Porous Cores in Small Thermoacoustic Devices for Building Applications

energies

Article Porous Cores in Small Thermoacoustic Devices for Building Applications

Fabio Auriemma 1,* , Elio Di Giulio 2, Marialuisa Napolitano 2 and Raﬀaele Dragonetti 2

1 Department of Mechanical and Industrial Engineering, Tallinn University of Technology—TalTech, 19086 Tallinn, Estonia 2 Department of Industrial Engineering, University of Naples Federico II, 80125 Naples, Italy; [email protected] (E.D.G.); [email protected] (M.N.); raﬀ[email protected] (R.D.) * Correspondence: [email protected]

Received: 5 May 2020; Accepted: 2 June 2020; Published: 8 June 2020

Abstract: The thermoacoustic behavior of different typologies of porous cores is studied in this paper with the goal of finding the most suitable solution for small thermoacoustic devices, including solar driven air coolers and generators, which can be used in future buildings. Cores provided with circular pores, with rectangular slits and with arrays of parallel cylindrical pins are investigated. For the type of applications in focus, the main design constraints are represented by the reduced amount of the input heat power and the size limitations of the device. In this paper, a numerical procedure has been implemented to assess the behavior of the different core typologies. For a fixed input heat power, the maximum acoustic power delivered by each core is computed and the corresponding engine configuration (length of the resonator and position of the core) is provided. It has been found that cores with parallel pins provide the largest amount of acoustic power with the smallest resonator length. This conclusion has been confirmed by experiments where additive manufactured cores have been tested in a small, light-driven, thermoacoustic prime mover.

Keywords: thermoacoustic heat engine; solar energy; solar cooling

1. Introduction This paper addresses the main technical problems involved with the use of small thermoacoustic heat engines—prime movers and heat pumps—in building applications. Different destinations in future buildings, including net zero-energy buildings [1], can be foreseen for thermoacoustic devices, especially solar driven air coolers and electric generators with small input powers [2,3]. In case of coolers, two heat engines are required: the first one is a prime mover which uses heat from solar radiation to generate acoustic power; the second one is a heat pump which uses the acoustic power to subtract heat from surrounding air [4]. For electric energy generation, a linear generator, a reversed loudspeaker or a piezoelectric device can be combined with the prime mover to perform the acoustic-to-electric power conversion [5,6]. This kind of solutions can help reducing the total emissions of future buildings. Thermoacoustic heat engines (TAHE) exploit the thermal diffusion process occurring in internal porous cores in order to pump mechanical energy into the working fluid or to extract heat from it [7]. These cores act as thermal reservoirs, providing heat exchanges between the solid surface of the pores and the oscillating fluid. Heat exchanges take place within the thermal boundary layer (δk), which has sub-millimeter thickness and requires a similar order magnitude of the pore size. However, the thickness of the viscous boundary layer (δv) has also same order magnitude as the pore size. As a consequence, large viscous losses take place in the fluid within the porous core. This condition, typically utilized in sound absorptive elements, is undesired in thermoacoustic devices which, inevitably, suffer from substantial viscous losses [8,9].

Energies 2020, 13, 2941; doi:10.3390/en13112941 www.mdpi.com/journal/energies Energies 2020, 13, 2941 2 of 19

