WO2025208120A1

WO2025208120A1 - Systems and methods for improved waste heat source recovery and engine architecture

Info

Publication number: WO2025208120A1
Application number: PCT/US2025/022187
Authority: WO
Inventors: Stephen Smith
Original assignee: Phasic Energy Co
Current assignee: Phasic Energy Co
Priority date: 2024-03-29
Filing date: 2025-03-28
Publication date: 2025-10-02
Anticipated expiration: 2026-09-29

Abstract

Described herein are systems, methods, computer-readable media, and techniques for recovering heat from a waste heat source, the system comprising a free piston Stirling engine, a topology-optimized heat transfer apparatus in thermal communication with the free piston Stirling engine, wherein the topology-optimized heat transfer apparatus comprises a continuously curved fluid flow passage, at least one inlet configured to receive a first fluid stream at a first temperature from the waste heat source, and at least one inlet configured to receive a second fluid stream at a second temperature lower than the first temperature, wherein heat transfer occurs within the topology-optimized heat transfer apparatus, and the free piston Stirling engine converts at least a portion of the transferred heat into mechanical energy, which is subsequently converted to electrical energy.

Description

SYSTEMS AND METHODS FOR IMPROVED WASTE HEAT SOURCE RECOVERY AND ENGINE ARCHITECTURE

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63/571,972, filed March 29, 2024, and U.S. Provisional Application No. 63/660,159, filed June 14, 2024, each of which is incorporated herein by reference in its entirety.

BACKGROUND

[0002] A Free Piston Stirling Engine (FPSE) is a variation of the traditional Stirling engine in which the pistons are not mechanically linked to a rotating crankshaft. Instead, they freely move back and forth within a cylinder, hence the term “free piston.” This allows the engine to have fewer moving parts, which may increase reliability and reduce maintenance.

[0003] The Free Piston Stirling Engine operates on the Stirling cycle, which consists of four main stages: heating and expansion, displacement, cooling and compression, and redisplacement. The working gas inside the engine (e.g., such as helium or hydrogen) is moved back and forth between the hot and cold ends of the engine by a displacer piston. The expansion and contraction of the gas caused by heating and cooling drives the power piston, creating a reciprocating motion.

SUMMARY

[0004] Recognized herein is a need for systems and methods that may efficiently recover and convert low-grade waste heat source from a variety of sources, including but not limited to industrial processes, commercial HVAC systems, building operations, and electronic equipment cooling loops, into usable energy. The present disclosure provides systems and methods for improved waste heat source recovery using topology-optimized Free Piston Stirling Engines (FPSE) and heat exchangers specifically configured for low-temperature differentials. These systems and methods address inefficiencies in current processes by implementing organic, topology-optimized heat transfer apparatus geometries that maximize heat flux distribution and thermal efficiency while minimizing pressure drops, thereby providing effective energy conversion even at temperature differentials as low as 30°C. [0005] In an aspect, a method for recovering heat from a waste heat source comprises providing a free piston Stirling engine and a topology-optimized heat transfer apparatus in thermal communication with the free piston Stirling engine, the heat transfer apparatus comprising a continuously curved fluid flow passage, directing a first fluid stream at a first temperature from the waste heat source into the heat transfer apparatus, directing a second fluid stream ata second, lower temperature into the heat transfer apparatus such that heat is transferred between the firstand second fluid streams, and operating the free piston Stirling engine to convert at least a portion of the transferred heat into mechanical energy and subsequently into electrical energy.

[0006] In some embodiments, the topology-optimized heat transfer apparatus results from a computational optimization process that produces a thermal resistance of not more than 2.0 xlO-⁴ m²K/W.

[0007] In some embodiments, the computational optimization process comprises providing one or more parameters of the free piston Stirling engine into a topology optimization algorithm, the parameters including at least one of an internal diameter of a pressure vessel, an internal diameter of a regenerator, a mean charge pressure, a piston diameter, a piston amplitude, or a compression space volume, generating via the topology optimization algorithm a continuously curved topology for the heat transfer apparatus in response to the one or more parameters, and sizing the continuously curved topology to accommodate a temperature differential such that the first fluid stream is at a higher temperature than the second fluid stream.

[0008] In some embodiments, the computational optimization process includes iteratively modifying at least one of wall thickness, surface curvature, or flow passage geometry in response to predicted thermal gradients during operation, with each iteration performed to reduce the total thermal resistance across the heat transfer apparatus.

[0009] In some embodiments, the second fluid stream is a coolant fluid in fluid communication with a cooling jacket of the free piston Stirling engine, and at least a portion of the coolant fluid exiting the free piston Stirling engine is directed to a cooling system to remove additional heat such that electrical power generation and fluid cooling occur concurrently from the same waste heat source.

[0010] In some embodiments, the heat transfer apparatus forms part of a closed-loop fluid circuit, and the first and second fluid streams are recirculated to at least one chiller and then returned to the heat transfer apparatus.

[0011] In some embodiments, the heat transfer apparatus is manufactured using an additive manufacturing technique that deposits successive layers of material to form at least one fluid flow passage having continuously varying cross-sectional profiles and a radius of curvature of at least 1 mm at every interior corner. [0012] In some embodiments, the heat transfer apparatus comprises a multi-level array of fluid flow passages arrayed in parallel or series such that each level maintains a controlled and quantifiable pressure drop of at most 0.5 bar across the apparatus.

[0013] In some embodiments, the method further comprises measuring, via at least one sensor, a real-time temperature difference between the first and second fluid streams within the heat transfer apparatus, adjusting at least one operating parameter selected from a flow rate of the first fluid stream, a flow rate of the second fluid stream, or a mean charge pressure of the free piston Stirling engine in response to the measured temperature differential, and maintaining a predetermined temperature differential range to improve the free piston Stirling engine’s efficiency in converting heat to electrical energy.

[0014] In some embodiments, the temperature differential between the first and second fluid streams is less than about 80°C.

[0015] In some embodiments, the first fluid stream is a low-grade waste heat source at a temperature of less than about 230°C, and the second fluid stream is arrayed to remove heat therefrom to create a temperature differential sufficient to operate the free piston Stirling engine.

[0016] In some embodiments, the first fluid stream comprises a refrigerant selected from R 134a, R410a, or ammonia, and the heat transfer apparatus is arrayed to receive the refrigerant from a waste heat source production system.

[0017] In some embodiments, the method further comprises performing one-dimensional and three-dimensional computational fluid dynamics (CFD) modeling prior to the computational optimization process, wherein the results of the CFD modeling establish parameters and objectives for the topology optimization algorithm.

[0018] In some embodiments, the free piston Stirling engine and the heat transfer apparatus are installed in a building that produces the waste heat source, the building comprising at least one of a residential building or a commercial building.

[0019] In some embodiments, the method further comprises performing one-dimensional CFD modeling on the free piston Stirling engine to determine initial parameters, performing three-dimensional CFD modeling to refine flow and thermal characteristics, applying a topology optimization process to produce an organic geometry with continuously curved fluid passages, and manufacturing the heat transfer apparatus via an additive manufacturing technique, wherein the apparatus has reduced thermal resistance for recovering waste heat source from a low-grade heat source. [0020] In some embodiments, the method further comprises defining at least 50 input parameters for the free piston Stirling engine, including at least one of a mean charge pressure, piston amplitude, heat exchanger channel geometry, or regenerator dimensions, iteratively adjusting at least one of these parameters during the one-dimensional and three- dimensional CFD modeling to refine predicted flow and thermal characteristics, incorporating the results into the topology optimization process to generate the continuously curved fluid passages, and verifying via additive manufacturing prototypes that the produced heat transfer apparatus achieves a thermal resistance of not more than about 2.0^10 ⁴ m²K/W such that efficient recovery of low-grade waste heat source below about 230°C is achieved. [0021] In an aspect, a system for recovering heat from a waste heat source comprises a free piston Stirling engine, a topology-optimized heat transfer apparatus in thermal communication with the free piston Stirling engine, the heat transfer apparatus comprising a continuously curved fluid flow passage, at least one inlet operable to receive a first fluid stream at a first temperature from the waste heat source, and at least one inlet operable to receive a second fluid stream at a second, lower temperature, wherein the free piston Stirling engine is operable to convert at least a portion of the heat transferred between the first and second fluid streams into mechanical energy and subsequently into electrical energy.

[0022] In some embodiments, the heat transfer apparatus results from a computational optimization process that produces a thermal resistance of not more than 2.0xl0"⁴ m²K/W. [0023] In some embodiments, the computational optimization process comprises providing one or more parameters of the free piston Stirling engine into a topology optimization algorithm, wherein the parameters include at least one of an internal diameter of a pressure vessel, an internal diameter of a regenerator, a mean charge pressure, a piston diameter, a piston amplitude, or a compression space volume, generating via the topology optimization algorithm a continuously curved topology for the apparatus in response to these parameters, and sizing the continuously curved topology to accommodate a temperature differential such that the first fluid stream is at a higher temperature than the second fluid stream.

[0024] In some embodiments, the computational optimization process includes iteratively modifying at least one of wall thickness, surface curvature, or flow passage geometry in response to predicted thermal gradients during operation, with each iteration performed to reduce the total thermal resistance across the heat transfer apparatus.

[0025] In some embodiments, the second fluid stream is a coolant fluid in fluid communication with a cooling jacket of the free piston Stirling engine, and at least a portion of the coolant fluid exiting the engine is directed to a cooling system to remove additional heat such that electrical power generation and fluid cooling occur concurrently from the same waste heat source.

[0026] In some embodiments, the heat transfer apparatus forms part of a closed-loop fluid circuit, and the first and second fluid streams are recirculated to at least one chiller and then returned to the apparatus.

[0027] In some embodiments, the heat transfer apparatus is manufactured using an additive manufacturing technique that deposits successive layers of material to form at least one fluid flow passage having continuously varying cross-sectional profiles and a radius of curvature of at least 1 mm at every interior corner.

[0028] In some embodiments, the heat transfer apparatus comprises a multi-level array of fluid flow passages arrayed in parallel or series such that each level maintains a controlled and quantifiable pressure drop of at most 0.5 bar across the apparatus.

[0029] In some embodiments, the system further comprises at least one sensor disposed to measure a real-time temperature difference between the first and second fluid streams within the apparatus and a controller operable to adjust at least one operating parameter selected from a flow rate of the first fluid stream, a flow rate of the second fluid stream, or a mean charge pressure of the free piston Stirling engine in response to the measured temperature difference, whereby the system is operable to maintain a predetermined temperature differential range to improve the engine’s efficiency in converting heat to electrical energy. [0030] In some embodiments, the temperature differential between the first and second fluid streams is less than about 80°C.

[0031] In some embodiments, the first fluid stream is a low-grade waste heat source at a temperature of less than about 230°C, and the second fluid stream is arrayed to remove heat therefrom to create a temperature differential sufficient to operate the free piston Stirling engine.

[0032] In some embodiments, the first fluid stream comprises a refrigerant selected from R 134a, R 410a, or ammonia, and the apparatus is arrayed to receive the refrigerant from a waste heat source production system.

[0033] In some embodiments, the system further comprises performing one-dimensional and three-dimensional computational fluid dynamics (CFD) simulations prior to the computational optimization process, wherein the results of the CFD modeling establish parameters and objectives for the topology optimization algorithm. [0034] In some embodiments, the free piston Stirling engine and the heat transfer apparatus are installed in a building that produces the waste heat source, the building comprising at least one of a residential building or a commercial building.

[0035] In some embodiments, the system further comprises a framework in which onedimensional CFD modeling is performed on the free piston Stirling engine to determine initial parameters, a three-dimensional CFD model is used to refine flow and thermal characteristics, a topology optimization module generates an organic geometry with continuously curved fluid passages, and an additive manufacturing subsystem fabricates the apparatus, whereby the resulting heat transfer apparatus has reduced thermal resistance for recovering waste heat source from a low-grade heat source.

[0036] In some embodiments, the system further comprises defining at least 50 input parameters for the free piston Stirling engine, including at least one of a mean charge pressure, piston amplitude, heat exchanger channel geometry, or regenerator dimensions, iteratively adjusting at least one of these parameters during the one-dimensional and three- dimensional CFD modeling to refine predicted flow and thermal characteristics, incorporating the results into the topology optimization process to generate the continuously curved fluid passages, and verifying via test prototypes that the produced apparatus achieves a thermal resistance of not more than about 2.0/ 10 ⁴ m²K/W such that efficient recovery of low-grade waste heat source below about 230°C is achieved.

[0037] In an aspect, a topology-optimized heat transfer apparatus comprises an amorphous, organically contoured body manufactured via an additive manufacturing technique, the body being substantially devoid of right angles and arrayed to provide continuously curved internal passages for fluid flow, a multi-level array of interconnected channels defined within the contoured body with each channel sized to reduce the total thermal resistance to below about 2.0/10 ⁴ m²K/W when receiving low-grade waste heat source, a minimum radiusof curvature of at least about 1 mm at every interior corner of the channels to mitigate flow stagnation and minimize pressure drop, and an overall geometry arrayed to be in thermal communication with a free piston Stirling engine, wherein the apparatus is obtained by a computational optimization process based on at least one engine parameter to provide recovery of low-grade heat for conversion into mechanical and subsequently electrical energy.

[0038] Another aspect of the present disclosure provides a non-transitory computer readable medium comprising machine executable code that, upon execution by one or more computer processors, implements any of the methods above or elsewhere herein. [0039] Another aspect of the present disclosure provides a system comprising one or more computer processors and computer memory coupled thereto. The computer memory comprises machine executable code that, upon execution by the one or more computer processors, implements any of the methods above or elsewhere herein.

[0040] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in this art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its several details are capable of modifications in various obvious respects, all without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.

INCORPORATION BY REFERENCE

[0041] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and/or take precedence over any such contradictory material.

BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The novel features of the disclosure are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present disclosure will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the disclosure are utilized, and the accompanying drawings of which:

[0043] FIG. 1 illustrates an example of a Free Piston Stirling Engine, in accordance with certain embodiments;

[0044] FIG. 2 illustrates an example of a Free Piston Stirling Engine, in accordance with certain embodiments;

[0045] FIG. 3 illustrates an example of a Free Piston Stirling Engine Heat Exchanger, in accordance with certain embodiments;

[0046] FIG. 4 illustrates an example of a Free Piston Stirling Engine Heat Exchanger, in accordance with certain embodiments; [0047] FIG. 5 illustrates an example of a Free Piston Stirling Engine Heat Exchanger, in accordance with certain embodiments;

[0048] FIG. 6 illustrates an example of a Free Piston Stirling Engine Regenerator, in accordance with certain embodiments;

[0049] FIG. 7 illustrates an example of a Free Piston Stirling Engine Heat Exchanger, in accordance with certain embodiments;

[0050] FIG. 8 illustrates an example of a Free Piston Stirling Engine, in accordance with certain embodiments;

[0051] FIG. 9 illustrates an example of a Free Piston Stirling Engine, in accordance with certain embodiments;

[0052] FIG. 10 shows a computer system that is programmed or otherwise configured to implement methods disclosed herein;

[0053] FIG. 11 illustrates an example of FPSE schematic, in accordance with certain embodiments;

[0054] FIG. 12 presents a schematic of the one-dimensional (ID) engine model developed within the Software (e.g., SAGE software) in accordance with certain embodiments;

[0055] FIG. 13 presents a schematic of displacer subcomponents, in accordance with certain embodiments;

[0056] FIG. 14 illustrates an example of convergence history of optimization of engine thermal efficiency, in accordance with certain embodiments;

[0057] FIG. 15 illustrates an example of energy production of FPSE over time, in accordance with certain embodiments;

[0058] FIG. 16 illustrates an example of FPSE heat exchanger fins, in accordance with certain embodiments;

[0059] FIG. 17 illustrates an example of heat exchanger fins geometry for CFD simulation, in accordance with certain embodiments;

[0060] FIG. 18A illustrates an example of a cross-section of FPSE fine mesh top section, in accordance with certain embodiments;

[0061] FIG. 18B illustrates an example of a cross-section of FPSE fine mesh middle section, in accordance with certain embodiments;

[0062] FIG. 18C illustrates an example of a cross-section of FPSE fine mesh bottom section, in accordance with certain embodiments;

[0063] FIG. 19A illustrates an example of models used for air, in accordance with certain embodiments; [0064] FIG. 19B illustrates an example of models used for solids, in accordance with certain embodiments;

[0065] FIG. 20 illustrates an example of boundary conditions specifications, in accordance with certain embodiments;

[0066] FIG. 21 illustrates an example of mesh sensitivity analysis results, in accordance with certain embodiments;

[0067] FIG. 22A illustrates a first example of a FPSE heat flux, in accordance with certain embodiments;

[0068] FIG. 22B illustrates a second example of a FPSE heat flux, in accordance with certain embodiments;

[0069] FIG. 23A illustrates a first example of the heat exchanger fins surface heat flux distribution from a top view perspective, in accordance with certain embodiments;

[0070] FIG. 23B illustrates a second example of the heat exchanger fins surface heat flux distribution from an angled perspective, in accordance with certain embodiments;

[0071] FIG. 24A illustrates an example of a FPSE air temperature contours at a side plane, in accordance with certain embodiments;

[0072] FIG. 24B illustrates an example of a FPSE air temperature contours at another side plane, in accordance with certain embodiments;

[0073] FIG. 24C illustrates an example of a FPSE air temperature velocity contours at a side plane, in accordance with certain embodiments;

[0074] FIG. 24D illustrates an example of a FPSE air pressure contours at a side plane, in accordance with certain embodiments;

[0075] FIG. 25 illustrates an example of a FPSE waste heat source air temperature contours, in accordance with certain embodiments;

[0076] FIG. 26 illustrates an example of a FPSE heat transfer contract surfaces, in accordance with certain embodiments;

[0077] FIG. 27 illustrates an example of architecture space envelope around cylinder used for topology optimization, in accordance with certain embodiments;

[0078] FIG. 28 illustrates an example of conformal mesh obtained for the air domain and solid fin domains, in accordance with certain embodiments;

[0079] FIG. 29A illustrates an example of resulting geometry for topology optimization (TO), in accordance with certain embodiments;

[0080] FIG. 29B illustrates an example of input geometry for full simulation with boundary layers (BL), in accordance with certain embodiments; [0081] FIG. 30A illustrates an example of a conventional finned heat exchanger, in accordance with certain embodiments;

[0082] FIG. 30B illustrates an example of a topology optimized finned heat exchanger, in accordance with certain embodiments;

[0083] FIG. 31 illustrates an example of a conventional finned heat exchanger, in accordance with certain embodiments;

[0084] FIG. 32 illustrates an example of a topology -optimized finned heat exchanger, in accordance with certain embodiments;

[0085] FIG. 33 illustrates an example of a Heat recovery system, in accordance with certain embodiments;

[0086] FIG. 34 illustrates an example of a Heat recovery system for a closed loop cooling system, in accordance with certain embodiments;

[0087] FIG. 35 illustrates an example of a Heat recovery system for an open loop cooling system, in accordance with certain embodiments;

[0088] FIG. 36 illustrates an example of periodic boundary conditions, in accordance with certain embodiments;

[0089] FIG. 37 illustrates an example of generated mesh for TO simulation domain, in accordance with certain embodiments;

[0090] FIG. 38 illustrates an example of TO solver setup parameters, in accordance with certain embodiments;

[0091] FIG. 39 illustrates an example of TO solver convergence history, in accordance with certain embodiments;

[0092] FIG. 40 illustrates an example of Resulting geometry from TO(left) and Input geometry for full simulation with BL (right), in accordance with certain embodiments;

[0093] FIG. 41 illustrates an example of Solver Residuals (top) and Convergence of total heat transfer value (bottom) , in accordance with certain embodiments;

[0094] FIG. 42 illustrates an example of Fins solid temperature (top), in accordance with certain embodiments;

[0095] FIG. 43 illustrates an example of Fins surface heat flux, in accordance with certain embodiments;

[0096] FIG. 44 illustrates an example of Contours of temperature at (top) side plane, in accordance with certain embodiments;

[0097] FIG. 45 illustrates an example of (bottom) top plane, in accordance with certain embodiments; [0098] FIG. 46 illustrates an example of Air gauge pressure contours at side plane, in accordance with certain embodiments;

[0099] FIG. 47 illustrates an example of Air velocity contours at side plane, in accordance with certain embodiments;

[0100] FIG. 48 illustrates an example Contours of temperature at the outlet of enclosure, in accordance with certain embodiments;

[0101] FIG. 49 illustrates a non-limiting example of a schematic of FPSE architecture configuration, in accordance with one or more embodiments herein.

[0102] FIG. 50 illustrates a non-limiting example of experimental data points from FPSE testing showing indicated power and displacer stroke, in accordance with one or more embodiments herein.

[0103] FIG. 51 illustrates a non-limiting example of experimental data points from FPSE testing, in accordance with one or more embodiments herein.

[0104] FIG. 52 illustrates a non-limiting example of a variable data set list for FPSE parameters, in accordance with one or more embodiments herein.

[0105] FIG. 53 illustrates a non-limiting example of a 3-D CAD geometry of an FPSE, in accordance with one or more embodiments herein.

[0106] FIG. 54 illustrates a non-limiting example of convergence history of thermal efficiency optimization, in accordance with one or more embodiments herein.

[0107] FIG. 55 illustrates a non-limiting example of discretization of an FPSE with polyhedral elements and overset technique, in accordance with one or more embodiments herein.

[0108] FIG. 56 illustrates an example Conformal mesh obtained for the air domain (yellow) and solid fins domain (blue), in accordance with certain embodiments;

[0109] FIG. 57 illustrates an example of a Waste heat source Recovery System (WHRS) integrated into a building, in accordance with certain embodiments;

[0110] FIG. 58 illustrates an example of a first view of topology optimized heat exchanger, in accordance with certain embodiments;

[0111] FIG. 59 illustrates an example of a second view of topology optimized heat exchanger, in accordance with certain embodiments;

[0112] FIG. 60 illustrates an example of a third view of topology optimized heat exchanger, in accordance with certain embodiments; and

[0113] FIG. 61 illustrates an example of a Waste heat source Recovery System (WHRS) integrated into a building, in accordance with certain embodiments. DETAILED DESCRIPTION

[0114] While certain embodiments of the systems and methods described herein have been presented and discussed, they are provided by way of example only. It will be apparent to those skilled in the art that numerous variations, changes, and substitutions may be made without departing from the scope of the disclosed technologies. It is not intended that the invention be limited by any specific examples provided. The descriptions and illustrations of embodiments are not meant to be construed in a limiting sense, and various alternatives or modifications may be employed in practicing the invention. It is therefore contemplated that the following claims shall cover any such alternatives, modifications, variations, or equivalents.

[0115] Unless the context requires otherwise, throughout this specification and the appended claims, the term “comprise” (and variations such as “comprises” and “comprising”) is to be interpreted as “including, but not limited to.”

[0116] As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise.

[0117] The term “or” is generally used in its inclusive sense (i.e., “and/or”) unless the context clearly dictates otherwise. Thus, a phrase such as “A or B” encompasses “A,” “B,” or “A and B.”

[0118] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, that term applies to each numerical value in the series.

[0119] Whenever the term “no more than,” “less than,” or “less than or equal to” precedes the first numerical value in a series of two or more numerical values, that term applies to each numerical value in the series.

[0120] Numerical ranges include the endpoints of the range, and each sub -range or intermediate value within the range is disclosed as though expressly set forth.

[0121] The term “about” or “about” refers to an acceptable degree of error for a given value, which may depend on measurement limitations or the context of use. It may mean within 1% to 20% of the stated value, within 1 or more standard deviations, or another appropriate tolerance recognized by those skilled in the art.

[0122] Terms such as “above,” “below,” “upper,” “lower,” “horizontal,” or “vertical” refer to the relative position or orientation of components as illustrated in example figures and are meant to include different orientations or reference frames unless expressly limited. [0123] When elements are described as being “on,” “onto,” “over,” “above,” or “connected,” such references may include direct or indirect contact or coupling unless otherwise specified by terms like “directly on” or “directly coupled.”

[0124] The operations of methods described may be performed in any feasible order, including concurrently, unless otherwise indicated. Nothing herein requires that operations be performed strictly in the sequence illustrated or described.

[0125] Numbered terms such as “first,” “second,” and “third” are used only to distinguish one element from another and do not indicate a specific order unless expressly stated.

[0126] Unless a term is expressly defined in this specification, it is intended to have its plain and ordinary meaning. Use of a term in one context does not limit it to that context unless clearly indicated.

[0127] Certain techniques or operations described may be implemented by hardware (e.g., dedicated circuitry such as ASICs or FPGAs), by software executed on one or more processors, or by a combination of both. References to hardware or software “modules” encompass both permanently and temporarily configured systems, including distributed computing architectures.

[0128] The term "continuously curved," as used herein, generally refers to any geometry or surface that transitions smoothly from one point to another without abrupt angles, edges, or breaks in curvature. In some instances, a geometry or surface is considered "continuously curved" if every interior corner or boundary maintains a minimum radius of curvature, such as at least about 1 mm, thereby avoiding any sharp or right-angled junctions. By way of example, a continuously curved fluid passage may consist of arcs, splines, or other curvilinear profiles that blend seamlessly, so that no portion of the passage includes an uninterrupted planar segment or a corner approximating 90° (or any other discontinuity). However, it may be understood that "continuously curved" is not strictly limited to circular or elliptical arcs and may include compound or irregular curves, provided that the flow path remains free of abrupt transitions that may create sharp edges or discrete comers.

[0129] The term "topology optimization," as used herein, generally refers to any computational or algorithmic process that iteratively modifies a given architecture space to optimize one or more performance objectives, such as minimizing thermal resistance, reducing weight, maximizing heat transfer, or optimizing fluid flow characteristics. In some embodiments, topology optimization involves systematically adding, removing, or reshaping material within a virtual 3D model based on input constraints (e.g., pressure drop limits, temperature gradients) and objective functions (e.g., maximizing heat dissipation, maintaining structural integrity). The method may incorporate gradient-based solvers, heuristic algorithms, genetic algorithms, or other numerical techniques to determine an “improved” geometry. By way of example, a topology optimization routine may begin with an initial block of material representing a heat exchanger and iteratively remove or reshape internal regions to achieve improved heat transfer performance, while respecting user-defined boundaries, physical constraints, or manufacturing limitations. However, as used herein, “topology optimization” is not limited to any particular solver, software, or algorithmic approach, provided that it involves modifying a structure’ s geometry in an iterative and goal- driven manner.

[0130] The term "low-grade waste heat source," as used herein, generally refers to thermal energy at a temperature that is insufficient for many conventional high-temperature processes but may still be effectively utilized by systems configured to recover heat from lower temperature sources. In some instances, low-grade waste heat source comprises fluids or gases at below about230°C, although other upper temperature thresholds (e.g., about 150°C or about 105°C) may be used depending on industrial or environmental contexts. For example, low-grade heat may include refrigerants discharged from air-conditioning units, cooling fluid from commercial buildings, or effluent from mild industrial processes. By way of illustration, a temperature range between about 70°C to about 105°C may be deemed “low- grade” in certain embodiments, where typical high-temperature heat-recovery devices fail to operate efficiently. However, it may be understood that “low-grade waste heat source” is not restricted to any exact numerical cutoff and may encompass any temperature deemed insufficient for direct use in conventional high-temperature or high-pressure heat applications.

[0131] The term "free piston Stirling engine," as used herein, generally refers to a Stirling- cycle heat engine in which pistons (including a power piston and/or a displacer) move reciprocally without being mechanically linked to a crankshaft. Instead, pressure oscillations of a contained working gas (e.g., helium, hydrogen, air, or nitrogen) drive piston motion. The absence of direct mechanical linkages allows for self-adjusting stroke lengths, reduced friction, and simpler sealing compared to traditional Stirling engines. A free piston Stirling engine typically comprises compression space, expansion space, and a regenerator, arrayed such that cyclical heating and cooling of the working gas produce reciprocating mechanical energy (which may be converted to electrical power via a linear alternator or other transducer). [0132] The term "closed-loop," as used herein, generally refers to a fluid circuit in which the fluid (e.g., a coolant or process fluid) is continuously recirculated between components without intentional release to the environment. In a closed-loop system, the same fluid flows through, for example, a heat transfer apparatus, a chiller, and a waste heat source, returning in a cyclical manner. This contrasts with an open-loop system, which may draw in fluid from an external source and discharge at least a portion of that fluid to a reservoir or to the atmosphere, not recirculating the same fluid indefinitely.

[0133] The term "flow passage geometry," as used herein, refers to the shape, dimensions, and spatial arrangement of any channel, conduit, or passage through which a fluid flows. A flow passage may be continuously curved, may include multi-level or nested channels, and may vary in cross-sectional area or radius of curvature. In some embodiments, the flow passage geometry is specifically topology -optimized to achieve improved heat transfer, lower pressure drop, or enhanced fluid distribution. The geometry may also encompass manifolds, branching pathways, and arrays of flow paths arrayed in parallel or series to suit desired thermal and fluidic performance.

[0134] The term "computational fluid dynamics (CFD)," as used herein, generally refers to numerical simulation methods used to analyze and predict fluid flow, heat transfer, pressure distributions, and related phenomena in a defined physical domain. CFD may include onedimensional (ID) modeling, which simplifies flow into lumped or linear segments, and three- dimensional (3D) modeling, which resolves spatial variations of temperature, velocity, pressure, or other fluid properties. CFD tools typically employ methods such as the finite volume, finite element, or finite difference approaches, possibly coupled with turbulence models, heat transfer correlations, and phase-change models to capture complex physics. Results from ID and/or 3D CFD simulations may inform architecture constraints and objective functions in a topology optimization process.

[0135] The term "mean charge pressure," as used herein, generally refers to the average or quasi-static gas pressure present within a sealed Stirling engine, particularly a free piston Stirling engine, when it is at rest or operating under stabilized conditions. In certain configurations, the mean charge pressure is regulated or preset by introducing or releasing the workinggas (e.g., helium) into the engine’s pressure vessel. This parameter may modify the power output, operating frequency, piston amplitude, and efficiency of the engine. For example, a higher mean charge pressure often correlates with increased power density, albeit sometimes at the expense of greater stress on internal components. [0136] The term "thermal resistance," as used herein, refers to a quantitative measure of the temperature difference (AT) required to induce a unit heat flow (q) across a specific material or system. It may be expressed in units such as m²K/W, indicating how a certain temperature gradient (in Kelvin or °C) translates into a given rate of heat transfer (in Watts) over a specified area. A lower thermal resistance indicates enhanced heat transfer or lower temperature gradients for a given heat flux. In the context of topology-optimized heat transfer apparatuses, reducing thermal resistance provides more efficient use of low-grade waste heat source, reducing temperature losses between the heat source and the engine or other downstream components.

[0137] The term “waste heat source,” as used herein, generally refers to thermal energy that is not fully utilized in its originating process and may otherwise be discharged or dissipated to the environment without recovery. In various embodiments, waste heat source may arise from industrial processes (e.g., chemical manufacturing, refining, metal casting), commercial equipment (e.g., HVAC units, refrigeration systems, kitchen appliances), transportation engines (e.g., internal combustion engines, turbines), building operations (e.g., heating, ventilation, and air conditioning), or electronic equipment (e.g. power supplies). Waste heat source may be present in gaseous, liquid, or two-phase forms, including hot air, steam, exhaust gases, heated water, refrigerant fluids, or other thermally elevated by-products, and may vary widely in temperature from near-ambient levels (e.g., about 25°C) up to higher temperatures (e.g., about 300°C or more). In some instances, “waste heat source” may be referred to as low-grade or medium-grade thermal energy, particularly where the temperature range is below about 230°C and may be inefficient for direct use in conventional high- temperature processes, yet still ly recoverable for heat exchange, mechanical work, or electrical power generation through specialized systems or devices.

[0138] As used herein, the term 'conventional' generally refers to systems, methods, components, or designs that follow established, common, standard, or prior art practices, particularly those known prior to or not incorporating the specific topology optimization techniques, low-temperature differential configurations, or other improvements described in the present disclosure. For example, when referring to components such as heat exchangers or fins, 'conventional' typically indicates designs lacking the specific topology optimization resulting in continuously curved passages or amorphous structures as described herein. In broader contexts, 'conventional' may also refer to established technologies or approaches (e.g., conventional Stirling engines with kinematic linkages, conventional high-temperature processes) where the distinction from the systems and methods of the present disclosure is clear from the context.

Overview

[0139] In one aspect, disclosed herein are waste heat source management systems for a waste heat source production system, comprising: waste heat source production devices; an waste heat source in thermal connection with the waste heat source production devices; a first conduit comprising a first stream, wherein the first stream comprises a first temperature; a second conduit comprising a second stream, wherein the second stream comprises a second temperature; a Free Piston Stirling Engine (FPSE) fluidically connected to the first conduit and the second conduit.

[0140] In another aspect, disclosed herein are waste heat source management systems for a waste heat source production system, comprising: waste heat source production devices, an waste heat source in thermal connection with the waste heat source production devices, a first conduit comprising a first stream, wherein the first stream comprises a first temperature, a second conduit comprising a second stream, wherein the second stream comprises a second temperature, a Free Piston Stirling Engine (FPSE) fluidically connected to the first conduit and the second conduit.

[0141] In some embodiments, the system further comprises a Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the first stream comprises a fluid, gas or solid. In some embodiments, the second stream comprises water. In some embodiments, a temperature of the first stream is equal to or greater than a temperature of the second stream. In some embodiments, the first stream comprises an output stream from the waste heat source production devices. In some embodiments, the second stream comprises an input stream to the waste heat source production devices. In some embodiments, the first conduit fluidically connects the waste heat source production devices to the FPSE. In some embodiments, the first stream enters the FPSE at a hot cylinder side. In some embodiments, the first stream transfers heat to the hot cylinder side. In some embodiments, the system further comprises a third conduit, wherein the third conduit comprises a third stream at a third temperature. In some embodiments, the third stream comprises a FPSE output stream. In some embodiments, the third conduit fluidically connects the FPSE to a heat exchanger. In some embodiments, the third stream comprises a temperature less than or equal to the first stream. In some embodiments, the system further comprises a fourth conduit, wherein the fourth conduit comprises a fourth stream at a fourth temperature. In some embodiments, the fourth conduit fluidically connects the heat exchanger to the FPSE. In some embodiments, the fourth temperature is equal to or less than the third temperature. In some embodiments, the fourth stream enters the FPSE at a cold cylinder side. The system of any of the preceding claims, further comprising a fifth conduit, wherein the fifth conduit comprises a fifth stream at a fifth temperature. In some embodiments, the fifth temperature is equal to or greater than the fourth temperature. In some embodiments, the fifth conduit fluidically connects the heat exchanger to a chiller. In some embodiments, the second conduit fluidically connects the chiller to the waste heat source production devices. In some embodiments, the second stream comprises a temperature equal to or less than the fourth stream. In some embodiments, the FPSE is configured to convert thermal energy from the first stream into mechanical energy, and then into electrical power. In some embodiments, the fourth stream is configured to create a temperature differential between the hot cylinder side and the cold cylinder side. In some embodiments, the second conduit is configured to fluidically connect the chiller to the waste heat source production devices at a side opposite to that of the first conduit. In some embodiments, the first conduit is configured to fluidically connect the waste heat source production devices to the heat exchanger at a side opposite the fifth conduit. In some embodiments, the third conduit is configured to fluidically connect the FPSE to the heat exchanger at a side opposite the fourth conduit. In some embodiments, the fourth conduit is configured to fluidically connect the FPSE to the heat exchanger. In some embodiments, a temperature differential between the fourth stream and the first stream generates a temperature differential between the hot cylinder side and the cold cylinder side. In some embodiments, the waste heat source production devices comprises one or more of electronic devices, and power supplies. In some embodiments, the system comprises a closed loop system. In some embodiments, the system comprises an open loop system.

[0142] In another aspect, disclosed herein are organic FPSE, the organic FPSE comprising a housing having a first end and a second end, wherein the first end and the second end are positioned along a displacement axis of the housing, and wherein the first end and the second end are separated by a travel length, a displacer positioned within the housing, wherein the displacer is reciprocally movable within the housing along the displacement axis and over at least a portion of the travel length, a piston positioned configured to apply a force to the displacer in a proximal direction of the housing, a heating head configured to add thermal energy to a working fluid, and a regenerator configured to recover and store thermal energy from a heated working fluid and transfer to a cooled working fluid, and at least one heat transfer apparatus having an organic topology. [0143] In some embodiments, the organic topology of the at least one heat transfer apparatus was generated by a topology algorithm, wherein the topology algorithm is configured to receive one or more parameters of the FPSE and generate the organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE. In some embodiments, the one or more parameters provided to the topology algorithm comprise engine architecture parameters. In some embodiments, the one or more parameters comprises one or more constraints. The organic FPSE of any preceding claim, further comprising a pressure vessel, wherein one or more parameters comprises an internal diameter of the pressure vessel Dpwall, wherein the Dpwall comprises about 2.621e-l [m]. In some embodiments, the one or more parameters comprises an internal diameter of the regenerator Dregen, wherein Dregen comprises about 1.747e-l [m]. In some embodiments, the one or more parameters comprises the FPSE operates at a mean charge pressure Pcharge, wherein Pcharge comprises between about 30 bar and about 80 bar. In some embodiments, the one or more parameters comprises a diameter of the piston Dpis, wherein Dpis comprises about 1.712e-l [m]. In some embodiments, the one or more parameters comprises an amplitude of the piston Xamp,pis, wherein Xamp,pis comprises about 7.953e-3 [m]. In some embodiments, the one or more parameters comprises a compression space volume Vcompression, wherein Vcompression comprises about 3.667e-4 [m^A3], In some embodiments, the optimized thermal efficiency (^thermal) of the FPSE comprises about 9.06%. In some embodiments, the optimized net power output (Wnet) of the FPSE comprises about 2000W. In some embodiments, the optimized heat input (Qin) of the FPSE comprises about 22060W. In some embodiments, the optimized heat output (Qout) of the FPSE comprises about 20060W. In some embodiments, the organic topology of the at least one heat transfer apparatus is substantially devoid of right angles. In some embodiments, the organic topology of the at least one heat transfer apparatus is substantially devoid of straight lines. In some embodiments, the organic topology of the at least one heat transfer apparatus comprises an amorphous shape. In some embodiments, the at least part of the organic topology comprises a form configured to encase a pressure vessel and for a heat transfer rate of up to about 0.67xthe max theoretical limit. In some embodiments, the FPSE improves a heat transfer rate (0.67/0.5 l)x that of a conventionally -finned FPSE.

