US11375320B2 - Thermoacoustic device and method of making the same - Google Patents
Thermoacoustic device and method of making the same Download PDFInfo
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- US11375320B2 US11375320B2 US16/556,228 US201916556228A US11375320B2 US 11375320 B2 US11375320 B2 US 11375320B2 US 201916556228 A US201916556228 A US 201916556228A US 11375320 B2 US11375320 B2 US 11375320B2
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04R—LOUDSPEAKERS, MICROPHONES, GRAMOPHONE PICK-UPS OR LIKE ACOUSTIC ELECTROMECHANICAL TRANSDUCERS; DEAF-AID SETS; PUBLIC ADDRESS SYSTEMS
- H04R23/00—Transducers other than those covered by groups H04R9/00 - H04R21/00
- H04R23/002—Transducers other than those covered by groups H04R9/00 - H04R21/00 using electrothermic-effect transducer
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- H—ELECTRICITY
- H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
- H05B—ELECTRIC HEATING; ELECTRIC LIGHT SOURCES NOT OTHERWISE PROVIDED FOR; CIRCUIT ARRANGEMENTS FOR ELECTRIC LIGHT SOURCES, IN GENERAL
- H05B1/00—Details of electric heating devices
- H05B1/02—Automatic switching arrangements specially adapted to apparatus ; Control of heating devices
- H05B1/0227—Applications
- H05B1/0288—Applications for non specified applications
- H05B1/0294—Planar elements
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F02—COMBUSTION ENGINES; HOT-GAS OR COMBUSTION-PRODUCT ENGINE PLANTS
- F02G—HOT GAS OR COMBUSTION-PRODUCT POSITIVE-DISPLACEMENT ENGINE PLANTS; USE OF WASTE HEAT OF COMBUSTION ENGINES; NOT OTHERWISE PROVIDED FOR
- F02G2243/00—Stirling type engines having closed regenerative thermodynamic cycles with flow controlled by volume changes
- F02G2243/30—Stirling type engines having closed regenerative thermodynamic cycles with flow controlled by volume changes having their pistons and displacers each in separate cylinders
- F02G2243/50—Stirling type engines having closed regenerative thermodynamic cycles with flow controlled by volume changes having their pistons and displacers each in separate cylinders having resonance tubes
- F02G2243/54—Stirling type engines having closed regenerative thermodynamic cycles with flow controlled by volume changes having their pistons and displacers each in separate cylinders having resonance tubes thermo-acoustic
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F25—REFRIGERATION OR COOLING; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS; MANUFACTURE OR STORAGE OF ICE; LIQUEFACTION SOLIDIFICATION OF GASES
- F25B—REFRIGERATION MACHINES, PLANTS OR SYSTEMS; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS
- F25B21/00—Machines, plants or systems, using electric or magnetic effects
- F25B21/02—Machines, plants or systems, using electric or magnetic effects using Peltier effect; using Nernst-Ettinghausen effect
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F25—REFRIGERATION OR COOLING; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS; MANUFACTURE OR STORAGE OF ICE; LIQUEFACTION SOLIDIFICATION OF GASES
- F25B—REFRIGERATION MACHINES, PLANTS OR SYSTEMS; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS
- F25B2309/00—Gas cycle refrigeration machines
- F25B2309/14—Compression machines, plants or systems characterised by the cycle used
- F25B2309/1402—Pulse-tube cycles with acoustic driver
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F25—REFRIGERATION OR COOLING; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS; MANUFACTURE OR STORAGE OF ICE; LIQUEFACTION SOLIDIFICATION OF GASES
- F25B—REFRIGERATION MACHINES, PLANTS OR SYSTEMS; COMBINED HEATING AND REFRIGERATION SYSTEMS; HEAT PUMP SYSTEMS
- F25B9/00—Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point
- F25B9/14—Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point characterised by the cycle used, e.g. Stirling cycle
- F25B9/145—Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point characterised by the cycle used, e.g. Stirling cycle pulse-tube cycle
Definitions
- thermoacoustic oscillations in thermally-driven fluids and gases has been known for centuries.
- a pressure wave travels in a confined gas-filled cavity while being provided heat, the amplitude of the pressure oscillations can grow unbounded. This self-sustaining process builds upon the dynamic instabilities that are intrinsic in the thermoacoustic process.
- thermoacoustics make key contributions to the theory of thermoacoustics by developing a fully analytical, quasi-one-dimensional, linear theory that provided excellent predictive capabilities. It was mostly Swift, at the end of the last century, who started a prolific series of studies dedicated to the design of various types of thermoacoustic engines based on Rott's theory. Since the development of the fundamental theory, many studies have explored practical applications of the thermoacoustic phenomenon with particular attention to the design of engines and refrigerators. However, to-date, thermoacoustic instabilities have been theorized and demonstrated only for fluids.
- thermoacoustic instability we first introduce the theoretical framework that uncovers the existence and the fundamental mechanism at the basis of the thermoacoustic instability in solids. Then, we provide numerical evidence to show that the instability can be effectively triggered and sustained. We anticipate that, although the fundamental physical mechanism resembles the thermoacoustic of fluids, the different nature of sound and heat propagation in solids produces noticeable differences in the theoretical formulations and in the practical implementations of the phenomenon.
- the fundamental system under investigation consists of a slender solid metal rod with circular cross section ( FIG. 1 ).