Thermoacoustic heat engines can be categorized into two groups, standing-wave and travelling-wave, according to the phase difference—nearly π/2 and nearly zero, respectively—between sound pressure and particle velocity. In other words, the first heat engine typology requires porous cores with Sh = O(1), whilst travelling-wave devices require Sh = O( 1), being Sh = d/2δv the Shear − wavenumber and d is the characteristic dimension of the pores. Only standing-wave devices are studied in this paper, for which the porous core is equivalently referred to as “stack”. By looking at the pore shape, an unlimited number of stack typologies can be found and studied. Practical solutions include mainly slit shaped pores, circular, triangular, honeycomb shapes and pin arrays. Fundamental studies in thermoacoustics, conducted by Rott [10] and Arnott et al. [11] on circular, square, rectangular and triangular pores point out that parallel sided pores exhibit better performance than other hollow pore geometries. The use of laminae in spiral configurations is one way to build cores with approximately parallel sided pores [7,12]. Semi-phenomenological models for porous materials have been used to study the thermoacoustic behavior of different cores in [13,14]. In another fundamental study [15], Swift et al. theoretically showed that cores with convex pores geometries, e.g., longitudinal pin array stacks, perform better than any hollow pore stack, including cores with parallel plates, due to the reduced amount of viscous losses when the working fluid has Prandtl number (Pr) < 1 (e.g., air). However, the approach used in [15] to assess the performance of pin cores is based on a figure of merit which is a rather qualitative approach, because neither the acoustic power generated nor the input heat power are accounted for. The literature related to pin array stacks is quite narrow because manufacturing issues have limited the use of this type of cores [15,16]. Among the main investigations, it is worth mentioning the studies related to cores with parallel pins, but displaced transversally to the fluid in order to reduce the heat transfer along the core [17,18]. Recently, the manufacturing issues of pin array cores have been largely overcome with the advent of additive manufacturing, as shown in the studies [19,20]. For this reason, the goal of this paper is to investigate additive manufactured cores with parallel pins and to provide a thorough comparison with traditional porous geometries. In this way, it is possible to detect which core typology is more suitable for building applications, where the input thermal power is limited and size constraints bind the length of the resonator. To this aim, the behavior of cores constituted by parallel pins, slits and circular tubes, but provided with equivalent thermo-viscous properties, has been modelled by means of one dimensional thermoacoustic simulations. An optimization procedure has been implemented where, per each core typology and for a given thermal input power, the resonator length and the position of the core which maximize the output power are found [21,22]. By comparison, it is possible to detect the best solution for the above mentioned thermoacoustic applications. It will be shown that cores made of circular tubes and slits greatly differ from cores which include parallel pins, providing less acoustic power and requiring longer resonator tubes. In a first approximation, the results provided here can be extended to more complex concave pore geometries, e.g., square and honeycomb pores, as well as to more complex convex pore arrangements, e.g., diamond pin cells and oblique pin cells. The numerical results are confirmed by an experimental comparison between different stacks in a light powered standing-wave prime mover. This study will be followed by upcoming investigations on additive manufactured lattice cells with oblique or predominantly parallel pins, including optimization of the pore geometries and the use functionally graded materials [23,24].

2. Materials and Methods

2.1. Theoretical Background

2.1.1. Standing Wave TAHE A schematic representation of a standing-wave TAHE prime mover is shown in Figure1. The device consists of: Energies 2020, 13, 2941 3 of 19

Energies 2020, 13, x FOR PEER REVIEW 3 of 20

A resonator tube with diameter Dr and length Lr.  •A resonator tube with diameter Dr and length Lr. A porous core (stack) of length Ls located at a distance La from the rigid wall of the resonator.  •A porous core (stack) of length Ls located at a distance La from the rigid wall of the resonator. A hot heat exchanger (HX) which heats up the first end of the stack, and a cold heat exchanger  •A hot heat exchanger (HX) which heats up the first end of the stack, and a cold heat exchanger CX (CX)( which) which cools cools down down the second the second end endof the of stack. the stack. Possibly, a utilizer. For instance, a thermoacoustic heat pump (i.e., reversed prime mover) for  •Possibly, a utilizer. For instance, a thermoacoustic heat pump (i.e., reversed prime mover) for air coolingair coolingapplications applications or a linear or a lineargenerator, generator, a reversed a reversed loudspeaker, loudspeaker, an impulse an impulse turbine turbine or a or a piezoelectricpiezoelectric device device for electric for electric energy energy conversion. conversion. The Thepresence presence of oftwo two microphones microphones ( (M1,M1, M2M2)) andand of of two two thermoacouples thermoacouples (TC1 (TC1and TC2and) isTC2) envisaged is envisagedin Figure in 1Figure to monitor, 1 to monitor, respectively, respectively, the acoustic the pressure acoustic at pressure specific locations at specific and locations the temperatures and the TH temperaturesand TC at TH the and hot andTC at cold the endshot and of the cold stack. ends of the stack. The Thecore coreacts as acts a series as a series of thermal of thermal reservoirs reservoirs at different at diff temperatures,erent temperatures, due to due the presence to the presence of the hotof the and hot cold and heat cold exchangers heat exchangers and to the and heat to the capacity heat capacity of the core of the itself. core At itself. the frequency At the frequency of the of first—mostthe first—most unstable—acoustic unstable—acoustic mode of mode the resonator, of the resonator, the heat the provided heat provided by the by hot the exchanger hot exchanger is is delivered,delivered, through through the the stack, stack, to to the the working fluidfluid during during the the phase phase of greatestof greatest condensation. condensation. Similarly, Similarly,the heat the taken heat awaytaken by away the coldby the exchanger cold exchanger is subtracted, is subtracted, through thethrough stack, the from stack, the workingfrom the fluid workingduring fluid the phaseduring of the greatest phase rarefaction. of greatest This rarefaction. condition, This usually condition, referred usually to as Rayleigh’s referred to criterion, as Rayleigh’sresults incriterion, thermal-to-mechanical results in thermal power‐to conversion‐mechanical in thepower form conversion of the amplification in the ofform sound of wavesthe at amplificationthe expense of sound of the heatwaves provided. at the expense The acoustic of the heat emission provided. takes The place acoustic if the thermal emission gradient takes place between if thethe thermal hot and gradient cold end between of the stack the hot is greaterand cold than end a of critical the stack value is ∆greaterTc. than a critical value ΔTc.

