+ΔTSHTF, and the average working fluid temperature TWF=TSHTF+ΔTHX can be determined.
With the temperatures established, it becomes possible to calculate the lost work, a formalism described, for example, by Swift [see G. W. Swift, Thermoacoustics: A unifying perspective for some engines and refrigerators, Acoustical Society of America, Melville, N.Y., pp. 135-141, 2002] and Bejan [see A. Bejan, Advanced Engineering Thermodynamics, 2nd ed., Wiley, N.Y., 1997], where power dissipation mechanisms are weighted by various ratios of the ambient temperature T0 to the temperatures where the dissipation occurs, to account for the dependence of machine inefficiency (the useful work not delivered by a heat engine, or the extra work needed to drive a refrigerator or heat pump) on the temperature at the location of the various dissipation mechanisms within the machine. The lost works due to tube thermo-viscous dissipation and for mesh/screen oscillatory heat exchange and flow loss are found by multiplying those power dissipations found earlier by T0/TWF:
W
LostTubeThermal
=W
TubeThermal
T
0
/T
WF,
W
LostTubeViscous
=W
TubeViscous
T
0
/T
WF,
W
LostScreenThermal
=W
ScreenThermal
T
0
/T
WF,
W
LostScreenViscous
=W
ScreenViscous
T
0
/T
WF.
The secondary heat transfer fluid pressure drop lost works and the heat-leak lost work are found by multiplying the secondary heat transfer fluid pressure drop powers dissipated at the heat exchanger, loop plumbing, and the FCU, and the magnitude of the heat-leak power into or out of the secondary heat transfer fluid (per heat exchanger), by T0/TSHTF:
W
LostSHTFDragHXElec
=W
SHTFDragHXElec
T
0
/T
SHTF,
W
LostSHTFDragLoopElec
=W
SHTFDragLoopElec
T
0
/T
SHTF,
W
LostSHTFDragFCUElec
=W
SHTFDragFCUElec
T
0
/T
SHTF,
W
LostLoopHeatLeak
=W
LoopHeatLeak
T
0
/T
SHTF.
The lost works associated with the temperature deficits ΔTFCU, ΔTSHTF, and ΔTHX are found by considering the thermoacoustic or Stirling machine as a whole. As the temperature deficits increase or decrease during the optimization of the heat exchanger under consideration—and the remaining parts of the machine have to pump heat over an increased or decreased, respectively, temperature span in the case of a refrigerator or heat pump, or generate useful work from a smaller or larger, respectively, temperature span in the case of a heat engine—it is assumed that the remaining parts of the machine have an efficiency that is a constant fraction of the Carnot efficiency. The difference in the work powers can be found to be, after some algebraic manipulation, to be in the case that the heat exchanger under consideration is the ambient heat exchanger:
and for the case that the heat exchanger under consideration is the non-ambient heat exchanger:
where WLostFCUΔT, WLostSHTFΔT, and WLostHXΔT, are the lost works associated respectively with the temperature deficits ΔTFCU, ΔTSHTF, and ΔTHX;
functions somewhat like a generalized signed efficiency relative to the Carnot efficiency, but which has a definition that is independent of which heat exchanger of the thermal core is under consideration and that is independent of the intended function of the machine, be it a refrigerator, a heat pump, a conventional heat engine that runs between a hot temperature and a lower ambient temperature, or an unconventional engine that runs between the ambient temperature and a lower cold temperature.
The total lost work WLostTOTAL shown in
To make sure that the tubes are structurally sound, rough analytic calculations of hoop, axial, and two-dimensional von Mises stresses are made. A buckling safety factor is estimated based on the inverse of an integral over the length of the tube of the product of the local slenderness ratio squared and the local longitudinal tube wall stress. Under most of the situations presented below, near where total lost work is minimized, the strength and stiffness of the tubes are sufficient; but where they become questionable, a finite element method calculation should be performed.
0in=
HX is −1.889 kW (negative value indicating heat leaving the machine) at an ambient bath temperature T0 of 305 K, and the cold heat input per stage
$in is +1.208 kW at a cold bath temperature T$ of 248 K. These values give an ηR=2.454, which is held constant throughout the optimization. The heat exchanger frontal dimensions are 178.9 mm×178.9 mm, for what may be considered to be a relatively medium heat exchanger power density of 59.0 kW/m2. The acoustic pressure amplitude is 110.0 kPa and the acoustic volumetric velocity amplitude is 0.03418 m3/s. An external fan coil unit that rejects the refrigerator's exhaust heat to the ambient environment is assumed to be relatively close to the refrigeration unit, with a round-trip loop length of 8 m of nominal ¾″ Schedule 40 PVC pipe (ID=20.9 mm), the FCU thermal resistance being 0.82 K/kW and with a manufacturer's provided data pressure drop of 2690 Pa (0.9 ft-wg) for a volume flow rate of 524 ml/s (8.3 gal/min). Heat leak between the secondary heat exchanger and the ambient is beneficial for ambient heat exchangers so no lost work from heat leak is included in the lost work tally. The secondary heat transfer fluid is assumed to be an ethylene glycol antifreeze mixed 50% with water (properties at 313 K: density=1080 kg·m−3, viscosity=2.3 mPa·s, heat capacity=3454 J·kg−1·K−1, thermal conductivity=0.424 W·m−1·K−1).