- the rod is subject to a temperature (spatial) gradient applied on its outer surface at a prescribed location, while the remaining sections have adiabatic boundary conditions.
- thermoelasticity an elastic wave traveling through a solid medium is accompanied by a thermal wave, and viceversa.
- the thermal wave follows from the thermoelastic coupling which produces local temperature fluctuations (around an average constant temperature T 0 ) as a result of a propagating stress wave.
- the elastic wave When the elastic wave is not actively sustained by an external mechanical source, it attenuates and disappears over a few wavelengths due to the presence of dissipative mechanisms (such as, material damping); in this case the system has a positive decay rate (or, equivalently, a negative growth rate).
- the mechanical wave In the ideal case of an undamped thermoelastic system, the mechanical wave does not attenuate but, nevertheless, it maintains bounded amplitude. In such situation, the total energy of the system is conserved (energy is continuously exchanged between the thermal and mechanical waves) and the stress wave exhibits a zero decay rate (or, equivalently, a zero growth rate).
- thermoelastic response can become unstable.
- the initial mechanical perturbation can grow unbounded due to the coupling between the mechanical and the thermal response.
- thermoacoustic device includes a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage.
- the stage further includes a first cooling component on a second terminus of the stage.
- a thermal conductivity of the stage is higher than a thermal conductivity of the bar.
- a heat capacity of the stage is higher than a heat capacity of the bar.
- thermoacoustic device including a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage. Additionally, the stage includes a first cooling component on a second terminus of the stage. A thermal conductivity of the stage is higher than a thermal conductivity of the bar. A heat capacity of the stage is higher than a heat capacity of the bar, and the bar forms a closed loop. Moreover, the thermoacoustic device includes a second cooling component on the bar, wherein the second cooling component is configured to cool to a same temperature as the first cooling component.
- thermoacoustic device including a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage. Additionally, the stage includes a first cooling component on a second terminus of the stage. A thermal conductivity of the stage is higher than a thermal conductivity of the bar. A heat capacity of the stage is higher than a heat capacity of the bar. Moreover, the bar includes a material wherein the material does not oxidize at temperatures ranging from ⁇ 100° C. to 2000° C. Further, the material remains a solid at temperatures ranging from ⁇ 100° C. to 2000° C.
- FIG. 1( a ) illustrates a system exhibiting thermoacoustic response.
- FIG. 1( b ) illustrates idealized reference temperature profile produced along the rod.
- FIG. 2( a ) illustrates thermodynamic cycle of a Lagrangian particle in the S-segment during an acoustic/elastic cycle.
- FIG. 2( b ) illustrates time averaged volume change work along the length of the rod showing that the net work is generated in the stage.
- FIG. 2( c ) illustrates evolution of an infinitesimal volume element during the different phases of the thermodynamic cycle.
- FIG. 2( d ) illustrates time history of the axial displacement fluctuation at the end of the rod for the fixed mass configuration.
- FIG. 2( e ) illustrates a table presenting a comparison of the results between the quasi-1D theory and the numerical FE 3D model.
- FIG. 3( a ) illustrates growth ratio versus the location of the stage non-dimensionalized by the length L of the rod.
- FIG. 3( b ) illustrates growth ratio versus the penetration thickness non-dimensionalized by the rod radius R.
- FIG. 4( a ) illustrates a multi-stage configuration.
- FIG. 4( b ) illustrates undamped time response at the moving end of a fixed-mass rod.
- FIG. 4( c ) illustrates 1% damped time response at the moving end of a fixed-mass rod.
- FIG. 5( a ) illustrates a looped rod.
- FIG. 5( b ) illustrates a resonance rod.
- FIG. 5( c ) illustrates temperature profile of the looped rod.
- FIG. 5( d ) illustrates temperature profile of the resonance rod.
- FIG. 6 illustrates mode shapes of the looped and the resonance rod and the naming convention for modes.
- FIG. 7 illustrates a semilog plot of the growth ratio versus the nondimensional radius for the Loop-1 mode in the looped rod and the Res-II mode in the resonance rod.
- FIG. 8 illustrates plot of the growth ratio versus the normalized stage location for the resonance rod Res-II.
- FIG. 9 illustrates plot of phase difference between engative stress and particle velocity for a resonance rod ‘Res-II’ versus a looped rod ‘Loop-1’.
- FIG. 10( a ) illustrates cycle-averaged heat flux for the looped rod.
- FIG. 10( b ) illustrates cycle-averaged mechanical power for the looped rod.
- FIG. 10( c ) illustrates cycle-averaged heat flux for the resonance rod.
- FIG. 10( d ) illustrates cycle-averaged mechanical power for the resonance rod.
- FIG. 11 illustrates relative difference of the growth rates estimates from energy budgets for the standing wave configuration and traveling wave configuration.
- FIG. 12( a ) illustrates an acoustic energy budget (LHS) for the traveling wave configuration.
- FIG. 12( b ) illustrates an acoustic energy budget (RHS) for the traveling wave configuration.
- FIG. 12( c ) illustrates an acoustic energy budget (LHS) for the standing wave configuration.
- FIG. 12( d ) illustrates an acoustic energy budget (RHS) for the standing wave configuration.
- FIG. 13 illustrates efficiencies of the traveling wave configuration and the standing wave configuration at various temperature differences.