La Ls

Utilizer Dr HX CX

TC1 TC2 M1 M2

FigureFigure 1. Schematic 1. Schematic of an of open an open standing standing-wave‐wave THAE THAE prime prime mover. mover. In an open standing-wave thermoacoustic device, the resonator tube is basically a quarter-wave In an open standing‐wave thermoacoustic device, the resonator tube is basically a quarter‐wave resonator and the operating frequency is c/4L, being c the speed of sound. Sound pressure gradient and resonator and the operating frequency is c/4L, being c the speed of sound. Sound pressure gradient particle velocity have ~π/2 phase shift and, in order to meet the Rayleigh’s criterion, imperfect heat and particle velocity have ~ π/2 phase shift and, in order to meet the Rayleigh’s criterion, imperfect transfer between the fluid and the porous material of the core is necessary. In fact, if the thermal contact heat transfer between the fluid and the porous material of the core is necessary. In fact, if the thermal was perfect, the phase shift would result in zero net heat transfer from any particular location of the contact was perfect, the phase shift would result in zero net heat transfer from any particular location stack to the fluid parcels, thus no net work would be converted. Imperfect thermal contact is realized of the stack to the fluid parcels, thus no net work would be converted. Imperfect thermal contact is by using stacks with a hydraulic radius of the pores which is roughly a few times larger than the realized by using stacks with a hydraulic radius of the pores which is roughly a few times larger than thermal penetration length. Contrariwise, in travelling-wave thermoacoustic devices, where pressure the thermal penetration length. Contrariwise, in travelling‐wave thermoacoustic devices, where and particle velocity are in phase, perfect thermal contact is required, therefore the hydraulic radius of pressure and particle velocity are in phase, perfect thermal contact is required, therefore the hydraulic the pores is similar to or smaller than the thermal penetration length. radius of the pores is similar to or smaller than the thermal penetration length. 2.1.2. Momentum and Thermal Diffusion 2.1.2. Momentum and Thermal Diffusion Thermal and viscous penetration depths are, respectively, defined as δk = √2k/ω and Thermalp and viscous penetration depths are, respectively, defined as   2  and δv = 2µ/ω, where k is the thermal diffusivity of the medium and µ is the dynamic viscosity.

 δk 2and δ, vwhereare the κ is distances the thermal from diffusivity the solid of wall the medium where heat and and μ is momentumthe dynamic canviscosity. diﬀuse δκ laterallyand T T δν areduring the distances the time from/π, beingthe solidthe wall period where of heat the acoustic and momentum oscillation can [25 diffuse]. At longer laterally distances, during the the ﬂuid timeresponds T/π, being adiabatically T the period to of the the acoustic acoustic perturbation. oscillation Instead,[25]. At longer in proximity distances, of the the walls, fluid the responds response is adiabaticallyisothermal. to At the intermediate acoustic perturbation. distances, a few Instead, times largerin proximity than δk, theof responsethe walls, is somehowthe response in between is isothermal. At intermediate distances, a few times larger than δκ, the response is somehow in between (imperfect thermal contact), which is the condition required by standing‐wave thermoacoustic