[0144] In another aspect, disclosed herein are method of designing a heat transfer apparatus for a FPSE, the method comprising providing one or more parameters of the FPSE into a topology algorithm, wherein the topology algorithm is configured to generate an organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE.

[0145] In some embodiments, the topology algorithm is configured to generate the organic topology tailored to a specific application of the heat transfer apparatus. In some embodiments, the specific application of the heat transfer apparatus is in a waste heat source production system environment. In some embodiments, the method further comprises generating manufacturing instructions derived from the organic topology. In some embodiments, the method further comprises manufacturing the heat transfer apparatus based on the generated manufacturing instructions, wherein the architecture of the apparatus being guided by the organic topology. In some embodiments, the manufacturing comprises an additive manufacturing process. In some embodiments, the additive manufacturing process comprises Direct Metal Laser Sintering (DMLS). In some embodiments, the manufacturing comprises a subtractive manufacturing process. In some embodiments, the method further comprises discretizing a fluid domain into a plurality of components. In some embodiments, each component of the plurality of components comprises a control volume. In some embodiments, the method further comprises defining mass, temperature, and pressure for each control volume. In some embodiments, the boundaries (e.g., nodes) between each control volume are used to represent and determine the mass flow rate between each control volume. In some embodiments, the method further comprises solving of differential equations for conservation of mass, momentum, and energy within the discretized fluid domain. In some embodiments, the system of differential equations is solved numerically. In some embodiments, solving the differential equations provides a detailed prediction of FPSE's performance under specified operating conditions. In some embodiments, the solver, upon reaching a stabilized state, attains a maximum thermal efficiency of 9.06% for a heat source temperature of 80°C and a heatsink temperature of 5°C. In some embodiments, the FPSE comprises a Carnot efficiency of about 42.5%. In some embodiments, the topology algorithm iteratively adjusts the distribution of material within a predefined domain based on gradients, seeking the improved structure to meet the defined objective, such as maximizing heat dissipation or minimizing pressure loss, with the aid of adjoint solver capabilities. In some embodiments, the topology algorithm incorporates a one-dimensional, third-order modeling, simulation, and optimization of the FPSE. In some embodiments, the method further comprises discretizing an FPSE domain into a plurality of building blocks. In some embodiments, each building block represents elemental components of the FPSE such as heat exchangers, regenerators, and pistons. In some embodiments, each building block comprises a localized self-contained entity. In some embodiments, the entire FPSE model comprises a summation of each component building blocks interconnected via mass flow rate, heat transfer, force, and pressure connectors. In some embodiments, the power and displacer pistons are represented as rigid moving components that cause volume displacement in compression and expansion spaces. In some embodiments, all components of the FPSE including heat exchangers, pistons, and working spaces are incorporated in the model. In some embodiments, the method further comprises specifying an average operating pressure for the FPSE using a pressure source. In some embodiments, the method further comprises connecting the endpoints of the system to specified heat sources In some embodiments, the configuration is used to estimate non-productive energy losses (e.g., parasitic losses). In some embodiments, a total of 75 input parameters for 9 components and their subcomponents are defined. In some embodiments, the method further comprises specifying the optimization variables. In some embodiments, the optimization variables comprise one or more of an internal diameter of the pressure vessel, an internal diameter of the regenerator, an FPSE mean charge pressure, a piston diameter, a piston amplitude, a volume of compression space, a width, height, and length of cooling head channel, a width, height, and length of heating head channel, a regenerator length, a regenerator wrapped foil gap and thickness, an amplitude and spring stiffness of the displacer, and a volume of expansion space. In some embodiments, the optimization variables are coupled with constraints. In some embodiments, the constraints ensure geometric and thermodynamic viability of architecture and an objective function. In some embodiments, the thermal efficiency, defined as the ratio of net work output to heat input (^ thermal = w_net/Q_in), is considered as the objective function. In some embodiments, the method further comprises ensuring a regenerator's diameter is larger than a diameter of the displacer rod. In some embodiments, the method further comprises ensuring a diameter of the pressure vessel is larger than the diameter of the regenerator. In some embodiments, the method further comprises ensuring that the form factor is maintained at a reasonable level, such that the flow distribution losses are minimal. In some embodiments, the method further comprises ensuring a diameter of the piston is larger than the diameter of the displacer rod. In some embodiments, the method further comprises ensuring a diameter of the piston has about a same diameter as the regenerator. In some embodiments, the method further comprises ensuring a sufficient volume in the dead space of the compression region to prevent any collision. In some embodiments, the method further comprises ensuring a sufficient volume in the dead space of the expansion region to prevent any collision. In some embodiments, the method further comprises ensuring a displacer operates freely by maintaining the components of a phasor force at zero. The method of any of the preceding claims, further ensuring a required power output is obtained from the FPSE. In some embodiments, the method further comprises running the one-dimensional code to simulate the energy generation over a 48-hour period, resulting in a total energy production of 98kWh, under the assumption of fixed heat source and sink temperatures and availability of heat source for 24 hours per day. In some embodiments, the discretization further comprises using polyhedral meshing to create conformal mesh interfaces between the parts In some embodiments, the contacting faces between different parts share a same boundary face topology. In some embodiments, the method further comprises performing simulations using four different base cell sizes to ensure mesh size sensitivity and check for mesh convergence, resultingin more accurate and faster simulations due to the elimination of the need for face interpolation on contacting patches. In some embodiments, the method further comprises establishing five layers on interfaces between air and solid components to accurately capture thermal boundary layer. In some embodiments, the method further comprises identifying four distinct simulation domains in the CAD geometry, wherein the four distinct simulation domains comprises specifically the stainless steel enclosure, air, copper fins, and an Inconel half-cylinder. In some embodiments, the method further comprises assigning each region a specific simulation model based on the material. In some embodiments, the boundary conditions are set such that the enclosure has adiabatic walls. In some embodiments, the inner wall of a half cylinder has a convective heat transfer coefficient of 860 W/m²K at 300°C. In some embodiments, the inlets for each pipe are set with parameters such as: Mass flow rate of 0.003184 kg/s, Temperature of 650°C, and Pressure of 101,325 Pa. In some embodiments, the outlets are set as pressure outlets. In some embodiments, the at least one enclosure surface comprises a roughness of about 0.05 mm. In some embodiments, the average air temperature within the system is calculated to be about 502.3°C. In some embodiments, a maximum theoretical heat transfer value comprises about 3566W. In some embodiments, the 1830W is extracted from the hot air to the fins. In some embodiments, the topology algorithm utilizes gradient-based Topology (TO) for heat transfer and pressure optimization. In some embodiments, the TO process begins by defining the architecture space envelope as well as the flow/thermal objective functions together with TO input parameters. In some embodiments, the TO process utilizes conformal mesh to ensure conservation of heat transfer values. In some embodiments, the surface mesh between all parts on the interface shares points at the interface to conserve heat transfer between parts. In some embodiments, the smaller mesh cells result in finer fin surface. In some embodiments, the smaller mesh cells result in larger fin surface area. In some embodiments, the method further comprises geometry preparation for full CFD simulation with Boundary Layers (BL). In some embodiments, the geometry preparation comprises smoothing out initial derived part geometry, using a surface wrapper to create a watertight surface, importing the watertight surface of the optimized geometry to the full case, subtracting the watertight surface from the air domain, imprinting it to the half cylinder geometry, and ensuring all required interfaces are created properly. In some embodiments, the jagged geometry obtained from TO is smoothed out and then re-imported back for CFD simulation for validation and verification purposes.

[0146] In another aspect, disclosed herein are methods of designing a heat transfer apparatus for a FPSE to be integrated into a waste heat source production system, the method comprising providing one or more parameters of the FPSE into a one-dimensional (ID) Computational Fluid Dynamics (CFD) model, providing at least one output parameter of the ID CFD model into a three-dimensional (3D) Computational Fluid Dynamics (CFD) model, providing at least one output parameter of the 3D CFD model into a topology algorithm, and generating the heat transfer apparatus comprising an organic topology tailored for a waste heat source production system.

[0147] In another aspect, disclosed herein are organic FPSE for a waste heat source production system environment, the organic FPSE comprising a housing having a first end and a second end, wherein the first end and the second end are positioned along a displacement axis of the housing, and wherein the first end and the second end are separated by a travel length, a displacer positioned within the housing, wherein the displacer is reciprocally movable within the housing along the displacement axis and over at least a portion of the travel length, a piston positioned configured to apply a force to the displacer in a proximal direction of the housing, a heating head configured to receive thermal energy from an waste heat source and add thermal energy to a working fluid, and a regenerator configured to recover and store thermal energy from a heated working fluid and transfer to a cooled working fluid, and at least one heat transfer apparatus having an organic topology tailored for the waste heat source production system environment.

[0148] In another aspect, disclosed herein are method of designing a heat transfer apparatus for a FPSE to be integrated into a waste heat source production system, the method comprising providing one or more parameters of the FPSE into a topology algorithm, wherein the topology algorithm is configured to generate an organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE, and generating the organic topology tailored for the waste heat source production system.

[0149] In another aspect disclosure herein are waste heat source management systems for a commercial building, comprising: a heat exchanger, an evaporator in fluidic connection with the heat exchanger, a condenser in fluidic connection with the heat exchanger, a first conduit comprising a first stream, wherein the first stream comprises a first temperature, a second conduit comprising a second stream, wherein the second stream comprises a second temperature, and a third conduit comprising a third stream, wherein the third stream comprises a third temperature, and a Free Piston Stirling Engine (FPSE).

[0150] In some embodiments, the system further comprises the third conduit thermally connected to the first conduit. In some embodiments, the system further comprises the third conduit fluidically connected to a heating head of the FPSE. In some embodiments, the system further comprises the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the system further comprises a cooling head of the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the first stream comprises an HVAC refrigerant stream. In some embodiments, the second stream comprises a cooling fluid. In some embodiments, the third stream comprises a heating fluid. In some embodiments, a temperature of the first stream is equal to or greater than a temperature of the third stream. In some embodiments, a temperature of the third stream is equal to or greater than a temperature of the second stream. In some embodiments, the first stream comprises an output stream from the HVAC unit. In some embodiments, the first stream comprises an output stream from the evaporator. In some embodiments, the first stream comprises an input stream to the heat exchanger. In some embodiments, the first stream comprises an output stream from the heat exchanger. In some embodiments, the first stream comprises an input stream to the condenser. In some embodiments, the first stream comprises an output stream from the condenser. In some embodiments, the first stream comprises an input stream to the evaporator. In some embodiments, the first stream comprises an output stream from the evaporator. In some embodiments, the first stream comprises an input stream to a heat exchanger. In some embodiments, the second stream comprises an output stream from a heat sink. In some embodiments, the second stream comprises an input stream to a cooling head of the FPSE. In some embodiments, the second stream comprises an output stream from a cooling head of the FPSE. In some embodiments, the second stream comprises an input stream to a heat sink. In some embodiments, the second stream comprises an output stream from a heat sink. In some embodiments, the third stream comprises an output stream from a heating head of a FPSE. In some embodiments, the third stream comprises an input stream to the heat exchanger. In some embodiments, the third stream comprises an output stream from the heat exchanger. In some embodiments, the third stream comprises an input stream to the heating head of the FPSE. In some embodiments, the second stream enters the FPSE at a cold cylinder side. In some embodiments, the second stream removes heat from the cold cylinder side. In some embodiments, the third stream enters the FPSE at a hot cylinder side. In some embodiments, the third stream transfers heat to the hot cylinder side. In some embodiments, the FPSE is configured to convert thermal energy from the third stream into mechanical energy, and then into electrical power. In some embodiments, the system further comprises a temperature differential between the hot cylinder side and the cold cylinder side. In some embodiments, the first conduit is configured to fluidically connect the condenser to the evaporator and to the heat exchanger. In some embodiments, the second conduit is configured to fluidically connect the heat sink to the cooling head. In some embodiments, the third conduit is configured to fluidically connect the heat exchanger to the heating head. In some embodiments, the system comprises a closed loop system. In some embodiments, the system comprises an open loop system. In some embodiments, the second stream comprises water. In some embodiments, the second stream comprises air. In some embodiments, the WHRS system is placed outside a commercial building. In some embodiments, the WHRS system is placed on top of a commercial building. In some embodiments, the WHRS system is placed inside a commercial building. In some embodiments, the WHRS system is fluidically connected to a commercial building. In some embodiments, the WHRS system is thermally connected to a commercial building.

Heat Recovery System

[0151] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a heat recovery system (HRS) configured to provide various benefits, including generating electricity from low-grade heat sources (e.g., below 230°C), enhancing energy efficiency of pre-existing energy conversion systems (e.g., residential, commercial, industrial), and reducing greenhouse gas emissions.

[0152] The HRS may comprise a HRS efficiency up to about about 23%.

[0153] In some cases, the formula for calculating the HRS efficiency comprises equation 1 : (Equation 1) [0154] In equation 1, “T|” comprises HRS efficiency, “I” comprises Input Energy in Waste heat source, and “O” comprises Output from Recovered Heat.

[0155] In some cases, the HRS efficiency comprises between about 0% to about 100%. In some cases, the HRS efficiency comprises between about 0% to about 5%, about 0% to about 10%, about 0% to about 15%, about 0% to about 20%, about 0% to about 25%, about 0% to about 30%, about 0% to about 35%, about 0% to about 40%, about 0% to about 45%, about 0% to about 50%, about 0% to about 100%, about 5% to about 10%, about 5% to about 15%, about 5% to about 20%, about 5% to about 25%, about 5% to about 30%, about 5% to about 35%, about 5% to about 40%, about 5% to about 45%, about 5% to about 50%, about 5% to about 100%, about 10% to about 15%, about 10% to about 20%, about 10% to about 25%, about 10% to about 30%, about 10% to about 35%, about 10% to about 40%, about 10% to about 45%, about 10% to about 50%, about 10% to about 100%, about 15% to about 20%, about 15% to about 25%, about 15% to about 30%, about 15% to about 35%, about 15% to about 40%, about 15% to about 45%, about 15% to about 50%, about 15% to about 100%, about 20% to about 25%, about 20% to about 30%, about 20% to about 35%, about 20% to about 40%, about20% to about 45%, about 20% to about 50%, about 20% to about 100%, about 25% to about 30%, about 25% to about 35%, about 25% to about 40%, about 25% to about 45%, about 25% to about 50%, about 25% to about 100%, about 30% to about 35%, about 30% to about 40%, about 30% to about 45%, about 30% to about 50%, about 30% to about 100%, about 35%to about 40%, about 35% to about 45%, about 35% to about 50%, about 35% to about 100%, about40% to about45%, about 40% to about 50%, about 40% to about 100%, about 45% to about 50%, about 45% to about 100%, or about 50% to about 100%. In some cases, the HRS efficiency comprises between about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%. In some cases, the HRS efficiency comprises between at least about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, or about 50%. In some cases, the HRS efficiency comprises between at most about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%.

[0156] In some cases, the HRS efficiency comprises about up to about 23% efficiency. In some cases, the HRS efficiency comprises aboutup to about 30% efficiency. In some cases, the HRS efficiency comprises about up to about 45% efficiency. In some cases, the HRS efficiency comprises about up to about 60% efficiency. For example, the input energy in waste heat source may be from an waste heat source. In further examples, the waste heat source comprises waste heat source gases, hot liquids, or other low-temperature heat streams. In even further examples, the HRS is configured to be heated by waste heat source from a waste heat source production system.

[0157] In some cases, the HRS efficiency is dependent on a feature of an waste heat source. In some instances, the HRS efficiency may be dependent on an waste heat source temperature and flow rate.

[0158] In some instances, the HRS efficiency comprises up to about 23% wherein the waste heat source comprises a temperature up to about 500°C. In some instances, the HRS efficiency comprises up to about 40% wherein the waste heat source comprises a temperature up to about 500°C. In some instances, the HRS efficiency comprises between about 7.5% to about 10%, wherein a heat source temperature comprises about 80°C and a heat sink temperature comprises about 5°C. For example, the HRS efficiency may comprises between about 7.5% and 10%, wherein the waste heat source comprises a temperature of up to about 80°C and the environment comprises a temperature of up to about 5°C.

[0159] In some cases, the HRS efficiency is dependent on a temperature differential between an waste heat source and an environment. In some instances, the HRS efficiency comprises between about 1% to about 50%, wherein the waste heat source comprises a temperature of up to about 500°C and the environment comprises a temperature of between about -90°C to about 60°C.

[0160] In some cases, the HRS efficiency is dependent on the heat exchanger efficiency. In some instances, the HRS efficiency comprises between about 20% and 40%, wherein the heat exchanger efficiency comprises between about 50% and 90%.

[0161] In some cases, the HRS efficiency is dependent on a size of an waste heat source generator (e.g., building or structure that produces the waste heat source). In some instances, the HRS efficiency comprises between about 1% to about 50% wherein the waste heat source generator comprises an industrial building (e.g., up to about 5 million sq. ft.). For example, the HRS efficiency may comprise up to about 45% efficiency wherein the waste heat source generator comprises an industrial building (e.g., up to about 5 million sq. ft.). In further examples, the HRS efficiency may comprise about 23% efficiency, wherein the waste heat source generator comprises an industrial building.

[0162] In some instances, the HRS efficiency comprises between about 1% to about 50% wherein the HRS comprises a domestic or residential building (e.g., up to about 2 million sq. ft.). For example, the HRS efficiency may comprise up to about 45% efficiency wherein the waste heat source generator comprises a domestic or residential building. In further examples, the HRS efficiency may comprise 23% efficiency, wherein the waste heat source generator comprises a domestic or residential building.

[0163] The HRS may comprise a HRS energy production capability of up to about 250 kilowatt hours/day (kWh/day). In some cases, the HRS energy production capability comprises between about 10 - 40 kilowatt hours/day. In some cases, the HRS energy production capability comprises an average of about 30 kWh/day. In some cases, the HRS energy production capability comprises up to about 8 kWh/day, wherein the waste heat source is available for 8 hours/day and a FPSE produces up to about 1 kW of power. In some cases, the HRS energy production capability comprises up to about 6000 kWh/day, wherein the waste heat source is available for 24 hours/day and a FPSE produces about 250 kW of power.

[0164] In some cases, the HRS energy production capability comprises between about 0 kWh/day to about 6,000 kWh/day. In some cases, the HRS energy production capability comprises between about 0 kWh/day to about 4 kWh/day, about 0 kWh/day to about 8 kWh/day, about 0 kWh/day to about 10 kWh/day, about 0 kWh/day to about 20 kWh/day, about 0 kWh/day to about 30 kWh/day, about 0 kWh/day to about 60 kWh/day, about 0 kWh/day to about 100 kWh/day, about 0 kWh/day to about 1 ,000 kWh/day, about 0 kWh/day to about 2,600 kWh/day, about 0 kWh/day to about 4,200 kWh/day, about 0 kWh/day to about 6,000 kWh/day, about 4 kWh/day to about 8 kWh/day, about 4 kWh/day to about 10 kWh/day, about 4 kWh/day to about 20 kWh/day, about 4 kWh/day to about 30 kWh/day, about 4 kWh/day to about 60 kWh/day, about 4 kWh/day to about 100 kWh/day, about 4 kWh/day to about 1,000 kWh/day, about 4 kWh/day to about 2,600 kWh/day, about 4 kWh/day to about 4,200 kWh/day, about 4 kWh/day to about 6,000 kWh/day, about 8 kWh/day to about 10 kWh/day, about 8 kWh/day to about 20 kWh/day, about 8 kWh/day to about 30 kWh/day, about 8 kWh/day to about 60 kWh/day, about 8 kWh/day to about 100 kWh/day, about 8 kWh/day to about 1,000 kWh/day, about 8 kWh/day to about 2,600 kWh/day, about 8 kWh/day to about 4,200 kWh/day, about 8 kWh/day to about 6,000 kWh/day, about 10 kWh/day to about 20 kWh/day, about 10 kWh/day to about 30 kWh/day, about 10 kWh/day to about 60 kWh/day, about 10 kWh/day to about 100 kWh/day, about 10 kWh/day to about 1,000 kWh/day, about 10 kWh/day to about 2,600 kWh/day, about 10 kWh/day to about 4,200 kWh/day, about 10 kWh/day to about 6,000 kWh/day, about 20 kWh/day to about 30 kWh/day, about 20 kWh/day to about 60 kWh/day, about 20 kWh/day to about 100 kWh/day, about 20 kWh/day to about 1,000 kWh/day, about 20 kWh/day to about 2,600 kWh/day, about 20 kWh/day to about 4,200 kWh/day, about 20 kWh/day to about 6,000 kWh/day, about 30 kWh/day to about 60 kWh/day, about 30 kWh/day to about

100 kWh/day, about 30 kWh/day to about 1,000 kWh/day, about 30 kWh/day to about 2,600 kWh/day, about 30 kWh/day to about 4,200 kWh/day, about 30 kWh/day to about 6,000 kWh/day, about 60 kWh/day to about 100 kWh/day, about 60 kWh/day to about 1,000 kWh/day, about 60 kWh/day to about 2,600 kWh/day, about 60 kWh/day to about 4,200 kWh/day, about 60 kWh/day to about 6,000 kWh/day, about 100 kWh/day to about 1,000 kWh/day, about 100 kWh/day to about 2,600 kWh/day, about 100 kWh/day to about 4,200 kWh/day, about 100 kWh/day to about 6,000 kWh/day, about 1,000 kWh/day to about 2,600 kWh/day, about 1,000 kWh/day to about 4,200 kWh/day, about 1,000 kWh/day to about 6,000 kWh/day, about 2,600 kWh/day to about 4,200 kWh/day, about 2,600 kWh/day to about 6,000 kWh/day, or about 4,200 kWh/day to about 6,000 kWh/day. In some cases, the HRS energy production capability comprises between about 0 kWh/day, about 4 kWh/day, about 8 kWh/day, about 10 kWh/day, about 20 kWh/day, about 30 kWh/day, about 60 kWh/day, about 100 kWh/day, about 1,000 kWh/day, about 2,600 kWh/day, about 4,200 kWh/day, or about 6,000 kWh/day. In some cases, the HRS energy production capability comprises between about at least about 0 kWh/day, about 4 kWh/day, about 8 kWh/day, about 10 kWh/day, about 20 kWh/day, about 30 kWh/day, about 60 kWh/day, about 100 kWh/day, about 1,000 kWh/day, about 2,600 kWh/day, or about 4,200 kWh/day. In some cases, the HRS energy production capability comprises between about at most about 4 kWh/day, about 8 kWh/day, about 10 kWh/day, about 20 kWh/day, about 30 kWh/day, about 60 kWh/day, about 100 kWh/day, about 1,000 kWh/day, about 2,600 kWh/day, about 4,200 kWh/day, or about 6,000 kWh/day.

[0165] The systems disclosed herein may improve over systems in the art by providing, in some cases, a Free Piston Stirling Engine (FSPE) configured to provide various benefits, including integration into a system where waste heat source is generated and extraction of heat from various waste heat source (e.g., waste heat source gases, hot liquids, or other low- temperature heat streams), high efficiency, low maintenance, low noise, small space occupation, easily integrate, high power, fuel flexibility, cost saving, durability, regulatory and incentive opportunities, and scalable.

[0166] In some cases, the FPSE may be specifically configured for waste heat source recovery from low temperature heat sources (e.g., below 230°C). In some cases, the FPSE

101 is configured for waste heat source recovery in domestic applications (e.g., waste heat source production systems). [0167] FIG. 1 shows a Free Piston Stirling Engine (FPSE). In FIG. 1, an exemplary FPSE 101 is shown. The FPSE 101 may comprise an improved thermodynamic heat engine 101. In some cases, the FPSE 101 comprises a first heat exchanger 102 and a second heat exchanger 103.

[0168] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a FPSE configured to provide various benefits, including advanced materials with a maximized thermal conductivity, and maximized durability.

[0169] In some cases, the advanced materials with a maximized thermal conductivity comprise nano-structured substances. In some instances, a surface treatment may comprise the nano-structed substance. In some instances, the nano-structured substance may comprise A1₂O₃.

[0170] In some cases, the nano-structured substance may be introduced at an inlet to an enclosure. In some instances, the inlet to the enclosure may comprise an waste heat source at an entry to a hot cylinder. In some instances, the nano-structured substance may be configured to increase heat transfer with a topology optimized heat exchanger. For example, a magnetic filter may separate that nanoparticle at the discharge of enclosure wherein the waste heat source exits the enclosure. In further examples, the nanofluid may be reused for a new cycle.

[0171] In some instances, the nano-structures substances comprise a thermal conductivity

Watts between about 0.15 W/mKs meter x Kelvin ) and about 400 W/mK at room temperature and room pressure. In some instances, the nano-structures substances comprise a thermal conductivity of greater than about 400 W/mK at room temperature and room pressure. [0172] In some cases, the advanced materials with a maximized thermal conductivity comprise composites. In some cases, the advanced materials with a high thermal conductivity comprise alloys. In some cases, the advanced materials comprise a thermal conductivity of up to about 400 W/mK

[0173] FIG. 2 shows a Free Piston Stirling Engine (FPSE). In FIG. 2, an FPSE 200 is shown. The FPSE 200 may comprise an improved thermodynamic heat engine 200. The FPSE 200 may comprise a housing 209. In some cases, the housing 209 comprises a first end 207. In some cases, the housing 209 comprises a second end 208. In some cases, the first end 207 of the housing 209 comprises a location vertically adjacent the second end 208 of the housing 209. In some cases, the first end 207 of the housing 209 comprises a location parallel adjacent to the second end 208 of the housing 209. In some cases, the first end 207 of the housing 209 comprises a location horizontally adjacent to the second end 208 of the housing 209.

[0174] The FPSE 200 may comprise a middle section of the housing 209. In some cases, the middle section of the housing 209 comprises a location positioned between the first end 207 of the housing 209 and the second end 208 of the housing. In some cases, the middle section of the housing 209 comprises a location vertically adjacent to both the first end 207 of the housing 209 and the second end 208 of the housing. In some cases, the middle section of the housing 209 comprises a location parallel adjacent to both the first end 207 of the housing 209 and the second end 208 of the housing. In some cases, the middle section of the housing 209 comprises a location horizontally adjacent to both the first end 207 of the housing 209 and the second end 208 of the housing.

[0175] The FPSE 200 may comprise a displacer 206 positioned within the housing 209. In some cases, the first end 207 of the housing 209 and the second end 208 of the housing 209 may be positioned along a displacement axis 210 of the housing 209. In some instances, the first end 207 and the second end 208 are separated by a travel length. In some cases, the displacer 206 is reciprocally movable within the housing 209 along the displacement axis 210 and over at least a portion of the travel length. In some instances, the displacement axis 210 may comprise a travel length of the displacer. In further examples, the first end 207 and the second end 208 are separated by the travel length of the displacer.

[0176] The FPSE 200 may comprise a heating head 203 disposed within the housing 209. In some cases, the first end 207 of the housing 209 comprises the heating head 203. In some cases, the second end of the housing 209 comprises the heating head 203. In some cases, a middle section of the housing 209 comprises the heating head 203.

[0177] The FPSE 200 may comprise a regenerator 202 disposed within the housing 209. In some cases, the middle section of the housing 209 comprises the regenerator 202. In some cases, the first end 207 ofthe housing 209 comprises the regenerator 202. In some cases, the second end 208 of the housing 209 comprises the regenerator 202.

[0178] The FPSE 200 may comprise a power piston 211 disposed within the housing 209. In some cases, the power piston 211 is positioned to apply a force to the displacer 206 in a proximal direction ofthe housing 209. In some cases, the second end 208 of the housing 209 comprises the power piston 211. In some cases, the first end 207 of the housing 209 comprises the power piston 211. In some cases, the middle section of the housing 209 comprises the power piston 211. [0179] The FPSE 200 may comprise a cooling head 201 disposed within the housing 209. In some cases, the second end 208 of the housing 209 comprisesthe cooling head 211. In some cases, the first 207 of the housing 209 comprises the cooling head 211. In some cases, the middle section of the housing 209 comprises the cooling head 211.

[0180] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a Free Piston Stirling Engine (FPSE) configured to provide various benefits, including an ability to operate at low temperature differentials (e.g., less than about 30°C), utilize waste heat source from low grade temperature steam (e.g., providing flexibility in heat stream selection), maximize efficiency at low temperature differentials (e.g., between heat source and heat sink), minimize thermal stress on engine components, and minimize operating costs.

[0181] The FPSE may be configured to operate under low-temperature differentials between a heat source and a heat sink. In some cases, the heat source comprises a heating head. In some instances, the heating head is configured to receive heat from an waste heat source. For example, the waste heat source may comprise an waste heat source (e.g., from a waste heat source production system). In even further examples, the temperature of the heating head may comprise about the temperature of the waste heat source (e.g., below about 80°C). [0182] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a FPSE configured to provide various benefits, including operating at low temperatures (e.g., below 230°C).

[0183] FIG. 8 illustrates an example of a Free Piston Stirling Engine. In the example of FIG. 8, the FPSE 800 comprises a hot cylinder 807. The hot cylinder 807 may comprise a heated working fluid. In some cases, the working fluid has been heated by a heating head 801. In some instances, the working fluid in the hot cylinder 807 comprises a temperature of between about 75°C to about 80°C. In some instances, the working fluid in the hot cylinder 807 comprises a temperature of up to about 75 °C. In some instances, the working fluid in the hot cylinder 807 comprises a temperature of greater than about 75°C. For example, the working fluid in the hot cylinder 807 comprises a temperature of up to about 230°C.

[0184] In the example of FIG. 8, the FPSE 800 comprises a cold cylinder 806. The cold cylinder 806 may comprise a cooled working fluid. In some instances, the cooled working fluid has been cooled by a cooling head 802. In some instances, the working fluid in the cold cylinder 806 comprises a temperature of between about 5°C to about 12°C.

[0185] In some instances, the working fluid in the cold cylinder 806 comprises a temperature of about 10°C. In some instances, the working fluid in the cold cylinder 806 comprises a temperature of up to about 10°C. In some instances, the working fluid in the cold cylinder 806 comprises a temperature of greater than about 10°C.

[0186] In the example of FIG. 8, the FPSE 800 comprises a regenerator 803. The regenerator 803 may be configured to release heat or remove heat from a working fluid. In some instances, the regenerator comprises a temperature of between about 40°C to about 60°C. In some instances, the regenerator comprises a temperature of up to about 60°C. In some instances, the regenerator comprises a temperature of greater than 60°C.

[0187] In some cases, the heat sink comprises a cooling head. In some instances, the cooling head comprises a water jacket. In some instances, the cooling head is configured to release heat to an environment. In even further examples, the temperature of the cooling head may comprise a temperature up to about 15 °C. In even further examples, the temperature of the cooling head may comprise a temperature of between about 5°C to about 7°C. In even further examples, the temperature of the cooling head may comprise a temperature of between about 12°C to about 15°C.

[0188] In some cases, the FPSE is configured to operate under a temperature differential of less than about 500°C. In some cases, the FPSE is configured to operate under a temperature differential of less than about 60°C. In some cases, the FPSE is configured to operate under a temperature differential of less than about 30°C.

[0189] In some cases, the FPSE is configured to operate under a temperature differential between about 0°C to about 500°C. In some cases, the FPSE is configured to operate under a temperature differential between about 0°C to about 10°C, about 0°C to about 20°C, about 0°C to about 30°C, about 0°C to about 40°C, about 0°C to about 50°C, about 0°C to about 60°C, about 0°C to about 70°C, about 0°C to about 80°C, about 0°C to about 100°C, about 0°C to about 250°C, about 0°C to about 500°C, about 10°C to about 20°C, about 10°C to about 30°C, about 10°C to about 40°C, about 10°C to about 50°C, about 10°C to about 60°C, about 10°C to about 70°C, about 10°C to about 80°C, about 10°C to about 100°C, about 10°C to about 250°C, about l0°C to about 500°C, about 20°C to about 30°C, about 20°C to about 40°C, about 20°C to about 50°C, about 20°C to about 60°C, about 20°C to about 70°C, about 20°C to about 80°C, about 20°C to about 100°C, about 20°C to about 250°C, about 20°C to about 500°C, about 30°C to about 40°C, about 30°C to about 50°C, about 30°C to about 60°C, about 30°C to about 70°C, about 30°C to about 80°C, about 30°C to about 100°C, about 30°C to about 250°C, about 30°C to about 500°C, about 40°C to about 50°C, about 40°C to about 60°C, about 40°C to about 70°C, about 40°C to about 80°C, about 40°C to about 100°C, about 40°C to about 250°C, about 40°C to about 500°C, about 50°C to about 60°C, about 50°C to about 70°C, about 50°C to about 80°C, about 50°C to about 100°C, about 50°C to about 250°C, about 50°C to about 500°C, about 60°C to about 70°C, about 60°C to about 80°C, about 60°C to about 100°C, about 60°C to about 250°C, about 60°C to about 500°C, about 70°C to about 80°C, about 70°C to about 100°C, about 70°C to about 250°C, about 70°C to about 500°C, about 80°C to about 100°C, about 80°C to about 250°C, about 80°C to about 500°C, about 100°C to about 250°C, about 100°C to about 500°C, or about 250°C to about 500°C. In some cases, the FPSE is configured to operate under a temperature differential between about 0°C, about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, about 250°C, or about 500°C. In some cases, the FPSE is configured to operate under a temperature differential between at least about 0°C, about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, or about 250°C. In some cases, the FPSE is configured to operate under a temperature differential between at most about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, about 250°C, or about 500°C.

[0190] The FPSE may be configured to maintain up to about a 45% efficiency at low temperature differentials. In some cases, the FPSE is configured to maintain up to about 45% efficiency at temperature differentials between about 0°C to about 500°C. In some cases, the FPSE is configured to maintain up to about 45% efficiency at temperature differentials between about 0°C to about 10°C, about 0°C to about 20°C, about 0°C to about 30°C, about 0°C to about 40°C, about 0°C to about 50°C, about 0°C to about 60°C, about 0°C to about 70°C, about 0°C to about 80°C, about 0°C to about 100°C, about 0°C to about 250°C, about 0°C to about 500°C, about 10°C to about 20°C, about 10°C to about 30°C, about 10°C to about 40°C, about 10°C to about 50°C, about 10°C to about 60°C, about 10°C to about 70°C, about 10°C to about 80°C, about 10°C to about 100°C, about 10°C to about 250°C, about 10°C to about 500°C, about 20°C to about 30°C, about 20°C to about 40°C, about 20°C to about 50°C, about 20°C to about 60°C, about 20°C to about 70°C, about 20°C to about 80°C, about 20°C to about 100°C, about 20°C to about 250°C, about 20°C to about 500°C, about 30°C to about 40°C, about 30°C to about 50°C, about 30°C to about 60°C, about 30°C to about 70°C, about 30°C to about 80°C, about 30°C to about 100°C, about 30°C to about 250°C, about 30°C to about 500°C, about 40°C to about 50°C, about 40°C to about 60°C, about 40°C to about 70°C, about 40°C to about 80°C, about 40°C to about 100°C, about 40°C to about 250°C, about 40°C to about 500°C, about 50°C to about 60°C, about 50°C to about 70°C, about 50°C to about 80°C, about 50°C to about 100°C, about 50°C to about 250°C, about 50°C to about 500°C, about 60°C to about 70°C, about 60°C to about 80°C, about 60°C to about 100°C, about 60°C to about 250°C, about 60°C to about 500°C, about 70°C to about 80°C, about 70°C to about 100°C, about 70°C to about 250°C, about 70°C to about 500°C, about 80°C to about 100°C, about 80°C to about 250°C, about 80°C to about 500°C, about 100°C to about 250°C, about 100°C to about 500°C, or about 250°C to about 500°C. In some cases, the FPSE is configured to maintain up to about a 60% efficiency at temperature differentials between about 0°C, about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, about 250°C, or about 500°C. In some cases, the FPSE is configured to maintain up to about a 60% efficiency at temperature differentials between at least about 0°C, about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, or about 250°C. In some cases, the FPSE is configured to maintain up to about a 60% efficiency at temperature differentials between at most about 10°C, about 20°C, about 30°C, about 40°C, about 50°C, about 60°C, about 70°C, about 80°C, about 100°C, about 250°C, or about 500°C.

[0191] In some cases, the FPSE efficiency comprises between about 0% to about 100%. In some cases, the FPSE efficiency comprises between about 0% to about 5%, about 0% to about 10%, about 0% to about 15%, about 0% to about 20%, about 0% to about 25%, about 0% to about 30%, about 0% to about 35%, about 0% to about 40%, about 0% to about 45%, about 0% to about 50%, about 0% to about 100%, about 5% to about 10%, about 5% to about 15%, about 5% to about 20%, about 5% to about 25%, about 5% to about 30%, about 5% to about 35%, about 5% to about 40%, about 5% to about 45%, about 5% to about 50%, about 5% to about 100%, about 10% to about 15%, about 10% to about 20%, about 10% to about 25%, about 10% to about 30%, about 10% to about 35%, about 10% to about 40%, about 10% to about45%, about 10% to about 50%, about 10% to about 100%, about 15% to about 20%, about 15% to about 25%, about 15% to about 30%, about 15% to about 35%, about 15% to about 40%, about 15% to about 45%, about 15% to about 50%, about 15% to about 100%, about 20% to about 25%, about 20% to about 30%, about 20% to about 35%, about 20% to about 40%, about 20% to about 45%, about 20% to about 50%, about 20% to about 100%, about 25% to about 30%, about 25% to about 35%, about 25% to about 40%, about 25% to about 45%, about 25% to about 50%, about 25% to about 100%, about 30% to about 35%, about 30% to about 40%, about 30% to about 45%, about 30% to about 50%, about 30% to about 100%, about 35% to about 40%, about 35% to about 45%, about 35% to about 50%, about 35% to about 100%, about 40% to about 45%, about 40% to about 50%, about 40% to about 100%, about 45% to about 50%, about 45% to about 100%, or about 50% to about 100%. In some cases, the FPSE efficiency comprises between about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%. In some cases, the FPSE efficiency comprises between at least about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, or about 50%. In some cases, the FPSE efficiency comprises between at most about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%.