The symbols in HX| for chosen representative tube spacings D where the tube length L and the secondary heat transfer fluid mass flow rate M were varied to find the minimum in the total lost work WLostTOTAL (filled circles) at each selected D. The lines are cubic spline interpolations to connect the symbols. Also shown connected by dashed cubic spline lines are the tube length L (symbol L), an approximate buckling safety factor B (symbol B), and the secondary heat transfer fluid mass flow rate M normalized by (divided by) |
HX|/cp (symbol M*). The optimal spacing D is found to be 0.398 mm, for a minimum in the total lost work WLostTOTAL=140.3 W (WLostTOTAL/|
HX|=7.43%), with L=7.77 mm, and M=0.0899 kg/s (M cp/|
HX|=0.1643 K−1). The linear scale in
The tube thermal lost work WLostTubeThermal (open hourglass symbols) due to oscillatory temperature swings in the working fluid decreases at small D but the lost work from tube viscous loss WLostTubeViscous (filled hourglasses) increases at small D. The minimum in the total lost work occurs near where these two curves cross. Also important at small D is the lost electrical work from pumping the secondary heat transfer fluid over the small tubes WLostSHTFDragHXElec (open hexagons). Because of this loss, the optimization lowers the secondary heat transfer fluid mass flow M (normalized M symbol M*). But the lower M increases the secondary heat transfer fluid temperature deficit ΔTSHTF and its associated lost work WLostSHTFΔT (filled diamonds), which becomes a major loss component that represents the increased work needed by the refrigerator to operate over an increased temperature span as its ambient heat exchanger temperature rises above the ambient bath temperature.
At large D, the tubes collectively have insufficient surface area to make good thermal contact to the working fluid, which increases the temperature deficit ΔTHX, causing the refrigerator to have to pump its exhaust heat to a higher temperature, increasing its associated lost work WLostHXΔT (filled hexagons), the major loss component that lowers the machine efficiency. Additionally, the poor thermal contact increases tube thermal lost work WLostTubeThermal because of increasing oscillating temperature swings in the working fluid within the tubes. With a relatively open path for the secondary heat transfer fluid at large D around the tubes within the heat exchanger under consideration, the optimization allows for a larger M, which drops the losses due to the temperature deficit ΔTSHTF. And although the lost works due to pumping the secondary heat transfer fluid through the external plumbing loop WLostSHTFDragLoopElec (open diamonds) and the FCU WLostSHTFDragFCUElec (open squares) increase at large D, they still remain small and nearly negligible. The lost work from the thermal resistance of the FCU WLostFCUΔT (filled squares), however, is more important than its pumping loss, although it remains largely independent of D. Buckling safety factor of the relatively slender tubes near the minimum in total lost work (symbol B), rather than tube strength (graph of von Mises stress not shown, but is everywhere less than 20 MPa), is marginal and should be checked with a more thorough finite element calculation. It can be improved with a thicker tube.
HX|=7.50%), optimal tube length of L=7.35 mm, and mass flow rate M=0.0895 kg/s (M cp/|
HX|=0.1637 K−1). However, a second minimum in the total lost work appears at D=4.12 mm, with WLostTOTAL=156.8 W (WLostTOTAL/|
HX|=8.30%), optimal tube length at the minimum of L=2.78 mm, and mass flow rate M=0.1513 kg/s (M cp/|
HX|32 0.277 K−1). By providing good thermal contact to the working fluid, the mesh/screens allow good performance over a range of much larger tube spacing D. In effect, because the mesh/screen makes better thermal contact to the working fluid than do the tubes at large D, the optimization brings the length L of the tubes down to decrease the losses on the tubes, leaving them just long enough to allow for sufficient passage of the secondary heat transfer fluid over the tubes. This breaks down at higher D where the mesh/screens have insufficient thermal conduction across the wider tube mouths, and the loss from the temperature deficit ΔTHX increases. For this case, the first minimum is slightly lower than the second minimum. Nevertheless, the second minimum with tubes of larger diameter and shorter length may be desirable due to manufacturing considerations.