- thermoacoustic equations for a homogeneous isotropic solid in an Eulerian reference frame are written as:
- Eqs. (1) and (2) are the conservation of momentum and energy, respectively.
- ⁇ is the material density
- E is the Young's modulus
- ⁇ is the Poisson's ratio
- ⁇ is the thermoelastic expansion coefficient
- c ⁇ is the specific heat at constant strain
- ⁇ is the thermal conductivity of the medium
- v i is the particle velocity in the x i direction
- ⁇ ij 2 ⁇ ij +[ ⁇ L e v ⁇ (2 ⁇ +3 ⁇ L )( T ⁇ T 0 )] ⁇ ij , (3)
- ⁇ and ⁇ L are the Lame constants
- ⁇ jj is the strain tensor
- T 0 is the mean temperature
- ⁇ ij is the Kronecker delta.
- thermoacoustic instability The fundamental element for the onset of the thermoacoustic instability is the application of a thermal gradient.
- the gradient is applied by using a stack element which enforces a linear temperature gradient over a selected portion of the domain. The remaining sections are kept under adiabatic conditions.
- the S-segment was the region underneath the stage, where the spatial temperature gradient was applied and heat exchange could take place.
- An important consideration must be drawn at this point.
- the interface between the stage and the rod should be highly conductive from a thermal standpoint, while providing negligible shear rigidity. This is a challenging condition to satisfy in mechanical systems and highlights a complexity that must be overcome to perform an experimental validation.
- the governing equations can be solved in order to show that the dynamic response of the solid accepts thermoacoustically unstable solutions.
- ⁇ G ⁇ ⁇ ⁇ E ⁇ ⁇ ⁇ c ⁇ ⁇ ( 1 - 2 ⁇ ⁇ v ) is the Grüneisen constant, i is the imaginary unit, û, ⁇ circumflex over (v) ⁇ and ⁇ circumflex over (T) ⁇ are the fluctuations of the particle displacement, particle velocity, and temperature averaged over the cross section of the rod. For brevity, they will be referred to as fluctuation terms in the following.
- the ⁇ H ⁇ circumflex over (T) ⁇ term in Eqn. 6 accounts for the thermal conduction in the radial direction, and it is the term that renders the theory quasi-1D.
- the function ⁇ H is given by:
- ⁇ H ⁇ ⁇ ⁇ ⁇ ⁇ top ⁇ ⁇ J 1 ⁇ ( ⁇ top ) J 0 ⁇ ( ⁇ top ) i ⁇ ⁇ ⁇ top ⁇ ⁇ J 1 ⁇ ( ⁇ top ) J 0 ⁇ ( ⁇ top ) - R 2 ⁇ k 2 x h ⁇ x ⁇ x c 0 elsewhere , ( 7 )
- J n ( ⁇ ) are Bessel functions of the first kind
- ⁇ is a dimensionless complex radial coordinate given by
- ⁇ top - 2 ⁇ i ⁇ R ⁇ k , where R is the radius of the rod.
- the thermal penetration thickness ⁇ k is defined as
- ⁇ k 2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ c ⁇ , and physically represents the depth along the radial direction (measured from the isothermal boundary) that heat diffuses through.
- the one-dimensional model was used to perform a stability eigenvalue analysis.
- the strength of the instability in classical thermoacoustics (often quantified in terms of the ratio ⁇ / ⁇ ) depends, among the many parameters, on the location of the thermal gradient.
- This location is also function of the wavelength of the acoustic mode that triggers the instability, and therefore of the specific (mechanical) boundary conditions.
- thermoacoustic theory provided a first important conclusion of this study, that is confirming the existence of thermoacoustic instabilities in solids as well as their conceptual affinity with the analogous phenomenon in fluids.
- the mechanical work transfer rate or, equivalently, the volume-change work per unit volume may be defined as
- w . - ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ t , where ⁇ and ⁇ are the total axial stress (i.e. including both mechanical and thermal components) and strain, respectively.
- ⁇ and ⁇ are the total axial stress (i.e. including both mechanical and thermal components) and strain, respectively.
- FIG. 2( a ) shows the ⁇ ⁇ diagram where the area enclosed in the curve represents the work per unit volume done by the infinitesimal volume element in one cycle.
- FIG. 2( c ) shows a schematic representation of the thermo-mechanical process taking place over an entire vibration cycle.
- the infinitesimal volume element When the infinitesimal volume element is compressed, it is displaced along the x direction while its temperature increases (step 1). As the element reaches a new location, heat transfer takes place between the element and its environment. Assuming that in this new position the element temperature is lower than the surrounding temperature, then the environment provides heat to the element causing its expansion. In this case, the element does net work dW (step 2) due to volume change. Similarly, when the element expands (step 3), the process repeats analogously with the element moving backwards towards the opposite extreme where it encounters surrounding areas at lower temperature so that heat is now extracted from the particle (and provided to the stage). In this case, work dW′ is done on the element due to its contraction (step 4). The net work generated during one cycle is dW-dW′.
- FIG. 2( d ) shows the time history of the axial displacement fluctuation u′ at the free end of the rod.
- the time response is evidently growing in time therefore showing clear signs of instability.
- the growth rate was estimated by either a logarithmic increment approach or an exponential fit on the envelope of the response.
- the logarithmic increment approach returns ⁇ as:
- thermoacoustic phenomenon in both solids and fluids we note similarities as well as important differences between the underlying mechanisms. These differences are mostly rooted in the form of the constitutive relations of the two media.