[0192] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a heat exchanger which has been configured based on generative topology optimization configured to provide various benefits, including uniform heat flux distribution across the hot and cold cylinders for a maximized efficiency heat transfer process with minimum pressure drop. In some cases, the heat exchangers comprise a architecture configured for improved efficiency of heat transfer process at low temperature differentials (e.g., below 30°C). In some cases, the FPSE comprises a heat exchanger. In some cases, the FPSE comprises a plurality of heat exchangers.

[0193] FIG. 2 illustrates an example of a Free Piston Stirling Engine. The FPSE 200 may comprise a first heat exchanger 205. In some cases, the first heat exchanger 205 may comprise a heating head 203 and a plurality of fins. In some cases, the first heat exchanger 205 may comprise at least of portion of displacer 206. In some cases, the at least a portion of displacer 206 may comprise a hot tip. In some cases, the heating head 203 is positioned at the second end 208 of the housing 209. In some cases, the heating head 203 is positioned adjacent to the displacer 206. For example, the heating head 203 may be positioned adjacent to a hot cylinder. In further examples, the heating head 203 may be positioned adjacent to a hot tip.

[0194] The FPSE 200 may comprise a second heat exchanger 204. In some cases, the second heat exchanger 204 may comprise a cooling head 201 and a plurality of fins. In some cases, the second heat exchanger 205 may comprise at least of portion of displacer 206. In some cases, the at least a portion of displacer 206 may comprise a cold tip or cold cylinder. In some cases, the cooling head 201 is positioned at the first end 207 of the housing 209. In some cases, the cooling head 201 is positioned adjacent to the displacer 206. For example, the cooling head 201 may be positioned adjacent to a cold tip. In further examples, the cooling head 201 may be positioned adjacent to a cold cylinder.

[0195] The FPSE 200 may comprise a third heat exchanger 202. In some cases, the third heat exchanger 202 may comprise a regenerator 202 and a plurality of fins. In some cases, the third heat exchanger 202 may comprise at least of portion of displacer 206. In some cases, at least the portion of displacer 206 may comprise an intermediate displacer section. For example, the intermediate displacer section may comprise a position between the cold tip and the hot tip.

[0196] In some cases, the regenerator 202 is positioned between a first end 207 of the housing 209 and a second end 208 of housing 209. In some cases, the regenerator 202 is positioned adjacent to the displacer 206. For example, the regenerator 202 maybe positioned adjacent to the intermediate displacer section. In some cases, the regenerator 202 is positioned between the first heat exchanger 205 and the second heat exchanger 204. In some instances, the regenerator 202 is positioned between the heating head 203 and the cooling head 201.

[0197] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a heat transfer process configured to provide various benefits, including efficiently managing thermal gradients in the FPSE hot cylinder, cold cylinder, and regenerator.

[0198] Thermal gradient may comprise a rate of change of temperature within a component of the FPSE. In some cases, thermal gradient may comprise a rate of change of temperature within the hot cylinder. In some cases, thermal gradient may comprise a rate of change of temperature within the cold cylinder. In some cases, thermal gradient may comprise a rate of change of temperature within the regenerator.

[0199] In some instances, the hot cylinder comprises the area wherein the working fluid is heated. In some instances, the cold cylinder comprises the area wherein the working fluid is cooled. In some instances, the regenerator comprises the area wherein the working fluid travels from the hot cylinder to the cold cylinder (e.g., and vice-a-versa). The heat exchangers disclosed herein may be configured to minimize a pressure drop between the hot cylinder and the cold cylinder.

[0200] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, heat exchanger materials configured to provide various benefits, including maximized thermal conductivity for efficient heat transfer, corrosion resistance to withstand various types of fluids, maximized mechanical properties for durability and pressure tolerance, ease of manufacturing and maintenance for practicality, and cost-effectiveness for economic feasibility. [0201] The heat exchangers disclosed herein may be constructed using additive manufacturing techniques and advanced materials with a maximized thermal conductivity and durability.

[0202] In some cases, the advanced materials with a maximized thermal conductivity comprise nano-structured substances. In some instances, a surface treatment may comprise the nano-structed substance. In some instances, the nano-structured substance may comprise A1₂O₃.

[0203] In some cases, the nano-structured substance may be introduced at an inlet to an enclosure. In some instances, the inlet to the enclosure may comprise an waste heat source at an entry to a hot cylinder. In some instances, the nano-structured substance may be configured to increase heat transfer with a topology optimized heat exchanger. For example, a magnetic filter may separate a nanoparticle at the discharge of enclosure where the waste heat source exits the enclosure. In further examples, a nanofluid may be reused for a new cycle.

[0204] In some instances, the nano-structures substances comprise a thermal conductivity between about 0.15 W/mK and about 400 W/mK at room temperature and room pressure. In some instances, the nano-structures substances comprise a thermal conductivity of greater than about 400 W/mK at room temperature and room pressure.

[0205] In some cases, the advanced materials with a maximized thermal conductivity comprise composites. In some cases, the advanced materials with a high thermal conductivity comprise alloys. In some cases, the advanced materials comprise a thermal conductivity of up to about 400 W/mK.

[0206] The heat exchangers disclosed herein may comprise conventional, unconventional, or complex geometries. In some cases, the heat exchangers comprise complex geometries (e.g., shown in FIG. 3, FIG. 4, FIG. 5, and FIG. 7). In some cases, the heat exchanger architecture comprises an organic form heat exchanger. In some cases, the organic form heat exchanger comprises a topology optimized heat exchanger. For example, the heat exchanger may be topology optimized utilizing advanced computational fluid dynamics simulations and adjoint methodology.

[0207] In some cases, the heat exchanger comprises a topology optimized width. In some instances, the heat exchanger comprises a topology optimized exterior width. In some instances, the heat exchanger comprises a topology optimized interior width. In some cases, the heat exchanger comprises a topology optimized length. In some instances, the heat exchanger comprises a topology optimized exterior length. In some instances, the heat exchanger comprises a topology optimized interior length. In some cases, the heat exchanger comprises a topology optimized depth. In some instances, the heat exchanger comprises a topology optimized exterior depth. In some instances, the heat exchanger comprises a topology optimized interior depth. In some cases, the heat exchanger comprises a topology optimized area.

[0208] In some cases, the heat exchanger comprises an inner layer and an outer layer. In some instances, the inner layer comprises a cooling head, regenerator, heating head, power piston, displacer, and a working fluid. In some instances, the inner layer comprises a plurality of fins. In some instances, the heat exchanger inner layer comprises an organic shape. For example, the organic shape may comprise a shape configured to maximize heat transfer and minimize pressure loss. In some instances, the heat exchanger inner layer comprises topology optimized dimensions.

[0209] In some instances, the outer layer comprises an external housing and a plurality of fins. In some instances, the outer layer comprises a cooling head, regenerator, heating head, power piston, displacer, and a working fluid. In some instances, the heat exchanger outer layer comprises an organic shape. In some instances, the heat exchanger outer layer comprises topology optimized dimensions.

[0210] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, heat exchanger surface treatments configured to provide various benefits, including maximized heat transfer, maximized resistance to corrosion, minimized fouling, and maximized fluid dynamics.

[0211] The heat exchangers disclosed herein may comprise a surface treatment on the outer layer or inner layer. In some cases, the heat exchangers comprise surface morphology changes. In some instances, the surface treatment may comprise epoxy coating (e.g., for corrosion resistance), nickel plating (e.g., for durability), anodizing (e.g., for aluminum heat exchangers), ceramic coating for heat resistance, electroless nickel plating for hardness and longevity, and PTFE (e.g., Teflon®) coating (e.g., for high-temperature and chemical resistance applications).

[0212] In some cases, the surface treatment may be configured to reduce fouling. In some instances, the surface treatment may be configured to prevent fouling. In some cases, the surface treatment may be configured to increase resistance to corrosion. In some instances, the surface treatment may be configured to prevent corrosion. In some cases, the surface treatment may be configured to maximize a heat transfer surface area. In some cases, the surface treatment may be configured to increase a heat transfer rate. In some cases, the surface treatment may be configured to reduce pressure drop.

[0213] The heat exchangers disclosed herein may be configured to maximize efficiency under varying external heat stream conditions. The heat source may comprise the heating head. In some cases, the heat heating head comprises a temperature below about 80°C. In some instances, the heating head is configured to receive an waste heat source. For example, the waste heat source comprises an waste heat source. In further examples, the waste heat source may comprise waste heat source gases, hot liquids, or other low-temperature heat streams. In even further examples, the waste heat source comprises a temperature below about 80°C. In even further examples, the waste heat source comprises a temperature below about 230°C.

[0214] In some cases, the formula for calculating the heat exchanger efficiency comprises equation 2 (e.g., described above) In some cases, T_in;H comprises an initial temperature of a hot fluid. In some cases, T_0Ut,H = final temperature of the hot fluid. In some cases, T_in;C = initial temperature of the cold fluid.

[0215] In some cases, the hot fluid comprises a temperature between about 75°C and about 80°C. In some cases, the cold fluid comprises a temperature between about 5°C and about 12°C. For example, the heat exchanger may comprise up to about 45% efficiency.

[0216] In some cases, the heat exchanger efficiency comprises between about 0% to about 100%. In some cases, the heat exchanger efficiency comprises between about 0% to about 5%, about 0% to about 10%, about 0% to about 15%, about 0% to about 20%, about 0% to about 25%, about 0% to about 30%, about 0% to about 35%, about 0% to about 40%, about 0% to about 45%, about 0% to about 50%, about 0% to about 100%, about 5% to about 10%, about 5% to about 15%, about 5% to about 20%, about 5% to about 25%, about 5% to about 30%, about 5% to about 35%, about 5% to about 40%, about 5% to about 45%, about 5% to about 50%, about 5% to about 100%, about 10% to about 15%, about 10% to about 20%, about 10% to about 25%, about 10% to about 30%, about 10% to about 35%, about 10% to about 40%, about 10% to about 45%, about 10% to about 50%, about 10% to about 100%, about 15% to about 20%, about 15% to about 25%, about 15% to about 30%, about 15% to about 35%, about 15% to about 40%, about 15% to about 45%, about 15% to about 50%, about 15% to about 100%, about 20% to about 25%, about 20% to about 30%, about 20% to about 35%, about 20% to about 40%, about 20% to about 45%, about 20% to about 50%, about 20% to about 100%, about 25% to about 30%, about 25% to about 35%, about 25% to about 40%, about25% to about 45%, about 25% to about 50%, about 25% to about 100%, about 30% to about 35%, about 30% to about 40%, about 30% to about 45%, about 30% to about 50%, about 30% to about 100%, about 35% to about 40%, about 35% to about 45%, about 35% to about 50%, about 35% to about 100%, about 40% to about 45%, about 40% to about 50%, about40% to about 100%, about45% to about 50%, about 45% to about 100%, or about 50% to about 100%. In some cases, the heat exchanger efficiency comprises between about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%. In some cases, the heat exchanger efficiency comprises between at least about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, or about 50%. In some cases, the heat exchanger efficiency comprises between at most about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%.

[0217] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a plurality of fins configured to provide various benefits, including an organic layout and structure to maximize heat dissipation and thermal efficiency.

[0218] The heat exchangers disclosed herein may comprise fin-type heat exchangers. In some cases, the fin-type heat exchangers comprise a plurality of fins. In some cases, the plurality of fins comprises a plurality of organic fins a varying size, shape, and distribution. In some cases, the plurality of organic fins is configured to maximize the efficiency of thermal pathways.

[0219] The heat exchangers disclosed herein may comprise an outer layer (e.g., external surface). In some cases, the outer layer of the heat exchanger may comprise the plurality of fins. In some instances, the plurality of fins is configured to contact an ambient environment. For example, a working fluid may transfer heat to the plurality of fins. In further examples, the plurality of fins may transfer heat to an ambient environment. In some instances, the plurality of fins is configured to contact a heat stream. For example, and waste heat source may transfer heat to the plurality of fins. In further examples, the plurality of fins may transfer heat to a heating head.

[0220] The plurality of fins disclosed herein may comprise a architecture from an iterative manner that mimics an evolution-like processes. In some cases, the plurality of fins architecture stems from generative topology optimization. In some instances, the plurality of fins comprise a plurality of topology optimized organic fins (e.g., topology optimized fins). [0221] In some cases, the plurality of fins comprises an organic shape. In some instances, the organic shape comprises any shape configured to efficiently transfer heat from a heat source to a working fluid (e.g., on a hot side of the engine). In some instances, the organic shape comprises any shape configured to efficiently transfer heat from a working fluid to a cooling source (e.g., on a cold side of the engine).

[0222] In some instances, the organic shape comprises any shape configured to maximize a contact surface area between the working fluid and the plurality of fins.

[0223] In some instances, the plurality of fins comprise a shape or cross section comprising a circle, equilateral triangle, isosceles triangle, scalene triangle, right triangle, square, rectangle, pentagon, hexagon, heptagon, octagon, nonagon, decagon, parallelogram, rhombus, trapezoid, oval, star, heart, crescent, sphere, cube, cylinder, cone, square pyramid, triangular pyramid, rectangular prism, triangular prism, hexagonal prism, tetrahedron, octahedron, dodecahedron, icosahedron, torus, hemisphere, or ellipsoid.

[0224] FIG. 3 shows a Free Piston Stirling Engine Heat Exchanger. In the example of FIG.

3, a first encasement 301 from the first heat exchanger 102 from the FSPE 101 of FIG. 1 is shown. In FIG. 3, the first encasement 301 comprises a plurality of fins 302. In FIG. 3, the plurality of fins 302 comprises one or more organic fins. For example, the one or more organic fins 302 may comprise a cylindrical shape. In FIG. 3, the plurality of fins comprises a spacing between one fin to another fin. Furthermore, in the example of FIG. 3, the plurality of fins comprises layers of fins. For example, in FIG. 3, the first encasement 301 comprises about 9 layers of organic fins. Moreover, the first encasement 301 comprises an external cooling head.

[0225] FIG. 4 shows a Free Piston Stirling Engine Heat Exchanger. In the example of FIG.

4, a second encasement 401 from the second heat exchanger 103 from the FSPE 101 of FIG. 1 is shown. In FIG. 4, the second encasement 401 comprises a plurality of fins 402. In FIG. 4, the plurality of fins 402 comprises one or more organic fins. For example, the one or more organic fins 402 may comprise a rectangular shape. In FIG. 4, the plurality of fins comprises a spacing between one fin to another fin. For example, the spacing may comprise a plurality of recessed rectangles. Furthermore, in the example of FIG. 4, the plurality of fins comprises layers of fins. For example, in FIG. 4, the second encasement 401 comprises about 9 layers of organic fins. Moreover, the second encasement 401 comprises an external heating head. [0226] The heat exchangers disclosed herein may comprise a plurality of fins organically distributed for effective heat storage. In some cases, the plurality of fins is configured to store heat from a heat stream and release heat to the heating head. In some instances, the organic distribution of the plurality of fins is configured to maximize a contact surface area between the waste heat source and the plurality of fins. In some instances, the organic distribution of the plurality of fins is configured to maximize a contact surface area between the plurality of fins and the heating head.

[0227] The plurality of fins may comprise a maximized thermal mass configured for efficient heat storage. The heat exchangers disclosed herein may comprise a plurality of fins organically distributed for effective heat release. In some cases, the plurality of fins is configured to receive heat from a cooling head and release heat to the environment. In some instances, the organic distribution of the plurality of fins is configured to maximize a contact surface area between the environment and the plurality of fins. In some instances, the organic distribution of the plurality of fins is configured to maximize a contact surface area between the plurality of fins and the cooling head.

[0228] The plurality of fins may comprise a maximized fin surface area configured for efficient heat release. In some cases, the plurality of fins may comprise a maximized fin thickness configured for efficient heat release. In some cases, the plurality of fins may comprise a maximized fin length configured for efficient heat release. In some cases, the plurality of fins may comprise a maximized fin width configured for efficient heat release. [0229] The plurality of fins disclosed herein may comprise a bounding envelope size configured to provide maximized heat storage or maximized heat release. In some cases, the bounding envelope size comprises the physical dimensions within each fin of the plurality of fins may fit. In some instances, each fin of the plurality of fins comprises a topology optimized bounding envelope size.

[0230] The plurality of fins disclosed herein may comprise a hierarchical arrangement of fins of different spacings allowed to enhance heat dissipation or heat storage. In some cases, the spacing comprises a distance between two adjacent fins. In some instances, the spacing between fins throughout the heat exchanger is uniform. For example, each fin in the plurality of fins may comprise an equidistant spacing between each fin. In some instances, the spacing throughout the heat exchanger is variable. For example, each fin in the plurality of fins may comprise a different distance spacing between each fin as between some or all other fins. [0231] The plurality of fins may be disposed within a heat exchanger. FIG. 7 shows a Free Piston Stirling Engine Heat Exchanger. In the example of FIG. 7, a second encasement 701 from the second heat exchanger 204 from the FSPE 200 of FIG. 2 is shown. In FIG. 7, the second encasement 701 comprises a plurality of fins 702. In FIG. 7, the plurality of fins 702 comprises one or more organic fins. For example, the one or more organic fins 702 may comprise a porous material. In further examples, the porous material may comprise wire mesh, foil (e.g., made from copper, steel, nickel, etc.). Moreover, the second encasement 701 comprises an internal heating head.

[0232] In FIG. 7, the plurality of fins 702 may comprise a plurality of sizes. For example, one organic fin may comprise a diameter less than another organic fin. Furthermore, the plurality of fins may comprise a pattern. For example, an organic fin may comprise a diameter less or more than an adjacent organic fin around a circumference of a heat exchanger. In further examples, the plurality of fins comprises a topology optimized diameter.

[0233] The plurality of fins disclosed herein may comprise a hierarchical arrangement of fins of different sizes configured to maximize heat dissipation and maximize heat exchanger efficiency.

[0234] In some cases, the hierarchical arrangement of fins comprises an organic distribution of fins. In some instances, the organic distribution comprises non-uniform or varied architecture patterns in the placement, distribution, or sizing of the fins (e.g., mimicking natural or “organic” systems).

[0235] FIG. 5 shows a Free Piston Stirling Engine Heat Exchanger. In the example of FIG. 5, a first encasement 501 from the first heat exchanger 205 from the FSPE 200 of FIG. 2 is shown. In FIG. 5, the first encasement 501 comprises a plurality of fins 502. In FIG. 5, the plurality of fins 502 comprises one or more organic fins. For example, the one or more organic fins 502 may comprise a length equivalent to a length of the first encasement 501. Moreover, the first encasement 501 comprises an internal cooling head.

[0236] The plurality of fins may comprise a distribution configured for maximum heat flux. In some cases, the plurality of fins may comprise an organic distribution configured for uniform temperature distribution. In some cases, the organic distribution maximizes surface area. In some instances, the plurality of fins may comprise organic fins. For example, the organic fins may comprise a topology optimized surface area.

[0237] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, fin materials configured to provide various benefits, including maximized thermal conductivity for efficient heat transfer, maximized corrosion resistance to withstand various types of fluids, maximized mechanical properties for durability and pressure tolerance, ease of manufacturing and maintenance for practicality, and cost-effectiveness for economic feasibility. In some cases, the fin material comprises nano-structured substances, composites, or alloys. In some cases, the fin material comprises copper. In some cases, the fin material comprises aluminum. In some cases, the plurality of fins comprises any material comprising a thermal conductivity up to about 205 W/mK at room temperature and room pressure.

[0238] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a Free Piston Stirling Engine (FPSE) configured to provide various benefits, including the topology optimize heating head (e.g., referred to as heating head or heat source) configured to transfer heat from a heat stream to a working fluid inside the FPSE.

[0239] The FPSE 200 may comprise a heating head 203 configured to add thermal energy to a working fluid. In some cases, the working fluid is in thermal contact with the heating head 203. In some cases, the heating head is configured to add an optimized amount of heat to the working fluid.

[0240] FIG. 9 shows a Free Piston Stirling Engine. In some cases, the FPSE 900 comprises a topology optimized heating head 901. In some cases, the topology optimized heating head 901 is configured to add heat to a working fluid. In some cases, the topology optimized heating head 901 may be configured to maintain isothermal conditions as a working fluid (e.g., an waste heat source) expands and pressure decreases.

[0241] The heating heads disclosed herein may comprises a architecture from an iterative manner that mimics an evolution-like processes. In some cases, the heating head architecture stems from generative topology optimization.

[0242] In some cases, the heating head comprises an organic shape. In some cases, the organic shape comprises any shape configured to efficiently transfer heat from a heat source to a working gas (e.g., on a hot side of the engine). In some cases, the organic shape comprises any shape configured to efficiently transfer heat from a plurality of fins to a working gas (e.g., on a hot side of the engine).

[0243] In some cases, the organic shape comprises any shape configured to maximize a contact surface area between the working gas and the heating head. In some instances, the contact surface area between the working gas and the heating head comprises a topology optimized contact surface area.

[0244] In some cases, the organic shape comprises any shape configured to maximize a contact surface area between the heating head and a plurality of fins. In some instances, the contact surface area between the heating head and the plurality of fins comprises a topology optimized contact surface area. [0245] In some cases, the heating head comprises a shape or cross section comprising a circle, equilateral triangle, isosceles triangle, scalene triangle, right triangle, square, rectangle, pentagon, hexagon, heptagon, octagon, nonagon, decagon, parallelogram, rhombus, trapezoid, oval, star, heart, crescent, sphere, cube, cylinder, cone, square pyramid, triangular pyramid, rectangular prism, triangular prism, hexagonal prism, tetrahedron, octahedron, dodecahedron, icosahedron, torus, hemisphere, or ellipsoid.

[0246] In some cases, the topology optimized heating head 901 comprises a architecture of complex structures that may enhance heat transfer within the topology optimized heating head 901. For example, the topology optimized heating head 901 is configured to increase efficiency of heat from the external heat source to the working fluid. For example, the efficiency transfer of heat from the external heat stream to the working fluid may comprise greater than 10%.

[0247] In some cases, the topology optimized heating head 901 is configured to maximize thermal performance of the FSPE. In some instances, the topology optimized heating head 901 may comprise an optimized internal structure. In further examples, the optimized internal structure may comprise improved heat distribution. In even further examples, the topology optimized heating head may comprise uniform temperature profiles (e.g., this, in turn, enhances thermal performance of the FPSE).

[0248] In some cases, the topology optimized heating head 901 may comprise topology optimized fins. In some instances, the topology optimized fins may comprise topology optimized internal fins. For example, the topology optimized internal fins may be more efficient than conventional fins. In some instances, the topology optimized fins may comprise topology optimized external fins. For example, the topology optimized external fins may be more efficient than conventional fins.

[0249] The heat exchangers disclosed herein may be configured to maximize heat transfer. In some cases, the heat exchangers are configured to ensure uniform heat flux distribution from a heat stream to a working fluid. In some cases, the heat exchangers are configured to maximize a uniform heat flux distribution from the heating head to the hot cylinder. In some cases, the heat exchanger comprises a topology optimized surface to volume ratio.

[0250] The heating heads disclosed herein may be configured to maximize efficiency under varying external heat stream conditions. In some cases, the heating head comprises a topology optimized heating head 901 configured to maximize efficiency by minimizing thermal losses. In some instances, the topology optimized heating head 901 is configured to maximize efficiency by maximizing the utilization of the heat from the external stream (e.g., waste heat source).

[0251] In some cases, the formula for calculating the heating head efficiency comprises equation 2 (e.g., as described above) In some cases, T_in;H comprises an initial temperature of an waste heat source. In some cases, T_0Ut,H = final temperature of the waste heat source. In some cases, T_in;C = initial temperature of the cold fluid (e.g., working fluid). In some cases, the waste heat source comprises a temperature between about 75°C and about 80°C. In some cases, the working fluid comprises a temperature between about 75°C and about 80°C. For example, the heating head may comprise up to about 100% efficiency.

[0252] In some cases, the heating head efficiency comprises between about 0% to about 100%. In some cases, the heating head efficiency comprises between about 0% to about 5%, about 0% to about 10%, about 0% to about 15%, about 0% to about 20%, about 0% to about 25%, about 0% to about 30%, about 0% to about 35%, about 0% to about 40%, about 0% to about 45%, about 0% to about 50%, about 0% to about 100%, about 5% to about 10%, about 5% to about 15%, about 5% to about 20%, about 5% to about 25%, about 5% to about 30%, about 5% to about 35%, about 5% to about 40%, about 5% to about 45%, about 5% to about 50%, about 5% to about 100%, about 10% to about 15%, about 10% to about 20%, about 10% to about 25%, about 10% to about 30%, about 10% to about 35%, about 10% to about 40%, about 10% to about45%, about 10% to about 50%, about 10% to about 100%, about 15% to about 20%, about 15% to about 25%, about 15% to about 30%, about 15% to about 35%, about 15% to about 40%, about 15% to about 45%, about 15% to about 50%, about 15% to about 100%, about 20% to about 25%, about 20% to about 30%, about 20% to about 35%, about 20% to about 40%, about 20% to about 45%, about 20% to about 50%, about 20% to about 100%, about 25% to about 30%, about 25% to about 35%, about 25% to about 40%, about 25% to about45%, about 25% to about 50%, about 25% to about 100%, about 30% to about 35%, about 30% to about 40%, about 30% to about 45%, about 30% to about 50%, about 30% to about 100%, about 35% to about 40%, about 35% to about 45%, about 35% to about 50%, about 35% to about 100%, about 40% to about 45%, about 40% to about 50%, about 40% to about 100%, about 45% to about 50%, about 45% to about 100%, or about 50% to about 100%. In some cases, the heating head efficiency comprises between about 0%, about 5%, about 10%, about 15%, about 20%, about25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%. In some cases, the heating head efficiency comprises between at least about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, or about 50%. In some cases, the heating head efficiency comprises between at most about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about40%, about 45%, about 50%, or about 100%.

[0253] In some cases, the heat exchangers are configured to maximize a uniform heat flux distribution from the heating head to the plurality of fins. In some cases, the heat exchangers comprise a topology optimized surface to volume ratio between the heating head to the plurality of fins.

[0254] The heating heads disclosed herein may comprise a compact architecture configured for applications in both industrial and non-industrial settings. In some cases, the topology optimized heating head 901 is configured to minimize material usage. In some instances, the topology optimized heating head 901 minimizes material addition and maintains structural integrity. In some instances, the topology optimized heating head 901 comprises a minimized weight of the heating head (e.g., contributing to a more lightweight and ly cost- effective architecture). For example, the topology optimized heating head 901 may comprise dimensions no greater than a conventional heating head.

[0255] In some cases, the topology optimized heating head 901 may be configured for heat stream characteristics. In some instances, the heat stream may comprise a low grade heat stream (e.g., below 230°C).

[0256] In some cases, the topology optimized heating head 901 is configured to minimize thermal gradients. In some instances, the topology optimized heating head 901 may comprise minimized thermal gradients within the topology optimized heating head 901. In further examples, the topology optimized heating head 901 may comprise optimized (e.g., smoother) temperature profiles contributing to more stable and predictable engine operation.

[0257] In some cases, the topology optimized heating head 901 is configured to integrate with system constraints. In some instances, the topology optimized heating head 901 may be configured for various system constraints (e.g., according to available space or structural requirements ensuring that topology optimized heating head architecture 901 aligns with the overall system architecture and constraints).

[0258] In some cases, the topology optimized heating head 901 is configured to provide maximized reliability and durability. In some instances, the topology optimized heating head 901 may be configured for minimize stress concentrations and maximized structural robustness of the topology optimized heating head 901 (e.g., leading to enhanced reliability and durability over the operational life of the FPSE). [0259] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a heat exchanger configured to provide various benefits, including a cooling head (e.g., also referred to as cooling head or heat sink) that facilitates heat rejection to a lower-temperature medium (e.g., often ambient air or another cooling fluid such as water).

[0260] The FPSE 200 may comprise a cooling head 201 configured to release thermal energy from the working fluid. In some cases, the working fluid is in thermal contact with the cooling head 201. In some instances, the cooling head 201 comprises a thermal conductivity of up to about 400 W/mK at room temperature and room pressure. In some instances, the cooling head 201 comprises a topology optimized surface area. In some cases, the cooling head 201 comprises a heat sink. In some instances, the temperature differential between the heat sink and the working fluid comprises up to about 56°C. . In some instances, the temperature differential between the heat sink and the working fluid comprises greater than about 56°C. In some cases, the cooling head 201 comprises a heat sink. In some instances, the heat sink 201 comprises a liquid-cooled heat sink. For example, the liquid-cooled heat sink 201 may comprise a water-cooled heat sink. In some instances, the heat sink comprises a gas-cooled heat sink. For example, the gas-cooled heat sink 201 may comprise an aircooled heat sink.

[0261] The cooling heads disclosed herein may comprises a architecture from an iterative manner that mimics an evolution-like processes. In some cases, the cooling head architecture stems from generative topology optimization.

[0262] In some cases, the cooling head comprises an organic shape. In some cases, the organic shape comprises any shape configured to efficiently transfer heat from a working fluid to the cooling head (e.g., on the cold side of the engine). In some cases, the organic shape comprises any shape configured to efficiently transfer heat from the cooling head to the plurality of fins (e.g., on the cold side of the engine).

[0263] In some cases, the organic shape comprises any shape configured to maximize a contact surface area between the working gas and the cooling head. In some instances, the contact surface area between the working gas and the cooling head comprises a topology optimized contact surface area.

[0264] In some cases, the organic shape comprises any shape configured to maximize a contact surface area between the cooling head and a plurality of fins. In some instances, the contact surface area between the cooling head and the plurality of fins comprises a topology optimized surface area. [0265] In some cases, the cooling head comprises a shape or cross section comprising a circle, equilateral triangle, isosceles triangle, scalene triangle, right triangle, square, rectangle, pentagon, hexagon, heptagon, octagon, nonagon, decagon, parallelogram, rhombus, trapezoid, oval, star, heart, crescent, sphere, cube, cylinder, cone, square pyramid, triangular pyramid, rectangular prism, triangular prism, hexagonal prism, tetrahedron, octahedron, dodecahedron, icosahedron, torus, hemisphere, or ellipsoid.

[0266] In some cases, the heat exchangers are configured to ensure uniform heat flux distribution from a working fluid to a heat sink. In some cases, the heat flux distribution forms the working fluid to the heat sink comprises a uniform heat flux distribution from the working fluid to the heat sink. . In some cases, the heat exchangers are configured to maximize a uniform heat flux distribution from the waterjacketto the cold cylinder. In some instances, the heat exchanger comprises a topology optimized surface to volume ratio between the water jacket to the cold cylinder of FPSE.

[0267] The cooling heads disclosed herein may be configured to maximize efficiency under varying external heat stream conditions. FIG. 9 shows a Free Piston Stirling Engine. In some cases, the FPSE 900 comprises a topology optimized water cooling head 902. In some cases, the topology optimized cooling head 902 is configured to remove heat from a working fluid. In some cases, the topology optimized cooling head 902 may be configured to maintain isothermal conditions as a working fluid (e.g., an waste heat source) compresses and pressure increases.

[0268] In some cases, the topology optimized cooling head 902 is configured to provide maximized heat rejection to an ambient environment. In some instances, the topology optimized cooling head 902 comprises an external heat sink. For example, the topology optimized cooling head 902 may be configured to maximize the heat transfer between a working fluid and a plurality of fins (e.g., improving the efficiency of heat rejection to the environment).

[0269] In some cases, the topology optimized cooling head 902 is configured to provide maximized heat dissipation. In some instances, the topology optimized cooling head 902 may be comprise of intricate internal structures configured to maximize heat dissipation and maximize the efficiency of the cooling process. For example, the internal structure may comprise a topology optimized internal structure. In further examples, the internal structures may comprise optimized surface morphology.

[0270] In some cases, the topology optimized cooling head 902 is configured to provide maximized thermal performance. In some instances, the topology optimized cooling head 902 may be configured for a maximized efficiency heat exchange between the working fluid and the cooling head (e.g., resultingin maximized thermal performance and heat rejection). [0271] In some cases, the topology optimized cooling head 902 is configured to provide increased cooling efficiency. In some instances, the optimization process is configured to optimize the internal configuration of the topology optimized cooling head 902 which increases heat transfer rates. For example, increased heat transfer rates may be configured for maximized cooling efficiency and maximized temperature control within the FPSE.

[0272] In some cases, the topology optimized cooling head 902 comprises a architecture of complex structures configured to maximize heat transfer from the working fluid to the plurality of fins (e.g., and to the environment).

[0273] In some cases, the formula for calculating the heating head efficiency comprises equation 2 (e.g., as described above) In some instances, T_in;H comprises an initial temperature of a working fluid. In some instances, T_0Ut,H = final temperature of the working fluid. In some instances, T_in;C = initial temperature of the cooling head (e.g., working fluid).

[0274] In some cases, the working fluid comprises a temperature between about 5°C and about 12°C. In some cases, the cooling head comprises a temperature between about 10°C and about 17°C. For example, the cooling head may comprise up to 100% efficiency.

[0275] In some cases, the cooling head efficiency comprises between about 0% to about 100%. In some cases, the cooling head efficiency comprisesbetween about 0% to about 5%, about 0% to about 10%, about 0% to about 15%, about 0% to about 20%, about 0% to about 25%, about 0% to about 30%, about 0% to about 35%, about 0% to about 40%, about 0% to about 45%, about 0% to about 50%, about 0% to about 100%, about 5% to about 10%, about 5% to about 15%, about 5% to about 20%, about 5% to about 25%, about 5% to about 30%, about 5% to about 35%, about 5% to about 40%, about 5% to about 45%, about 5% to about 50%, about 5% to about 100%, about 10% to about 15%, about 10% to about 20%, about 10% to about 25%, about 10% to about 30%, about 10% to about 35%, about 10% to about 40%, about 10% to about45%, about 10% to about 50%, about 10% to about 100%, about 15% to about 20%, about 15% to about 25%, about 15% to about 30%, about 15% to about 35%, about 15% to about 40%, about 15% to about 45%, about 15% to about 50%, about 15% to about 100%, about 20% to about 25%, about 20% to about 30%, about 20% to about 35%, about 20% to about 40%, about 20% to about 45%, about 20% to about 50%, about 20% to about 100%, about 25% to about 30%, about 25% to about 35%, about 25% to about 40%, about 25% to about45%, about 25% to about 50%, about 25% to about 100%, about 30% to about 35%, about 30% to about 40%, about 30% to about 45%, about 30% to about 50%, about 30% to about 100%, about 35% to about 40%, about 35% to about 45%, about 35% to about 50%, about 35% to about 100%, about 40% to about 45%, about 40% to about 50%, about 40% to about 100%, about 45% to about 50%, about 45% to about 100%, or about 50% to about 100%. In some cases, the cooling head efficiency comprises between about 0%, about 5%, about 10%, about 15%, about20%, about25%, about 30%, about 35%, about 40%, about 45%, about 50%, or about 100%. In some cases, the cooling head efficiency comprises between at least about 0%, about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about 40%, about 45%, or about 50%. In some cases, the cooling head efficiency comprises between at most about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, about 35%, about40%, about 45%, about 50%, or about 100%.

[0276] In some cases, the heat exchangers are configured to maximize a uniform heat flux distribution from the water jacket to the plurality of fins. In some cases, the heat exchangers comprise a topology optimized surface to volume ratio between the water jacket to the plurality of fins.

[0277] The coolers disclosed herein may comprise a compact architecture configured for applications in both industrial and non-industrial settings. In some cases, the topology optimized cooling head 902 comprises a minimized weight and minimized material usage. In some instances, the optimization process minimizes material addition and maintains structural integrity. In some instances, the optimization process reduces the weight of the cooling head 902 (e.g., contributing to a more lightweight and ly cost-effective architecture). For example, the topology optimized cooling head 902 may comprise a weight no greater than a weight of a conventional cooling head of a same power capacity, or a dimension no greater than a conventional cooling head of the same power capacity.

[0278] In some cases, the topology optimized cooling head 902 may be configured for cooling source characteristics. In some instances, the cooling source may comprise a cooling liquid (e.g., water). In some instances, the cooling source may comprise a cooling gas (e.g., air).

[0279] In some cases, the topology optimized cooling head 902 is configured to minimize thermal gradients. In some instances, the topology optimized cooling head 902 is configured to minimize thermal gradients within the topology optimized cooling head 902. In further examples, the topology optimized cooling head 902 may comprise smoother temperature profiles contributing to more stable and predictable engine operation. [0280] In some cases, the topology optimized cooling head 902 is configured to integrate with system constraints. In some instances, the topology optimized cooling head 902 may be configured for various system constraints (e.g., such as available space or structural requirements, ensuring that the topology optimized cooling head 902 aligns with the overall system architecture and constraints).

[0281] In some cases, the topology optimized cooling head 902 is configured to provide enhanced reliability and durability. In some instances, the topology optimized cooling head 902 may minimize stress concentrations and maximize structural robustness, (e.g., leading to enhanced reliability and durability over the operational life of the FPSE).

[0282] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a regenerator configured to provide various benefits, including storing and releasing heat as the working fluid cycles between a hot heat exchanger and cold heat exchanger, and a maximized overall efficiency of the FPSE (e.g., by reducing heat losses).

[0283] The regenerator may comprise a architecture from an iterative manner that mimics an evolution-like processes. In some cases, the regenerator architecture stems from generative topology optimization. In some cases, the generative topology optimization is configured to produce a topology optimized regenerator.

[0284] In some cases, the regenerator comprises an organic shape. In some cases, the organic shape comprises any shape configured to efficiently transfer heat from a working fluid to the cooling head (e.g., on the cold side of the engine). In some cases, the organic shape comprises any shape configured to efficiently absorb heat from a working fluid. In some cases, the organic shape comprises any shape configured to efficiently release heat from a working fluid.

[0285] In some cases, the organic shape comprises any shape configured to maximize a contact surface area between the working fluid and the regenerator. In some instances, the contact surface area between the working fluid and the regenerator comprises a topology optimized contact surface area.

[0286] In some cases, the topology optimized regenerator 903 comprises variable geometry. In some instances, the variable geometry comprises a geometry similar to shape memory alloys.

[0287] In some cases, the regenerator comprises a shape or cross section comprising a circle, equilateral triangle, isosceles triangle, scalene triangle, right triangle, square, rectangle, pentagon, hexagon, heptagon, octagon, nonagon, decagon, parallelogram, rhombus, trapezoid, oval, star, heart, crescent, sphere, cube, cylinder, cone, square pyramid, triangular pyramid, rectangular prism, triangular prism, hexagonal prism, tetrahedron, octahedron, dodecahedron, icosahedron, torus, hemisphere, or ellipsoid.