Energies 2020, 13, x FOR PEER REVIEW 4 of 20 Energies 2020, 13, 2941 4 of 19 2 devices. δκ and δν are linked by means of the Prandtl number of the fluid, Pr        Pr      2 (imperfect thermal  contact),. which is the condition required by standing-wave thermoacoustic devices. 2 δk andViscousδv are linked and bythermal means ofpenetration the Prandtl numberdepths ofare the comparable fluid, Pr = vlengths,/k = (δv /whichδk) . implies that thermoacousticViscous and thermal devices penetration are affected depths by substantial are comparable viscous lengths, losses, which especially implies that in the thermoacoustic porous core. devicesHowever, are ainff ectedmany by fluids, substantial Pr is less viscous than losses,one, which especially means in that the δ porousν is smaller core. than However, δκ, although in many the fluids,sizes Prremainis less comparable. than one, which For meansinstance, that Prδ v=is 0.71 smaller in air than at 20δk ,°C although and slightly the sizes decreases remain comparable.up to 350 °C. ForThis instance, circumstancePr = 0.71 motivates in air at 20the◦ Cuse and of slightly cores provided decreases with up to convex—rather 350 ◦C. This circumstance than concave—radii motivates of thecurvature use of cores of the provided pores. withIn Figure convex—rather 2, the thermal than and concave—radii momentum of diffusion curvature regions of the pores.are schematized In Figure2 ,in thethe thermal cross section and momentum of two different diffusion cores: regions one provided are schematized with convex in the pores cross (Figure section of2a) two and di onefferent with cores:concave one pores provided (Figure with 2b). convex In both pores cases, (Figure the radii2a) andare supposedly one with concave identical pores and (Figure a fluid2 isb). chosen In both with ν κ ν cases,Pr <1, the which radii means are supposedly δ < δ . Since identical the momentum and a fluid diffusion is chosen occurs with in Pra region< 1, which of fluid means of thicknessδv < δk .δ Sinceand thethe momentumthermal diffusion diffusion in a region occurs of in thickness a region of δκ, fluid the ratio of thickness Aκ/Aν betweenδv and the the thermal thermal and diff viscoususion diffusion areas is higher in case of convex pores. As a consequence, for convex pores, the total volume in a region of thickness δk, the ratio Ak/Av between the thermal and viscous diffusion areas is higher ininvolved case of convex in thermal pores. diffusion, As a consequence, Vκ, is sensibly for convex larger pores,than that the involved total volume in viscous involved losses, in thermal Vν. The circumstance represented in Figure 2a is the best‐case scenario (pin array pores) for cores with convex diffusion, Vk, is sensibly larger than that involved in viscous losses, Vv. The circumstance represented insurfaces, Figure2a whilst is the best-case the case scenarioin Figure (pin 2b ( array circular pores) tube for pores) cores withis the convex worst‐ surfaces,case scenario whilst for the cores case with in Figureconcave2b (circular surfaces. tube When pores) the iscore the worst-caseis constituted scenario by parallel for cores plates, with the concave pore surfaces.surfaces Whenare flat the and coreconcave is constituted at the vertexes, by parallel thus plates, the amount the pore of viscous surfaces losses are flat reduces, and concave but not at as the much vertexes, as for pin thus cores. the amount of viscous losses reduces, but not as much as for pin cores.

Solid material

Momentum diffusion region

Thermal diffusion (a) region (b)

Figure 2. Thermal diffusion region (parallel red lines) and viscous diffusion region (blue areas) in Figure 2. Thermal diﬀusion region (parallel red lines) and viscous diﬀusion region (blue areas) in cores cores provided with convex (a) and concave cores (b). provided with convex (a) and concave cores (b).