- Both the longitudinal mode and the transverse heat transfer are pivotal quantities in thermal-induced oscillations of either fluids or solids.
- the longitudinal mode sustains the stable vibration and provides the necessary energy flow, while the transverse heat transfer controls the heat and momentum exchange between the medium and the stage/stack.
- the growth rate of the mechanical oscillations is affected by several parameters including the amplitude of the temperature gradient, the location of the stage, the thermal penetration thickness, and the energy dissipation in the system.
- the amplitude of the temperature gradient is affected by several parameters including the amplitude of the temperature gradient, the location of the stage, the thermal penetration thickness, and the energy dissipation in the system.
- the effect of the temperature gradient is straightforward because higher gradients result in higher growth rate.
- the location of the stage relates to the phase lag between the particle velocity and the temperature fluctuations, which is one of the main driver to achieve the instability.
- the optimal location of the stack in a tube with closed ends is about one-forth the tube length, measured from the hot end.
- the optimal location of the stage is at the midspan for the fixed-free boundary condition, and at the mass end for the fixed-mass boundary condition ( FIG. 3 a ).
- ⁇ k 2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ c ⁇ indicates the distance, measured from the isothermal boundary, that heat can diffuse through. Solid particles that are outside this thermal layer do not experience radial temperature fluctuations and therefore do not contribute to building the instability.
- the value of the thermal penetration thickness ⁇ k or more specifically, the ratio of ⁇ k /R is a key parameter for the design of the system. Theoretically, the optimal value of this parameter is attained when the rod radius is equal to ⁇ k . In fluids, good performance can be obtained for values of 2 ⁇ k to 3 ⁇ k .
- the damping contributes to the negative real part of the system eigenvalue, therefore effectively counteracting the thermoacoustic growth rate (which, as shown above, requires a positive real part).
- the thermally induced growth i.e. the thermoacoustic effect
- this condition translates into the ratio
- the damping ratio ⁇ is generally very small (on the order of 1% for aluminum).
- ⁇ ⁇ becomes one or two orders of magnitude lower than the damping ratio ⁇ . Therefore, despite the relatively low intrinsic damping of the material the growth is effectively impeded.
- thermodynamic cycle in fluids is done by thermal expansion at high pressure (or stress, in the case of solids) and compression at low pressure.
- Thermal deformation in fluids and solids can occur on largely disparate spatial scales. This behavior mostly reflects the difference in the material parameters involved in the constitutive laws with particular regard to the Young's modulus and the thermal expansion coefficient.
- a solid exhibits a lower sensitivity to thermal-induced deformations which ultimately limits the net work produced during each cycle, therefore directly affecting the growth rate of the system.
- thermoacoustics in solids is more sensitive to dissipative mechanisms because of the lower net work produced in one cycle.
- multi-stage multiple stage
- this approach simply uses a series of stages uniformly distributed along the rod. The separation distance between two consecutive stages must be small enough, compared to the fundamental wavelength of the standing mode, in order to not alter the phase lag between the temperature and velocity fields.
- FIG. 4 shows the time averaged mechanical work ⁇ dot over (w) ⁇ along the rod.
- the elements in each stage do net work in each cycle. Although the segments between stages are reactive (because the non-uniform T 0 still perturbs the phase), their small size does not alter the overall trend.
- the positive growth rate obtained on the damped system shows that thermoacoustic oscillations can be successfully obtained in a damped solid if a multi-stage configuration is used.
- FIGS. 4 b and 4 c show the time response of the axial displacement fluctuation at the mass-end for both the undamped and the damped rods.
- the quasi-1D theory is still predictive but not as accurate. The reason for this discrepancy can be attributed to the effect of axial heat conduction.
- the net axial heat flux For the single stage configuration, the net axial heat flux
- thermoacoustic response in solids.
- the next logical step in the development of this new branch of thermoacoustics consists in the design of an experiment capable of validating the SS-TA effect and of quantifying the performance.
- the most significant challenge that the authors envision consists in the ability to fabricate an efficient interface (stage-medium) capable of high thermal conductivity and negligible shear force.
- stage-medium efficient interface
- thermoacoustic systems it is relatively simple to create a fluid/solid interface with high heat capacity ratio which is a condition conducive to a strong TA response.
- the absolute difference between the heat capacities of the constitutive elements i.e. the stage and the operating medium
- the absolute difference between the heat capacities of the constitutive elements i.e. the stage and the operating medium
- the stage should have a sufficiently large volume compared to the SS-TA operating medium (in the present case the aluminum rod) in order to behave as an efficient thermal reservoir.
- High thermal conductivity at the interface is also needed to approximate an effective isothermal boundary condition while a zero-shear-force contact would be necessary to allow the free vibration of the solid medium with respect to the stage.
- Such an interface could be approximated by fabricating the stage out of a highly conductive medium (e.g. copper) and using a thermally conductive silver paste as coupler between the stage and the solid rod.
- a highly conductive medium e.g. copper
- thermally conductive silver paste as coupler between the stage and the solid rod.
- this design tends to reduce the thermal transfer at the interface (compared to the conductivity of copper) and therefore it would either reduce the efficiency or require larger temperature gradients to drive the TA engine. Nonetheless, we believe that optimal interface conditions could be achieved by engineering the material properties of the solid so to obtain tailored thermo-mechanical characteristics.