[0288] FIG. 9 shows a Free Piston Stirling Engine. In some cases, the FPSE 900 comprises a topology optimized regenerator 903. In some cases, the topology optimized regenerator 903 is configured to connect the topology optimized heating head 901 to the topology optimized cooling head 902. In some cases, the topology optimized regenerator 903 is configured to connect the hot cylinder to the cold cylinder.

[0289] In some cases, the topology optimized regenerator 903 is configured to absorb and store heat from a working fluid. In some instances, the topology optimized regenerator 903 is configured to receive a working fluid during a heating phase of a Stirling cycle. For example, the heating phase comprises a working fluid moved from a cold cylinder to a hot cylinder e.g., the working fluid passes through the regenerator). In further examples, the topology optimized regenerator 903 absorbs and stores an optimized amount of heat from a hot working fluid.

[0290] In some cases, the topology optimized regenerator 903 is configured to release heat to a working fluid. In some instances, the topology optimized regenerator 903 is configured to receive a working fluid during a cooling phase of the Stirling cycle. For example, the cooling phase comprises a working fluid moved from a hot cylinder to a cold cylinder (e.g., the working fluid passes through the regenerator). In further examples, the topology optimized regenerator 903 releases an optimized amount of heat to the cold working fluid.

[0291] In some cases, the regenerator comprises a thermal conductivity of up to 400 W/mK at room temperature and room pressure. In some cases, the regenerator comprises a topology optimized surface area. In some cases, the temperature differential between the regenerator and the working fluid comprises between about 0°C to about 80°C.

[0292] FIG. 6 illustrates an example of a Free Piston Stirling Engine Regenerator. In some cases, the regenerator comprises a plurality of fins. In FIG. 6, the regenerator 202 comprises a plurality of fins 602. In some instances, in FIG. 6, the plurality of fins 602 are organically distributed for effective heat storage and heat release (e.g., to maintain the engine’s operational efficiency).

[0293] In some cases, the topology optimized regenerator 903 is configured to provide maximized thermal performance. In some instances, the topology optimized regenerator 903 is configured to provide a maximized effective transfer of heat between the hot and cold ends of the regenerator 903 (e.g., resulting in maximized thermal performance and maximized efficiency in the FPSE).

[0294] In some cases, the topology optimized regenerator 903 is configured to provide minimized thermal gradients. In some instances, the topology optimized regenerator 903 is configured to provide a uniform heat exchange and prevent localized temperature variations. [0295] In some cases, the topology optimized regenerator 903 is configured to maximize heat exchange. In some instances, the topology optimized regenerator 903 may comprise intricate internal structures to maximize heat exchange efficiency during a cyclic compression and expansion processes.

[0296] In some cases, the topology optimized regenerator 903 is configured to provide maximized heat storage capacity. In some instances, the topology optimized regenerator 903 structure maximizes heat storage capacity (e.g., allowing it to store more heat during the hot phase and release more heat during the cold phase of the FPSE cycle). For example, the topology optimized regenerator 903 may comprise a maximized thermal mass (mCP). In further examples, the topology optimized regenerator 903 may comprise a topology optimized thermal mass (mCP).

[0297] The topology optimized regenerator 903 may comprise a porous matrix regenerator. In some cases, the porous matrix regenerators may comprise materials with interconnected pores in various geometric shapes. In some instances, the pores may comprise a topology optimized shape. In further examples, the pores may comprise a topology optimized diameter. In even further examples, the pores may comprise a topology optimized length. [0298] FIG. 6 shows a Free Piston Stirling Regenerator. In the example of FIG. 6, the heat exchanger 202 from the FSPE 200 of FIG. 2 is shown. In some cases, the heat exchanger 202 comprises a regenerator. In some cases, the regenerator 202 comprises optimum porosity. In some instances, the topology optimized regenerator 903 may comprise a porosity configured to maximize heat transfer and minimize pressure drop (e.g., optimizing its overall performance).

[0299] In some cases, the regenerator is configured to minimize a pressure drop. In some instances, the regenerator comprises a pressure drop no greater than about 3 kPa. In some instances, the regenerator comprises a pressure drop no greater than about 10 kPa. In some instances, the regenerator comprises a pressure drop greater than about 10 kPa.

[0300] In some cases, the porosity comprises a ratio of void volume to total volume in the regenerator material. In some instances, the ratio of void volume to total volume in the regenerator material may comprise a topology optimized ratio of void volume to total volume.

[0301] In some cases, the topology optimized regenerator 903 comprises a mesh or screen regenerator. In some instances, the mesh or screen regenerator may comprise a cylindrical or annular mesh structure.

[0302] In some cases, the topology optimized regenerator 903 comprises a stacked disc regenerators. In some instances, the stacked disc regenerators may comprise thin discs stacked together in a cylindrical or annular arrangement.

[0303] In some cases, the topology optimized regenerator 903 comprises a honeycomb regenerator. In some instances, the honeycomb regenerator may feature a hexagonal cell structure for efficient heat exchange.

[0304] In some cases, the topology optimized regenerator 903 comprises a random fiber regenerator. For example, the random fiber regenerators may utilize randomly oriented fibers or filaments for enhanced heat transfer.

[0305] In some cases, the topology optimized regenerator 903 comprises a structured matrix regenerator. For example, the structured matrix regenerators may involve engineered matrices with specific geometric patterns to optimize heat transfer and fluid flow.

[0306] In some cases, the topology optimized regenerator 903 comprises a fixed-plate regenerator. For example, the fixed-plate regenerators may consist of stacked fixed plates or fins creating channels for working fluid flow.

[0307] In some cases, the topology optimized regenerator 903 comprises a vortex-flow regenerator. For example, the vortex-flow regenerators may induce vortex flow through helical or spiral channels to enhance heat transfer in Free Piston Stirling Engines.

[0308] In some cases, the topology optimized regenerator 903 is configured to provide maximized adaptability to various working fluids. In some instances, the topology optimized regenerator 903 architecture may be adapted to work efficiently with different working fluids, allowing for flexibility in choosing the most suitable fluid for specific applications.

[0309] In some cases, the topology optimized regenerator 903 is configured to adjust its characteristics in response to the temperature and flow pattern of the heat flow. In some instances, the regenerator may comprise a material configured to increase pore size when receiving heat. In some instances, the regenerator may comprise a material configured to decrease pore size when storing heat.

[0310] In some cases, the topology optimized regenerator 903 is configured to adjust its characteristics in response to the hot phase of the Stirling engine cycle (e.g., where the working fluid undergoes expansion as it absorbs heat from the external heat stream). In some instances, the topology optimized regenerator 903 is configured to adjust its characteristics to absorb heat from the working fluid.

[0311] In some cases, the topology optimized regenerator 903 is configured to adjust its characteristics in response to the transition phase of the Stirling engine cycle (e.g., where the working fluid remains about at constant volume after the hot phase). In some instances, the topology optimized regenerator 903 is configured to adjust its characteristics to undergo a change in temperature.

[0312] In some cases, the topology optimized regenerator 903 is configured to adjust its characteristics in response to the cold phase of the Stirling engine cycle (e.g., where the working fluid undergoes compression as it releases heat to the external heat sink). In some instances, the topology optimized regenerator 903 is configured to adjust its characteristics to supply heat to the working fluid, releasing the stored thermal energy to the working fluid as it passes through during compression.

[0313] In some cases, the topology optimized regenerator 903 is configured to adjust its characteristics in response to the transition phase of the Stirling engine cycle (e.g., where the working fluid remains about at constant volume after the cold phase). In some instances, the topology optimized regenerator 903 is configured to adjust its characteristics to undergo a change in temperature.

[0314] The systems disclosed herein may improve over systems in the art by providing, in some cases, a working fluid configured to provide various benefits, including a gas mixture specially formulation to maximize thermodynamic performance at lower temperatures (e.g., below 230°C), and thermal conductivity (e.g., compared to conventional pure gases used in similar applications).

[0315] The working fluid may comprise a novel gas mixture. In some cases, the mixture may comprise a formulation optimized for use in a FPSE. In some cases, the working fluid may comprise a formulation configured to maximize a power output in the Stirling cycle with an waste heat source input. In some cases, the working fluid is configured to generate a higher power output than a conventional working fluid (e.g. , helium) at a same heat input. In some instances, the working fluid comprises a higher thermal conductivity. In some instances, the working fluid comprises a lower viscosity.

[0316] In some cases, the working fluid comprises any gas. In some instances, the gas comprises helium, hydrogen, air, or a combination thereof. In some cases, the working fluid comprises a thermal conductivity of at least about 0.15 W/mK at room temperature and room pressure. In some cases, the working fluid comprises a thermal conductivity of less than 0.15 W/mK at room temperature and room pressure. In some cases, the working fluid comprises a specific heat capacity of at least about 5.193 J/g°C at constant pressure (Cp). In some cases, the working fluid comprises a specific heat capacity of less than about 5. 193 J/g°C at constant pressure (Cp).

[0317] The systems disclosed herein may improve over systems in the art by providing, in some cases, a minimized dead volume to provide various benefits, including increased efficiency, improved power output, enhanced compression ratio, optimized heat exchange, smoother operation, reduced mechanical stress, enhanced control and compact FSPE architecture.

[0318] The FPSE may comprise an optimized engine architecture. In some cases, the FPSE may comprise a minimized dead volume. In some cases, the dead volume comprises space within the engine's cylinders that is not effectively used during the compression and expansion phases of the working fluid (e.g., typically a gas).

[0319] In some instances, the minimized dead volume may be, at least in part, a result of utilizing advanced manufacturing techniques (AM) that integrates multiple parts into one. For example, the heating head head, heating head fins and regenerator may be integrated into one. In alternative examples, the cooling head, cooling head fins, and regenerator may be integrated into one. In some cases, the FPSE comprises a topology optimized dead volume. In some cases, the FPSE comprises a topology optimized swept volume.

[0320] The systems disclosed herein may improve over systems in the art by providing, in some cases, a displacer configured to provide various benefits, including enhanced efficiency, reduced weight, improved durability and lifespan, optimized size, and compatibility with advanced manufacturing techniques (e.g., such as 3D printing).

[0321] FIG. 8 illustrates an example of a Free Piston Stirling Engine. In the example of FIG. 8, the FPSE 800 comprises a displacer 805.

[0322] In some cases, the displacer 805 is configured to move the bulk of a working fluid located between displacer and cold tip (e.g., as the piston expands the gas). In some instances, the temperature of the working fluid decreases (e.g., based on ideal gas law). For example, as gas temperature decreases it absorbs or lifts heat through cold tip causing a cold tip temperature to also decrease.

[0323] In some cases, the displacer 805 is configured to move the bulk of a working fluid located between the displacer and the piston (e.g., as the piston compresses the gas). In some instances, the temperature of the working fluid increases (e.g., based on ideal gas law). For example, as the gas temperature increases, it rejects heat through the heat exchanger to the environment.

[0324] In some cases, the displacer 805 is configured to facilitate the continuous operation of the FPSE. In some instances, by moving the cool gas back to the hot end for heating and expansion, the displacer provides the cycle to repeat, continuously converting heat energy into mechanical work.

[0325] In some cases, the displacer 805 is configured to move the heated gas from the hot end to the cold end of the engine. In some instances, the displacer 805 moving cools the working gas, preparing it for the next part of the Stirling cycle.

[0326] In some cases, a displacer 805 material comprises stainless steel, graphite, ceramic, composite, or a combination thereof. In some cases, a displacer material comprises a low- conductivity, heat-resistant material (e.g., such as stainless steel, ceramic, or graphite), to minimize heat loss and withstand the high temperatures of the hot end of the engine.

[0327] In some cases, a shape of the displacer is cylindrical, allowing it to comfortably fit within the cylindrical body of the engine, thereby promoting an effective back-and-forth movement of the working fluid.

[0328] In some cases, a displacer structure is made hollow, which reduces its weight and, therefore, the amount of energy necessary for moving it. In some cases, a dimensions of the displacer are carefully selected to ensure it fits inside the engine cylinder while leaving adequate clearance for the passage of the working fluid around its edges.

[0329] In some cases, the displacer might exhibit porosity or be dotted with small holes, facilitating the flow of the working fluid through the displacer, rather than merely around it. In some cases, the displacer 805 comprises a topology optimized structure. In some cases, the displacer 805 comprises a topology optimized volume. In some cases, the displacer 805 comprises a cold tip. In some cases, a cold tip temperature comprises between about 5°C to about 12°C temperature. In some cases, the cold tip temperature comprises greater than about 5°C. In some cases, the cold tip temperature comprises less than about 80°C.

[0330] In some cases, the displacer 805 comprises a hot tip. In some cases, the hot tip temperature comprises between about 75°C to about 80°C temperature. In some cases, the hot tip temperature comprises greater than about 75 °C. In some cases, the cold tip temperature comprises less than about 80°C.

[0331] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a power piston configured to provide various benefits, including increased efficiency, weight reduction, improved durability and lifespan, size optimization, and compatibility with advanced manufacturing techniques such as 3D printing.

[0332] In some cases, the power piston generates mechanical power. FIG. 8 illustrates an example of a Free Piston Stirling Engine. In the example of FIG. 8, the FPSE 800 comprises a power piston 804.

[0333] In some cases, the power piston 804 is configured to reciprocate back and forth. In some instances, the power piston motion 804 may be transferred to a load via a spring-mass system or directly used to generate electricity.

[0334] In some cases, the power piston 804 is configured to move as a response to the expansion of a working gas (e.g., such as hydrogen or helium) enclosed in a sealed space. In some instances, when the gas is heated by an external heat stream at the hot end, it expands, pushingthe power piston 804. For example, this motion may be transferred to a load via a spring-mass system or directly used to generate electricity.

[0335] In some cases, the power piston 804 is configured to move in response to the contraction of the cooling working gas. In some instances, the cooling of the gas creates a vacuum effect, pulling the power piston 804 back to its original position. In further examples, the motion of this piston, similar to the heating and expansion phase, may be used to perform work.

[0336] In some cases, the power piston 804 comprises a topology optimized material. In some instances, the topology optimized material comprises one or more of steel (e.g., for its strength and heat resistance), aluminum (e.g., for its light weight and good heat transfer capability), cast iron (e.g., for its excellent wear resistance and lubrication qualities), ceramics or composite materials (e.g., for high temperature operations), and bronze (e.g., for its high resistance to wear and excellent sliding properties). In some cases, the power piston 804 comprises a topology optimized structure.

[0337] In some cases, the power piston 804 comprises a topology optimized material. In some instances, the topology optimized material comprises one or more of steel (e.g., for its strength and heat resistance), aluminum (e.g., for its light weight and good heat transfer capability), cast iron (e.g., for its excellent wear resistance and lubrication qualities), ceramics or composite materials (e.g., for high temperature operations), and bronze (e.g., for its high resistance to wear and excellent sliding properties). In some cases, the power piston 804 comprises a topology optimized structure.

[0338] The systems disclosed herein may improve over systems in the art by providing, in some cases, an FPSE configured to repeat a cyclic process. [0339] In some cases, the FPSE continues to repeat the cyclic process indefinitely. In some cases, the cyclic process comprises cyclically compress and expand the working fluid, extracting heat from the waste heat source and converting it into mechanical power indefinitely.

Heat Source

[0340] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing a heat exchanger configured to provide various benefits, including utilize waste heat source from low grade temperature streams (e.g., below 230°C) and maximizing efficiency.

[0341] In some cases, the heat stream comprises a waste heat source. In some instances, the waste heat source may comprise a refrigerant. For example, the refrigerant may comprise R- 134a (Tetrafluoroethane, commonly used in automotive air conditioning systems and other cooling applications), R-410a (a blend of Difluorom ethane and Pentafluoroethane, commonly used in residential and commercial air conditioning systems), or ammonia.

[0342] In even further examples, the waste heat source may comprise waste heat source from a plurality of computers, waste heat source production systems, or other equipment. In further examples, the refrigerant may originate from different heating and cooling systems such as central air conditioning systems, refrigeration units in supermarkets, industrial cooling systems, or heat pumps.

[0343] In some cases, the heat stream comprises a temperature between about 75°C to about 85°C. In some instances, the heat stream comprises a temperature of about 80°C.

[0344] In some cases, the heat stream comprises a thermal conductivity up to about 0.2 W/mK at room temperature and room pressure.

Product By Process

[0345] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some cases, a method of designing a heat transfer apparatus configured to provide numerous benefits such as enhanced efficiency and precision in heat transfer architecture through integration of one-dimensional and three-dimensional Computational Fluid Dynamics modeling and topology optimization, capable of accommodating specific engine architecture parameters and optimizing for use in specific locations like waste heat source production systems.

[0346] In some embodiments, a method of designing a heat transfer apparatus for a FPSE is provided. In some cases, the heat transfer apparatus may be configured by one or more processes. In some instances, the one or more processes may comprise one-dimensional (ID) Computational Fluid Dynamics (CFD) modeling of a FPSE. In some instances, the one or more processes may comprise three-dimensional (3D) CFD modeling of a FPSE. In some instances, the one or more processes may comprise topology optimization (TO). In some instances, the one or more processes may comprise integrating two or more of onedimensional (ID) CFD modeling, three-dimensional (3D) CFD modeling, and topology optimization (TO). In some instances, the one or more processes may comprise integrating three or more of one-dimensional (ID) CFD modeling, three-dimensional (3D) CFD modeling, and topology optimization (TO). In some instances, the one or more processes may comprise integrating one-dimensional (ID) CFD modeling, three-dimensional (3D) CFD modeling, and topology optimization (TO).

[0347] In some instances, the architecture method may follow an ordered approach, beginning with one-dimensional (ID) Computational Fluid Dynamics (CFD) modeling, proceeding to three-dimensional (3D) CFD modeling, and culminating in topology optimization (TO). However, the flexibility of the methods disclosed allows for alternatives to this sequence. For example, three-dimensional (3D) CFD modeling may be executed prior to one-dimensional (ID) CFD modeling depending on the specific requirements or characteristics of the FPSE. Similarly, topology optimization (TO) may be initiated at various stages in the architecture process, either after the completion of ID CFD modeling and prior to 3D CFD modeling, or after the execution of both ID and 3D CFD modeling. Furthermore, each of these processes may operate synergistically to handle at least one engine architecture parameter, contributing to an optimized FPSE architecture tailored for its specific location, such as a waste heat source production system.

[0348] In some instances, the one or more processes may be configured to receive at least one engine architecture parameter configured for a FPSE. In some instances, the one or more processes may be configured to create a topology optimized FPSE for use in a specific location. For example, the specific location may comprise a waste heat source production system. In some instances, the one or more processes may comprise integrating three- dimensional (3D) modeling, three-dimensional (3D) Computational Fluid Dynamics (CFD), and topology optimization (TO).

[0349] In some instances, each process within the method, namely one-dimensional (ID) Computational Fluid Dynamics (CFD) modeling, three-dimensional (3D) CFD modeling, and topology optimization (TO), may serve as a dynamic input and output to each other, promoting an iterative and integrated approach to the architecture. For example, the results derived from the ID CFD modeling, for instance, may serve as valuable input for the 3D CFD modeling, helping to refine and add granularity to the latter process. In further examples, the outputs of the 3D CFD modeling may feed into the topology optimization, providing data that informs the optimization process, ultimately fine-tuning the architecture to best meet the parameters of the FPSE. In even further examples, the outcomes from the topology optimization may be looped back into the ID and 3D CFD modeling processes, influencing subsequent iterations and ly highlighting areas for further refinement or improvement. In further examples, the interconnected mechanism allows for a more holistic and adaptive architecture approach, yielding a heat transfer apparatus that is attuned to the specific requirements and conditions of its intended application.

ID Third Order Modeling

[0350] In some embodiments, the method comprises modeling, simulating, and optimizing a Free Piston Stirling Engine (FPSE) using third-order modeling (Nodal analysis). In some cases, the method comprises modeling, simulating, and optimizing a Free Piston Stirling Engine (FPSE) using one-dimensional (ID) third-order modeling (Nodal analysis).

[0351] In some cases, the method comprises defining control volumes. In some instances, control volumes are defined by dividing the engine's components into multiple segments. For example, Each segment contains a certain mass, with nodes serving as the boundaries of these volumes and set pressure and temperature.

[0352] In some cases, the method further comprises discretizing an FPSE domain into a plurality of building blocks. In some instances, each building block represents elemental components of the FPSE such as heat exchangers, regenerators, and pistons. In some instances, each building block comprises a localized self-contained entity. In some instances, the entire FPSE model comprises a summation of each component building blocks interconnected via mass flow rate, heat transfer, force, and pressure connectors. In some instances, the power and displacer pistons are represented as rigid moving components that cause volume displacement in compression and expansion spaces. In some instances, all components of the FPSE including heat exchangers, pistons, and working spaces are incorporated in the model. In some instances, the method further comprises specifying an average operating pressure for the FPSE using a pressure source. In some instances, the method further comprising connecting the endpoints of the system to specified heat sources. In some instances, the method further comprises the configuration is used to estimate nonproductive energy losses (e.g., parasitic losses). In some cases, the third-order modeling further comprises discretizing a fluid domain into a plurality of components. In some instances, the third-order modeling further comprises each component of the plurality of components comprises a control volume. For example, the third-order modeling further comprises defining mass, temperature, and pressure for each control volume. In some instances, the third-order modeling further comprises boundaries (e.g., nodes) between each control volume are used to represent and determine the mass flow rate between each control volume. In some cases, the method comprises each component of the engine, such as expansion and compression spaces, heat exchangers, and gaps, is divided into interconnected cells. In some instances, these cells interact to form a matrix for each variable within each component, considering both spatial and time discretization.

[0353] In some cases, the method comprises formulating governing equations. In some instances, the governing equations are formulated in accordance with the conservation principles of mass, momentum, and energy. For example, the governing equations may account for any non-idealities during the engine simulation.

[0354] In some cases, the method comprises simplifying differential equations into a onedimensional (ID) format. In some instances, the ID equations are then solved numerically with small incremental time steps and mathematical stabilization techniques.

[0355] In some cases, the method comprises calculating the precise distribution of pressure, temperature, and mass in the engine at each time step. In some instances, the method comprises further solving of differential equations for conservation of mass, momentum, and energy within the discretized fluid domain. In some instances, the third-order modeling further comprises solving the system of differential equations numerically. In some instances, the method further comprises solving the differential equations provides a detailed prediction of FPSE's performance under specified operating conditions.

[0356] In some cases, the method further comprises, upon reaching a stabilized state, attains a maximum thermal efficiency of between about 9.0% to about 10.0% for a heat source temperature of between about 70°C and about 85°C and a heatsink temperature of between about 1°C and about 10°C. In some cases, the third-order modeling further FPSE comprises a Carnot efficiency of between about 40% to about 50%.

[0357] In some cases, the method comprises a conducting a simulation using software like SAGE. In this model, each component of the engine is portrayed as a building block to construct the complete engine model.

Input Parameters

[0358] In some embodiments, the method comprises defining input parameters for a third order modeling of a FPSE. [0359] In some embodiments, the method comprises defining about 1 parameter, about 2 parameters, about 3 parameters, about 4 parameters, about 5 parameters, about 10 parameters, about 15 parameters, about 20 parameters, about 25 parameters, about 30 parameters, about 35 parameters, about 40 parameters, about 45 parameters, about 50 parameters, about 55 parameters, about 60 parameters, about 65 parameters, about 70 parameters, about 75 parameters, about 80 parameters, about 85 parameters, about 90 parameters, about 95 parameters, about 100 parameters, or any sub-range in-between. In some cases, the method comprises defining between about 75 different input parameters.

[0360] In some embodiments, the method comprises defining about 1 parameter to about 2,000 parameters. In some embodiments, the method comprises defining about 1 parameter to about 10 parameters, about 1 parameter to about 25 parameters, about 1 parameter to about 50 parameters, about 1 parameter to about 75 parameters, about 1 parameter to about 100 parameters, about 1 parameter to about 250 parameters, about 1 parameter to about 500 parameters, about 1 parameter to about 750 parameters, about 1 parameter to about 1,000 parameters, about 1 parameterto about 1,500 parameters, about 1 parameter to about 2,000 parameters, about 10 parameters to about 25 parameters, about 10 parameters to about 50 parameters, about 10 parameters to about 75 parameters, about 10 parameters to about 100 parameters, about 10 parameters to about250 parameters, about 10 parameters to about 500 parameters, about 10 parameters to about 750 parameters, about 10 parameters to about 1,000 parameters, about 10 parameters to about 1,500 parameters, about 10 parameters to about 2,000 parameters, about 25 parameters to about 50 parameters, about 25 parameters to about 75 parameters, about 25 parameters to about 100 parameters, about 25 parameters to about 250 parameters, about 25 parameters to about 500 parameters, about 25 parameters to about 750 parameters, about 25 parameters to about 1,000 parameters, about 25 parameters to about 1,500 parameters, about 25 parameters to about 2,000 parameters, about 50 parameters to about 75 parameters, about 50 parameters to about 100 parameters, about 50 parameters to about 250 parameters, about 50 parameters to about 500 parameters, about 50 parameters to about 750 parameters, about 50 parameters to about 1,000 parameters, about 50 parameters to about 1,500 parameters, about 50 parameters to about 2,000 parameters, about 75 parameters to about 100 parameters, about 75 parameters to about 250 parameters, about 75 parameters to about 500 parameters, about 75 parameters to about 750 parameters, about 75 parameters to about 1,000 parameters, about 75 parameters to about 1,500 parameters, about 75 parameters to about 2,000 parameters, about 100 parameters to about 250 parameters, about 100 parameters to about 500 parameters, about 100 parameters to about 750 parameters, about 100 parameters to about 1,000 parameters, about 100 parameters to about 1,500 parameters, about 100 parameters to about 2,000 parameters, about 250 parameters to about 500 parameters, about 250 parameters to about 750 parameters, about 250 parameters to about 1,000 parameters, about 250 parameters to about 1,500 parameters, about 250 parameters to about2,000 parameters, about 500 parameters to about 750 parameters, about 500 parameters to about 1,000 parameters, about 500 parameters to about 1,500 parameters, about 500 parameters to about 2,000 parameters, about 750 parameters to about 1,000 parameters, about 750 parameters to about 1,500 parameters, about 750 parameters to about 2,000 parameters, about 1,000 parameters to about 1,500 parameters, about 1,000 parameters to about 2,000 parameters, or about 1,500 parameters to about 2,000 parameters. In some embodiments, the method comprises defining about 1 parameter, about 10 parameters, about 25 parameters, about 50 parameters, about 75 parameters, about 100 parameters, about 250 parameters, about 500 parameters, about 750 parameters, about 1,000 parameters, about 1,500 parameters, or about2,000 parameters. In some embodiments, the method comprises defining at least about 1 parameter, about 10 parameters, about 25 parameters, about 50 parameters, about 75 parameters, about 100 parameters, about 250 parameters, about 500 parameters, about 750 parameters, about 1,000 parameters, or about 1,500 parameters. In some embodiments, the method comprises defining at most about 10 parameters, about 25 parameters, about 50 parameters, about 75 parameters, about 100 parameters, about 250 parameters, about 500 parameters, about 750 parameters, about 1,000 parameters, about 1,500 parameters, or about 2,000 parameters.

[0361] In some instances, the input parameters may comprise one or more of a mean average engine pressure, piston amplitude, phase angle, geometrical parameters, dynamic variables, the type of working gas, or a combination thereof. In some instances, the input parameters may comprise one or more of the internal diameter of the pressure vessel, the engine's mean charge pressure, piston diameter, piston amplitude, compression space volume, cooling head channel dimensions (e.g., width, height, and length), heating head channel dimensions, regenerator dimensions (length, wrapped foil gap, and thickness), displacer amplitude, displacer spring stiffness, and expansion space volume.

[0362] In some instances, the input parameters may comprise one or more of displacer rod diameter, power piston length, power piston angle, number of cooling head heat exchanger channels, number ofheating head HE channels, cooling head and heating head fin thickness, surface roughness for heating head, cooling head, and regenerator, power piston mass, displacer piston mass, heat exchangers, walls, engine operating frequency, working gas, heat sink temperature, heat source temperature.

[0363] In some embodiments, the input parameters contributing to the FPSE's operation include beneficial operational and physical aspects. In some cases, the operational and physical input parameters may comprise one or more of the internal diameter of the pressure vessel, the engine's mean charge pressure, piston diameter, piston amplitude, compression space volume, cooling head channel dimensions (e.g., width, height, and length), heating head channel dimensions, regenerator dimensions (length, wrapped foil gap, and thickness), displacer amplitude, displacer spring stiffness, and expansion space volume.

[0364] In some cases, the method comprises defining between about 0 and about 100 input parameters input parameters for one or more components of the FPSE. In some cases, the method comprises defining between about 0 and about 100 input parameters for each component. In some instances, the method comprises defining between about 0 and about 100 input parameters for about 1 component, about 2 components, about 3 components, about 4 components, about 5 components, about 6 components, about 7 components, about 8 components, about 9 components, about 10 components, about 11 components, about 12 components, about 13 components, about 14 components, about 15 components, about 16 components, about 17 components, about 18 components, about 19 components, about 20 components, about 21 components, about 22 components, about 23 components, about 24 components, about 25 components, about 26 components, about 27 components, about 28 components, about 29 components, about 30 components, about 31 components, about 32 components, about 33 components, about 34 components, about 35 components, about 36 components, about 37 components, about 38 components, about 39 components, about 40 components, about 41 components, about 42 components, about 43 components, about 44 components, about 45 components, about 46 components, about 47 components, about 48 components, about 49 components, about 50 components, or more. For example, the one or more components of the FPSE may comprise a regenerator, heating head, cooling head, piston, pressure shell, fluid channels, working spaces, displacer, heat exchanger fins, or a combination thereof.

[0365] In some cases, the method comprises diving the fluid domain inside the engine is into multiple control volumes or nodes. For example, each node may comprise a unique mass, temperature, and pressure. In some instances, the method comprises, for each node, differential equations representing the conservation of mass, momentum, and energy are established based on the fundamental laws of physics governing the engine's operation. For example, the differential equations are solved numerically using specific software tools.

SAGE

[0366] In some embodiments, the method comprises using software (e.g., SAGE software) to represent and calculate each individual component of the engine and their interactions. For example, the software may be configured to adjust the defined input parameters within certain ranges (e.g., improving the engine's thermal efficiency).

[0367] In some cases, the software (e.g., SAGE software) is configured to define input parameters within specified ranges using Software (e.g., SAGE software), a steady-periodic Stirling cycle architecture, and simulation program. In some instances, the optimization process adjusts the input parameters to enhance the engine's thermal efficiency, a metric defined as the ratio of net work output to input heat.

[0368] In some embodiments, the method further comprises defining optimization variables (e.g., such as input parameters), constraints to ensure geometric and thermodynamic architecture viability, and an objective function. In some cases, the method further comprises defining optimization variables (e.g., such as input parameters), constraints to ensure geometric and thermodynamic architecture viability, an objective function, or a combination thereof within the software (e.g., SAGE software). In some instances, the objective function comprises thermal efficiency, defined as the ratio of net work output to heat input (r|_thermal = w_net/Q_in).

[0369] In some embodiments, the method further comprises defining one or more constraints. In some cases, the one or more constraints may comprise one or more of ensuring that regenerator diameter is larger than displacer rod diameter, ensuring that pressure vessel diameter is larger than regenerator diameter, ensuring that form factor is reasonable with minimal flow distribution losses, ensuring that piston diameter is larger than displacer rod diameter, ensuring that piston diameter has almost the same diameter as regenerator, ensuring that there are enough volume in dead space of compression region to avoid collision, ensuring that there are enough volume in dead space of expansion region to avoid collision, ensuring that displacer runs freely by making components of phasor force zero, ensuring that required power output is obtained.

[0370] In some instances, the Displacer rod diameter comprises between about 0.005 [m] and about 0.02 [m]. In some instances, the Power piston length may comprise between about 0.025 [m] and about 0.1 [m]. For example, the Power piston angle may comprise between about 0 [deg] and 0 [deg]. In some instances, the number of cooling head and heating head Heat Exchanger channels may each comprise between about 1000 and about 4000. For example, the Cooling head and heating head fin thickness may comprise between about 2.5e- 4 [m] and le-3 [m], In some instances, the Surface roughness for the heating head, cooling head, and regenerator may comprise between about 5e-4 [m] and 2e-3 [m], In some instances, the Power piston mass comprises between about 0.5 [kg] and 2 [kg]. In some instances, the Displacer piston mass may comprise between about 0.125 [kg] and 0.5 [kg]. In some instances, the Engine operating frequency may comprise between about 15 [Hz] and 60 [Hz], In some instances, the Heat sink temperature may comprise between about 2.5 [C] and 10 [C], In some instances, the Heat source temperature may comprise between about 40 [C] and 160 [C],

[0371] In some embodiments, the method further comprises producing a plurality of output parameters. In some cases, the plurality of output parameters may comprise thermal efficiency, net power output (Wnet), input heat (Qin), heat out (Qout), or a combination thereof. In some instances, the output parameters are obtained from the software. For example, the output parameters may provide valuable insights into the performance of the engine under various operating conditions. In further examples, the output parameters may deliver a comprehensive framework for simulating and optimizing FPSE operation. In even further examples, the output parameters may promote efficient utilization of low-grade heat sources towards sustainable energy management and carbon footprint reduction.

[0372] In some cases, the organic FPSE further comprises a pressure vessel. In some instances, the one or more parameters comprises an internal diameter of the pressure vessel (Dpwall). For example, the Dpwall may comprise between about 2.000e-l [m] to about 3.000e-l [m], In further examples, the Dpwall may comprise at least 2.000e-l [m], In further examples, the Dpwall may comprise at most 3.000e-l [m], In further examples, the Dpwall may comprise between about 2.400e-l [m] and 2.800e-l [m], In further examples, the Dpwall may comprise between about 2.600e-l [m] and 2.700e-l [m], In further examples, the Dpwall may comprise between about 2.620e-l [m] and 2.630e-l [m], [0373] In further examples, the Dpwall increases with increase of power output. In even further examples, the Dpwall decreases with reduction of power output.

[0374] In some cases, one or more parameters comprises an internal diameter of the regenerator Dregen. In some instances, the Dregen comprises between about 1.000e-l [m] and about 2.000 e-1 [m], In some instances, the Dregen comprises between about 1.500e-l [m] and about 1.800 e-1 [m], In some instances, the Dregen comprises between about 1.700e-l [m] and about 1.800 e-1 [m]. In some instances, the Dregen comprises between about 1.740e-l [m] and about 1.750 e-1 [m],

[0375] In some instances, the Dregen comprises between about 9.5e-2 [m] to about 6e-l [m], In some instances, the Dregen comprises at most 1.750e-l [m]. In some instances, the Dregen comprises at least 1.740e-l [m],

[0376] In some cases, one or more parameters comprises a mean charge pressure (Pcharge) for FPSE operation. In some instances, the Pcharge comprises between about 50 bar and 90 bar. In some instances, the Pcharge comprises between about 40bar to about 80 bar. In some instances, the Pcharge comprises at most about 40bar. In some instances, the Pcharge comprises at least about 40bar. In some instances, the Pcharge comprises about 0 bar, about 5 bar, about 10 bar, about 15 bar, about 20 bar, about 25 bar, about 30 bar, about 35 bar, about 30 bar, about 30 bar, about 50 bar, about 55 bar, about 60 bar, about 65 bar, about 80 bar, about 75 bar, about 80 bar, about 85 bar, about 90 bar, about 95 bar, or about 100 bar. [0377] In some cases, one or more parameters comprises a diameter of the piston (Dpis).

[0378] In some instances, the Dpis comprises between about 1 .000e-l [m] and about 2.000 e- 1 [m]. In some instances, the Dpis comprises between about 1 ,500e-l [m] and about 1 .800 e- 1 [m]. In some instances, the Dpis comprises between about 1 ,700e-l [m] and about 1 .800 e- 1 [m]. In some instances, the Dregen comprises between about 1.710e-l [m] and about 1.720 e-1 [m]. In some instances, Dpis comprises between about 9.318e-2 [m] to about 5.88e-l [m]. In some instances, Dpis comprises at most about 1 ,720e-l [m]. In some instances, Dpis comprises at least about 1.710e-l [m],

[0379] In some cases, one or more parameters comprises an amplitude of the piston Xamp,pis. In some instances, the Xamp,pis comprises between about 7.000e-3 [m] and about 8.000e-3 [m]. In some instances, the Xamp,pis comprises between about 7.500e-3 [m] and about 8.000e-3 [m]. In some instances, the Xamp, pis comprises between about 7.700e-3 [m] and about 8.000e-3 [m]. In some instances, the Xamp, pis comprises between about 7.900e-3 [m] and about 8.000e-3 [m]. In some instances, the Xamp, pis comprises between about 7.950e-3 [m] and about 7.960e-3 [m],

[0380] In some instances, Xamp, pis comprises between about2.586e-3 [m] to about 1 ,074e-2 [m]. In some instances, the Xamp, pis comprises at most about 7.960e-3 [m]. In some instances, the Xamp, pis comprises at least about 7.950e-3 [m],

[0381] In some cases, one or more parameters comprises a compression space volume Vcompression. In some instances, Vcompression comprises between about 3.000e-4 [m^A3] and about 4.000e-4 [m^A3], In some instances, Vcompression comprises between about 3.500e-4 [m^A3] and about 3.800e-4 [m^A3], In some instances, Vcompression comprises between about3.600e-4 [m^A3] and about 3.700e-4 [m^A3], In some instances, Vcompression comprises between about 3.660e-4 [m^A3] and about 3.670e-4 [m^A3], In some instances,

Vcompression comprises between about 3.993E-5 [m^A3] to about 2.02e-3 [m^A3], In some instances, Vcompression comprises at least about 3.993E-5 [m^A3], In some instances,

Vcompression comprises at most about 3.993E-5 [m^A3],

[0382] In some cases, the optimized thermal efficiency (rjthermal) of the FPSE comprises between about 5.00% to about 15.00 %. In some cases, the optimized thermal efficiency

(rjthermal) of the FPSE comprises about 5 00%, about 6.00%, about 7.00%, about 8.00%, about 9.00%, about 10.00%, about 11.00%, about 12.00%, about 13.00%, about 14.00%, about 15.00%, about 16.00%, about 17.00%, about 18.00%, about 19.00%, about 20.00%, or any sub-range in-between. In some instances, the optimized thermal efficiency (^thermal) of the FPSE comprises between about 8.0% to about 14.0%. In some instances, the optimized thermal efficiency (^thermal) of the FPSE comprises at least about 8.0%. In some instances, the optimized thermal efficiency (rjthermal) of the FPSE comprises at most about 14.0%. [0383] In some cases, the optimized net power output (Wnet) of the FPSE comprises between about 0W to about 5000W. In some cases, the optimized net power output (Wnet) of the FPSE comprises about 0W, about 100W, about 600W, about 1100W, about 1600W, about 2100W, about 2600 W, about 3100W, about 3600W, about 4100W, about 4600W, or about 5100W, or any sub-range in-between. In some instances, the optimized net power output (Wnet) of the FPSE comprises between about 1000W to about 3000W. In some instances, the optimized net power output (Wnet) of the FPSE comprises at least about 1000W. In some instances, the optimized net power output (Wnet) of the FPSE comprises at most about 3000W.