2.1.3.2.1.3. Equivalent Equivalent Cores Cores InIn order order to to compare compare the the thermoacoustic thermoacoustic performance performance of of cores cores provided provided with with di differentfferent types types of of pores,pores, the the geometrical geometrical parameters parameters must must be be chosen chosen in in order order to to have have cores cores which which are are “equivalent” “equivalent” from from thethe thermoacoustic thermoacoustic point point of view.of view. A possible A possible choice choice is enforcing is enforcing the same the hydraulicsame hydraulic radius forradius the cores,for the conventionallycores, conventionally defined as defined the ratio as of the the channel’sratio of the cross-sectional channel’s cross area‐ tosectional its perimeter, area toHR its= Aperimeter,/Π [19]. 𝛬′ MoreHR=A/ generally,Π [19]. More the thermal generally, characteristic the thermal length characteristicΛ0 can be length used instead can ofbeHR used. This instead parameter of HR was. This firstparameter introduced was by first Champoux introduced and by Allard Champoux to study and thermal Allard effects to study inside thermal porous material, effects inside following porous a primarymaterial, work following by Johnson a primary et al. relatedwork by to Johnson viscous eetff ectsal. related [26,27]. to The viscous thermal effects characteristic [26,27]. The length thermal is characteristic length is defined as 𝛬 2𝑉 /𝐴 , where 𝑉 is the volume of fluid and 𝐴 is defined as Λ0 = 2V f luid/Awet, where V f luid is the volume of fluid and Awet is the wet area, namely the areathe of wet the area, solid namely skeleton the of area the coreof the which solid isskeleton in touch of with the core the fluid. which is in touch with the fluid. 𝛬′ ImposingImposing the the same same thermal thermal characteristic characteristic length lengthΛ0 for diff erentfor different cores represents cores represents the first condition the first tocondition have equivalent to have cores, equivalent since the cores, wet since area is the responsible wet area foris responsible thermal diff forusion thermal and the diffusion volume and of the the fluidvolume moving of the within fluid the moving core is strictlywithin linkedthe core to theis strictly viscous linked losses. to Based the viscous on the definition losses. Based of Λ 0on, it isthe simpledefinition to obtain of 𝛬′ thermal, it is simple characteristic to obtain length thermal for characteristic circle, plate and length pin for geometries, circle, plate as reported and pin geometries, in Table1. as reportedHowever, in the Table sole 1. condition of equal Λ’ is not sufficient to completely define the geometries of the cores.However, By assuming the sole that condition the manufacturing of equal Λ’ is capabilities not sufficient allow to completely us to produce define similar the geometries thicknesses of forthe the cores. solid By parts assuming of different that cores,the manufacturing it is reasonable capabilities to impose allow a second us to condition produce to similar have equivalent thicknesses cores,for the that solid is 2r partsp = 2 hof= different2d. As a cores, consequence, it is reasonable the distance to impose between a second adjacent condition pins and to the have thickness equivalent of thecores, solid that skeleton is 2𝑟 (for2ℎ2𝑑 slits and. As circular a consequence, tubes) are the also distance characterized. between As adjacent shown pins in Table and1 ,the these thickness two of the solid skeleton (for slits and circular tubes) are also characterized. As shown in Table 1, these

Energies 2020, 13, x FOR PEER REVIEW 5 of 20 EnergiesEnergies 2020 2020, 13, 13, x, FORx FOR PEER PEER REVIEW REVIEW 5 of5 of20 20 EnergiesEnergies 20202020, 13, x, FOR13, 2941PEER REVIEW 5 of 20 5 of 19 two geometrical parameters are related to the porosity of the core, 𝜙𝑉/𝑉, with 𝑉 being two geometrical parameters are related to the porosity of the core, 𝜙𝑉/𝑉, with 𝑉 being twothe geometrical total volume parameters of the core. are related to the porosity of the core, 𝜙𝑉 /𝑉 , with 𝑉 being thethe total total volume volume of of the the core. core. the total volume of the core. geometrical parameters are related to the porosity of the core, φ = V f luid/Vtotal, with Vtotal being the Table 1. Geometrical and viscous‐thermal relationships between the cores studied in this paper. TableTable 1. 1.Geometrical Geometrical and and viscous viscous‐thermal‐thermal relationships relationships between between the the cores cores studied studied in in this this paper. paper. totalTable volume 1. Geometrical of the core.and viscous‐thermal relationships between the cores studied in this paper. Minimum MinimumMinimum Distance MinimumCharacteristic TableCell Type 1. Geometrical andΛ viscous-thermal’ 𝜙 relationshipsDistance between Characteristic the cores studied in this paper. Cell Type Λ’ 𝜙 betweenDistance Pins CharacteristicDimension of Cell Type Λ’ 𝜙 between Pins Dimension of Solid Skeleton between Pins SolidDimensionSolid Skeleton Skeleton of Minimum Characteristic Cell Type Λ’ φ Distance between Pins Solid Skeleton Dimension of Solid Skeleton