- the solid state design is particularly well suited for piezoelectric energy conversion. Either ceramics or flexible piezoelectric elements can be easily bonded on the solid element in order to perform energy extraction and conversion.
- the SS-TA presents an important advantage. In SS-TA the acoustic energy is already generated in the form of elastic energy within the solid medium and it can be converted directly via the piezoelectric effect. On the contrary, fluid-based systems require an additional intermediate conversion from acoustic to mechanical energy that further limits the efficiency. It is also worth noting that, with the advent of additive manufacturing, the SS-TA can enable an alternative energy extraction approach if the host medium could be built by combining both active and passive materials fully integrated in a single medium.
- thermoacoustic oscillations in solids.
- the theory served as a starting point to develop a quasi-1D linearized model to perform stability analysis and characterize the effect of different design parameters, as well as a nonlinear 3D model.
- the occurrence of the thermoacoustic phenomenon was illustrated for a sample system consisting in a metal rod. Both models were used to simulate the response of the system and to quantify the instability.
- a multi-stage configuration was proposed in order to overcome the effect of structural damping, which is one of the main differences with respect to the thermoacoustics of fluids.
- thermoacoustics of solids were developed with key insights into the underlying mechanisms leading to self-sustained oscillations in thermally-driven solid systems. It is envisioned that the physical phenomenon explored in this study could serve as the fundamental principle to develop a new generation of solid state thermoacoustic engines and refrigerators.
- FIG. 5 we consider two configurations ( FIG. 5 ) in which a ring-shaped slender metal rod with circular cross section is under investigation. Specifically, they are called the looped rod ( FIGS. 5( a ) and 5( c ) ) and the resonance rod ( FIGS. 1( b ) and 1( d ) ).
- the rod experiences an externally imposed axial thermal gradient applied via isothermal conditions on its outer surface at a certain location, while the remaining exposed surfaces are adiabatic.
- the difference between the two configurations lies in the imposition of a displacement/velocity node ( FIG. 1( d ) ), which is used in the resonance rod to suppress the traveling wave mode.
- the displacement node could be realized by constraining the rod with a clamp at a proper location ( FIG. 1( b ) ).
- the coupled thermoacoustic response induced by the external thermal gradient and the initial mechanical excitation is investigated.
- the initial mechanical excitation could grow with time as a result of the coupling between the mechanical and thermal response provided a sufficient temperature gradient is imposed on the outer boundary of a solid rod at a proper location. This phenomenon is identified as the thermoacoustic response of solids.
- a stage element is used to impose a thermal gradient on the surface of the looped rod ( FIG. 5( a ) ).
- the specific location of the stage element in this case is irrelevant due to the periodicity of the system.
- the segment surrounded by the stage is named S-segment, which experiences a spatial temperature gradient (from T c to T h ) due to the externally enforced temperature distribution.
- the interface between the stage and the S-segment is ideally assumed to have a high thermal conductivity, which assures the isothermal boundary conditions along with a zero shear stiffness.
- the stage is considered as a thermal reservoir so that the temperature fluctuation on the surface of S-segment is assumed to be zero (isothermal).
- a Thermal Buffer Segment (TBS) next to the thermal gradient provides a thermal buffer between T h and room temperature T c .
- the temperature drop in the TBS is caused by the secondary cold heat exchanger (SHX, FIG. 5( a ) ) located at x b .
- SHX secondary cold heat exchanger
- we calculated the growth ratio of a standing wave mode in the resonance rod with the same wavelength ( ⁇ L) and frequency (approx.
- the dimensionless growth ratio ⁇ / ⁇ is used as a metric of the SSTA engine's ability to convert heat into mechanical energy; such normalization accounts for the fact that thermoacoustic engines operating at high frequencies naturally exhibit high growth rates and vice versa.
- the inherent structural damping is commonly expressed as a fraction of the frequency of the oscillations, i.e. the damping ratio; the latter is widely used to quantify the frequency-dependent loss/dissipative effect in solids.
- the optimal growth ratio was found by gradually varying the radius R of the looped rod. We used the dimensionless radius R/ ⁇ k to represent the effect of geometry, where ⁇ k was assumed to be constant at the operating frequency
- the $Loop-I curve in FIG. 7 shows the growth ratio ⁇ / ⁇ vs. the dimensionless radius R/ ⁇ k of a full-wavelength traveling wave mode.
- FIG. 7 shows that as R>> ⁇ k , all the curves, whether the looped or the resonance rod, reach zero due to the weakened thermal contact between the solid medium and the stage.
- R/ ⁇ k reaches zero (shaded grey region)
- the stage is very strongly thermally coupled with the elastic wave.
- the traveling wave mode dominates.
- the stability curves also tell that the traveling wave engine has about 4 times higher growth ratio in the limit R/ ⁇ k ⁇ 0, compared to the standing wave resonance rod (Res-II, case A) in which maximal growth ratio is obtained (at R/ ⁇ k ⁇ 2).
- the noteworthy improvement on growth ratio is essential to the design of more robust solid state thermoacoustics devices.
- thermoacoustics the phase delay between pressure and crossectional averaged velocity is an essential controlling parameter of thermoacoustic energy conversion.