[0384] In some instances, the optimized heat input (Qin) of the FPSE comprises about 0 kW, about 1 kW, about 2 kW, about 3 kW, about 4 kW, about 5 kW, about 6 kW, about 7 kW, about 8 kW, about 9 kW, about 10 kW, about 11 kW, about 12 kW, about 13 kW, about 14 kW, about 15 kW, about 16 kW, about 17 kW, about 18 kW, about 19 kW, about 20 kW, about 21 kW, about 22 kW, about 23 kW, about 24 kW, about 25 kW, about 26 kW, about 27 kW, about 28 kW, about 29 kW, about 30 kW, about 31 kW, about 32 kW, about 33 kW, about 34 kW, about 35 kW, about 36 kW, about 37 kW, about 38 kW, about 39 kW, about 40 kW, about 41 kW, about 42 kW, about 43 kW, about 44 kW, about 45 kW, about 46 kW, about 47 kW, about 48 kW, about 49 kW, to about 50 kW, or any sub-range in-between. In some cases, the optimized heat input (Qin) of the FPSE comprises between about 22kW to about 23 kW. In some instances, the optimized heat input (Qin) of the FPSE comprises between about 7.0 kW to about 35.0 kW. In some instances, the optimized heat input (Qin) of the FPSE comprises at least about 7.0 kW. In some instances, the optimized heat input (Qin) of the FPSE comprises at most about 35.00 kW.

[0385] In some cases, the optimized heat output (Qout) of the FPSE comprises between about 20.60 kW and about 21.00 kW. In some instances, the optimized heat output (Qout) of the FPSE comprises between about 6.00kW to about 32.00kW. In some instances, the optimized heat output (Qout) of the FPSE comprises about 0 kW, about 1 kW, about 2 kW, about 3 kW, about 4 kW, about 5 kW, about 6 kW, about 7 kW, about 8 kW, about 9 kW, about 10 kW, about 11 kW, about 12 kW, about 13 kW, about 14 kW, about 15 kW, about 16 kW, about 17 kW, about 18 kW, about 19 kW, about 20 kW, about 21 kW, about 22 kW, about 23 kW, about 24 kW, about 25 kW, about 26 kW, about 27 kW, about 28 kW, about 29 kW, about 30 kW, about 31 kW, about 32 kW, about 33 kW, about 34 kW, about 35 kW, about 36 kW, about 37 kW, about 38 kW, about 39 kW, about 40 kW, about 41 kW, about 42 kW, about 43 kW, about 44 kW, about 45 kW, about 46 kW, about 47 kW, about 48 kW, about 49 kW, to about 50 kW, or any sub-range in-between.

[0386] In some embodiments, the method further comprises running the one-dimensional code to simulate the energy generation over a period ranging from 24 to 72 hours, resulting in a total energy production ranging from 49kWh to 147kWh, under the assumption of fixed heat source and sink temperatures and availability of heat source for a duration varying from 12 to 36 hours per day. In some instances, the FPSE product produced through this method offers enhanced efficiency in harnessing low-grade heat sources, serving as an essential tool for sustainable energy management and carbon footprint reduction.

3D Third Order Modeling

[0387] In some embodiments, the method comprises modeling, simulating, and optimizing a Free Piston Stirling Engine (FPSE) using third-order modeling (Nodal analysis). In some cases, the method comprises modeling, simulating, and optimizing a Free Piston Stirling Engine (FPSE) using three-dimensional (3D) third-order modeling (Nodal analysis).

[0388] In some embodiments, the method comprises the initial preparation of a comprehensive 3D model architecture of the Free Piston Stirling Engine (FPSE). In some cases, the 3D model architecture is inclusive of all its internal and external components. In some instances, the method comprises computational fluid dynamics (CFD) analysis, which simulates the flow of fluid, heat transfer, and forces acting within the engine. In further embodiments, the method encompasses the preparation of the geometry of the model, including dimensions and details of components such as the heat exchanger fins and cylinders.

[0389] In additional embodiments, the method includes discretization of the 3D model using a polyhedral meshing technique to ensure accurate simulations. In some embodiments, the method comprises the discretization further comprising using polyhedral meshing to create conformal mesh interfaces between the parts. In some embodiments, the method comprises the contacting faces between different parts share a same boundary face topology. In some embodiments, the method further comprises performing simulations using four different base cell sizes to ensure mesh size sensitivity and check for mesh convergence. In some cases, the method increases the speed and accuracy of simulations due to the elimination of the need for face interpolation on contacting patches. In some embodiments, the method further comprises establishing between about one and about ten layers on interfaces between air and solid components to accurately capture thermal boundary layer. In some embodiments, the method further comprises identifying between about one and about 10 distinct simulation domains in the CAD geometry. In some cases, the method further comprises identifying between about four distinct simulation domains in the CAD geometry. In some instances, the four distinct simulation domains comprises specifically the stainless steel enclosure, air, copper fins, and an Inconel half-cylinder. In some embodiments, the method further comprises assigning each region a specific simulation model based on the material. In some embodiments, the method comprises setting boundary conditions. In some cases, the boundary conditions are set such that the enclosure has adiabatic walls.

[0390] In some embodiments, the method comprises the inner wall of a half cylinder has a convective heat transfer coefficient of between about 800 W/m²K at 300°C to about 1000 W/m²K at 300°C. In some cases, the method comprises the inner wall of a half cylinder has a convective heat transfer coefficient of between about 800 W/m²K at 300°C to about 900 W/m²K at 300°C. In some instances, the method comprises the inner wall of a half cylinder has a convective heat transfer coefficient of between about 850 W/m²K at 300°C to about 875 W/m²K at 300°C. For example, the method comprises the inner wall of a half cylinder has a convective heat transfer coefficient of between about 855 W/m²K at 300°C to about 865 W/m²K at 300°C.

[0391] In some embodiments, the method comprises inlets for each pipe are set with parameters. In some cases, the inlet pipe parameters may comprise a mass flowrate between about 0.001592 kg/s and about 0.006368 kg/s, temperature between about 325°C and 1300°C, or pressure between about 50,662.5 Pa and 202,650 Pa. In some instances, the inlet pipe parameters may comprise a mass flow rate between about 0.002388 kg/s and about 0.004776 kg/s, temperature between about 487.5°C and 975°C, or pressure between about 75,993.75 Pa and 151,987.5 Pa. For example, the method comprises inlets for each pipe are set with parameters. In some cases, the inlet pipe parameters may comprise a mass flow rate between about 0.002866 kg/s and about 0.003502 kg/s, temperature between about 585°C and 715°C, or pressure between about 91,192.5 Pa and 111,457.5 Pa.

[0392] In some embodiments, the method comprises pre-processing where each simulation domain, based on the material involved, is assigned a specific simulation model. In some cases, the method assigns appropriate boundary conditions to different parts of the FPSE model. In yet other embodiments, the method utilizes CFD simulations to predict heat transfer, flow patterns, and other performance details of the FPSE. In some instances, the method comprises post-processing activities such as conducting a mesh independence study to ensure the reliability of the simulation and analyzing results. In certain embodiments, the method involves visualization and analysis of the simulation results, including temperature distribution, heat flux, air velocity, and pressure contours. In other embodiments, the method comprises the use of a Topology Optimization (TO) algorithm to iteratively optimize the architecture for improved heat dissipation or pressure loss minimization.

[0393] In some embodiments, the method comprises efficient heat transfer from the hot air to the fins, with about 1830W of heat extracted, signifying a high level of efficiency. In other embodiments, the method involves effective temperature regulation, with the air temperature falling within a narrow band around 400°C after passing over the fins. In further embodiments, the method includes regulation of airflow within the engine, demonstrated by a steady mass flow rate, which helps maintain desired temperature levels and optimize engine performance. In some cases, the method ensures an even distribution of heat flux across the external and internal surfaces of the fins, preventing localized heat concentration and thereby enhancing overall engine efficiency. In subsequent embodiments, the method includes conductinga mesh independence study to validate the reliability of the simulation, ensuring that outcomes may be modified by variations in the mesh size. Finally, in some embodiments, the method involves the employment of topology optimization in enhancing the heat exchanger architecture, thereby boosting the overall performance of the Free Piston Stirling Engine (FPSE).

Topology Algorithm

[0394] The systems, the methods, and the techniques disclosed herein may improve over systems in the art by providing, in some embodiments, the organic topology of the at least one heat transfer apparatus was generated by a topology algorithm. In some instances, the topology algorithm is configured to receive one or more parameters of the FPSE. For example, the topology algorithm may be configured to generate the organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE. [0395] In some embodiments, the method comprises preparation of the model. In some cases, the preparation of the model comprises the initial model preparation involves setting up predefined solver parameters within a simulation software (e.g., such as STAR-CCM+). In some cases, the method comprises defining the geometry and meshing, establishing boundary conditions, and initializing the material properties.

[0396] In some embodiments, the method comprises sensitivity analysis of TO parameters. In some cases, the method comprises, wherein the initial testing of the model results in suboptimal performance, a parametric study is initiated to assess the modify of various TO solver parameters on the objective function (total heat transfer rate in this case). In some instances, the analysis explores parameters such as topology holes and source strength, penalty value, intensity of the optimized fin surface smoothing, and step size.

[0397] In some embodiments, the method comprises initialization of the TO process. In some cases, after the sensitivity analysis and parametric studies, the TO algorithm is initialized. In some instances, the algorithm iteratively adjusts the distribution of material within the component (like the fins) based on the objective function.

[0398] In some cases, the method comprises geometry optimization. In some instances, based on the TO results, the 3D geometry of the component is optimized to improve heat dissipation or minimize pressure loss. For example, the resulting geometry usually has an organic shape that provides efficient heat transfer performance.

[0399] In some cases, the method comprises post-processing & CFD verification. In some instances, after obtaining the optimized geometry, the topology optimized heat transfer apparatus undergoes smoothing and is reintroduced into the simulation for validation and verification through Computational Fluid Dynamics (CFD) simulations. For example, this may refine the engine's performance.

[0400] In some cases, the method comprises evaluation and Analysis. In some instances, the final step comprises evaluation and analysis of the results obtained from TO and CFD. For example, the effectiveness and efficiency of the topology -optimized components are then assessed, providing valuable insights into the performance enhancement they bring about. [0401] In some cases, the organic topology of the at least one heat transfer apparatus is substantially devoid of right angles. In some cases, the organic topology of the at least one heat transfer apparatus is substantially devoid of straight lines. In some cases, the organic topology of the at least one heat transfer apparatus comprises an amorphous shape.

[0402] In some cases, at least part of the organic topology comprises a form configured to encase a pressure vessel. In some cases, at least part of the organic topology comprises a form configured to shield a high-pressure chamber, envelop a pressure container, surround a compressed gas cylinder, cover a high-pressure vessel, or envelop a hydraulic reservoir. [0403] In some instances, at least part of the organic topology comprises a form configured for a heat transfer rate up to a maximum theoretical limit. In some instances, with the assumption of maximum theoretical heat transfer limit between two temperature differentials being 1, the heat transfer comprises about 0.50 to about 0.75 of the max limit. In some instances, with the assumption of maximum theoretical heat transfer limit between two temperature differentials being 1, the heat transfer comprises about 0.10, about 0.20, about 0.30, about 0.40, about 0.50, about 0.60, about 0.70, about 0.80, about 0.90, to about 1.00 of the max limit, or any sub-range in between.

[0404] For example, the heat transfer may comprise at least about 0.52, 0.53, 0.54, 0.55, 0.56, 0.57, 0.58, 0.59, 0.60, 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.69, 0.70, 0.71, 0.72, 0.73, 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.80, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.89, 0.90, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, to 0.99 of the max theoretical limit.

[0405] In some embodiments, the method comprises modeling, simulating, and optimizing a Free Piston Stirling Engine (FPSE) using a topology optimization algorithm,

[0406] In some embodiments, the topology algorithm utilizes gradient-based Topology (TO) for heat transfer and pressure optimization. In some embodiments, the TO process begins by defining the architecture space envelope as well as the flow/thermal objective functions together with TO input parameters. In some embodiments, the TO process utilizes conformal mesh to ensure conservation of heat transfer values. In some embodiments, the surface mesh between all parts on the interface shares points at the interface to conserve heat transfer between parts. In some embodiments, the smaller mesh cells result in finer fin surface. In some embodiments, the smaller mesh cells result in larger fin surface area. In some embodiments, the method further comprises geometry preparation for full CFD simulation with Boundary Layers (BL). In some embodiments, the geometry preparation comprises smoothing out initial derived part geometry, using a surface wrapper to create a watertight surface, importing the watertight surface of the optimized geometry to the full case, subtracting the watertight surface from the air domain, imprinting it to the half cylinder geometry, and ensuring all required interfaces are created properly. In some embodiments, the jagged geometry obtained from TO is smoothed out and then re-imported back for CFD simulation for validation and verification purposes.

[0407] In some cases, the method comprises providing one or more parameters of the FPSE into a topology algorithm. In some instances the topology algorithm is configured to generate an organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE. In some cases, the topology algorithm is configured to generate the organic topology tailored to a specific application of the heat transfer apparatus. In some instances, the specific application may comprise for use in waste heat source production systems.

[0408] In some instances, the topology algorithm iteratively adjusts the distribution of material within a predefined domain based on gradients, seeking the improved structure to meet the defined objective, such as maximizing heat dissipation or minimizing pressure loss, with the aid of adjoint solver capabilities.

[0409] In some embodiments, the topology optimization (TO) algorithm is configured is to augment heat transfer from hot air to the engine's fins. In some instances, the TO algorithm is configured to enhance the heat transfer up to about 1830W (e.g., from a max from a theoretical value of about 3566W). This amplification of heat extraction capabilities highlights the FPSE's to capitalize on low-grade heat sources more efficiently.

Product

[0410] FIG. 31 illustrates a cross section of an FPSE, labeled as 3100. In some cases, FIG. 31 provides an overview of a conventional engine with specific components earmarked for topology optimization.

[0411] In some embodiments, component 3101 of the FPSE may be subject to topology optimization, in some cases, component 3101 may comprise elements such as a regenerator, heating head, cooling head, or heat exchanger fins.

[0412] In some embodiments, component 3102 may be selected for topology optimization. In some cases, component 3102 may comprise components like a regenerator, heating head, cooling head, or heat exchanger fins.

[0413] In some cases, component s 103 may be selected for topology optimization. In some instances, component s 103 may comprise elements like a regenerator, heating head, cooling head, or heat exchanger fins. [0414] In some embodiments, component 3104 may be subject to topology optimization. In some cases, component 3104 may incorporate components such as a regenerator, heating head, cooling head, or heat exchanger fins.

[0415] In some embodiments, component 3105 of the FPSE may be optimized using topology optimization methods. In some cases, component 3105 might comprise elements like a regenerator, heating head, cooling head, or heat exchanger fins.

[0416] FIG. 32 illustrates a topology-optimized component of an FPSE 3200. In some cases, component 3200 is the result of implementing topology optimization techniques on one or more of the parts 3101-3105 as shown in FIG. 31 . For instance, the optimized FPSE component 3200 may comprise a topology-optimized version of part 3401 (e.g., which may comprise a regenerator, heating head, cooling head, or heat exchanger fins). In further examples, after topology optimization, component 3200 represents an improved architecture with enhanced performance characteristics.

[0417] In further examples, Similar optimization techniques may also be applied to other parts of the FPSE, such as parts 3102, 3103, 3104, and 3105. In even further examples, each of these parts may comprise components like a regenerator, heating head, cooling head, or heat exchanger fins, which may be subject to topology optimization (e.g., to achieve operational efficiency).

[0418] In some cases, the method further comprises generating manufacturing instructions derived from the organic topology. In some instances, the method further comprises manufacturing the heat transfer apparatus (e.g., such as component 3200) based on the generated manufacturing instructions. In some cases, the architecture of the apparatus is guided by the organic topology. In some instances, the manufacturing comprises an additive manufacturing process. For example, the additive manufacturing process may comprise Direct Metal Laser Sintering (DMLS).

[0419] FIG. 58 illustrates an example of a first view of topology optimized heat exchanger. In some embodiments, the first view may comprise a top view the topology optimized heat exchanger. In some cases, the first view may comprise a top view of an external surface of the topology optimized heat exchanger. FIG. 59 illustrates an example of a second view of topology optimized heat exchanger. In some embodiments, the second view may comprise a side view the topology optimized heat exchanger. In some cases, the second view may comprise a side view of an external surface of the topology optimized heat exchanger.

[0420] FIG. 60 illustrates an example of a third view of topology optimized heat exchanger. In some embodiments, the third view may comprise a bottom view the topology optimized heat exchanger. In some cases, the third view may comprise a bottom view of an external surface of the topology optimized heat exchanger. In some cases, the third view may comprise a bottom view of an internal surface of the topology optimized heat exchanger. [0421] In some embodiments, the topology optimized heat exchanger may comprise a symmetrical pattern of holes. In some cases, the symmetrical pattern of holes may correlate to natural heat, pressure, temperature, or other contours of an input heat stream. In some instances, the input heat stream may comprise a output from a building. For example, the building may comprise a commercial or residential building.

[0422] In some cases, the topology optimized heat exchanger may comprise a plurality of heat exchanger fins. In some instances, the plurality of heat exchanger fins may be configured to encase one or more of a heating head, cooling head, regenerator, pressure shell, or other engine components.

[0423] In some cases, the topology optimized heat exchanger may comprise a plurality of organic shapes. In some cases, the topology optimized heat exchanger may comprise a plurality of non-planar surfaces. In some cases, the topology optimized heat exchanger may comprise a plurality of angles. In some cases, the topology optimized heat exchanger may comprise a plurality of recessed surfaces. In some cases, the topology optimized heat exchanger may comprise a plurality of protruding surfaces.

[0424] In some embodiments, the topology optimized heat exchanger may comprise an organic topology of the at least one heat transfer apparatus is substantially devoid of right angles. In some embodiments, the topology optimized heat exchanger may comprise an organic topology of the at least one heat transfer apparatus is substantially devoid of straight lines. In some embodiments, the topology optimized heat exchanger may comprise organic topology of the at least one heat transfer apparatus comprises an amorphous shape. In some embodiments, the topology optimized heat exchanger may comprise at least part of the organic topology comprises a form configured to encase a pressure vessel and for a heat transfer rate of up to about 0.67x the max theoretical limit. In some embodiments, the topology optimized heat exchanger may comprise FPSE improves a heat transfer rate (0.67/0.5 l)x that of a conventionally -finned FPSE.

Heat Recovery Systems

[0425] Referring to FIG. 33, an example of a Heat recovery system (HRS) 3300 is presented, integrating an FPSE with the cooling process of waste heat source production system equipment with a fluid. This creates a cycle that not only dissipates heat from the waste heat source production system but also generates additional electricity usingthe heat absorbed by the cooling fluid from the waste heat source production system equipment as a heat source. [0426] In system 3300. electricity is transferred to the waste heat source production system 3304 via an input electricity connection 3301.

[0427] Further, the system comprises a connection 3307 between the waste heat source production system 3304 and engine 3302. In some instances, via connection 3307 one or more waste heat source flows to the engine 3302.

[0428] Further, the system comprises a connection 3303 between the engine and a heat exchanger 3305. In some cases, the heat exchanger 3305 may cool down the waste heat source.

[0429] Further, the system comprises a connection 3308 between the engine 3302 and the heat exchanger 3305. In some cases, the waste heat source (e.g., cooled waste heat source) flows from the heat exchanger 3305 to the chiller 3309 via connection 3308.

[0430] Further the system comprises a connection 3310 between the chiller 3309 and the waste heat source production system 3304. In some cases, the chiller 3309 outputs, for example, cooled waste heat source to the waste heat source production system 3304 via connection 3310. In some instances, the cooled waste heat source picks up heat from the waste heat source production system 3304 to produce heated waste heat source. For example, the heated waste heat source may flow to the engine 3302 via connection 3307.

[0431] In some cases, the heat exchanger 3305 may be configured to perform a heat exchange with an ambient environment. In some instances, the heat exchanger 3305 may be configured to perform a heat exchange with an adjacent system like a heating, ventilation, and air conditioning (HVAC) system, an industrial process that generates excess heat, a cooling fluid such as water or refrigerant, or a geothermal energy source underground.

[0432] In some cases, the engine may convert thermal energy carried by the fluid into mechanical energy. In some instances, the engine may convert the mechanical energy into electrical power. For example, the engine may enhance overall energy efficiency.

[0433] In some cases, a chilling fluid (e.g., water) is circulated through the cold side of the FPSE to create a temperature differential. In some instances, the chilling fluid cooling down the working gas. For example, the chilling fluid may cool down the working gas while the FPSE converts excess thermal energy into electricity.

[0434] In some cases, the HRS 3300 (as depicted in FIG. 33) comprises several advantages, such as reduced energy consumption by reusing waste heat source for power generation. In some instances, the HRS 3300 may reduce energy consumption by up to about lOOkWh or more, under the assumption of fixed heat source and sink temperatures and availability of heat source for 24 hours per day.

[0435] FIG. 34 illustrates an example of a Heat recovery system (HRS) for a closed loop cooling system. In some embodiments, the HRS comprises a closed loop system 3400. In some cases, the closed loop system 3400 comprises a first recirculation loop 3406. In some instances, closed loop system 3400 comprises a chiller 3403. For example, the chiller 3403 may output cold fluid to engine 3401 via connection 3404.

[0436] In some cases, the engine 3401 comprises a coolinghead 3405. In some instances, a cold fluid flows into the cooling head 3405 of the engine. For example, the cold fluid may cool the working gas inside the engine 3401 .

[0437] In some instances, closed loop system 3400 comprises a chiller 3403. For example, a warm fluid may flow out of engine 3401 via connection 3402 to chiller 3403. In further examples, first recirculation loop 3406 recirculates continuously. In some cases, the closed loop system 3400 comprises a second recirculation loop 3412. In some instances, engine 3401 further comprises a heating head 3410. In some instances, a computing system 3407 is connected to heating head 3409 via connection 3408. For example, a waste heat source may flow out of computing system 3407 into heating head 3409 via connection 3408. In further examples, the waste heat source may transfer heat to the working gas inside engine 3401 . [0438] In some instances, heating head 3409 is connected to a chiller 3411 via connection 3410. For example, the waste heat source may flow out of heating head 3409 into chiller 3411 via connection 3410. In some instances, the waste heat source may enter the heating head 3409 at a first temperature range. In some instances, the waste heat source may exit the heating head 3409 at a second temperature range. For example, the second temperature may be less than or equal to the first. In further examples, the first temperature range may comprise between about 75°C to about 80°C and the second temperature may comprise a maximum temperature of about 80°C.

[0439] In some instances, waste heat source may enter the chiller 3411 via connection 3410 at the second temperature range. For example, the chiller 3411 may cool the waste heat source such that waste heat source exits the chiller 3411 at a third temperature range. In further examples, the third temperature range may be less than or equal to the second temperature range.

[0440] In some instances, the chiller 3411 may be connected to computing system 3407 via connection 3412. For example, the waste heat source at the third temperature range may travel via fluid connection 3412 to the computing system 3407. In further examples, the second recirculation loop 3413 recirculates continuously.

[0441] In some embodiments, the computing system 3407 comprises a waste heat source production system. In some cases, the waste heat source comprises a fluid. In some instances, the first temperature range comprises between about 75°C and about 105°C. In some instances, the second temperature range comprises at most about 105°C . In some instances, the third temperature range comprises between about 15°C and about 50°C.

[0442] FIG. 35 illustrates an example of a Heat recovery system for an open loop cooling system, in accordance with certain embodiments.

[0443] In some embodiments, the HRS comprises an open loop cooling system 3500.

[0444] In some cases, the open loop cooling system 3500 comprises an input fluidic connection 3503. In some instances, the fluidic connection 3503 may comprise a connection between an input stream source 3502 and engine 3501. For example, the engine 3501 comprises a cooling head 3504. In further examples, the input stream source 3502 may comprise a cooling fluid source. For example, the cooling fluid source may comprise a water source. In further examples, the water source may comprise an underground water source, rainwater collection system, a municipal water supply, a well, a lake, a river, a reservoir, a desalination plant, a recycled or reclaimed water system, a snowmelt collection system, a natural spring, or a combination thereof.

[0445] In some instances, the cooling fluid flows into the cooling head 3504 of the engine 3501. For example, the cooling fluid may cool the working gas inside the engine 3501.

[0446] In some cases, the open loop cooling system comprises an output fluidic connection 3505. In some instances, the fluidic connection 3505 may comprise a connection between the engine 3501 and an external unit 3506. For example, the cooling fluid may flow out of engine 3501 via connection 3502 to the external unit 3506. In further examples, the external unit 3506 may comprise an external cooling tower, additional heat exchanger (e.g., such as a fin to heat exchanger pushing ambient fan through it). In further examples, the external unit 3506 may comprise an underground water source, rainwater collection system, a municipal water supply, a well, a lake, a river, a reservoir, a desalination plant, a recycled or reclaimed water system, a snowmelt collection system, a natural spring, or a combination thereof.

[0447] In some instances, the cooling fluid may enter the cooling head 3504 at a first cooling fluid temperature range. In some instances, the cooling fluid may exit the cooling head 3504 at a second cooling fluid temperature range. For example, the second cooling fluid temperature range may be greater than or equal to the first cooling fluid temperature range. In further examples, the first cooling fluid temperature range may comprise a temperature of between about 10°C and about 15°C. In further examples, the second cooling temperature range may comprise between about 20°C and 25°C.

[0448] In some cases, the open loop cooling system 3500 comprises a recirculation loop 3513. In some instances, engine 3501 further comprises a heating head 3509. In some instances, computing system 3507 is connected to heating head 3509 via connection 3508. For example, the waste heat source may flow out of computing system 3507 into heating head 3509 via connection 3508. In further examples, the waste heat source may transfer heat to the working gas inside engine 3501.

[0449] In some instances, heating head 3509 is connected to waste heat source chiller 3511 via connection 3510. For example, the waste heat source may flow out of heating head 3508 into a chiller 3511 via connection 3510. In some instances, the waste heat source may enter the heating head 3509 at a first waste heat source temperature range. In some instances, the waste heat source may exit the heating head 3509 at a second waste heat source temperature range. For example, the second waste heat source temperature range may be less than or equal to the first waste heat source temperature range. In further examples, the first waste heat source temperature range may comprise between about 75°C -80°C and the second waste heat source temperature range may comprise a maximum temperature of about 80°C. [0450] In some instances, the waste heat source may enter the chiller 3511 via connection 3510 at the second waste heat source temperature range. For example, the waste heat source chiller 3511 may cool the waste heat source such that waste heat source exits the chiller 3511 at a third temperature range. In further examples, the third waste heat source temperature range may be less than or equal to the second waste heat source temperature range.

[0451] In some instances, the waste heat source chiller 3511 may be connected to computing system 3507 via connection 3512. For example, the waste heat source at the third waste heat source temperature range may travel via fluid connection 3512 to the computing system 3507. In further examples, the second recirculation loop 3513 recirculates.

[0452] In some embodiments, the computing system 3507 comprises a waste heat source production system. In some cases, the waste heat source comprises a fluid. In some instances, the first temperature range comprises between about 75°C and about 105°C. In some instances, the second temperature range comprises at most about 105°C . In some instances, the third temperature range comprises between about 15°C and about 50°C. [0453] In some embodiments, the cooling fluid may comprise any liquid, gas or solid configured to remove heat from a working gas. In some cases, the cooling fluid may comprise water. In some cases, the cooling fluid may comprise air, refrigerants like Freon, ethylene glycol, propylene glycol, dielectric liquids, mineral oil, synthetic oils, helium gas, or even a mixture of water and antifreeze. In some cases, the waste heat source may comprise phase-change materials that absorb heat through melting and solidification.

Buildings

[0454] FIG. 57 illustrates an example of a Waste heat source Recovery System (WHRS) integrated into a building. In some embodiments, the building may comprise a commercial building. In some embodiments, the building may comprise a residential building.

[0455] In another aspect disclosure herein are waste heat source management systems for a commercial building, comprising: a heat exchanger, an evaporator in fluidic connection with the heat exchanger, a condenser in fluidic connection with the heat exchanger, a first conduit comprising a first stream, wherein the first stream comprises a first temperature, a second conduit comprising a second stream, wherein the second stream comprises a second temperature, and a third conduit comprising a third stream, wherein the third stream comprises a third temperature, and a Free Piston Stirling Engine (FPSE).

[0456] In some embodiments, the system further comprises the third conduit thermally connected to the first conduit. In some embodiments, the system further comprises the third conduit fluidically connected to a heating head of the FPSE. In some embodiments, the system further comprises the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the system further comprises a cooling head of the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the first stream comprises an HVAC refrigerant stream. In some embodiments, the second stream comprises a cooling fluid. In some embodiments, the third stream comprises a heating fluid. In some embodiments, a temperature of the first stream is equal to or greater than a temperature of the third stream. In some embodiments, a temperature of the third stream is equal to or greater than a temperature of the second stream. In some embodiments, the first stream comprises an output stream from the HVAC unit. In some embodiments, the first stream comprises an output stream from the evaporator. In some embodiments, the first stream comprises an input stream to the heat exchanger. In some embodiments, the first stream comprises an output stream from the heat exchanger. In some embodiments, the first stream comprises an input stream to the condenser. In some embodiments, the first stream comprises an output stream from the condenser. In some embodiments, the first stream comprises an input stream to the evaporator. In some embodiments, the first stream comprises an output stream from the evaporator. In some embodiments, the first stream comprises an input stream to a heat exchanger. In some embodiments, the second stream comprises an output stream from a heat sink. In some embodiments, the second stream comprises an input stream to a cooling head of the FPSE. In some embodiments, the second stream comprises an output stream from a cooling head of the FPSE. In some embodiments, the second stream comprises an input stream to a heat sink. In some embodiments, the second stream comprises an output stream from a heat sink. In some embodiments, the third stream comprises an output stream from a heating head of a FPSE. In some embodiments, the third stream comprises an input stream to the heat exchanger. In some embodiments, the third stream comprises an output stream from the heat exchanger. In some embodiments, the third stream comprises an input stream to the heating head of the FPSE. In some embodiments, the second stream enters the FPSE at a cold cylinder side. In some embodiments, the second stream removes heat from the cold cylinder side. In some embodiments, the third stream enters the FPSE at a hot cylinder side. In some embodiments, the third stream transfers heat to the hot cylinder side. In some embodiments, the FPSE is configured to convert thermal energy from the third stream into mechanical energy, and then into electrical power. In some embodiments, the system further comprises a temperature differential between the hot cylinder side and the cold cylinder side. In some embodiments, the first conduit is configured to fluidically connect the condenser to the evaporator and to the heat exchanger. In some embodiments, the second conduit is configured to fluidically connect the heat sink to the cooling head. In some embodiments, the third conduit is configured to fluidically connect the heat exchanger to the heating head. In some embodiments, the system comprises a closed loop system. In some embodiments, the system comprises an open loop system. In some embodiments, the second stream comprises water. In some embodiments, the second stream comprises air. In some embodiments, the WHRS system is placed outside a commercial building. In some embodiments, the WHRS system is placed on top of a commercial building. In some embodiments, the WHRS system is placed inside a commercial building. In some embodiments, the WHRS system is fluidically connected to a commercial building. In some embodiments, the WHRS system is thermally connected to a commercial building.

[0457] In some embodiments, the WHRS system is placed outside a residential building. In some embodiments, the WHRS system is placed on top of a residential building. In some embodiments, the WHRS system is placed inside a residential building. In some embodiments, the WHRS system is fluidically connected to a residential building. In some embodiments, the WHRS system is thermally connected to a residential building. [0458] FIG. 61 illustrates an example of a Waste heat source Recovery System (WHRS) integrated into a building. The system in FIG. 61 may comprise a waste heat source management system (WHMS) for a building. In some embodiments, the WHMS may comprise a heat exchanger. In some embodiments, the WHMS may comprise an evaporator in fluidic connection with the heat exchanger. In some embodiments, the WHMS may comprise a condenser in fluidic connection with the heat exchanger. In some embodiments, the WHMS may comprise a first conduit comprising a first stream, wherein the first stream comprises a first temperature. In some embodiments, the WHMS may comprise a second conduit comprising a second stream, wherein the second stream comprises a second temperature. In some embodiments, the WHMS may comprise a third conduit comprising a third stream, wherein the third stream comprises a third temperature. In some embodiments, the WHMS may comprise a Free Piston Stirling Engine (FPSE).

[0459] In some embodiments, the system further comprises an expansion valve in fluidic connection with condenser and the evaporator. In some embodiments, the system further comprises condenser input comprises a heat exchanger output. In some embodiments, the system further comprises condenser input comprises a warm refrigerant. In some embodiments, the system further comprise expansion valve is configured to receive a condenser output. In some embodiments, the system further comprises condenser output comprises a lower temperature than the condenser input. In some embodiments, the system further comprises condenser output comprises a cold refrigerant. In some embodiments, the system further comprises expansion valve input comprises a condenser output. In some embodiments, the system further comprises expansion valve output comprises a chilled refrigerant. In some embodiments, the system further comprises expansion valve output comprises a lower temperature than the expansion valve input. In some embodiments, the system further comprises a compressor in fluidic connection with the heat exchanger and the evaporator. In some embodiments, the system further comprises compressor is configured to receive an evaporator output and produce a heat exchanger input. In some embodiments, the system further comprises compressor input comprises a lower temperature than a compressor output. In some embodiments, the system further comprises heat exchanger input comprises hot refrigerant. In some embodiments, the system further comprises heat exchanger output comprises warm refrigerant. In some embodiments, the system further comprises heat exchanger output comprises a heating head input. In some embodiments, the system further comprises a heating head output comprises a heat exchanger input. In some embodiments, the system further comprises heating heat output comprises a lower temperature than the heating head input. In some embodiments, the system further comprises evaporator comprises a building input. In some embodiments, the system further comprises building input comprises cold air. In some embodiments, the system further comprises building output comprises an evaporator input. In some embodiments, the system further comprises building output comprises hot air. In some embodiments, the system further comprises the third conduit thermally connected to the first conduit. In some embodiments, the system further comprises the third conduit fluidically connected to a heating head of the FPSE. In some embodiments, the system further comprises the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the system further comprises a cooling head of the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit. In some embodiments, the system further comprises the first stream comprises an HVAC refrigerant stream. In some embodiments, the system further comprises the second stream comprises a cooling fluid. In some embodiments, the system further comprises the third stream comprises a heating fluid. In some embodiments, the system further comprises a temperature of the first stream is equal to or greater than a temperature of the third stream. In some embodiments, the system further comprises a temperature of the third stream is equal to or greater than a temperature of the second stream. In some embodiments, the system further comprises the first stream comprises an output stream from the HVAC unit. In some embodiments, the system further comprises the first stream comprises an output stream from the evaporator. In some embodiments, the system further comprises the first stream comprises an input stream to the heat exchanger. In some embodiments, the system further comprises the first stream comprises an output stream from the heat exchanger. In some embodiments, the system further comprises the first stream comprises an input stream to the condenser. In some embodiments, the system further comprises the first stream comprises an output stream from the condenser. In some embodiments, the system further comprises the first stream comprises an input stream to the evaporator. In some embodiments, the system further comprises the first stream comprises an output stream from the evaporator. In some embodiments, the system further comprises the first stream comprises an input stream to a heat exchanger. In some embodiments, the system further comprises the second stream comprises an output stream from a heat sink. In some embodiments, the system further comprises the second stream comprises an input stream to a cooling head of the FPSE. In some embodiments, the system further comprises the second stream comprises an output stream from a cooling head of the FPSE. In some embodiments, the system further comprises the second stream comprises an input stream to a heat sink. In some embodiments, the system further comprises the second stream comprises an output stream from a heat sink. In some embodiments, the system further comprises the third stream comprises an output stream from a heating head of a FPSE. In some embodiments, the system further comprises the third stream comprises an input stream to the heat exchanger. In some embodiments, the system further comprises the third stream comprises an output stream from the heat exchanger. In some embodiments, the system further comprises the third stream comprises an input stream to the heating head of the FPSE. In some embodiments, the system further comprises the second stream enters the FPSE at a cold cylinder side. In some embodiments, the system further comprises the second stream removes heat from the cold cylinder side. In some embodiments, the system further comprises the third stream enters the FPSE at a hot cylinder side. In some embodiments, the system further comprises the third stream transfers heat to the hot cylinder side. In some embodiments, the system further comprises the FPSE is configured to convert thermal energy from the third stream into mechanical energy, and then into electrical power. In some embodiments, the system further comprises a temperature differential between the hot cylinder side and the cold cylinder side. In some embodiments, the system further comprises the first conduit is configured to fluidically connect the condenser to the evaporator and to the heat exchanger. In some embodiments, the system further comprises the second conduit is configured to fluidically connect the heat sink to the cooling head. In some embodiments, the system further comprises the third conduit is configured to fluidically connect the heat exchanger to the heating head. In some embodiments, the system further comprises the system comprises a closed loop system. In some embodiments, the system further comprises the system comprises an open loop system. In some embodiments, the system further comprises the second stream comprises water. In some embodiments, the system further comprises the second stream comprises air. In some embodiments, the system further comprises the WHRS system is placed outside a commercial building. In some embodiments, the system further comprises the WHRS system is placed on top of a commercial building. In some embodiments, the system further comprises the WHRS system is placed inside a commercial building. In some embodiments, the system further comprises the WHRS system is fluidically connected to a commercial building. In some embodiments, the system further comprises the WHRS system is thermally connected to a commercial building.