Particularly in the case of the second minimum, it may be beneficial to place mesh/screens on only the face of the monocoque tube bundle that faces the regenerator, leaving no mesh/screen on the face facing the open duct, because of joining losses that occur at transitions between regions of nearly adiabatic conditions in the working fluid and nearly isothermal regions. If the working fluid penetration depth is small compared to the local tube hydraulic radius, which scales with D, as could easily be the case near the second minimum, the working fluid space within the tube could become nearly adiabatic. Should this space be enclosed by two nearly isothermal mesh/screens there is the potential for two additional joining loss transitions. Thus it may be better to give up the advantage of duct laminar flow promotion that a mesh/screen on the duct facing face of the heat exchanger would provide in order to avoid an additional joining loss on each side of that mesh/screen. Conceptually, that mesh/screen could be doubled up with another mesh/screen next to the regenerator to not affect the total thermal contact to the working fluid. Similarly, it may be advantageous to favor the placement of conductive porous plug material within the tubes to be near the regenerator end of the tubes to avoid creating unnecessary adiabatic-isothermal transitions when the tube hydraulic radius becomes large compared to the thermal penetration depth.
0in=
HX is −4.811 kW at an ambient bath temperature T0 of 396 K, and the hot heat input per stage
$in is 5.210 kW at a hot bath temperature T$ of 490 K, giving an ηR=0.398. The heat exchanger frontal dimensions are 100.0 mm×100.0 mm, for what may be considered to be a relatively high heat exchanger power density of 481 kW/m2, eight times that of the refrigerator example. The acoustic pressure amplitude is 165.6 kPa and the acoustic volumetric velocity amplitude is 0.04339 m3/s. The same external fan coil unit used in the refrigerator examples of
HX|=3.36%), at D=2.48 mm, L=3.49 mm, and M=0.1358 kg/s (M cp/|
HX|=0.1129 K−1), is lower than the first minimum, with WLostTOTAL=180.6 W (WLostTOTAL/|QHX|=3.75%), at D=0.441 mm, L=2.66 mm, and M=0.658 kg/s (M cp/|
HX|=0.0513 K−1).
0in=
HX is −67.105 kW at an ambient bath temperature T0 of 300 K, and the hot heat input per stage
$in is 88.642 kW at a hot bath temperature T$ of 700 K, giving an ηR=0.425. The heat exchanger frontal dimensions are 1.00 m×1.00 m, for a heat exchanger power density of 67 kW/m2, similar to that of the refrigerator example, although at 5.5 times lower mean pressure. The acoustic pressure amplitude is 30 kPa and the acoustic volumetric velocity amplitude is 2.388 m3/s. An external fan coil unit of thermal resistance 0.040 K/kW, with a reference pressure drop of 2690 Pa at a volume flow rate of 7.3 L/s, and a round-trip 20 m loop length of nominal 1.5″ pipe (ID=40.9 mm) is assumed. The secondary heat transfer fluid is again assumed to be an ethylene glycol antifreeze mixed 50% with water (properties at 325 K: density=1055 kg·m−3, viscosity=1.7 mPa·s, heat capacity=3493 J·kg−1·K−1, thermal conductivity=0.426 W·m−1·K−1). In this case what could have been the first minimum disappears, although there is a hint of its remnant at around D=0.30 mm, where WLostTOTAL=4909 W (WLostTOTAL/|
HX|=7.32%), L=2.06 mm, and M=0.265 kg/s (M cp/|
HX|=0.01381 K−1). The only true minimum in the total lost work occurs with WLostTOTAL=2928 W (WLostTOTAL/|
HX|=4.36%), at D=7.56 mm, L=3.81 mm, and M=1.134 kg/s (M cp/|
HX|=0.0590 K−1).
It can be seen that although qualitatively similar, the optimizations of the dimensions of the ambient monocoque heat exchangers with screens for the three cases shown in
Turning away from these ambient heat exchangers and back to the refrigerator, $in=
HX is 1.2076 kW at a cold load bath temperature T$ of 248 K, and the ambient heat input per stage
0in is −1.8887 kW at an ambient bath temperature T0 of 305 K, giving a ηR=2.454, as before. The heat exchanger frontal dimensions are again 178.9 mm×178.9 mm, for a heat exchanger power density of 37.7 kW/m2. The acoustic pressure amplitude is 110 kPa and the acoustic volumetric velocity amplitude is 0.0304 m3/s at the cold heat exchanger under consideration. As before, an external fan coil unit of thermal resistance 0.82 K/kW, with a reference pressure drop of 2690 Pa at a volume flow rate of 524 ml/s is assumed. The assumed application is for a split unit commercial food freezer, so the secondary heat transfer fluid in this case is selected to be a potassium formate based solution which is non-toxic, non-flammable, and has relatively low viscosity at freezer temperatures (TYFOXIT F40 properties at 243 K: density=1356 kg·m−3, viscosity=13.98 mPa·s, heat capacity=2650 J·kg−1·K−1, thermal conductivity=0.436 W·m−1·K−1).