- Re[e i( ⁇ t+ ⁇ ⁇ ] and particle velocity v
- Re[e i( ⁇ t+ ⁇ v )], where ⁇ ⁇ and ⁇ v denote the phases of ⁇ and v respectively, ⁇ ⁇ v ⁇ ⁇ .
- a negative stress in solids indicates compression which is equivalent to a positive pressure in fluids.
- Re[e i( ⁇ t+ ⁇ ⁇ + ⁇ /2) ] sin ⁇ and v T
- TWC is not existent.
- the non-zero growth rate ⁇ will cause a small phase shift, which makes the phase difference ⁇ close to but not exactly 90°.
- the mode shape is much similar to that of a resonance rod because SWC is still dominant and the phase difference is close to 90°.
- the displacement nodes may exist intrinsically in the system without clamped points.
- the looped rod is sufficiently thin (R ⁇ k ) the traveling wave component plays a dominant role.
- the phase delay decreases to 30° at most.
- the time history of the displacement along the looped rod shows that, as R ⁇ k (small phase difference), the wave mode is dominated by TWC.
- a cycle-averaged heat flux in the axial direction is generated in the S-segment due to its heat exchange with the stage. Neglecting the axial thermal conductivity, the transport of entropy fluctuations due to the fluctuating velocity v 1 (subscript 1 for a first order fluctuating term in time) is the only way heat can be transported along the axial direction, and it is expressed in the time domain as
- ⁇ dot over (q) ⁇ 2 can be expressed in terms of T 1 , v 1 and ⁇ 1 .
- the counterparts of these three quantities in frequency domain ⁇ circumflex over (T) ⁇ , ⁇ circumflex over (v) ⁇ , and ⁇ circumflex over ( ⁇ ) ⁇ can be extracted from the eigenfunctions of the eigenvalue problem.
- the instantaneous mechanical power carried by the wave is defined as
- This quantity physically represents the rate per unit area at which work is done by an element onto its neighbor. It can be also called ‘work flux’ because it shows the work flow in the medium as well. When an element is compressed ( ⁇ >0), it ‘pushes’ its neighbor so that a positive work is done on the adjacent element. A notable fact is that there is a directionality to I 2 , which depends on the direction of v 1 .
- the work source can be further defined as the gradient of the mechanical power as
- w 2 ⁇ ⁇ _ 1 ⁇ x ⁇ v 1 + ⁇ v 1 ⁇ x ⁇ ⁇ _ 1 ( 9 )
- the first term of w 2 vanishes after applying cycle-averaging, because according to the momentum conservation, ⁇ 1 / ⁇ x and v 1 are 90° out of phase under the assumption that the small phase difference caused by the non-zero ⁇ can be neglected due to: ⁇ / ⁇ 1.
- the remaining term is equivalent to
- 10( a ) and 10( c ) illustrate that heat flux only exists in the S-segment and that wave-induced transport of heat occurs from the hot to the cold heat exchanger.
- the negative values in the S-segment in (a) and (c) are due to the fact that the hot exchanger is on the right side of the cold one, so heat flows to the negative x direction in that case.
- the non-zero spatial gradient in ⁇ tilde over (Q) ⁇ in the S-segment proves that there is heat exchange happening on the boundary of this segment because the heat flux in the axial direction is not balanced on its own.
- FIG. 10( d ) shows the mechanical power in the standing wave engine.
- the positive slope of ⁇ in the S-segment elucidates the fact that the work generated in this region is positive, as discussed above. This amount of work drops along the axial direction in the remaining segments at the spatial rate of d ⁇ /dx. The work drop in the hot and cold segments balances the accumulation of energy because there is no radial energy exchange in these sections. Clearly, if there is no energy growth, the slope of ⁇ should be zero in these sections, as also discussed above.
- the work flow in the traveling wave engine has a very large value, which is due to the fact that negative stress ⁇ and particle velocity v have a phase difference much smaller than 90° ( FIG. 9 ).
- the slope of ⁇ is negative in the S-segment, because it is balancing the positive work created by ⁇ against the temperature gradient in the TBS.
- the volumetric integration of the work source w i.e. the spatial integration of W along the rod, should be zero because, globally, their is no energy output in the system. All the energy converted from the heat in the S-segment should eventually lead to a uniformly distributed perturbation energy growth. More discussions will be addressed in the following paragraphs.
- E 2 1 2 ⁇ ⁇ ⁇ ⁇ v 1 2 + 1 2 ⁇ 1 E ⁇ ( 1 + ⁇ G ⁇ T 0 ) ⁇ ⁇ _ 1 2 ( 15 )
- I 2 ⁇ _ 1 ⁇ v 1 ( 16 )
- R 2 ⁇ 1 + ⁇ G ⁇ T 0 ⁇ dT 0 dx ⁇ I 2 ( 17 )
- P 2 - D 2 ⁇ 1 + ⁇ G ⁇ T 0 ⁇ 1 R ⁇ ⁇ ⁇ ⁇ ⁇ c ⁇ ⁇ q 1 ⁇ q _ 1 ( 18 ) ⁇ 2 , I 2 , 2 , 2 , and 2 are the second order energy norm, work flux, energy redistribution term, thermoacoustic production and dissipation, respectively.
- ⁇ EB P ⁇ - D ⁇ - ( ⁇ I ⁇ ⁇ x + R ⁇ ) 2 ⁇ E ⁇ ( 25 )
- the growth rates ⁇ EB calculated from Eq. (25) are within 0.4% from the direct output of the eigenvalue problem in both the standing wave and the traveling wave configurations, which validates the consistency of the derivations in this paragraph.