[0460] In some embodiments, the system further comprises the WHRS system is placed outside a residential building. In some embodiments, the system further comprises the WHRS system is placed on top of a residential building. In some embodiments, the system further comprises the WHRS system is placed inside a residential building. In some embodiments, the system further comprises the WHRS system is fluidically connected to a residential building. In some embodiments, the system further comprises the WHRS system is thermally connected to a residential building.

[0461] In some embodiments, the WHRS described herein may be configured for any building. In some cases, the building may comprise a commercial building. In some instances, the commercial building may comprise office buildings, shopping malls, warehouses, or restaurants.

[0462] In some cases, the building may comprise a residential building. In some instances, the residential building may comprise apartments, single-family homes, duplexes, townhouses, or condominiums.

[0463] In some embodiments, the WHRS described herein maybe configured for any device or system powered by electricity. In some cases, the device or system powered by electricity may comprise computers, HVAC systems, refrigeration units, and industrial machinery. In some instances, the waste heat source may comprise heat radiated from microprocessors, excess heat from HVAC operations, heat released during refrigeration cycles, and thermal energy exhausted from heavy machinery.

[0464] In some embodiments, the WHRS described herein maybe configured for any device or system configured to be powered by an engine. In some cases, the system configured to be powered by an engine may comprise trucks, construction equipment, generators, or farming equipment.

[0465] In some embodiments, the WHRS described herein maybe configured for any device or system capable of producing waste heat source. In some cases, the system capable of producing waste heat source may comprise industrial furnaces, power plants, or manufacturing facilities. In some instances, the waste heat source may comprise hot exhaust gases from industrial furnaces, steam from power plants, thermal residues from manufacturing processes, or heat generated by electronic equipment.

[0466] In some embodiments, the WHRS described herein may be configured for nuclear power plants, space stations, satellites, or lunar bases. In some instances, the waste heat source may comprise surplus thermal energy from nuclear reactions, heat from life support systems in space stations, solar irradiation absorbed by satellites, or heat released from energy systems in lunar bases. [0467] In some embodiments, the WHRS described herein may be configured for any motorized vehicle. In some cases, the motorized vehicle may comprise a car, airplane, boat, motorcycle, train, or bus. In some instances, the waste heat source may comprise heat generated from internal combustion engines, heat from aerodynamic friction in airplanes, engine heat in boats, heat produced by motorcycle engines, heat from train locomotives, or heat released by bus engines.

WHR Technology

[0468] In some embodiments, a majority of conventional Waste heat source Recovery (WHR) technologies are rendered either inoperative, offer suboptimal performance, or necessitate expansive heat transfer areas when confronted with low-grade waste heat sources and minimal temperature differentials in various facilities. In some cases, development of a heat engine for waste heat source to power conversion has been pursued to overcome these limitations. In some instances, this development takes advantage of the Stirling cycle. For example, the Stirling cycle is recognized in terms of thermodynamic efficiency compared to the Carnot cycle under certain conditions.

[0469] In some embodiments, emphasis has been placed on enhancing the architecture of heat exchangers and employing highly conductive working gases to augment the rate of heat transfer in Stirling engines for WHR. In some cases, heat exchanger enhancement has been accomplished via a methodology of Topology Optimization (TO). In some instances, TO is used in conjunction with Additive Manufacturing (AM). For example, the use of TO and AM results in a more efficient and uniform heat transfer rate. As an example, this simultaneously reduces losses associated with uneven flow distribution in heat exchangers.

[0470] In some embodiments, the application of Additive Manufacturing (AM) facilitates the integration of multiple components into a unified structure. In some cases, this integration minimizes dead volume and associated losses. In some instances, this leads to the creation of a compact and efficient engine architecture.

[0471] In some embodiments, working gases are investigated for optimizing engine performance. In some cases, a variety of pure and mixed gases are investigated. In some instances, these gases are compared to traditional working mediums. For example, the investigation aims to identify improved solutions for use in the heat engine.

[0472] In some embodiments, among the range of available Stirling engines, Free Piston Stirling Engines (FPSE) present distinct advantages. In some cases, these advantages include high efficiency, low maintenance requirements, modular architecture, compactness, quiet operation, fuel flexibility, self-starting features, hermetically sealed enclosure, longevity, and low noise operation. In some instances, FIG. l 1 presents a schematic of a typical FPSE (1100). For example, the primary working areas and components (reference numerals #1101- 1113) include compression and expansion spaces, an internal heat exchanger (comprising a heating head, cooling head, and regenerator), displacer and power pistons, flexures, and electrical components (like a linear generator, mover laminations).

[0473] In some embodiments, a layered approach has been developed to architecture and optimize the FPSE for waste heat source recovery (WHR). In some cases, the approach is applied for WHR in heat production facilities. In some instances, the approach incorporates both 1-D modeling and 3-D Computational Fluid Dynamics (CFD) simulations and optimization.

[0474] In some embodiments, the 1-D modeling approach provides comprehensive parametric studies and optimization of input parameters and working gases. In some cases, this identifies input parameters and working gases with the highest impact on performance. In some instances, 1-D modeling provides rapid and efficient turnarounds. For example, this allows for the exploration of an extensive architecture space in a timely manner. As an example, by reducing the architecture space using 1 -D optimization, the most efficient configurations that meet the imposed constraints and boundary conditions are chosen for further investigation using 3-D CFD simulations.

[0475] In some embodiments, 3-D Computational Fluid Dynamics (CFD) has been utilized. In some cases, CFD investigates inefficiencies in heat exchange and pressure losses in the regenerator, heating head, and cooling head. In some instances, CFD is used for topology optimization (TO) of heat exchangers and the regenerator. For example, the TO enhances the heat transfer rate for the most promising candidates identified from 1-D modeling.

Modeling of Heat Engine

[0476] In some embodiments, a Free Piston Stirling Engine (FPSE) configuration does not utilize mechanical linkage among pistons, diverging from traditional Stirling engines. In some cases, all elements exhibiting reciprocating motion operate independently, facilitated by variations in engine pressure. In some instances, the incorporation of a suitably configured mass-spring resonance mechanism allows the pistons' oscillating movements to become autonomously adapted to the engine's operating parameters. For example, the engine's thermodynamic and gas dynamic properties are entwined with the spring-piston dynamics. As an example, this requires integrated consideration for accurate modeling of engine behavior, rendering separate modeling methods not feasible. [0477] In some embodiments, three distinct modeling strategies exist for the proposed engine, characterized by their precision and intricacy. In some cases, Primary modeling involves definitive solutions for specialized cases of sinusoidal volume fluctuations and isothermal hot and cold spaces, drawing upon Schmidt's investigation, but does not account for losses directly. In some instances, Intermediate modeling leverages an adiabatic examination that deducts losses induced by heat transfer and fluid dynamics, building on a tailored version of Schmidt's study and necessitating non-linear time integration of model equations. For example, Intermediate modeling postulates adiabatic expansion and compression regions. As an example, Advanced modeling (Nodal analysis) provides the most precise method among these strategies. In some instances, Advanced modeling utilizes control volumes or nodes to resolve one-dimensional governing equations directly, depicting concurrent energy and fluid shifts by solving conservation of mass, momentum, and energy, thereby including engine non-idealities during simulation.

[0478] In some embodiments, Advanced modeling (Nodal analysis) is performed to attain accurate estimations of the engine's performance under imposed boundary conditions. In some cases, this modeling is performed in steps including: 1) Discretization of the fluid domain in different components into several control volumes which contain mass, temperature and pressure, 2) Setting up differential equations for conservation of mass, momentum and energy, and 3) Solving the system of equations numerically. In some instances, nodes are defined as boundaries between control volume cells and define the mass flow rate across each cell.

[0479] In some embodiments, the aforementioned modeling methods operate on a onedimensional approach. In some cases, flow characteristics are modeled singularly along the longitudinal axis of each component. In some instances, differential equations are reduced to one dimension and numerically solved in minute incremental time steps. For example, mathematical stabilization techniques are employed to determine the pressure, temperature, and mass distribution in the engine after each respective time step.

[0480] In some embodiments, a software tool configured for steady -periodic Stirling cycle architecture and simulation is utilized for the advanced (one-dimensional, third-order) modeling and optimization of the proposed engine. In some cases, this software represents each engine component as a building block, with the total model being an assembly of these interconnected component blocks via connectors for mass flow rate, heat transfer, force, and pressure. In some instances, each building block signifies fundamental engine components, such as heat exchangers, regenerators, and pistons, operating as localized, self-contained entities. For example, both power and displacer pistons are rigid, moving parts that generate volume displacement in compression and expansion spaces. As an example, all working spaces (expansion and compression), alongside heat exchangers and gaps, are segmented into multiple interconnected cells through which components interact, and a matrix is subsequently generated for each variable in each component, alongside space and time discretization (see FIG. 12). In some instances, FIG. 12 provides a schematic (1200) of the developed one-dimensional engine model within the software environment, showing vital components including heat exchangers, pistons, and working spaces. For example, a pressure source is used to define the mean average engine pressure, while endpoints are linked to a point heat source and heat sink for estimating parasitic losses.

[0481] In some embodiments, components in the model (see FIG. 12) have associated subcomponents, facilitating a thorough and accurate modeling of the heat engine's physics. In some cases, the displacer is composed of multiple subcomponents to simulate the engine's dynamic behavior as well as loss through gaps (see FIG. 13, 1300).

[0482] In some embodiments, each primary and subcomponent requires the definition of particular input parameters for modeling and optimization. In some cases, these include boundary conditions, pressure value, engine operation frequency, piston amplitude, phase angle, geometric parameters, dynamic variables, working gas, and materials. In some instances, about 75 input parameters are specified across 9 components and their corresponding subcomponents. For example, given the substantial number of input parameters and the requirement for accurate timing coordination between power and displacer pistons to avoid collision, optimization is employed in lieu of manual input settings. [0483] In some embodiments, the optimization process mandates the specification of an objective function. In some cases, maximizing thermal efficiency is set as the objective function. In some instances, thermal efficiency is defined by equation 1 : thermal = Wnet/Qin (1), where Wnet represents the total power output, and Qin is the heat input to the engine.

[0484] In some embodiments, the optimization process includes optimization variables. In some cases, Table 1 illustrates non-limiting examples of optimization variables and their broad ranges.

[0485] Table 1: Example Variable Parameter Ranges for Modeling

[0486] In some embodiments, the optimization process includes constraints.

[0487] Table 2 presents non-limiting examples of constraints imposed on an optimizer to ensure a viable engine architecture geometrically and thermodynamically. Table 2: Example Modeling Constraints

[0488] Table 3 presents non-limiting examples of values or ranges for other input parameters for different components that are kept fixed or varied during optimization.

Table 3: Example Fixed or Ranged Input Parameters for Modeling

[0489] With all the information listed above, the optimization was conducted to maximize FPSE thermal efficiency subject to the constraints with convergence history shown in FIG. 54. As evident after about 50 iterations the solver reaches a stabilized status with maximum thermal efficiency of 8.1% for heat source temperature of 80°C and heatsink temperature of 20°C.

[0490] In some embodiments, optimization is conducted to maximize FPSE thermal efficiency using inputs and constraints such as those described. In some cases, the convergence history for one optimization run is shown in FIG. 54 (1400). In some instances, as one non-limiting example result illustrated by FIG. 54, after about 50 iterations (1401) the solver reaches a stabilized status achieving a thermal efficiency (1402) of about 8.1%. For example, this specific 8.1% efficiency result corresponds to particular example conditions including a heat source temperature of 80°C (or between about 75°C and 85°C) and a heatsink temperature of 20°C (or between about 15°C and 25°C). As an example, other optimization runs under different conditions, such as a heatsink temperature of 5°C (or between about 0°C and 10°C), yield different efficiencies, such as about 9.06%.

[0491] In some embodiments, an optimization process yields a set of optimized input variables and corresponding output parameters. In some cases, one particular set of optimized input variables derived during such an optimization, along with the corresponding calculated output parameters for the specific, non-limiting example achieving 8.1% efficiency, is listed in Table 4. [0492] . In some embodiments, an optimization process yields a set of resulting input variables and corresponding output parameters. In some cases, Table 4 provides an illustrative set of modeled input and output parameters obtained from one specific, nonlimiting simulation example.

Table 4. Illustrative Set of Modeled Input and Output Parameters [0493] In some embodiments, efficiency results are evaluated relative to theoretical limits. In some instances, the specific temperature differential shown in the non-limiting example of Table 4 (80°C source, 20°C sink) corresponds to a calculated Carnot efficiency of about 16.9%. In some cases, this indicates that, for this illustrative case, the developed FPSE exhibited about 52.01% of the theoretical Carnot efficiency. For example, this level of performance relative to the Carnot limit is higher than that of competitive technologies such as Organic Rankine Cycle (ORC) under certain low-temperature differential conditions. As an example, the FPSE's inherent thermodynamic characteristics provide Waste heat source Recovery (WHR) even at very low temperatures (e.g., source temperatures below 100°C, or below 230°C) with ly high thermal efficiency. In some instances, based on the illustrative 1000W net power output (Wnet) example from Table 4, assuming continuous 24/7/365 operation, the estimated annual energy production is calculated to be about 8760 kWh.

3-D CFD Simulation of Heat Engine

[0494] In some embodiments, (one-dimensional, third-order) modeling for the proposed engine delivers assessments and supports early architecture stages through parametric analyses and optimization. In some cases, this approach only accounts for flow fluctuations along the axial direction and does not address impacts of nonuniform flow distribution, conjugate heat transfer between the solid matrix and fluid, or the effect of dead zones and abrupt geometrical alterations within flowregions. In some instances, three-dimensional (3D) Computational Fluid Dynamics (CFD) addresses these limitations, ensuring the architecture of all engine components meets specified criteria. For example, this method is an instrument used for understanding and reducing losses within the engine. As an example, primary, intermediate, and one-dimensional modeling approaches use empirical coefficients for heat transfer and flow friction derived from particular engine calibrations, constraining their applicability to engines with corresponding calibration data. In some instances, CFD operates without tuning coefficients to align with experimental data, possessing broad applicability and delivering prediction precision. For example, the 3D CFD tool is utilized for full engine simulation, wherein simulation and optimization of individual components are conducted, ensuring internal and external heat exchangers, as well as the regenerator, demonstrate performance levels comparable to one-dimensional results and respective counterparts.

[0495] In some embodiments, 3-D CFD simulations are conducted on the external acceptor fins of a standard engine configuration to demonstrate the efficacy of the topology-optimized (TO) heat exchanger and enhancements in heat transfer. In some cases, such fins augment the heat transfer rate to the hot cylinder of the proposed engine. In some instances, CFD is executed for a baseline case with conventional fins, followed by an optimized case predicated on TO.

[0496] In some embodiments, TO constitutes a computational method that optimizes the arrangement of fluids and structures within a specified architecture space, amplifying fluid flow properties like minimizing pressure drop or maximizing flow uniformity and heat transfer under defined boundary conditions and constraints. In some cases, it uses variational principles and sensitivity analysis, often harnessing adjoint-based optimization techniques, to iteratively modify the fluid or structure distribution, thereby achieving designs that optimize fluid dynamic performance and energy efficiency. In some instances, by repeatedly adjusting the distribution of material within the predefined domain based on these gradients, the algorithm identifies a structure that fulfills the stated objective, such as maximizing heat dissipation or minimizing pressure loss. For example, the benefits of using gradient-based TO for heat transfer and pressure optimization are multiple; it enables the automatic discovery of architecture solutions, enhances performance, and often diminishes material usage and weight. As an example, the method is computationally efficient compared to manual optimization techniques, as it directly uses sensitivity information to steer the optimization process, thereby reducing the number of architecture iterations required to reach a solution. [0497] FIG. 16 shows a non-limiting example of the three-dimensional schematic of a typical engine configuration along with an isolated view of the external acceptor fins. In some cases, these conventional fins are uniformly distributed around the hot cylinder's periphery. In some instances, they are constructed from copper.

[0498] In some embodiments, CFD simulations are conducted using STAR-CCM+ software forthe upper section of the geometry demonstrated in FIG. 16. In some cases, the top part of this engine type is enclosed within a casing to facilitate heat exchange with a hot medium, such as combustion gases or hot air. In some instances, the external acceptor fins are covered by an enclosure featuring four inlet and four outlet pipes (each with a diameter of 0.0349m) for hot air, as depicted in FIG. 17. For example, hot air enters the enclosure via side inlet pipes and, after exchanging heat with the exterior acceptor fins, is discharged from the top pipes.

[0499] In some embodiments, the use of polyhedral meshing achieves conformal mesh interfaces among the components. In some cases, this method ensures that interfaces between distinct parts share a consistent boundary face topology. In some instances, benefits of this technique include accuracy, simulation speed, and the elimination of face interpolation on contact patches. For example, simulations are performed using four varied base cell sizes to address mesh size sensitivity and convergence. As an example, five layers are appended to the interfaces between air and solid components (fins, cylinder, enclosure) to capture the thermal boundary layer, as shown in FIG. 18.

[0500] In some embodiments, within the CAD geometry displayed in FIG. 18, four distinct simulation domains are defined — namely, the enclosure (stainless steel), air, fins (copper), and the half-cylinder (Inconel). In some cases, material properties are sourced from the STAR-CCM+ material library, except for Inconel, which is procured based on standard material specifications. In some instances, each region is assigned a specific simulation model based on the designated material, as highlighted in FIG. 19.

[0501] In some embodiments, boundary conditions for the CFD simulation are established with reference to FIG. 20. In some cases, the enclosure is assigned as an adiabatic wall, and the inner wall of the half-cy Under is assigned a convective heat transfer coefficient with a value of 860 W/m^A2-K at 300°C (these values reflect known operating conditions of the engine and data from the manufacturer). In some instances, the inlets (indicated by red arrows) are assigned a mass flow rate of 0.003184 kg/s per inlet, and temperature and pressure of 650°C and 101325Pa, respectively. For example, for the outlets (indicated by blue arrows), a pressure boundary condition is specified. As an example, the working fluid inside the enclosure is air at 101325 Pa, and a surface roughness of 0.05 mm is assigned to all faces with a wall boundary condition.

[0502] FIG. 21 shows a non-limiting example of the mesh independence study, where the quantity of mesh cells is increased, and the mean temperature of the inner shell convective surface is monitored. In some cases, FIG. 22 outlines contours of surface temperature, FIG. 23 presents heat flux, and FIG. 24 A and 24B present contours of air temperature at various section planes. In some instances, air velocity and pressure contours are shown in FIGs. 24C and 24D, respectively, while the mean outlet air temperature from the enclosure is depicted in FIG. 25. For example, the average outlet air temperature is 502.3 °C. As an example, the CFD results demonstrate that from the maximal theoretical heat transfer value of 3566W, about 1830W is extracted from the hot air to the fins, as outlined in FIG. 26.

[0503] In some embodiments, given that the conventional fins transfer about 50% of the maximum possible heat, a topology -optimized (TO) fin architecture is utilized, which augments heat extraction from the hot air stream within the enclosure. In some cases, this is accomplished through the use of adjoint solver capabilities within STAR-CCM+, by defining flow/thermal objective functions together with input parameters, as shown in FIG. 27. In some instances, all other boundary conditions and setup remain identical to those in the baseline case. For example, considering computational resources, a periodic boundary condition is applied, and one-eighth of the model domain is simulated, as displayed in FIG. 36. As an example, with a cell size of 0.5 mm in the optimization region, the TO domain is discretized with a total number of about 4,000,000 cells, as illustrated in FIG. 37.

[0504] In some embodiments, the case is set up with pre-established solver parameters recommended by STAR-CCM+. In some cases, initial test cases result in specific optimization performance characteristics, with fluctuations in heat transfer values followed by minimal improvements, even with a large number of optimization iterations. In some instances, a parametric study examines the impact of various TO solver parameters on the objective function, defined as the total heat transfer rate from air to the exterior heat exchanger. For example, this sensitivity analysis includes analysis of the effects of topology holes and source strength, penalty value, intensity of reconstructed surface smoothing, and step size, with the last parameter exhibiting the most impact on optimization results. As an example, results are obtained with no hole formulation and simplified geometry processing, while penalty value and intensity of reconstructed surface smoothing had minimal impact on the results. In some instances, step size demonstrates an inverse correlation to mesh size, where a doubling of step size occurs when cell size is halved. For example, step sizes above a threshold lead to solver divergence, requiring a balance in the selection of these two parameters. FIG. 38 shows a non-limiting example of the solver setting parameters used for TO simulation based on preceding parametric studies.

[0505] In some embodiments, the solver is operated, and the total heat transfer value to the half cylinder and solid volume ratio are monitored, as depicted in FIG. 39. In some cases, the total heat transfer drops after about 2900 iterations due to the step size value.

[0506] FIG. 40 shows a non-limiting example of the jagged and smoothed geometry of the TO-configured heat exchanger. In some embodiments, since the TO solver is not configured to accommodate Boundary Layers (BL) for simulation, CFD is rerun for the optimized heat exchanger using smoothed geometry that includes BL.

[0507]

[0508] In some embodiments, owing to the shape of the TO-based heat exchanger, conformal meshing is utilized, as illustrated in FIG. 56. In some cases, surface mesh between all parts on the interface shares points at the interface, conserving heat transfer between parts. In some instances, the base mesh cell size affects the optimization results; smaller mesh cells yield a fin surface with a larger surface area. [0509] In some embodiments, the CFD setup remains consistent with the previous sections. In some cases, the convergence history of heat transfer and residuals is depicted in FIG. 41. [0510] FIG. 42 shows non-limiting examples of contours of surface temperature, while FIG. 43 presents the heat flux. In some cases, contours of temperature are presented in FIG. 44 and FIG. 45 at various section planes. In some instances, a uniform heat flux distribution is observed with the TO architecture heat exchanger (ref. FIG. 43). For example, pressure contours and air velocity vectors are presented in FIG. 46 and FIG. 47 respectively. As an example, the average outlet air temperature from the enclosure is 463.4 °C, as shown in Figure 48.

[0511] In some embodiments, the transition to 3-D CFD simulations results in performance changes, providing refinement of internal and external heat exchangers. In some cases, through TO, the development of organically shaped HE is possible. In some instances, this results in an enhancement in heat transfer, with a value of 2258W, which is a 23% increase compared to conventional finned heat exchangers (1830W). For example, the results serve for proof of concept.

[0512] In some embodiments, validation of the CFD model is a step to ensure accuracy. In some cases, comparing CFD outcomes with experimental data validates that the model represents physical phenomena. In some instances, this process establishes confidence in the CFD predictions, allowing their application in architecture and decision-making processes. For example, before the complete engine CFD simulation is carried out, a benchmark CFD model is developed and validated against experimental data. As an example, while experimental data for this type of engine is limited, an experimental investigation provided by NASA offers resources. In some instances, the examined engine, referred to as RE-1000, is a Research Engine with a total power output of about IkW, tested under diverse operating conditions and hardware configurations (781 test points). For example, RE-1000 is depicted in FIG. 49 with its components.

[0513] FIG. 50 depicts an example of graphical data representing engine performance characteristics. Here, the graphical data may illustrate relationships between different operational parameters under various temperature conditions. In this example, the primary component is the graphical data display, configured to show indicated power and displacer stroke. It includes a left graph plotting Indicated Power (INDPR, W) and a right graph plotting Displacer Stroke (DISPST). Both the left and right graphs share a common horizontal axis, which represents an unspecified operational parameter varying from 400 to 1100 (units not labeled). The left graph has a vertical axis for power (400-1100 W), and the right graph has a vertical axis for stroke (1.8-2.6, units likely cm). Additionally, a legend is shown, indicating four distinct data series corresponding to different heating head temperatures (600°C, 550°C, 500°C, 450°C), each plotted with unique symbols on both the left and right graphs. Here, the graphical data shows how indicated power (on the left graph) and displacer stroke (on the right graph) generally increase as the value on the horizontal axis increases. As an example, this figure displays how these performance metrics (power and stroke) shift depending on the heating head temperature condition, with higher temperatures (e.g., the 600°C series) generally resulting in higher power and stroke values compared to lower temperatures (e.g., the 450°C series) for a given value on the horizontal axis. In this case, the distinct plotted lines for each temperature illustrate the sensitivity of indicated power and displacer stroke to the heating head operating temperature. Overall, FIG. 50 demonstrates how graphical data, including the relationships shown in the left and right graphs across different temperature series, may be utilized to visualize engine performance trends and parameter sensitivities. These depicted relationships reflect how operational parameters interact within the disclosed technology.

[0514] FIG. 51 shows an example of a data record display from an engine test. Here, the data record display may provide a snapshot of numerous measured and calculated parameters for a specific operating point (Reading 1010) of a free-piston Stirling engine. In this example, the primary component is the data record display, configured to present test identification information (e.g., NASA Lewis Sensitivity Test Data, RE-1000 Engine, Test D603, REC/RDG numbers, date/time) and list operational parameters. The display may also include sections detailing input conditions such as Power Input (Amps, Volts), Engine Charge Pressure (PRESUP, MEANSP, MEANCP), and working Fluid (Helium). Additionally, multiple data groups are shown, serving to report measured values like Gas Temperatures, Surface Temperatures, cooling system data (Dashpot Cooling, Cooling head Cooling), Vibration, Phase Angles, and Engine Speed, along with Calculated Parameters, Remote Calculations, and Dynamic Calculations. Here, the data record display presents specific numerical values for the listed parameters corresponding to the Reading 1010 test condition. For example, pressures under 'Engine Charge Pressure' may be around 7000-7500 kPa, temperatures listed under 'Gas Temperatures' and 'Surface Temperatures' may range from approx. 30°C to over 600°C, and calculated external efficiency under 'Calculated Parameters' is shown as 25.4%. As an example, this figure displays cooling system performance via flow rates and temperature differences shown under 'Heat to Dashpot Cooling' and 'Heat to Cooling head'. It also shows dynamic characteristics like piston/displacer strokes under 'Remote Calculations' and 'Dynamic Calculations' (approx. 2.3-2.6 cm) and pressure amplitudes/drops under 'Dynamic Calculations'. In this case, the specific values shown throughout the various data groups provide a detailed characterization of the engine's state and performance during this particular test run (Reading 1010), operating at a frequency near 30.1 Hz as listed under 'Engine Speed'. Overall, FIG. 51 demonstrates how a data record display, presenting numerous specific parameter values from input conditions to measured and calculated results, maybe utilized to document the detailed operational state of an engine during a specific test. This detailed snapshot reflects the type of data collected for analyzing the performance of the disclosed technology.

[0515] FIG. 52 depicts an example of a table showing nominal parameter values for a system employing a working fluid under specified conditions. Here, the table may summarize various operational scenarios or test series identified by reading numbers for sensitivity testing. In this example, the primary component is the table, configured to list nominal values corresponding to different test runs or series identified in the 'Reading Number' column. The table includes columns specifying the 'Gas' type (e.g., 'He' or 'N2'), 'Heating head' temperature in °C, 'Cooling head' temperature in °C, 'Pressure' in MPa, and 'Stroke' length in cm.

Additionally, columns headed 'Disp.' (representing Displacer setting), 'Regen.' (representing Regenerator configuration), and 'Pist.' (representing Piston type, e.g., 'Std' or 'Light') are shown, serving to document the engine hardware configuration used in different test series. Here, the table indicates how specific parameters listed under column headers like 'Pressure' or 'Stroke' were systematically varied ('Var') across different sets of reading numbers while others were held at nominal values. For instance, the 'Pressure' column shows variations among 7.0, 5.5, or 4.0 MPa for certain runs, and the 'Stroke' column indicates variations near 1.8 cm to 2.0 cm for others. As an example, this figure displays how the 'Heating head' temperature may be adjusted across runs (e.g., listed as 600, 550, 500, 450 °C in the 'Heating head' column) or the 'Cooling head' temperature adjusted (e.g., listed as 25, 40, 55 °C in the 'Cooling head' column) for specific test series identified by their 'Reading Number'. In this case, component configurations listed under the 'Disp.', 'Regen.', and 'Pist.' columns indicate alternative hardware arrangements were tested, such as selectable settings (T or 'Var 1, 2') for the displacer and regenerator, or different piston types ('Std' or 'Light'). Overall, FIG. 52 demonstrates how the nominal operating conditions listed in the table, organized under columns for gas type, temperatures, pressure, stroke, and component settings, may be defined and systematically varied for engine sensitivity testing. These documented arrangements (and their variations indicated by 'Var' within the columns) reflect the scope of parametric analyses performed relative to the disclosed technology.

3-1) Geometry

[0516] In some embodiments, a CAD model of RE-1000 exists for CFD simulation (see FIG. 53), based on available geometrical data. In some cases, the CAD model includes all components such as the pressure vessel, power and displacer pistons, heating head tubes, and cooling head. In some instances, the model further incorporates the displacer rod, bounce space, all gaps, clearances, centering ports (which provide communication between the displacer gas spring and bounce space), and dead volumes.

[0517] In some embodiments, for discretization of the RE-1000, polyhedral meshes are utilized, known for their flexibility in fitting complex geometries and providing a balance between accuracy and computational efficiency. In some cases, within Simcenter STAR- CCM+, polyhedral meshes achieve high-quality simulations with fewer cells compared to purely tetrahedral meshes, leading to faster convergence and reduced computational cost. In some instances, to accommodate the movement of power and displacer pistons, an overset meshing technique is used, which is a numerical technique to handle complex moving geometries and multiple interacting objects within a flow field. For example, this method employs multiple overlapping grids, where each grid moves independently, allowing for flexibility in simulating dynamic and complex systems. As an example, an overset mesh consists of a main (background) grid and one or more overlapping (foreground) grids that overlap and exchange data at their boundaries through interpolation, allowing independent movement while maintaining simulation continuity and accuracy. FIG. 55 presents the generated overset mesh at various locations across the RE-1000.

[0518] In some embodiments, a benchmark model configuration exists. In some cases, the discretization employs the overset mesh technique, and all gaps and clearances are accounted for in the grid generation process. In some instances, pre-processing of the RE-1000 model setup is established, wherein displacement equations for the power and displacer pistons are converted into velocities and assigned to relevant components to emulate the harmonic motion of these components. For example, evaluations are performed for alternative working fluids using 1 -D and 3-D approaches to enhance heat transfer rates and reduce pressure drops. As an example, simulations for the entire engine are conducted, initially focusing on the original geometry incorporating results from prior analyses for conventional finned-style heating head, cooling head, and regenerator configurations. As another example, following the demonstrated effectiveness of TO in enhancing heat transfer processes, all mentioned heat exchangers are optimized based on TO principles.

[0519] In some embodiments, building on the successes of CFD optimizations, various surface morphologies are investigated to augment the heat transfer process. In some cases, this initiative surpasses the limitations of conventional heat exchangers by introducing surfaces configured to maximize heat absorption and dissipation, thereby enhancing overall engine efficiency. In some instances, while the working gas (i.e., helium) is utilized for its high thermal conductivity and low viscosity, new gas mixtures promising enhanced performance are explored. For example, these novel mixtures are evaluated for their thermal conductivity, viscosity, and overall compatibility with the engine architecture. As an example, this evaluation identifies a working fluid to further enhance engine performance. [0520] In some embodiments, the heat engine represents a advancement in waste heat source utilization, transforming a traditionally overlooked byproduct into a valuable energy source. In some cases, this innovation bolsters energy efficiency in various applications. In some instances, it contributes to environmental sustainability. For example, it reduces dependence on traditional fossil fuels. As an example, it decreases greenhouse gas emissions.

[0521] The systems disclosed herein may improve over systems in the art by providing, in some cases, a HRS configured to reduce a carbon dioxide emission. In some cases, the WSHR system is configured to utilize an waste heat source that comprises about 0.1% to about 15% greenhouse gases (e.g., Carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O), fluorinated gases, ozone (O₃), water vapor (H₂O). In some instances, the waste heat source comprises more than 15% greenhouse gases. In some instances, the WSHR system is configured to utilize an waste heat source that comprises about 1. 1 billion tons of CO₂/year. For example, the HRS may be configured to prevent 1 billion tons of CO₂/year for up to 100 years or more from atmospheric emission.

[0522] The systems disclosed herein may improve over systems in the art by providing, in some cases, a HRS configured to produce energy for storage.

[0523] In some cases, the HRS comprises an energy storage device. In some instances, the energy storage device comprises 1) batteries for storing electricity from the grid or renewable sources, 2) flywheels that store energy in a rotating mass, 3) thermal energy storage systems like ice storage or hot water storage tanks, 4) Superconducting Magnetic Energy Storage (SMES) systems that store energy in a magnetic field, 5) capacitors and super capacitors for storing energy in an electric field, 6) Pumped Hydroelectric Storage used for grid storage where energy is stored in the energy of water, 7) Compressed Air Energy Storage (CAES) which uses off-peak electricity to compress air for later use, or 8) Hydrogen Storage where electricity is used to split water into hydrogen and oxygen, wherein the hydrogen is stored for later use in a fuel cell.

[0524] In some cases, the HRS comprises a plurality of batteries configured for energy storage. In some instances, the plurality of batteries is attached to the HRS. In some instances, the HRS is configured to provide up to about 250 kilowatt hours/day to the plurality of batteries.

[0525] In some cases, the HRS comprises a plurality of solar panels configured for energy storage. In some instances, the plurality of solar panels is attached to the HRS. For example, the HRS is configured to provide up to about 250 kilowatt hours/day to the plurality of solar panels.

Artificial Intelligence

[0526] In some cases, the HRS comprises an artificial intelligence (Al) model. In some cases, the Al model is configured to measure energy needs.

[0527] In some cases, the Al model is configured to collecting data from a plurality of HRS. In some instances, the Al model is configured to via machine learning train on the data from the plurality of HRS. In some instances, the Al model is configured to predict power output of HRS for different time horizons. For example, the time horizon may comprise a short term, medium term, or long term. In some cases, Al and ML may be used for predicting malfunction in the unit and scheduling in-advance maintenance and part replacement.

[0528] In some cases, machine learning (ML) may generally involve identifying and recognizing patterns in existing data in order to facilitate making predictions for subsequent data. ML may include a ML model (which may include, for example, a ML algorithm). Machine learning, whether analytical or statistical in nature, may provide deductive or abductive inference based on real or simulated data. The ML model may be a trained model. ML techniques may comprise one or more supervised, semi-supervised, self-supervised, or unsupervised ML techniques. For example, an ML model may be a trained model that is trained through supervised learning (e.g., various parameters are determined as weights or scaling factors). ML may comprise one or more of regression analysis, regularization, classification, dimensionality reduction, ensemble learning, meta learning, association rule learning, cluster analysis, anomaly detection, deep learning, or ultra-deep learning. ML may comprise: k-means, k-means clustering, k-nearest neighbors, learning vector quantization, linear regression, non-linear regression, least squares regression, partial least squares regression, logistic regression, stepwise regression, multivariate adaptive regression splines, ridge regression, principal component regression, least absolute shrinkage and selection operation (LASSO), least angle regression, canonical correlation analysis, factor analysis, independent component analysis, linear discriminant analysis, multidimensional scaling, nonnegative matrix factorization, principal components analysis, principal coordinates analysis, projection pursuit, Sammon mapping, t-distributed stochastic neighbor embedding, AdaBoosting, boosting, gradient boosting, bootstrap aggregation, ensemble averaging, decision trees, conditional decision trees, boosted decision trees, gradient boosted decision trees, random forests, stacked generalization, Bayesian networks, Bayesian belief networks, naive Bayes, Gaussian naive Bayes, multinomial naive Bayes, hidden Markov models, hierarchical hidden Markov models, support vector machines, encoders, decoders, autoencoders, stacked auto-encoders, perceptrons, multi-layer perceptrons, artificial neural networks, feedforward neural networks, convolutional neural networks, recurrent neural networks, long short-term memory, deep belief networks, deep Boltzmann machines, deep convolutional neural networks, deep recurrent neural networks, large language models, vision transformers, or generative adversarial networks.

[0529] Training the ML model may include, in some cases, selecting one or more untrained data models to train using a training data set. The selected untrained data models may include any type of untrained ML models for supervised, semi-supervised, self-supervised, or unsupervised machine learning. The selected untrained data models may be specified based upon input (e.g., user input) specifying relevant parameters to use as predicted variables or other variables to use as explanatory variables. For example, the selected untrained data models may be specified to generate an output (e.g., a prediction) based upon the input. Conditions for training the ML model from the selected untrained data models may likewise be selected, such as limits on the ML model complexity or limits on the ML model refinement past a certain point. The ML model may be trained (e.g., via a computer system such as a server) using the training data set. In some cases, a first subset of the training data set may be selected to train the ML model. The selected untrained data models may then be trained on the first subset of training data set using appropriate ML techniques, based upon the type of ML model selected and any conditions specified for training the ML model. In some cases, due to the processing power requirements of training the ML model, the selected untrained data models may be trained using additional computing resources (e.g., cloud computing resources). Such training may be performed iteratively to develop desired model characteristics. [0530] In some cases, the performance of a trained ML model may be assessed, for instance, by applying the model to a subset of data to generate predictions. The model's performance, based on these predictions, may inform further development or refinement. For example, additional training may be performed to refine the ML model, which may comprise retraining using different data subsets or adjusting model parameters to align with operational requirements. This iterative process of training and refinement may be employed to develop models with certain performance characteristics. Once developed, the ML model may be stored for present or future use. The ML model may be stored as sets of parameter values or weights configured for analysis of further input (e.g., further relevant parameters, explanatory variables, user interaction data, etc.). This stored representation may also include analysis logic or performance indicators. In some cases, a plurality of ML models may be stored, ly configured for generating predictions under different sets of input data conditions. The ML model(s) may be stored in a database (e.g., associated with a server).