A round-trip 50 m loop length is assumed. The heat leak into a long cold loop can be important. Insulation of 6″ (236 mm) OD and thermal conductivity 0.037 W·m−1·K−1 is assumed to surround a Schedule 40 PVC pipe of thermal conductivity 0.19 W·m−1·K−1. The curves and points of HX|=24.0%), is at D=4.60 mm, L=2.88 mm, and M=0.1418 kg/s (M cp/|
HX|=0.311 K−1), while the slightly lower second minimum of total lost work for the ½″ pipe of WLostTOTAL=286.2 W (WLostTOTAL/|
HX|=23.7%) is at D=4.05 mm, L=2.28 mm, and M=0.1078 kg/s (M cp/|
HX|=0.237 K−1)—showing a dependence, but only a small dependence, of lost work and heat exchanger dimensions on the loop pipe assumptions. The higher, first minimum for the ¾″ pipe of WLostTOTAL=326.8 W (WLostTOTAL/|QHX|=27.1%), occurs at D=0.419 mm, L=6.67 mm, and M=0.0629 kg/s (M cp/|
HX|=0.1380 K−1).
It is interesting to note that the lost work from the heat leak WLostLoopHeatLeak (open inverted triangles) shows a weak dependence on tube spacing D, because at smaller D the optimization drives down the secondary heat transfer fluid mass flow rate M to compensate for the higher flow loss over a tight packing of smaller tubes, which in turn lowers the cold temperature that the refrigerator must reach to cool the external cold box at a fixed temperature T$ of 248 K, which lowers the average temperature of the secondary heat transfer fluid within the external loop pipe, and thus increases slightly the heat leak at smaller D.
The optimizations of
HX|=0.01361 K−1), for a minimum in the total lost work WLostTOTAL of 7.69 kW (WLostTOTAL/|
HX|=11.5%).
In
In some embodiments, the mesh/screen or similar porous planar structure joined to the first mouth end and/or the second mouth end comprise a multi-layered mesh/screen. In
Tube Waist Region: The tubes of the monocoque tube bundle are tapered at both ends of each tube portion with each tube having mouths at each end that are wider in cross section than the cross section of the tube waist region between the two mouths in the general manner of an hourglass, Venturi tube or other suitable shape. The cross section of the tubes of the monocoque bundle varies between the respective two mouths and the region that has the narrowest cross section is defined as the tube waist region.
Pore Size: The mesh/screen, porous packing, or similar porous structures have minute openings or constrictions of complicated three dimensional shape through which working fluid may pass. Pore size herein is defined by the average pore hydraulic radius, which is the volume of working fluid within the mesh/screen, porous packing, or similar porous structure, divided by the surface area of the mesh/screen, porous packing, or similar porous structure in contact with the working fluid.
Tube Local Hydraulic Radius: As a function of local position along the tube's length, the local hydraulic radius is half of the tube's local radius for circular cross-section tubes, or the local tube cross sectional area divided by the local tube perimeter for tubes of non-circular cross-section.
Thermal Penetration Depth: Thermal penetration depth, δκ=√{square root over (2κ/(ρcpω))}, where κ is the thermal conductivity, ρ is the density, cp is the heat capacity of the working fluid at constant pressure, and ω is the angular frequency of the cycle.
The present invention has been described in the context of thermoacoustic and Stirling devices, but it may be useful in more conventional heat exchange applications where the working fluid flows in only one direction.
The foregoing detailed description has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit embodiments to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. The described embodiments were chosen to explain principles and practical applications, to thereby enable others skilled in the art to utilize various embodiments and with various modifications as are suited to the particular use contemplated. As will be clear to those of skill in the art, the illustrated and discussed embodiments of the present invention may be altered in various ways without departing from the scope or teaching of the present invention. As such, this disclosure should be interpreted broadly. It is intended that the scope be defined by the claims appended hereto.
Various patents, patent applications, and/or publications have been referred to in this application. The material contained in these patents, patent applications, and/or publications is incorporated in their entirety herein by reference.
This application is a Continuation of application Ser. No. 16,965,057, filed Jul. 27, 2020, which claims the benefit of U.S. PCT Application No. PCT/US2019/015832, filed Jan. 30, 2019, which claims the benefit of U.S. Provisional Application No. 62/624,170, filed Jan. 31, 2018, each of which are hereby incorporated by reference in its entirety.
This invention was made with government support under Grant No. N00014-98-1-0212 and N00014-03-1-0652, awarded by the Office of Naval Research and under Grant No. DE-FC26-04NT42113 and DE-AR0000130, awarded by the Department of Energy. The Government has certain rights in the invention.
| Number | Date | Country | |
|---|---|---|---|
| 62624170 | Jan 2018 | US |
| Number | Date | Country | |
|---|---|---|---|
| Parent | 16965057 | Jul 2020 | US |
| Child | 18077939 | US |