- ⁇ I ⁇ ⁇ x is the work source defined in the previous paragraphs, is an energy redistribution term. and are the thermoacoustic production and dissipation, respectively.
- the energy redistribution term in the acoustic energy budgets of solid thermoacoustics cannot be found in the fluid counterpart of the same equations. This term is absent in fluids because it is canceled in the algebraic derivations by expressing the variation of mean density according to the ideal gas law, as a function of the mean temperature gradient.
- solidstate thermoacoustics the heat-induced density variation is neglected and the impact of the temperature gradient is manifest in the stress-strain constitutive relation.
- FIG. 12 plots every term in the acoustic energy budgets (Eq. (19)) in the standing wave and traveling wave configurations, respectively.
- the energy conversion becomes different because of the existence of the TBS.
- the TBS creates a temperature drop, which makes the energy redistribution term non zero in this section.
- the shape of the work flux gradient is the mirror image of that of the energy redistribution term because the addition of these two terms should be the negative of the spatially uniform energy accumulation rate.
- a negative distribution in the S-segment is necessary to balance the positive redistributed work in the TBS so that the spatial integration is zero.
- efficiency is defined as the ratio of work done to thermal energy consumed.
- the rod since there is no energy harvesting element in the system, the rod has no work output.
- the thermal energy consumed is not available directly from the quasi-1D model because the evaluation of the radial heat conduction at the boundary is lacking.
- the heat flux ⁇ dot over (Q) ⁇ could be considered as uniform for a short stack, which is approximately equal to the consumed thermal energy.
- the efficiency ⁇ is expressed as
- the efficiency of SSTA can be improved by designing an inhomogeneous medium having optimized mechanical and thermal thermoacoustic properties.
- the traveling wave SSTA engine is found to be more efficient than its standing wave counterpart.
- this study confirms the theoretical existence of traveling wave thermoacoustics in a solid looped rod which could open the way to the next generation of highly-robust and ultracompact traveling wave thermoacoustic engines and refrigerators.
- thermoacoustic device includes a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage.
- the stage further includes a first cooling component on a second terminus of the stage.
- a thermal conductivity of the stage is higher than a thermal conductivity of the bar.
- a heat capacity of the stage is higher than a heat capacity of the bar.
- the bar comprises at least one of copper, iron, steel, lead, or a metal. In some embodiments, the bar comprises any solid. In some embodiments the bar is monolithic.
- the bar includes a material, wherein the material is not susceptible to oxidation at temperatures ranging from ⁇ 100° C. to 2000° C., and wherein the material remains a solid at temperatures ranging from ⁇ 100° C. to 2000° C.
- a first terminus of the bar is fixed, and a second terminus of the bar is free.
- the second terminus of the bar includes a solid mass, wherein a density of the solid mass is greater than a density of the bar.
- a first terminus of the bar is fixed, and a second terminus of the bar is fixed.
- a first terminus of the bar is fixed, and a second terminus of the bar is attached to a spring, wherein the spring is fixed.
- a first terminus and a second terminus of the bar are free from constraints.
- a temperature gradient between the first heating component and the first cooling component is 10° C./cm or higher. In some embodiments, a temperature gradient between the first heating component and the first cooling component is 20° C./cm or higher.
- the thermoacoustic device further includes at least one additional stage coupled to the bar, wherein the at least one additional stage includes a second heating component and a second cooling component.
- a temperature gradient between the second heating component and the second cooling component of the at least one additional stage is 10° C./cm or higher. In some embodiments, a temperature gradient between the second heating component and the second cooling of the at least one additional stage is 20° C./cm or higher.
- the thermoacoustic device further includes a piezoelectric material coupled to the bar.
- the first cooling component includes at least one of a thermoelectric cooler, dry ice, or liquid nitrogen.
- thermoacoustic device including a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage. Additionally, the stage includes a first cooling component on a second terminus of the stage. A thermal conductivity of the stage is higher than a thermal conductivity of the bar. A heat capacity of the stage is higher than a heat capacity of the bar, and the bar forms a closed loop. Moreover, the thermoacoustic device includes a second cooling component on the bar, wherein the second cooling component is configured to cool to a same temperature as the first cooling component.
- thermoacoustic device including a stage coupled to a bar, wherein the stage includes a first heating component on a first terminus of the stage. Additionally, the stage includes a first cooling component on a second terminus of the stage. A thermal conductivity of the stage is higher than a thermal conductivity of the bar. A heat capacity of the stage is higher than a heat capacity of the bar. Moreover, the bar includes a material wherein the material does not oxidize at temperatures ranging from ⁇ 100° C. to 2000° C. Further, the material remains a solid at temperatures ranging from ⁇ 100° C. to 2000° C.