[0531] In some embodiments, the topology-optimized heat transfer apparatus comprises fluid flow passages with specific geometric features configured to enhance fluid flow and heat transfer efficiency. In some cases, each internal corner of at least one fluid flow passage is configured with a radius of curvature of at least 1 mm. In some instances, embodiments allow the radius of curvature to vary from about 1 mm to 5 mm or more based on operational requirements. For example, maintaining this minimum radius of curvature helps reduce turbulence and pressure drop. As an example, the organic geometry naturally eliminates sharp corners and produces continuously curved surfaces with radii that typically exceed 1 mm. In some embodiments, these radii may extend up to 5 mm or higher at internal junctions, as confirmed by simulation studies. In some cases, the broadened minimum radius of curvature substantially reduces wear and erosion compared to conventional designs. [0532] In some embodiments, the topology -optimized heat transfer apparatus comprises a multi-level array of fluid flow passages arrayed strategically within the structure. In some cases, the fluid flow passages may be arrayed in parallel to distribute fluid evenly across the heat transfer surface. In some instances, the passages may also be arrayed in series to maximize residence time and heat transfer . For example, the multi-level array may include primary passages feeding into secondary passages that further branch into tertiary channels. As an example, one configuration typically maintains a pressure drop between 0.5 kPa and 3 kPa per level. In some embodiments, broader implementations allow pressure drops to range from about 0.3 kPa up to 5 kPa per level depending on architecture parameters. [0533] In some embodiments, the performance of the topology-optimized heat transfer apparatus is compared with that of a conventional shell-and-tube heat exchanger under equivalent operating conditions. In some cases, the shell-and-tube architecture represents widely used industrial heat exchange technology. In some instances, the topology-optimized architecture achieves at least a 20% increase in overall heat transfer efficiency compared to the conventional architecture. For example, certain embodiments demonstrate efficiency improvements ranging from 20% to 30%. As an example, comparative testing under identical fluid compositions, flow rates, and temperature differentials has shown a typical 23% increase in the heat transfer coefficient. In some embodiments, actual improvements may vary further based on specific operating conditions.

[0534] In some embodiments, performance metrics for the topology-optimized heat transfer apparatus are measured under standard flow conditions. In some cases, one set of conditions includes a first fluid stream temperature of 80°C with a tolerance of ± 5°C. In some instances, this corresponds to an operational range of about 75°C to 85°C for the first fluid stream. For example, a second fluid stream may be maintained at 20°C with a tolerance of ± 5°C. As an example, this yields an operational range of roughly 15°C to 25°C for the second fluid stream. In some embodiments, the first fluid stream flow rate is about 0.003184 kg/s with a variation of ± 10% per inlet pipe. In some cases, this flow rate may vary from about 0.0029 kg/s to 0.0035 kg/s. For example, the system pressure is typically set at 101,325 Pa with a tolerance of ± 1%. As an example, this corresponds to an operational range of roughly 100,000 to 102,000 Pa. In some instances, under these conditions, the apparatus achieves a heat transfer rate of at least 0.67 of the calculated theoretical maximum. In some embodiments, conventional designs typically achieve a rate of around 0.51, though these values may vary based on configuration.

[0535] In some embodiments, the waste heat source recovery system is configured to operate with a first fluid stream at temperatures ranging from about 75°C to 105°C. In some cases, this temperature range is typical for waste heat sources from applications such as industrial processes, and HVAC systems. In some instances, certain embodiments may extend this range to about 70°C to 110°C to accommodate varying conditions. For example, the system may also be configured to operate with a second fluid stream at temperatures between 10°C and 50°C. As an example, some designs might broaden the range of the second fluid stream to about 5°C to 60°C. In some cases, the temperature differential between these streams provides a sufficient thermal gradient to drive the Free Piston Stirling Engine efficiently. For example, this configuration helps maintain the apparatus within its improved operating parameters.

[0536] In some embodiments, the topology-optimized heat transfer apparatus incorporates fluid flow passages with continuously varying cross-sectional profiles. In some cases, these profiles are produced using additive manufacturing techniques that deposit successive layers of material based on a computationally optimized architecture. In some instances, the continuous variation in geometry eliminates abrupt transitions that may cause pressure losses. For example, this architecture approach helps maintain heat transfer efficiency throughout the exchanger. As an example, fluid channels may gradually transition from circular to elliptical to rectangular profiles along their length. In some cases, these transitions are engineered to maximize surface area contact with the fluid. For example, they are configured to minimize flow resistance and ensure optimized flow characteristics. As an example, conventional manufacturing techniques often cannot achieve such a wide range of smoothly varying profiles.

[0537] In some embodiments, the topology optimization process is directly correlated with specific geometric features that reduce thermal resistance. In some cases, the organic geometries produced by this process include internal corners that maintain a minimum radius of curvature from about 0.25 mm to about 1.75 mm. In some instances, maintaining this range of curvature ensures smooth fluid flow without creating stagnation zones. For example, computational studies indicate that an increase in the surface area-to-volume ratio from about 3.75% to about 43.75% corresponds to a thermal resistance reduction from about 7.5% to about 70%. As an example, the equation Rthermai = L/(k- A) demonstrates that maximizing the effective heat transfer area while minimizing the conduction length leads to lower thermal resistance, with cross-sectional areas varying from about 0. 125 mm² to about 8.75 mm² and taper angles from about 3.75° to about 26.25°.

[0538] In some embodiments, the first fluid stream is introduced into the apparatus at a temperature from about 18.75°C to about 131 ,25°C. In some cases, the Reynolds number for this stream ranges from about 500 to about 8750, and the pressure differential is maintained from about 0.125 kPa to about 5.25 kPa. In some instances, the second fluid stream is maintained at a temperature from about 2.5°C to about 87.5°C, with Reynolds numbers ranging from about 375 to about 7000 and pressure differentials from about 0.075 kPa to about 4.375 kPa. For example, the heat transfer coefficients for the first fluid stream range from about 500 W/m²K to about 1200W/m²K, while those for the second stream range from about 300 W/m²K to about 800 W/m²K. As an example, the thermal gradient across the

- I l l - apparatus is optimized to be from about 1.25°C/mm to about 35°C/mm, resulting in a heat flux uniformly distributed with variations typically not exceeding about ±26% from the mean value.

[0539] In some embodiments, the heat transfer apparatus is integrated with the free piston Stirling engine via a thermal interface that ensures efficient energy transfer. In some cases, this interface employs a flanged connection with a thermal contact resistance from about 2.5/ 10 ⁵ m²K/W to about 1.75/ 10 ⁴ m²K/W. In some instances, the mating surfaces are machined to a flatness tolerancefrom about 0.0125 mm to about 0.0875 mm, and thermally conductive interface materials with conductivities ranging from about 1.25 W/mK to about 8.75 W/mK are used. For example, the power piston has a diameter from about 4.28 cm to about29.96 cm and an amplitude from about 1.99 mm to about 13.91 mm. As an example, the linear alternator achieves a conversion efficiency from about 85% to about 92% and operates at a frequency from about 7.5 Hz to about 52.5 Hz, while the control system maintains thermal efficiency within about ±2% of the improved value, supporting a sustained electrical output from about 500 W to about 3500 W.

Computer systems

[0540] The present disclosure provides computer systems that are programmed to implement methods of the disclosure. FIG. 10 shows a computer system 1001 that is programmed or otherwise configured to control, monitor, or regulate the HRS (e.g., according to any of the systems, methods and techniques described herein). The computer system 1001 may control various aspects of the HRS (e.g., according to any of the systems, methods and techniques described herein) of the present disclosure, such as, for example, regulating energy production, monitoring energy used by an energy user, collect data, predict power output of HRS systems, predict malfunctions and schedule maintenance and part replacement. The computer system 1001 may be an electronic device of a user or a computer system that is remotely located with respect to the electronic device. The electronic device may be a mobile electronic device.

[0541] The computer system 1001 includes a central processing unit (CPU, also “processor” and “computer processor” herein) 1002, which maybe a single core or multi core processor, ora plurality of processors for parallel processing. The computer system 1001 also includes memory or memory location 1004 (e.g., random-access memory, read-only memory, flash memory), electronic storage unit 1003 (e.g., harddisk), communication interface 1005 (e.g., network adapter) for communicating with one or more other systems, and peripheral devices 1006, such as cache, other memory, data storage and/or electronic display adapters. The memory 1004, storage unit 1003, interface 1005 and peripheral devices 1006 are in communication with the CPU 1002 through a communication bus (solid lines), such as a motherboard. The storage unit 1003 may be a data storage unit (or data repository) for storing data. The computer system 1001 may be operatively coupled to a computer network (“network”) 1007 with the aid of the communication interface 1005. The network 1007 may be the Internet, an internet and/or extranet, or an intranet and/or extranet that is in communication with the Internet. The network 1007 in some cases, is a telecommunication and/or data network. The network 1007 may include one or more computer servers, which may provide distributed computing, such as cloud computing. The network 1007, in some cases, with the aid of the computer system 1001, may implement a peer-to-peer network, which may provide devices coupled to the computer system 1001 to behave as a client or a server.

[0542] The CPU 1002 may execute a sequence of machine-readable instructions, which may be embodied in a program or software. The instructions may be stored in a memory location, such as the memory 1004. The instructions may be directed to the CPU 1002, which may subsequently program or otherwise configure the CPU 1002 to implement methods of the present disclosure. Examples of operations performed by the CPU 1002 may include fetch, decode, execute, and writeback.

[0543] The CPU 1002 may be part of a circuit, such as an integrated circuit. One or more other components of the system 1001 may be included in the circuit. In some cases, the circuit is an application specific integrated circuit (ASIC).

[0544] The storage unit 1003 may store files, such as drivers, libraries, and saved programs. The storage unit 1003 may store user data, e.g., user preferences and user programs. The computer system 1001 in some cases, may include one or more additional data storage units that are external to the computer system 1001, such as located on a remote server that is in communication with the computer system 1001 through an intranet or the Internet.

[0545] The computer system 1001 may communicate with one or more remote computer systems through the network 1007. For instance, the computer system 1001 may communicate with a remote computer system of a user (e.g., control, monitor, or regulate the process for energy production (e.g., according to any of the systems, methods and techniques described herein). Examples of remote computer systems include personal computers (e.g., portable PC), slate or tablet PC’s (e.g., Apple® iPad, Samsung® Galaxy Tab), telephones, Smart phones (e.g., Apple® iPhone, Android-enabled device, Blackberry®), or personal digital assistants. The user may access the computer system 1001 via the network 1007. [0546] Methods as described herein may be implemented byway of machine (e.g., computer processor) executable code stored on an electronic storage location of the computer system 1001, such as, for example, on the memory 1004 or electronic storage unit 1003. The machine executable or machine readable code may be provided in the form of software. During use, the code may be executed by the processor 1002. In some cases, the code may be retrieved from the storage unit 1003 and stored on the memory 1004 for ready access by the processor 1002. In some situations, the electronic storage unit 1003 may be precluded, and machine-executable instructions are stored on memory 1004.

[0547] The code may be pre-compiled and configured for use with a machine having a processer adapted to execute the code or may be compiled during runtime. The code may be supplied in a programming language that may be selected to provide the code to execute in a pre-compiled or as-compiled fashion.

[0548] Aspects of the systems and methods disclosed herein, such as the computer system 1001, may be embodied in programming. Various aspects of the technology may be thought of as “products” or “articles of manufacture” typically in the form of machine (or processor) executable code and/or associated data that is carried on or embodied in a type of machine readable medium. Machine-executable code may be stored on an electronic storage unit, such as memory (e.g., read-only memory, random-access memory, flash memory) or a hard disk. “Storage” type media may include any or all of the tangible memory of the computers, processors or the like, or associated modules thereof, such as various semiconductor memories, tape drives, disk drives and the like, which may provide non-transitory storage at any time for the software programming. All or portions of the software may at times be communicated through the Internet or various other telecommunication networks. Such communications, for example, may provide loading of the software from one computer or processor into another, for example, from a management server or host computer into the computer platform of an application server. Thus, another type of media that may bear the software elements includes optical, electrical, and electromagnetic waves, such as used across physical interfaces between local devices, through wired and optical landline networks and over various air-links. The physical elements that carry such waves, such as wired or wireless links, optical links, or the like, also may be considered as media bearing the software. As used herein, unless restricted to non-transitory, tangible “storage” media, terms such as computer or machine “readable medium” refer to any medium that participates in providing instructions to a processor for execution. [0549] Hence, a machine readable medium, such as computer-executable code, may take many forms, including a tangible storage medium, a carrier wave medium or physical transmission medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in any computer(s) or the like, such as may be used to implement the databases, etc. shown in the drawings. Volatile storage media include dynamic memory, such as main memory of such a computer platform. Tangible transmission media include coaxial cables; copper wire and fiber optics, including the wires that comprise a bus within a computer system. Carrier-wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer- readable media therefore include for example: a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD or DVD-ROM, any other optical medium, punch cards paper tape, any other physical storage medium with patterns of holes, a RAM, a ROM, a PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave transporting data or instructions, cables or links transporting such a carrier wave, or any other medium from which a computer may read programming code and/or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution.

[0550] The computer system 1001 may include or be in communication with an electronic display 1008 that comprises a user interface (UI) 1009 for providing, for example, to control, monitor, or regulate the process for energy production (e.g., accordingto any of the systems, methods and techniques described herein). Examples of UFs include, without limitation, a graphical user interface (GUI) and web-based user interface.

[0551] Methods and systems of the present disclosure may be implemented by way of one or more algorithms. An algorithm may be implemented by way of software upon execution by the central processing unit 1002. The algorithm may, for example, may control, monitor, or regulate the process for energy production (e.g., according to any of the systems, methods and techniques described herein).

EMBODIMENTS

[0552] Embodiment Al. A waste heat source management system for a waste heat source production system, comprising waste heat source production devices, a waste heat source in thermal connection with the waste heat source production devices, a first conduit comprising a first stream, wherein the first stream comprises a first temperature, a second conduit comprising a second stream, wherein the second stream comprises a second temperature, a Free Piston Stirling Engine (FPSE) fluidically connected to the first conduit and the second conduit.

[0553] Embodiment A2. The system of embodiment Al, wherein the Free Piston Stirling Engine (FPSE) fluidically connected to the second conduit.

[0554] Embodiment A3. The system of Embodiment A2, wherein the first stream comprises a fluid, gas, or solid.

[0555] Embodiment A4. The system of Embodiment A3, wherein the second stream comprises water.

[0556] Embodiment A5. The system of Embodiment A4, wherein a temperature of the first stream is equal to or greater than a temperature of the second stream.

[0557] Embodiment A6. The system of Embodiment A5, wherein the first stream comprises an output stream from the waste heat source production devices.

[0558] Embodiment A7. The system of Embodiment A6, wherein the second stream comprises an input stream to the waste heat source production devices.

[0559] Embodiment A8. The system of Embodiment A7, wherein the first conduit fluidically connects the waste heat source production devices to the FPSE.

[0560] Embodiment A9. The system of Embodiment A8, wherein the first stream enters the FPSE at a hot cylinder side.

[0561] Embodiment Al 0. The system of Embodiment A9, wherein the first stream transfers heat to the hot cylinder side.

[0562] Embodiment Al 1. The system of Embodiment A10, further comprising a third conduit, wherein the third conduit comprises a third stream at a third temperature.

[0563] Embodiment A12. The system of Embodiment Al 1, wherein the third stream comprises an FPSE output stream.

[0564] Embodiment Al 3. The system of Embodiment Al 2, wherein the third conduit fluidically connects the FPSE to a heat exchanger.

[0565] Embodiment A14. The system of Embodiment A13, wherein the third stream comprises a temperature less than or equal to the first stream.

[0566] Embodiment Al 5. The system of Embodiment Al 4, further comprising a fourth conduit, wherein the fourth conduit comprises a fourth stream at a fourth temperature. [0567] Embodiment A16. The system of Embodiment Al 5, wherein the fourth conduit fluidically connects the heat exchanger to the FPSE.

[0568] Embodiment Al 7. The system of Embodiment Al 6, wherein the fourth temperature is equal to or less than the third temperature. [0569] Embodiment Al 8. The system of Embodiment Al 7, wherein the fourth stream enters the FPSE at a cold cylinder side.

[0570] Embodiment A19. The system of Embodiment Al 8, further comprising a fifth conduit, wherein the fifth conduit comprises a fifth stream at a fifth temperature.

[0571] Embodiment A20. The system of Embodiment Al 9, wherein the fifth temperature is equal to or greater than the fourth temperature.

[0572] Embodiment A21 . The system of Embodiment A20, wherein the fifth conduit fluidically connects the heat exchanger to a chiller.

[0573] Embodiment A22. The system of Embodiment A21, wherein the second conduit fluidically connects the chiller to the waste heat source production devices.

[0574] Embodiment A23. The system of Embodiment A22, wherein the second stream comprises a temperature equal to or less than the fourth stream.

[0575] Embodiment A24. The system of Embodiment A23, wherein the FPSE is configured to convert thermal energy from the first stream into mechanical energy, and then into electrical power.

[0576] Embodiment A25. The system of Embodiment A24, wherein the fourth stream is configured to create a temperature differential between the hot cylinder side and the cold cylinder side.

[0577] Embodiment A26. The system of Embodiment A25, wherein the second conduit is configured to fluidically connect the chiller to the waste heat source production devices at a side opposite to that of the first conduit.

[0578] Embodiment A21. The system of Embodiment A26, wherein the first conduit is configured to fluidically connect the waste heat source production devices to the heat exchanger at a side opposite the fifth conduit.

[0579] Embodiment A28. The system of Embodiment A27, wherein the third conduit is configured to fluidically connect the FPSE to the heat exchanger at a side opposite the fourth conduit.

[0580] Embodiment A29. The system of Embodiment A28, wherein the fourth conduit is configured to fluidically connect the FPSE to the heat exchanger.

[0581] Embodiment A30. The system of Embodiment A29, wherein a temperature differential between the fourth stream and the first stream generates a temperature differential between the hot cylinder side and the cold cylinder side. [0582] Embodiment A31 . The system of Embodiment A30, wherein the waste heat source comprises hot exhaust gases from industrial furnaces, steam from power plants, thermal residues from manufacturing processes, or heat generated by electronic equipment.

[0583] Embodiment A32. The system of Embodiment A31, wherein the system comprises a closed loop system.

[0584] Embodiment A33. The system of Embodiment A31, wherein the system comprises an open loop system.

[0585] Embodiment A34. The system of Embodiment A33, further comprising an organic FPSE, the organic FPSE comprising a housing having a first end and a second end, wherein the first end and the second end are positioned along a displacement axis of the housing, and wherein the first end and the second end are separated by a travel length, a displacer positioned within the housing, wherein the displacer is reciprocally movable within the housing along the displacement axis and over at least a portion of the travel length, a piston configured to apply a force to the displacer in a proximal direction of the housing, a heating head configured to add thermal energy to a working fluid, and a regenerator configured to recover and store thermal energy from a heated working fluid and transfer it to a cooled working fluid, and at least one heat transfer apparatus having an organic topology.

[0586] Embodiment A35. The system of Embodiment A34, wherein the organic topology of the at least one heat transfer apparatus was generated by a topology algorithm configured to receive one or more parameters of the FPSE and generate the organic topology based at least in part on the one or more parameters.

[0587] Embodiment A36. The system of Embodiment A35, wherein the one or more parameters comprise engine architecture parameters.

[0588] Embodiment A37. The system of Embodiment A36, wherein the one or more parameters comprise one or more constraints.

[0589] Embodiment A38. The system of Embodiment A37, wherein the one or more parameters comprise an internal diameter of the pressure vessel (Dpwall), wherein the Dpwall comprises about 0.2621 m.

[0590] Embodiment A39. The system of Embodiment A38, wherein the one or more parameters comprise an internal diameter of the regenerator (Dregen), wherein Dregen comprises about 0.1747 m.

[0591] Embodiment A40. The system of Embodiment A39, wherein the one or more parameters comprise a mean charge pressure (Pcharge), wherein Pcharge comprises about 80 bar to 30 bar. [0592] Embodiment A41 . The system of Embodiment A40, wherein the one or more parameters comprise a piston diameter (Dpis), wherein Dpis comprises about 0.1712 m. [0593] Embodiment A42. The system of Embodiment A41, wherein the one or more parameters comprise a piston amplitude (Xamp,pis), wherein Xamp,pis comprises about 0.007953 m.

[0594] Embodiment A43. The system of Embodiment A42, wherein the one or more parameters comprise a compression space volume (Vcompression), wherein Vcompression comprises about 0.0003667 m³.

[0595] Embodiment A44. The system of Embodiment A43, wherein the optimized thermal efficiency (^thermal) of the FPSE comprises about 9.06%.

[0596] Embodiment A45. The system of Embodiment A44, wherein the optimized net power output (Wnet) of the FPSE comprises about 2000 W.

[0597] Embodiment A46. The system of Embodiment A45, wherein the optimized heat input (Qin) of the FPSE comprises about 22060 W.

[0598] Embodiment A47. The system of Embodiment A46, wherein the optimized heat output (Qout) of the FPSE comprises about 20060 W.

[0599] Embodiment A48. The system of Embodiment A47, wherein the organic topology of the at least one heat transfer apparatus is substantially devoid of right angles.

[0600] Embodiment A49. The system of Embodiment A48, wherein the organic topology of the at least one heat transfer apparatus is substantially devoid of straight lines.

[0601] Embodiment A50. The system of Embodiment A49, wherein the organic topology comprises an amorphous shape.

[0602] Embodiment A51 . The system of Embodiment A50, wherein at least part of the organic topology is configured to encase a pressure vessel and to achieve a heat transfer rate of up to about 0.67* the maximum theoretical limit.

[0603] Embodiment A52. The system of Embodiment A51, wherein the FPSE improves a heattransfer rate by a factor of about (0.67/0.51) compared to a conventionally finned FPSE. [0604] Embodiment A53. The system of Embodiment A52, further comprising a method of designing the heat transfer apparatus, the method comprising providing one or more parameters of the FPSE to a topology algorithm configured to generate an organic topology. [0605] Embodiment A54. The system of Embodiment A53, wherein the topology algorithm generates the organic topology tailored to a specific application of the heat transfer apparatus. [0606] Embodiment A55. The system of Embodiment A54, wherein the specific application is in a waste heat source production system. [0607] Embodiment A56. The system of Embodiment A55, further comprising generating manufacturing instructions from the organic topology.

[0608] Embodiment A57. The system of Embodiment A56, further comprising manufacturing the heat transfer apparatus based on the generated instructions.

[0609] Embodiment A58. The system of Embodiment A57, wherein the manufacturing comprises an additive manufacturing process.

[0610] Embodiment A59. The system of Embodiment A58, wherein the additive manufacturing process comprises Direct Metal Laser Sintering (DMLS).

[0611] Embodiment A60. The system of Embodiment A59, wherein the manufacturing comprises a subtractive process.

[0612] Embodiment A61. The system of Embodiment A60, further comprising discretizing a fluid domain into a plurality of control volumes.

[0613] Embodiment A62. The system of Embodiment A61, wherein each control volume has defined mass, temperature, and pressure.

[0614] Embodiment A63. The system of Embodiment A62, wherein nodes between each control volume represent and determine mass flow rates.

[0615] Embodiment A64. The system of Embodiment A63, further comprising solving differential equations for conservation of mass, momentum, and energy numerically.

[0616] Embodiment A65. The system of Embodiment A64, wherein the numerical solver predicts FPSE performance under operating conditions.

[0617] Embodiment A66. The system of Embodiment A65, wherein the solver attains a maximum thermal efficiency of 9.06% for a heat source temperature of 80°C and a heat sink temperature of 5 °C.

[0618] Embodiment A67. The system of Embodiment A66, wherein the FPSE has a Carnot efficiency of about 42.5%.

[0619] Embodiment A68. The system of Embodiment A67, wherein the topology algorithm iteratively adjusts material distribution using adjoint solver capabilities.

[0620] Embodiment A69. The system of Embodiment A68, wherein the topology algorithm incorporates one-dimensional, third-order modeling, simulation, and optimization.

[0621] Embodiment A70. The system of Embodiment A69, wherein the FPSE domain is discretized into building blocks representing elemental components.

[0622] Embodiment A71. The system of Embodiment A70, wherein each building block is a localized self-contained entity. [0623] Embodiment A72. The system of Embodiment A71, wherein the full FPSEmodel is a summation of interconnected building blocks.

[0624] Embodiment A73. The system of Embodiment A72, wherein pistons are modeled as rigid moving components.

[0625] Embodiment A74. The system of Embodiment A73, wherein components including heat exchangers, pistons, and working spaces are modeled.

[0626] Embodiment A75. The system of Embodiment A74, wherein average operating pressure is specified using a pressure source.

[0627] Embodiment A76. The system of Embodiment A75, wherein endpoints are connected to specified heat sources.

[0628] Embodiment A77. The system of Embodiment A76, wherein parasitic losses are estimated.

[0629] Embodiment A78. The system of Embodiment A77, wherein a total of 75 input parameters are defined for 9 components.

[0630] Embodiment A79. The system of Embodiment A78, wherein optimization variables include geometric and thermodynamic parameters.

[0631] Embodiment A80. The system of Embodiment A79, wherein constraints ensure geometric and thermodynamic viability.

[0632] Embodiment A81. The system of Embodiment A80, wherein thermal efficiency is the objective function.

[0633] Embodiment A82. The system of Embodiment A81, wherein regenerator diameter is larger than displacer rod diameter.

[0634] Embodiment A83. The system of Embodiment A82, wherein the pressure vessel diameter is larger than the regenerator diameter.

[0635] Embodiment A84. The system of Embodiment A83, wherein piston diameter is larger than displacer rod diameter.

[0636] Embodiment A85. The system of Embodiment A84, wherein piston diameter is about equal to the regenerator diameter.

[0637] Embodiment A86. The system of Embodiment A85, wherein dead space in the compression region prevents collision.

[0638] Embodiment A87. The system of Embodiment A86, wherein dead space in the expansion region prevents collision.

[0639] Embodiment A88. The system of Embodiment A87, wherein the displacer operates freely by maintaining phasor force components at zero. [0640] Embodiment A89. The system of Embodiment A88, wherein required power output is ensured.

[0641] Embodiment A90. The system of Embodiment A89, wherein a one-dimensional code simulates 48-hour energy generation resulting in 98 kWh.

[0642] Embodiment A91. The system of Embodiment A90, wherein polyhedral meshing creates conformal mesh interfaces.

[0643] Embodiment A92. The system of Embodiment A91, wherein contacting faces share boundary face topology.

[0644] Embodiment A93. The system of Embodiment A92, wherein four base mesh sizes ensure mesh sensitivity and convergence.

[0645] Embodiment A94. The system of Embodiment A93, wherein five layers are established on air-solid interfaces to capture thermal boundary layers.

[0646] Embodiment A95. The system of Embodiment A94, wherein four simulation domains are identified: stainless steel, air, copper fins, and Inconel.

[0647] Embodiment A96. The system of Embodiment A95, wherein each domain is assigned a specific model based on material.

[0648] Embodiment A97. The system of Embodiment A96, wherein boundary conditions include adiabatic walls for the enclosure.

[0649] Embodiment A98. The system of Embodiment A97, wherein the inner wall of a half cylinder has a convective heat transfer coefficient of 860 W/m²K at 300°C.

[0650] Embodiment A99. The system of Embodiment A98, wherein inlets have parameters: mass flow 0.003184 kg/s, temperature 650°C, pressure 101,325 Pa.

[0651] Embodiment A100. The system of Embodiment A99, wherein outlets are set as pressure outlets.

[0652] Embodiment Al 01. The system of Embodiment Al 00, wherein at least one enclosure surface has roughness of about 0.05 mm.

[0653] Embodiment A102. The system of Embodiment A101, wherein average air temperature is calculated to be about 502.3 °C.

[0654] Embodiment Al 03. The system of Embodiment Al 02, wherein the maximum theoretical heat transfer is about 3566 W.

[0655] Embodiment Al 04. The system ofEmbodiment A103, wherein 1830 W is extracted from hot air to fins.

[0656] Embodiment Al 05. The system ofEmbodiment Al 04, wherein topology optimization uses gradient-based techniques for heat and pressure. [0657] Embodiment Al 06. The system of Embodiment Al 05, wherein topology optimization begins with defining architecture envelope and objectives.

[0658] Embodiment A107. The system of Embodiment A106, wherein conformal mesh ensures conservation of heat transfer values.

[0659] Embodiment Al 08. The system of Embodiment Al 07, wherein surface mesh shares interface points to conserve heat transfer.

[0660] Embodiment A109. The system of Embodiment A108, wherein smaller mesh cells result in finer fin surfaces.

[0661] Embodiment Al 10. The system of Embodiment Al 09, wherein smaller mesh cells result in larger fin surface area.

[0662] Embodiment Al l i. The system of Embodiment Al 10, wherein geometry preparation includes smoothing derived part geometry and creating a watertight surface.

[0663] Embodiment Al 12. The system of Embodiment Al 11, wherein the watertight surface is imported, subtracted from the air domain, and imprinted to the half -cylinder.

[0664] Embodiment Al 13. The system of Embodiment Al 12, wherein all required interfaces are created properly.

[0665] Embodiment Al 14. The system of Embodiment Al 13, wherein jagged geometry from topology optimization is smoothed and re-imported for CFD simulation.

[0666] Embodiment Al 15. A method of designing a heat transfer apparatus for a FPSE to be integrated into a waste heat source production system, the method comprising providing one or more parameters of the FPSE into a one-dimensional (ID) Computational Fluid Dynamics (CFD) model, providing at least one output parameter of the ID CFD model into a three- dimensional (3D) CFD model, providing at least one output parameter of the 3D CFD model into a topology algorithm, and generating the heat transfer apparatus comprising an organic topology tailored for a waste heat source production system.

[0667] Embodiment Al 16. An organic FPSE for a waste heat source production system environment, the organic FPSE comprising a housing having a first end and a second end, wherein the first end and the second end are positioned along a displacement axis of the housing, and wherein the first end and the second end are separated by a travel length, a displacer positioned within the housing, wherein the displacer is reciprocally movable within the housing along the displacement axis and over at least a portion of the travel length, a piston configured to apply a force to the displacer in a proximal direction of the housing, a heating head configured to receive thermal energy from a waste heat source and add thermal energy to a working fluid, and a regenerator configured to recover and store thermal energy from a heated working fluid and transfer it to a cooled working fluid, and at least one heat transfer apparatus having an organic topology tailored for the waste heat source production system environment.

[0668] Embodiment Al 17. A method of designing a heat transfer apparatus for a FPSE to be integrated into a waste heat source production system, the method comprising providing one or more parameters of the FPSE into a topology algorithm, wherein the topology algorithm is configured to generate an organic topology for the heat transfer apparatus based at least in part on the one or more parameters of the FPSE, and generating the organic topology tailored for the waste heat source production system.

[0669] Embodiment Al 18. A waste heat source management system for a commercial building, comprising: a heat exchanger, an evaporator in fluidic connection with the heat exchanger, a condenser in fluidic connection with the heat exchanger, a first conduit comprising a first stream, wherein the first stream comprises a first temperature, a second conduit comprising a second stream, wherein the second stream comprises a second temperature, a third conduit comprising a third stream, wherein the third stream comprises a third temperature, and a Free Piston Stirling Engine (FPSE).

[0670] Embodiment Al 19. The system of Embodiment Al 18, wherein the third conduit is thermally connected to the first conduit.

[0671] Embodiment A120. The system of Embodiment Al 19, wherein the third conduit is fluidically connected to a heating head of the FPSE.

[0672] Embodiment A121. The system of Embodiment A120, wherein the FPSE is fluidically connected to the second conduit.

[0673] Embodiment A122. The system ofEmbodiment A121, wherein a cooling head of the FPSE is fluidically connected to the second conduit.

[0674] Embodiment A123. The system of Embodiment A122, wherein the first stream comprises an HVAC refrigerant stream.

[0675] Embodiment A124. The system of Embodiment A123, wherein the second stream comprises a cooling fluid.

[0676] Embodiment A125. The system of Embodiment A124, wherein the third stream comprises a heating fluid.

[0677] Embodiment A126. The system ofEmbodiment A125, wherein a temperature of the first stream is equal to or greater than a temperature of the third stream.

[0678] Embodiment A127. The system ofEmbodiment A126, wherein a temperature of the third stream is equal to or greater than a temperature of the second stream. [0679] Embodiment A128. The system of Embodiment A127, wherein the first stream comprises an output stream from the HVAC unit.

[0680] Embodiment A129. The system of Embodiment A128, wherein the first stream comprises an output stream from the evaporator.

[0681] Embodiment A130. The system of Embodiment A129, wherein the first stream comprises an input stream to the heat exchanger.

[0682] Embodiment A131. The system of Embodiment Al 30, wherein the first stream comprises an output stream from the heat exchanger.

[0683] Embodiment A132. The system of Embodiment A131, wherein the first stream comprises an input stream to the condenser.

[0684] Embodiment A133. The system of Embodiment A132, wherein the first stream comprises an output stream from the condenser.

[0685] Embodiment A134. The system of Embodiment A133, wherein the first stream comprises an input stream to the evaporator.

[0686] Embodiment A135. The system of Embodiment Al 34, wherein the second stream comprises an output stream from a heat sink.

[0687] Embodiment A136. The system of Embodiment A135, wherein the second stream comprises an input stream to a cooling head of the FPSE.

[0688] Embodiment Al 37. The system of Embodiment Al 36, wherein the second stream comprises an output stream from a cooling head of the FPSE.

[0689] Embodiment A138. The system of Embodiment A137, wherein the third stream comprises an output stream from a heating head of the FPSE.

[0690] Embodiment A139. The system of Embodiment A138, wherein the third stream comprises an input stream to the heat exchanger.

[0691] Embodiment A140. The system of Embodiment A139, wherein the third stream comprises an output stream from the heat exchanger.

[0692] Embodiment A141. The system of Embodiment AMO, wherein the third stream comprises an input stream to the heating head of the FPSE.

[0693] Embodiment A142. The system of Embodiment A141, wherein the second stream enters the FPSE at a cold cylinder side.

[0694] Embodiment A143. The system of Embodiment A142, wherein the second stream removes heat from the cold cylinder side.

[0695] Embodiment Al 44. The system of Embodiment Al 43, wherein the third stream enters the FPSE at a hot cylinder side. [0696] Embodiment A145. The system of Embodiment A144, wherein the third stream transfers heat to the hot cylinder side.

[0697] Embodiment A146. The system of Embodiment A145, wherein the FPSE is configured to convert thermal energy from the third stream into mechanical energy, and then into electrical power.

[0698] Embodiment A147. The system of Embodiment A146, further comprising a temperature differential between the hot cylinder side and the cold cylinder side.

[0699] Embodiment A148. The system of Embodiment A147, wherein the first conduit is configured to fluidically connect the condenser to the evaporator and to the heat exchanger. [0700] Embodiment Al 49. The system of Embodiment Al 48, wherein the second conduit is configured to fluidically connect the heat sink to the cooling head.

[0701] Embodiment A150. The system of Embodiment A149, wherein the third conduit is configured to fluidically connect the heat exchanger to the heating head.

[0702] Embodiment A151. The system of Embodiment Al 50, wherein the system comprises a closed loop system.

[0703] Embodiment Al 52. The system of Embodiment Al 50, wherein the system comprises an open loop system.

[0704] Embodiment Al 53. The system of Embodiment Al 50, wherein the second stream comprises water.

[0705] Embodiment Al 54. The system of Embodiment Al 50, wherein the second stream comprises air.

[0706] Embodiment Al 55. The system ofEmbodiment A150, wherein the WHRS system is placed outside a commercial building.

[0707] Embodiment Al 56. The system ofEmbodiment Al 50, wherein the WHRS system is placed on top of a commercial building.

[0708] Embodiment Al 57. The system ofEmbodiment Al 50, wherein the WHRS system is placed inside a commercial building.

[0709] Embodiment Al 58. The system ofEmbodiment A150, wherein the WHRS system is fluidically connected to a commercial building.

[0710] Embodiment Al 59. The system ofEmbodiment Al 50, wherein the WHRS system is thermally connected to a commercial building.

[0711] Embodiment Al 60. The system ofEmbodiment Al 59, wherein the waste heat source management system comprises waste heat source production devices and waste heat source in thermal connection with the waste heat source production devices. [0712] Embodiment A161. The system of Embodiment A160, wherein the first conduit comprises a first stream at a first temperature and the second conduit comprises a second stream at a second temperature.

[0713] Embodiment A162. The system of Embodiment A161, wherein the Free Piston Stirling Engine (FPSE) is fluidically connected to both the first conduit and the second conduit.

[0714] Embodiment Al 63. The system of Embodiment Al 62, wherein the first stream comprises water.

[0715] Embodiment A164. The system of Embodiment A162, wherein the second stream comprises water.

[0716] Embodiment Al 65. The system of Embodiment Al 62, wherein a temperature of the first stream is equal to or greater than a temperature of the second stream.

[0717] Embodiment A166. The system of Embodiment A165, wherein the first stream comprises an output stream from the waste heat source production devices.

[0718] Embodiment A167. The system of Embodiment A166, wherein the second stream comprises an input stream to the waste heat source production devices.

[0719] Embodiment A168. The system of Embodiment A167, wherein the first conduit fluidically connects the waste heat source production devices to the FPSE.

[0720] Embodiment Al 69. The system of Embodiment Al 68, wherein the first stream enters the FPSE at a hot cylinder side.

[0721] Embodiment A170. The system of Embodiment A169, wherein the first stream transfers heat to the hot cylinder side.

[0722] Embodiment A171. The system of Embodiment A170, further comprising a third conduit comprising a third stream at a third temperature.

[0723] Embodiment A172. The system of Embodiment A171, wherein the third stream comprises an FPSE output stream.

[0724] Embodiment Al 73. The system of Embodiment Al 72, wherein the third conduit fluidically connects the FPSE to a heat exchanger.

[0725] Embodiment A174. The system of Embodiment A173, wherein the third stream comprises a temperature less than or equal to the first stream.

[0726] Embodiment Al 75. The system of Embodiment Al 74, further comprising a fourth conduit comprising a fourth stream at a fourth temperature.

[0727] Embodiment A176. The system of Embodiment A175, wherein the fourth conduit fluidically connects the heat exchanger to the FPSE. [0728] Embodiment Al 77. The system of Embodiment Al 76, wherein the fourth temperature is equal to or less than the third temperature.

[0729] Embodiment Al 78. The system of Embodiment Al 77, wherein the fourth stream enters the FPSE at a cold cylinder side.

[0730] Embodiment A179. The system of Embodiment A178, further comprising a fifth conduit comprising a fifth stream at a fifth temperature.