Abstract
Description
is the material derivative, T is the total temperature, and ev is the volumetric dilatation which is defined as ev=Σj=1 3εjj·Fb,i and .qg are the mechanical and thermal source terms, respectively. The stress-strain constitutive relation for a linear isotropic solid, including the Duhamel components of temperature induced strains, is given by:
σij=2μεij+[λL e v−α(2μ+3λL)(T−T 0)]δij, (3)
where μ and λL are the Lame constants, εjj is the strain tensor, T0 is the mean temperature, and δij is the Kronecker delta.
where
is the Grüneisen constant, i is the imaginary unit, û, {circumflex over (v)} and {circumflex over (T)} are the fluctuations of the particle displacement, particle velocity, and temperature averaged over the cross section of the rod. For brevity, they will be referred to as fluctuation terms in the following. The intermediate transformation iΛû={circumflex over (v)} avoids the use of quadratic terms in Λ, which ultimately enables the system to be fully linear. The αH{circumflex over (T)} term in Eqn. 6 accounts for the thermal conduction in the radial direction, and it is the term that renders the theory quasi-1D. The function αH is given by:
where Jn(⋅) are Bessel functions of the first kind, and ξ is a dimensionless complex radial coordinate given by
and thus, the dimensionless complex radius is
where R is the radius of the rod. The thermal penetration thickness δk is defined as
and physically represents the depth along the radial direction (measured from the isothermal boundary) that heat diffuses through.
where σ and ε are the total axial stress (i.e. including both mechanical and thermal components) and strain, respectively. During one acoustic/elastic cycle, the time averaged work transfer rate per unit volume is
where τ is the period of a cycle, and
which effectively results in the linearization of the momentum equation. Full nonlinear terms are instead retained in the energy equation.
where A1 and Ai are the amplitudes of the response at the time instant t1 and ti, and where t1 and ti are the start time and the time after (i−1) periods. Both approaches return β=0.212(rad/s). This value is found to be within 1% accuracy from the value obtained via the quasi-1D stability analysis, therefore confirming the validity of the 1D theory and of the corresponding simplifying assumptions.
while λ/8=6.86 m is beyond the total length of the rod L=1.8 m. Hence, in this case the optimal location of the stage approaches the end mass.
indicates the distance, measured from the isothermal boundary, that heat can diffuse through. Solid particles that are outside this thermal layer do not experience radial temperature fluctuations and therefore do not contribute to building the instability. The value of the thermal penetration thickness δk, or more specifically, the ratio of δk/R is a key parameter for the design of the system. Theoretically, the optimal value of this parameter is attained when the rod radius is equal to δk. In fluids, good performance can be obtained for values of 2δk to 3δk. Here below, we study the optimal value of this parameter for the two configurations above.
yields the highest growth ratio β/ω for both boundary conditions. The above analysis shows that the optimal values of xk/L and δk/R are quantitatively equivalent to their counterparts in fluids.
becomes one or two orders of magnitude lower than the damping ratio ζ. Therefore, despite the relatively low intrinsic damping of the material the growth is effectively impeded.
is mostly negligible other than at the edges of the stage (see
The $Loop-I curve in
The
The total heat flux through the cross section of the rod is
The second equality holds because the eigenfunctions are all cross-section-averaged quantities. We note that {dot over (Q)} is a function of the axial position x.
This quantity physically represents the rate per unit area at which work is done by an element onto its neighbor. It can be also called ‘work flux’ because it shows the work flow in the medium as well. When an element is compressed (
The total mechanical power through the cross section I of the rod is given by
The work source can be further defined as the gradient of the mechanical power as
By expanding Eq. (8), w2 can be further expressed as
The first term of w2 vanishes after applying cycle-averaging, because according to the momentum conservation, ∂σ1/∂x and v1 are 90° out of phase under the assumption that the small phase difference caused by the non-zero β can be neglected due to: β/ω<<1. The remaining term is equivalent to
i.e.
whose cycle average is consistent with the cycle-averaged volume change work.
where,
indicates the conductive heat flux at the medium-stage interface. Multiplying Eq. (12) by ρv1 and Eq. (13) by
ε2, I2, 2, 2, and 2 are the second order energy norm, work flux, energy redistribution term, thermoacoustic production and dissipation, respectively. Note that the work flux shown in Eq. (16) is consistent with the heuristic definition adopted. With the harmonic convention ( )1=e(β+iw)t({circumflex over ( )}) and the assumption β/ω<<1, taking the cycle averaging of Eq. (14) yields
Where {tilde over (ε)}, Ĩ, , , and are transformed from the cycle averages of the cross-sectionally-averaged second order terms in Eqs. 15-18, following the assumption of cycle averaging: <( )2>=e2βt({tilde over ( )}). They are expressed as:
The growth rate can be recovered via:
is the work source defined in the previous paragraphs, is an energy redistribution term. and are the thermoacoustic production and dissipation, respectively. The energy redistribution term in the acoustic energy budgets of solid thermoacoustics cannot be found in the fluid counterpart of the same equations. This term is absent in fluids because it is canceled in the algebraic derivations by expressing the variation of mean density according to the ideal gas law, as a function of the mean temperature gradient. On the other hand, in solidstate thermoacoustics, the heat-induced density variation is neglected and the impact of the temperature gradient is manifest in the stress-strain constitutive relation. It has been proved numerically that the spatial integration of this term is zero, so it does not produce or dissipate energy, but just redistributes it. In summary, it represents the work created by the acoustic flux acting against the temperature gradient.
peaks in the S-segment, and has a constant negative value out of the S-segment. As foreshadowed by the discussions in the previous paragraph, this distribution means that
adjusts itself so that β is uniform. In other words, energy is accumulated everywhere at the same rate.
is exactly the same as the rate of the energy accumulation to keep the condition of zero local net production.
Claims (19)
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