[0731] Embodiment Al 80. The system of Embodiment A179, wherein the fifth temperature is equal to or greater than the fourth temperature.

[0732] Embodiment A181. The system of Embodiment Al 80, wherein the fifth conduit fluidically connects the heat exchanger to a chiller.

[0733] Embodiment Al 82. The system of Embodiment Al 81, wherein the second conduit fluidically connects the chiller to the waste heat source production devices.

[0734] Embodiment Al 83. The system of Embodiment Al 82, wherein the second stream comprises a temperature equal to or less than the fourth stream.

[0735] Embodiment A184. The system of Embodiment A183, wherein the FPSE is configured to convert thermal energy from the first stream into mechanical energy, and then into electrical power.

[0736] Embodiment Al 85. The system of Embodiment A184, wherein the fourth stream is configured to create a temperature differential between the hot cylinder side and the cold cylinder side.

[0737] Embodiment Al 86. The system of Embodiment Al 85, wherein the second conduit is configured to fluidically connect the chiller to the waste heat source production devices at a side opposite to that of the first conduit.

[0738] Embodiment Al 87. The system of Embodiment Al 86, wherein the first conduit is configured to fluidically connect the waste heat source production devices to the heat exchanger at a side opposite the fifth conduit.

[0739] Embodiment A188. The system of Embodiment A187, wherein the third conduit is configured to fluidically connect the FPSE to the heat exchanger at a side opposite the fourth conduit.

[0740] Embodiment Al 89. The system of Embodiment Al 88, wherein the fourth conduit is configured to fluidically connect the FPSE to the heat exchanger.

[0741] Embodiment A190. The system of Embodiment A189, wherein a temperature differential between the fourth stream and the first stream generates a temperature differential between the hot cylinder side and the cold cylinder side. [0742] Embodiment Al 91. The system of Embodiment Al 90, wherein the waste heat source comprises hot exhaust gases from industrial furnaces, steam from power plants, thermal residues from manufacturing processes, or heat generated by electronic equipment.

[0743] Embodiment Al 92. The system of Embodiment Al 91, wherein the system comprises a closed loop system.

[0744] Embodiment Al 93. The system of Embodiment Al 91, wherein the system comprises an open loop system.

EXAMPLES

[0745] The following illustrative examples are representative of examples of the software applications, systems, and methods described herein and are not meant to be limiting in any way.

Example 1 — Heat recovery system Operation (HRS)

[0746] The HRS may include a Free Piston Stirling Engine connected to an waste heat source system (e.g., an waste heat source pipe). The FPSEhas an enclosure which covers a heating head. The FPSE enclosure may include one or more inlets and one or more outlets. A hot stream from the building waste heat source system may from various devices and/or pipes into the engine or into a collector. Furthermore, the hot stream (e.g., building waste heat source) may be guided from the various pipes or collector toward the enclosure for heat exchange. After the heat exchange the lower temperature waste heat source is directed toward either a chiller, another heat exchanger, or back to the waste heat source.

[0747] The building waste heat source has an energy depending on the constituents and a temperature of up to about 105°C. The waste heat source may be connected to a plurality of fins connected to a heating head on the Free Piston Stirling Engine.

[0748] The plurality of fins is made of copper or aluminum. The plurality of fins includes a topology optimized bounding envelope size and surface area. The plurality of fins has a thermal conductivity up to about 205 W/mK at room temperature and room pressure. The plurality of fins has a topology optimized thermal mass. The plurality of fins has a topology optimized spacing between each fin. The plurality of fins covers a portion of an external surface of the heating head, cooling head, and regenerator.

[0749] The plurality of fins may be configured to transfer the heat from the waste heat source to the heating head. The contact surface area between the heating head and the plurality of fins is topology optimized. The heat transfer rate is topology optimized between the waste heat source and the heating head. [0750] The heating head, for example, heating head 3200 from FIG. 32, which is also a heat exchanger with a topology optimized surface area increased by the use of the plurality of fins extracts even more heat from the working fluid, cooling it down. The heating head is made of nano-structured substances, composites, or alloys and topology optimized shape and dimensions. The heating head has a topology optimized thermal mass (mCP). The heating head is configured to add an optimized amount of heat to the working fluid. The contact surface area between the working fluid and the heating head is topology optimized. The heating head has athermal conductivity of up to about 400 W/mK at room temperature and room pressure. The temperature differential between the heating head and the working fluid comprises between about 0°C to about 10°C. The heating head comprises a topology optimized surface to volume ratio between the heating head and the working fluid. The heat transfer rate may comprise a topology optimized heat transfer rate.

[0751] The working fluid has an optimized formulation. The working fluid has a thermal conductivity of at least about 0.15 W/mK at room temperature and room pressure. The working fluid has a specific heat capacity of at least about is about 5.193 J/g°C at constant pressure (Cp).

[0752] The working fluid is heated by the heating head. The working fluid temperature increases to between about 75°C to about 80°C. As the working fluid expands due to heating, it pushes against the power piston, which is moved outwards. The work done on the power piston in this phase is converted into electrical power. As the piston reaches the end of its stroke, the working fluid enters the regenerator, which is a heat exchanger with a topology optimized surface area, (e.g., increasedby the presence of the fins). Here, the working fluid gives off up an optimized amount of heat to the regenerator.

[0753] The regenerator is made of porous material with pores of a topology optimized diameter. The regenerator, for example, regenerator 3200 from FIG. 32, has a topology optimized thermal mass (mCP) the optimized storage of heat is stored in the regenerator. The contact surface area between the working fluid and the regenerator comprises a topology optimized contact surface area. The regenerator has a thermal conductivity of up to about 400 W/mK at room temperature and room pressure. The temperature differential between the regenerator and the working fluid comprises between about 0°C to about 80°C.

[0754] The cooling head, which is also a heat exchanger, for example, cooling head 3200 from FIG. 32, with a topology optimized surface area increased by the use of a plurality of fins which increases the extraction of heat from the gas, cooling it down. The cooling head is made of nano-structured substances, composites, or alloys and a topology optimized shape. The cooling head has a topology optimized thermal mass (mCP). The cooling head is configured to remove an optimized amount of heat from the working fluid. The contact surface area between the working fluid and the cooling head comprises a topology optimized contact surface area. The cooling head has a thermal conductivity of up to about 400W/mK at room temperature and pressure. The temperature differential between the cooling head and the working fluid comprises between about 0°C to about 80°C. The cooling head comprises a topology optimized surface to volume ratio between the cooling head and the working fluid. The heat transfer rate may comprise an optimized heat transfer rate.

[0755] The plurality of fins may be configured to transfer the heat from the cooling head to the ambient environment. The contact surface area between the cooling head and the plurality of fins is a topology optimized surface area.

[0756] Due to the decrease in temperature, the gas contracts and the piston is moved back to its original position. A part of the work for compressing the gas is provided by a spring mechanism or magnetic interaction in the case of free piston Stirling engines. As the piston starts moving outwards again, the cold gas passes back through the regenerator and picks up heat stored there during the second step. This pre-heating of the gas reduces the amount of heat that needs to be input in the next heating phase. This cycle repeats continuously.

Example 2— Use of Third Order Modeling and Software (e.g., SAGE software) for Engine Performance Estimation

[0757] In an embodiment of the present invention, a method is disclosed for estimating the performance of a Free Piston Stirling Engine under set boundary conditions using Nodal Analysis, a third order modeling technique. The engine's components are divided into a multitude of control volumes that each contain mass, with defined pressure and temperature at the nodes acting as the boundaries of these volumes. Through this process, the method directly addresses one-dimensional governing equations pertaining to the conservation of mass, momentum, and energy, along with engine non-idealities during the simulation.

[0758] This method simplifies all differential equations into a ID form that are solved numerically using small incremental time steps and mathematical stabilization techniques. As a result, precise distribution of pressure, temperature, and mass in the engine at each time step may be calculated.

[0759] Furthermore, all components of the engine, including its expansion and compression spaces, heat exchangers, and gaps, are divided into interconnected cells where interaction occurs, allowing the creation of a matrix for each variable within each component, factoring in both space and time discretization. This facilitates an accurate estimation of the engine's performance under various conditions. a. Software (e.g., SAGE software)

[0760] In another embodiment, a software program, specifically SAGE software, is utilized to conduct one-dimensional third order modeling, simulation, and optimization of the full Free Piston Stirling Engine. In the context of SAGE, each component of the engine is depicted as a building block. The complete engine model is then assembled using these interconnected blocks. Connections between various components are designated by arrows and numbers, which represent different boundary conditions in terms of parameters such as mass flow rate, heat transfer, force, and pressure.

[0761] Third order modeling is utilized with the objective of obtaining the most precise estimation of engine performance under imposed boundary conditions. This process is executed in two primary steps: (1) The fluid domain in different components is discretized into numerous control volumes. Each volume contains mass, and temperature and pressure are defined atthe nodes which act as boundaries between cells, also defining the mass flow rate across each cell; (2) Differential equations representing the conservation of mass, momentum, and energy are established and subsequently solved to provide precise estimations of engine performance. b. Schematic Outline

[0762] Section: Engine Modeling and Optimization Using Software (e.g., SAGE software) [0763] FIG. 12 presents a schematic of the one-dimensional (ID) engine model 1200 developed within the Software (e.g., SAGE software). As depicted in FIG. 12, a onedimensional (ID) engine model 1200 is developed within the software. This schematic includes all essential components of the Free Piston Stirling Engine (FPSE), including the heat exchanger (HE), pistons, and working spaces.

[0764] The mean average engine pressure is designated with a pressure source. To assist in estimating the parasitic losses, point heat sources are strategically placed at the end points. This detailed model offers an elaborate representation of the engine, paving the way formore precise simulations and optimizations.

[0765] Continuing with FIG. 13, each component of the engine model includes several subcomponents to capture a precise and comprehensive modeling 1300 of the heat engine's physics. For instance, the displacer and cylinder are comprised of multiple subcomponents that accurately model the dynamic behavior of the engine and any losses that occur via the gaps. [0766] Each principal component, along with its respective subcomponents, necessitates the definition of certain input parameters. These parameters include boundary conditions, pressure values, engine operating frequency, piston amplitude, phase angle, geometrical parameters, dynamic variables, working gas, and materials. In total, 75 input parameters are defined across the nine components and their respective subcomponents.

[0767] Given the large number of input parameters and the precise timing between the power and displacer pistons to avoid collisions, an optimization process is utilized to simplify manual input insertion. This optimization process requires the specification of optimization variables, primarily the input parameters for the components. Additionally, it is essential to set constraints to ensure the geometric and thermodynamic feasibility of the architecture, as well as define an objective function for the optimization process.

[0768] In this embodiment, the objective function, defined as thermal efficiency, is aimed to be maximized. The formula for thermal efficiency (r|_thermal) is given by Equation 3 :

T|th = Wnet / Qin (Equation 3)

[0769] In this equation, W_net represents the net work output of the engine, while Qi_n is the total heat input to the engine. The variables considered for optimization in this study, and their respective ranges, are outlined in Table 5. The remaining input parameters are kept constant, their values derived from existing literature, engineering knowledge, and specific requirements of the engine for the intended application. These fixed inputs are detailed in the same table.

Table 5: Parameters Explored in Optimization

[0770] In a subsequent embodiment, to facilitate the development of an efficient and operational engine, particular constraints are applied to the optimization process. These constraints function as guides directing the optimization towards a viable engine architecture, and are detailed in Table 6. These conditions assist in preventing complications in engine performance while simultaneously satisfying the specified requirements of the engine's intended application.

Table 6: Assumed Operating Conditions for Optimization [0771] In another aspect of the embodiment, in addition to the variables deemed for optimization, numerous other input parameters were maintained constant throughout the process. These parameters, associated with various engine components, play pivotal roles in determining the engine's performance and overall operation. The constant values for these parameters, utilized throughout the optimization process, are compiled in Table 7. These parameters, established based on previous literature, engineering knowledge, and specific engine requirements, have been strategically chosen to guarantee a reliable and efficient engine architecture.

Table 7: Additional Modeling Input Parameters

[0772] In the subsequent phase of this embodiment, upon the definition of the objective function, variables, fixed parameters, and constraints, the optimization process was initiated. The objective of this process was to maximize the thermal efficiency of the Free Piston Stirling Engine (FPSE). The progression of this optimization process may be graphically represented using a convergence history graph 1400, as illustrated in FIG. 14.

[0773] As depicted in FIG. 14, the Y-axis 1402 corresponds to thermal efficiency, while the X-axis 1401 corresponds to the number of iterations. After about 35 iterations, the solver achieves a steady state, providing a maximum thermal efficiency of 9.06%. This is observed for a heat source temperature of 80°C and a heat sink temperature of 5°C. This convergence history graphically illustrates the progression of thermal efficiency enhancement with each iteration until it reaches an improved point. [0774] The results of the analysis, comprising certain input variables and corresponding output parameters, are presented in Table 8:

Table 8: Optimization Study Results

[0775] Considering the temperature differential between the heat source and heat sink, the theoretical maximum efficiency of a heat engine, known as the Carnot efficiency, was calculated to be about 21.3%. This calculation indicates that the developed Free Piston Stirling Engine (FPSE) reached about 42.5% of the Carnot efficiency. Notably, this efficiency level exceeds that of numerous competing technologies, such as the Organic Rankine Cycle (ORC), signifying a beneficial improvement in the field.

[0776] Additionally, it is beneficial to note that most Waste heat source Recovery (WHR) technologies, particularly those employing waste heat sources, tend to be inefficient at low temperatures, specifically as low as 80°C. However, due to the intrinsic thermodynamic properties of the Free Piston Stirling Engine (FPSE), it is capable of facilitating WHR even at these extremely low temperatures while maintaining notable high thermal efficiency. This indicates a beneficial advantage over traditional systems and provides further optimization in diverse applications.

[0777] In another aspect of this embodiment, a simulation was executed using the ID code to forecast energy production over a span of 48 hours. This simulation was conducted under the presumption of constant heat source and sink temperatures and the continuous availability of the heat source. The cumulative energy production during this period was about 98kWh, as visualized in FIG. 15.

[0778] FIG. 15 presents a graph 1500 that illustrates the energy production of the Free Piston Stirling Engine (FPSE) overtime, specifically across 48 hours. In FIG. 15, the y-axis 1501 denotes power (W) and the x-axis 1502 signifies time in hours. This period demonstrates a steady power output over a 48-hour duration, further emphasizing the reliable performance of the FPSE in continuous operation conditions.

Example 3 — 3D CFD Simulation of Heat Engine

[0779] In a further embodiment, a comprehensive method of analysis is utilized. The 3D CFD simulation allows for a more nuanced and detailed evaluation of the Free Piston Stirling Engine (FPSE). This approach is configured to encompass not only variations of flow along the axis direction, but also accommodates nonuniform flow and heat transfer distribution, abrupt changes in geometry within flow regions, and conjugate heat transfer interactions between the solid matrix, fluid, or dead zones. Therefore, these 3D models may provide a more accurate and comprehensive estimation of the intricate performance details of the full engine.

[0780] In the next phase of this embodiment, a three-dimensional Computational Fluid Dynamics (CFD) is leveraged. Also known as fourth-order modeling, CFD provides a more comprehensive and precise understanding of the engine's performance. Unlike other modeling approaches, CFD — grounded in the Navier-Stokes equations — extends its application scope and enhances predictive accuracy. It provides a robust tool for accurately quantifying and mitigating losses within the Free Piston Stirling Engine (FPSE) without needing to rely on empirically obtained coefficients for heat transfer and flow friction. This broadened application scope and increased precision makes it an invaluable tool in the analysis and architecture of efficient engines.

[0781] In this embodiment, the CFD tool was put into action for a full engine simulation. The preliminary steps included individual component simulation and optimization, ensuring that both the internal and external heat exchangers, along with the regenerator, displayed performance competitive with the ID results and their counterparts.

[0782] The initial efforts were channeled towards performing CFD simulations on the external acceptor fins. These components are used to augment the heat transfer rate to the Free Piston Stirling Engine's (FPSE) hot cylinder, as depicted in FIG. 16. These heat exchangers, fashioned in a conventional finned shape and distributed uniformly around the periphery of the hot cylinder, are composed of copper. FIG. 16 provides an illustration of the heat exchanger fins of the FPSE. c. Geometry

[0783] FIG. 16, in this embodiment, illustrates an example of heat exchanger fins geometry prepared for Computational Fluid Dynamics (CFD) simulation. It presents a cross-section of the Free Piston Stirling Engine (FPSE) 1601, emphasizing the location of the external acceptor fins 1603. The diagram further dissects the external fins 1603, revealing their internal and external surfaces as demonstrated in 1602.

[0784] The geometry considered for the CFD simulations focuses on the upper section of the engine structure. Simulations were executed using the STAR-CCM+ software. In this setup, the external acceptor fins are housed within an enclosure featuring multiple inlet and outlet pipes. This architecture facilitates the movement of hot air into and out of the enclosure as depicted in FIG. 16.

[0785] In more detail, the enclosure includes four inlet pipes positioned on the side section and four outlet pipes located at the top section. All these pipes have a standard diameter of about 0.0349 meters. This strategic arrangement of the pipes ensures an efficient flow of hot air, thereby optimizing heat transfer within the engine system. d. Discretization

[0786] FIG. 17, in a subsequent embodiment, displays an example of the heat exchanger fins geometry 1700 prepared for Computational Fluid Dynamics (CFD) simulation. In this diagram, the fins are shown to extend in x/y/z directions. The figure also highlights four inlet pipes, indicated as 1701, and four outlet pipes, denoted as 1702. For the purpose of these simulations, polyhedral meshing was utilized to create conformal mesh interfaces between different parts of the engine system. This method ensures that the contacting faces between various components share the same boundary face topology, eliminating the need for face interpolation on contacting patches. As a result, more accurate and expedited simulations are achieved.

[0787] The simulations were conducted using four distinct base cell sizes, assisting in checking for mesh size sensitivity and ensuring mesh convergence. To accurately capture the thermal boundary layer for precise simulation results, five layers were established on the interfaces between the air and solid components such as fins, the half cylinder, and the enclosure.

[0788] FIG. 18, in another embodiment, presents examples of a cross-section of the Free Piston Stirling Engine's (FPSE) fine mesh at the top 1801, middle 1802, and bottom sections 1803, respectively. These figures provide a clear visualization of the intricate polyhedral meshing applied to the system. This complex meshing methodology serves as a beneficial element contributing to the comprehensive and accurate results achieved in the Computational Fluid Dynamics (CFD) simulations. e. Pre-processing

[0789] In the Computer-Aided Architecture (CAD) geometry, as demonstrated in FIG. 16, four distinct simulation domains were identified in this embodiment. These include the enclosure made of stainless steel, the air, the fins composed of copper, and the half cylinder made of Inconel. With the exception of Inconel, all other material properties were sourced from the STAR-CCM+ materials library.

[0790] Each simulation domain was assigned a specific simulation model, which was contingent upon the material involved. FIG. 19A - FIG. 19B present examples of models used for air 1901 and solid 1902 domains. This figure provides a comprehensive overview of the case setup within the STAR-CCM+ software, outlining the various domains that constitute the entire engine system. f. Boundary Conditions

[0791] In the Computational Fluid Dynamics (CFD) simulation, as depicted in FIG. 20, specific boundary conditions were applied to the heat exchanger fins of the Free Piston Stirling Engine. As demonstrated in this figure, the enclosure walls were treated as adiabatic, indicating that no heat transfer occurs through them. The inner wall of the cylinder was assigned a convective heat transfer coefficient of 860 W/m^A2K at a temperature of 300°C. [0792] Each inlet pipe, represented by red arrows, had a predetermined mass flow rate of 0.003184 kg/s, a temperature of 650°C, and a pressure of 101,305 Pa. Conversely, the outlets, indicated by blue arrows, were designated as pressure outlets. It is beneficial to note that the working fluid within the enclosure is air, and its pressure was set at 101,305 Pa. [0793] In addition, all surfaces were assigned a degree of roughness and treated as walls, effectively replicating the physical attributes of the real engine components. This approach ensures an accurate and realistic simulation of the engine performance.

[0794] FIG. 20, in this further embodiment, also highlights the 3D geometry of the external fins 2001, the inlet pipes 2002, and a cross-section of the fins, enclosure, and inner wall of the cylinder 2003. These visuals contribute beneficially towards a comprehensive understanding of the engine's structure and the conditions under which its simulations are conducted. This robust visual representation ensures an in-depth analysis and evaluation of the Free Piston Stirling Engine. g. Post Processing

[0795] In this embodiment's initial step, a mesh independence study was undertaken. Throughout this process, the number of mesh cells was progressively increased, while concurrently monitoring the convective surface mean temperature on the inner shell, depicted in FIG. 21. This figure illustrates the geometry of the heat exchanger fins for the Computational Fluid Dynamics (CFD) simulation.

[0796] In the half-cylinder average temperature graph 2100 depicted in FIG. 21, the y-axis represents the inner shell convective surface mean temperature in Celsius, with a range from 566.3 to 566.8°C. Conversely, the x-axis 2101 signifies the size index of the mesh, varying between 0 and 2.

[0797] Table 9 presents the results of an analysis examining the sensitivity of simulation outcomes to mesh density.

[0798] As detailed in Table 9, the analysis comprised three different mesh resolutions: coarse, medium, and fine. The coarse mesh comprises 630309 cells, a temperature of 566.45°C, and a size index of 1.8. The medium mesh comprises 990522 cells, a temperature of 566.5 °C, and a size index of 1.5. The fine mesh comprises 309747 cells, a temperature of 566.6°C, and a size index of 1.

[0799] The purpose of this analysis was to verify that the simulation outcomes may be modified by mesh resolution above a certain density, thereby supporting the accuracy and robustness of the model.

Table 9: Mesh Independence Study Details

[0800] In a subsequent phase of this embodiment, a comprehensive analysis was conducted to examine the surface temperature and heat flux contours of the Free Piston Stirling Engine (FPSE). As demonstrated in FIG. 22A and FIG. 22B, contours of the surface temperature provide a cross-sectional view of various FPSE components. These visual representations offer valuable insights into the patterns of temperature distribution across the surfaces, thereby enhancing our understanding of the thermal performance of the engine under different conditions.

[0801] Complementing the temperature contours, the heat flux within the Free Piston Stirling Engine (FPSE) is represented in FIG. 23A and FIG. 23B. These figures display the heat flux contours and provide indispensable information about the rate and direction of heat transfer across the surface of the heat exchanger fins within the engine. This information may be observed from a top-down perspective (FIG. 23A) as well as an angled viewpoint (FIG. 23B).

[0802] A comprehensive understanding of the thermal behavior of the FPSE, beneficial for its performance optimization, may be derived from the combination of these surface temperature and heat flux contours. Moreover, FIG. 24A and FIG. 24B illustrate air temperature contours at various planes, further enriching the detailed thermal analysis of the FPSE. Studying these aspects in conjunction provides valuable insights into the heat management dynamics within the engine, a beneficial factor for enhancing its efficiency. h. Solid Parts Surface Temp

[0803] FIG. 22B, in this subsequent embodiment, illustrates a conventional finned heat exchanger, represented in 3D CAD as 2202. As depicted in this figure, the temperature of the heat exchanger ranges between 387°C and 444°C, with the inner surface of the heat exchanger predominantly within the region between 387°C to 415°C.

[0804] In contrast, FIG. 22A provides a 3D CAD representation, denoted as 2201, of a heat exchanger cylinder and its casing. As observed in depiction 2201, the temperature of the heat exchanger spans from 312°C to 488°C. A uniform temperature distribution is evident across the entire range on both the internal and external surfaces of the heat exchanger.

[0805] This indicates that the topology-optimized heat exchanger offers more consistent temperature regulation than its conventional counterpart, indicating a beneficial performance enhancement when operating under low-temperature conditions. More specifically, the area surrounding the cylinder and fins, identified as 2203, lies almost entirely in the 444°C to 488°C range. Additionally, the external surface of the fins, indicated as 2205, primarily falls in the 420-488°C range, mostly between 400°C and 444°C. The internal surface of the cylinder, represented as 2204, is mainly in the 313 °C to 420°C range.

[0806] These detailed temperature distributions contribute to a comprehensive understanding of thermal dynamics within the engine, providing beneficial insights for performance optimization.

Heat Flux Distribution

[0807] FIG. 23A and FIG. 23B, in this next phase of the embodiment, provide different perspectives of the heat flux distribution across the surface of the Free Piston Stirling Engine's (FPSE) heat exchanger fins. FIG. 23A offers an aerial view of this distribution, while FIG. 23B delivers an angled view, both accentuating the multidirectional nature of the heat flux.

[0808] As evidenced in both figures, the heat flux across the external surfaces of the fins, indicated as 2302, extends in the x, y, and z directions. It ranges from about -5e0.4 W/m^A2 to around 5e0.4 W/m^A2. More specifically, the heat flux distribution on the external surface predominantly lies between 0 W/m^A2 and about 5e0.4 W/m^A2. This consistent degree of heat transfer efficiency within this range signifies that the architecture and material of the fins effectively distribute heat in all directions. This even distribution assists in preventing localized heat concentration, thereby enhancing the overall engine efficiency.

[0809] Conversely, the heat flux distribution on the internal surfaces of the heat exchanger fins, referenced as 2301, spans between -5e0.4 W/m^A2 and about 0 W/m^A2 in the x, y, and z directions. This consistent heat transfer efficiency within this range further underscores the effectiveness of the fins' architecture and material in uniformly distributing heat, thereby boosting the engine's overall efficiency.

Air temperature contours

[0810] FIG. 24A, in a subsequent phase of this embodiment, presents a representation of air temperature contours at a side plane of the Free Piston Stirling Engine (FPSE) equipped with heat exchanger fins. The depicted contours demonstrate a temperature range from about 300°C to 650°C.

[0811] Specifically, these temperature contours, which spanfrom about 300°C to 650°C, are observed between the inner surface 2401 of the cylinder and the outer surface of the heat exchanger fins 2402.

[0812] Moreover, an area encircling the top surface of the heating head, denoted as 2403, displays air temperature contours lying between about 450°C and 570°C. This specific temperature range represents a zone of peak heat transfer, emphasizing the effectiveness of the heat exchanger fins in this region.

[0813] The diversity in air temperature contours along the sides and top surface of the heating head indicates a uniform heat distribution across these areas. This even temperature distribution is indicative of effective heat transfer within the FPSE, a beneficial factor in maintaining engine performance and efficiency. The data suggest that the heating head and heat exchanger fins efficiently dissipate heat to the surrounding air.

[0814] On the other hand, FIG. 24B presents air temperature contours at a side plane of the FPSE without heat exchanger fins. These contours span a temperature range from about 300°C to 650°C. The inner and outer boundary of the cylinder, 2404, is mostly within the blue range, indicating temperatures from 300°C to less than 475 °C. The region surrounding the cylinder, referenced as 2403, falls between 300°C and 650°C, with the majority of the area being between 475°C and 562°C. These observations further validate the importance of heat exchanger fins for improved thermal performance within the FPSE.

Air velocity and pressure contours at side plane

[0815] Air velocity and pressure contours are displayed in FIG. 24C and FIG. 25, respectively, in a further embodiment, while the average outlet air temperature from the enclosure is demonstrated in FIG. 26. The average air temperature was calculated in this study.

[0816] More specifically, FIG. 24C depicts an example of air velocity contours at a side plane of the Free Piston Stirling Engine (FPSE). The velocity magnitude in these contours ranges from about 0 m/s to about 10 m/s.

[0817] In more detail, these velocity contours between the inner surface 2401 of the heating head and the outer surface of the heat exchanger fins 2402 fall within a range of about 0 m/s to 10 m/s. Additionally, an area 2403 surrounding the top surface of the heat head displays velocity contours lying between about 0 m/s and about 6 m/s. Most notably, a majority of the region exhibits a velocity magnitude between about 0 m/s and 3 m/s.

[0818] These velocity contour ranges ly suggest a smooth and relatively slow airflow in most areas of the FPSE. This observation may indicate that the architecture provides efficient heat transfer while minimizing heat losses due to rapid air movement. It also points to a well- regulated airflow that helps maintain the desired temperature levels within the system, thereby contributing to the engine's overall efficiency.

[0819] FIG. 24D, in the next phase of this embodiment, provides an illustration of air pressure contours at a side plane of the Free Piston Stirling Engine (FPSE). In these contours, the pressure magnitude spans from about -8 Pa to about 16 Pa.

[0820] More specifically, the pressure contours between the inner surface 2401 of the heating head and the outer surface of the heat exchanger fins 2402 range from about -8 Pa to about 5 Pa. Additionally, there is an area 2403 surrounding the top surface of the heat head where the pressure contours range from about -8 Pa to roughly 4 Pa. Most notably, a majority of the region exhibits a pressure between about -8 Pa to 4 Pa.

[0821] This relatively narrow range of pressure values indicates a balanced and well- maintained pressure environment within the system. This equilibrium is beneficial to the FPSE's operation, as it may prevent unwanted fluctuations in flow characteristics, contributing to more efficient and stable heat transfer performance. Waste heat source Air Contours

[0822] FIG. 25, in a further embodiment, offers an illustrative representation (2500) of the temperature contours of the exhaust fluid air in the Free Piston Stirling Engine (FPSE). The temperature of the waste heat source air, divided into areas 2501 to 2504, falls within a range of about 386°C to 515°C.

[0823] Notably, a beneficial portion of the waste heat source air displays a temperature ranging from about 500°C to 515°C. Moreover, smaller sections within segments 2501 to 2504 exhibit temperatures closer to 450°C at the outer surface of the waste heat source air stream.

[0824] The Computational Fluid Dynamics (CFD) results demonstrated efficient heat transfer from the hot air to the fins. About 1830W of heat was extracted from a maximum theoretical heat transfer value. These findings suggest high heat transfer efficiency in the FPSE.

[0825] The substantial waste heat source air temperature range infers that the generated heat in the engine is efficiently utilized before being transferred. This efficient utilization and conversion of heat lead to improved performance and enhanced overall efficiency of the FPSE.

[0826] Smaller sections, indicating temperatures closer to 450°C at the outer surface of the waste heat source air stream, might represent areas of slightly lower heat transfer efficiency. The extraction of substantial heat from the hot air underscores this high efficiency, illustrating that more than half of the heat energy may be successfully transferred from the hot air to the fins. This effective heat utilization and transfer contribute to less fluid required for cooling, increased energy generation from a given amount of heat, and ly reduced cooling requirements and energy costs.

Rate of heating

[0827] FIG. 26, in a subsequent phase of this embodiment, offers an illustrative example of the heat transfer across different contact surfaces in the Free Piston Stirling Engine (FPSE), detailing heat transfer values for these various components. The rate of heating bar graph 2600 in FIG. 26 plots power (W) on the y-axis against different contact surfaces on the x- axis.

[0828] The theoretical power, indicated by 2601, exceeds 3500 W, representing the maximum power transfer. In contrast, 2602 represents the power at the air-fins contact surface, where power transfer exceeds 1500 W. Next, 2603 denotes the power from the air to the cylinder contact surfaces (which may include the heating head, cooling head, and regenerator), with power values ranging between 0 W and 300 W. Finally, 2604 indicates the power at the enclosure-half cylinder contact surface, where power ranges between 0 W and 200 W.

[0829] These varying power values across different contact surfaces indicate a well- configured and efficient heat transfer system within the FPSE. The high heat transfer power at the air-fins contact surface 2602 demonstrates that the fins effectively dissipate the heat from the air. The lower power values at the air-half cylinder contact surfaces 2603 and the enclosure-half cylinder contact surface 2604 suggest that these areas serve as thermal buffers, limiting heat loss from the engine. This controlled heat distribution contributes to the improved performance and high efficiency of the FPSE.

Example 4 — Topology Optimization

[0830] In a further embodiment titled "Enhanced Heat Extraction System (EHES)", FIG. 27 illustrates an exemplar of such a system. This diagram demonstrates the architecture space envelope 2701 around the cylinder 2702, which is utilized for topology optimization.

[0831] The depicted system integrates Topological Optimization (TO) with the heat transfer process of enclosure-based equipment. This unique integration allows for a process that not only expels heat from the components but also optimizes the efficiency of heat extraction. Within this setup, the heat absorbed by the acceptor fins from the enclosure equipment acts as a catalyst for the TO process.

[0832] In this specific instance, the TO algorithm adjusts the distribution of material within the fins in an iterative manner. It leverages the thermal energy within the fluid to attain the most suitable structure for heat dissipation or pressure loss minimization. Through this process, the fluid is cooled efficiently while the structure of the fins is simultaneously optimized, thereby improving the system's overall heat extraction efficiency. The hot air is circulated through the surfaces of the TO-enhanced fins, leading to a temperature differential for operation. After passage over the TO-optimized fins, the hot air is cooled, but not to a beneficial extent. Consequently, it passes through another set of acceptor fins for further cooling. These fins, while still warm, are cooling head than their initial state.

[0833] This additional cooling step further reduces the fluid temperature, preparing it for recirculation back to the enclosure’s heat extraction system. The slightly warmed fins also revert back to the original temperature, ready for a new cycle. This innovative approach offers several advantages, such as a decrease in energy consumption by reusing the fluid for efficient heat extraction. It also contributes to environmental sustainability by reducing the dependence on traditional energy sources for cooling and associated operational costs. [0834] This method embodies an efficient strategy for energy management and conservation in enclosure operations, striking a balance between operational efficiency and environmental responsibilities. i. Meshing

[0835] Due to the complex form and intensive optimization of the heat exchanger surface processing in this further embodiment, a conformal mesh was employed, despite the detailed part preparation process it necessitates. This type of mesh was chosen to ensure accurate conservation of heat transfer values. Utilizing non-conformal meshing processes may require less preparation, but at the expense of precision due to the mapping involved.

[0836] FIG. 28A presents an example 2801 of a conformal mesh for the air domain (represented in yellow) and the solid fin domains (depicted in blue). FIG. 28B highlights a close-up view 2802 of a portion of the conformal mesh for both the air domain and solid fin domains.

[0837] This conformal mesh was explicitly configured for the topology-optimized organic Heat Extractor (HE). As observed in the figures, the surface mesh between all parts on the interface shares points at the interface, ensuring conservation of heat transfer between the parts. Beneficially, the base mesh cell size has a considerable effect on the optimization results, where smaller mesh cells yield a finer fin surface and a larger surface area.

[0838] In both FIG. 28A and FIG. 28B, the conformal mesh for the air domain is represented by 2803, while the conformal mesh for the solid fins domain is represented by 2804. These figures provide a visual understanding of how the conformal mesh facilitates accurate heat transfer computations in the complex geometry of the topology -optimized heat exchanger.

Topology Optimization Pre-processing

[0839] Initially, in this subsequent embodiment, the settings for the simulation case were established using predefined solver parameters as suggested by the STAR-CCM+ platform. However, initial testing resulted in suboptimal optimization performance, characterized by pronounced fluctuations in heat transfer values and negligible improvements, even after numerous optimization iterations. Therefore, a parametric study was initiated to assess the modify of various topology optimization (TO) solver parameters on the objective function, defined as the total heat transfer rate from air to the acceptor Heat Exchanger (HE). This sensitivity analysis explored parameters such as topology holes and source strength, penalty value, intensity of the optimized fin surface smoothing, and step size. Of these parameters, step size had the most beneficial effect on the optimization results. [0840] Geometry preparation for a full Computational Fluid Dynamics (CFD) simulation involving a Boundary Layer (BL) followed specific steps: initially, smoothing the derived part geometry, followed by creating a watertight surface with a surface wrapper. Then, the watertight surface of the optimized geometry was imported into the full case, including the BL. Subsequently, the watertight surface was subtracted from the air domain and imprinted on the cylinder geometry. The final step ensured all necessary interfaces were correctly created. This detailed process guarantees a robust and accurate CFD simulation, beneficially contributing to the optimization of the FPSE.

Topology Optimization Post-processing

[0841] In the current stage of this embodiment, topology optimization (TO) may not include the fluid domain with boundary layers due to the elimination and addition of cells in the TO process. Thus, after the TO geometry is obtained, it is smoothed out and reintroduced for validation and verification via Computational Fluid Dynamics (CFD) simulations within the STAR-CCM+ platform. This progression to 3D CFD simulations represents a beneficial step towards enhancing the FPSE's performance by refining both the internal and external heat exchangers.

[0842] Resulting from topology optimization, organically shaped heat exchangers (HE) were developed. These heat exchangers may ly lead to an enhancement in heat transfer, the specific percentage of which remains to be determined, compared to traditional heat exchangers. This characteristic is beneficial for the low-temperature operation of the engine. Notably, these efficient and intricate heat exchangers have become feasible for production through additive manufacturing techniques such as Direct Metal Laser Sintering (DMLS). [0843] FIG. 30A presents a 3D CAD representation (3001) and a 3D CFD representation (3002) of a conventional finned heat exchanger. As shown in 3002, the temperature of the heat exchanger ranges between 289°C and 443°C, with the inner surface of the heat exchanger predominantly in the region between 389°C and 416°C.

[0844] Conversely, FIG. 30B displays a 3D CAD (3003) and a 3D CFD (3004) representation of a topology-optimized heat exchanger. As indicated in 3004, the temperature of the optimized heat exchanger spans from 234°C to 525°C. The internal and external surfaces of the heat exchanger display a uniform temperature distribution across the entire range, indicating enhanced and consistent temperature regulation compared to its conventional counterpart.

[0845] This improvement indicates that the employment of topologically-optimized heat exchangers may notably augment the performance of engines, particularly under low- temperature conditions. This highlights the beneficial of employing topology optimization in the architecture and efficiency enhancement of Free Piston Stirling Engines.

[0846] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. It is not intended that the invention be limited by the specific examples provided within the specification. While the invention has been described with reference to the aforementioned specification, the descriptions and illustrations of the embodiments herein are not meant to be construed in a limiting sense. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. Furthermore, it shall be understood that all aspects of the invention are not limited to the specific depictions, configurations or relative proportions set forth herein which depend upon a variety of conditions and variables. It may be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is therefore contemplated that the invention shall also cover any such alternatives, modifications, variations, or equivalents. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

CLAIMS WHAT IS CLAIMED IS:

1. An energy production system (EPS), the EPS comprising: a. a building; b. a Free Piston Stirling Engine (FPSE); and c. at least one topology optimized heat transfer apparatus.