2 3𝑦 𝜋𝑟 2√3𝑦 𝜋𝑟 𝜋 2√3𝑦 𝜋𝑟 2√3𝑦 𝜋𝑟 𝜋 2√3𝑦 𝜋𝑟 2√3𝑦 𝜋𝑟 𝑦 𝑟𝜋 𝑟 𝑦 𝑟 𝑟 2 3𝑦 2𝜋𝑟𝜋𝑟2 2 2√3𝑦 𝜋𝑟√ 2 2 𝑦 𝑟 𝜋2√ 3 q 𝑟 √(2 √32𝜋𝑟y πr) 2(2√33𝑦y0 πrp) 2√3 π 2𝜋𝑟0 p 2√23√𝑦3𝑦− 𝑦 𝑟 2√y3 = r 𝑟 − 2 0 0 rp 2πrp 2√3y 2 √3 2𝜋𝑟 2√3𝑦 0 2√3

b=y0-rp b=y0 -rp b=y 0-rp Parallel pins (hexagonal ParallelParallel pins pins (hexagonal (hexagonal Parallel pinspattern) (hexagonal pattern)pattern) pattern)

4𝑦 𝜋𝑟 4𝑦 𝜋𝑟 4𝑦 𝜋𝑟 4𝑦 𝜋𝑟2 2 √𝜋 4𝑦 𝜋𝑟2 2 4𝑦 𝜋𝑟(4y πr ) 𝜋√𝜋 𝑟 (4y1 πrp) 1 p 𝑦 𝑟√ √π 𝑟 4𝑦 𝜋𝑟2𝜋𝑟− 4𝑦 𝜋𝑟4𝑦− 𝑦 𝑟 𝑟 rp 2𝜋𝑟2πr 4𝑦 2 𝑦 𝑟 √𝜋2 y1 = r1 2𝜋𝑟 p 4𝑦 4y1 2 2 𝑟 𝑦 𝑟 2𝜋𝑟 4𝑦 2

Parallel pins (square ParallelParallelParallel pins pins pins (square (square (square pattern)pattern) Parallelpattern) pinspattern) (square pattern) h h h 2𝑎 2𝑎 2𝑎2𝑎 2ℎ 𝛬′ 2𝑎 2𝑎2a 2ℎ Λ0 2ℎ 𝛬′ 2𝑎 = Λ0 2a 22𝑎a 2𝑎+ 2ℎ𝛬′= + 2ℎ 2𝑎 2h φ 2a 2a 2𝑎 2𝑎2a 𝛬′ 2 2ℎh𝛬′ 2a 2h 2ℎ 𝛬′𝜙2𝑎 − 2𝑎 2ℎ 𝜙 2𝑎 2a 2𝑎𝛬′ 2ℎ 𝜙 2𝑎 2ℎ

InﬁniteInfinite rectangular rectangular slit slit 2𝑎 2ℎ InfiniteInfinite rectangular rectangular slit slit Infinite rectangular slit

𝜋𝑅 𝜋𝑅𝜋𝑅 2 𝑅 πR 2𝑑 𝑅 R 2 2𝑑 2d 2√𝜋𝑅32𝑑𝑅√3(d+R) 𝑅 2√23√3𝑑𝑅𝑑𝑅 2𝑑 2√3𝑑𝑅

Circular tubes (hexagonal CircularCircularCircular tubes tubes tubes (hexagonal (hexagonal (hexagonal pattern) Circular tubespattern)pattern)pattern) (hexagonal pattern) Cores provided with parallel pins typically have pin arrays distributed in hexagonal or in square CoresCores provided provided with with parallel parallel pins pins typically typically have have pin pin arrays arrays distributed distributed in in hexagonal hexagonal or or in in square square patterns.Cores providedThe first configurationwith parallel pins was typically studied byhave Swift pin [15],arrays while distributed the second in hexagonal one has beenor in squareused for patterns.patterns.Cores The The first providedfirst configuration configuration with was parallelwas studied studied pins by by Swift typically Swift [15], [15], while havewhile the the pin second second arrays one one distributed has has been been used used in for hexagonalfor or in square patterns.the experiments The first configurationin the present waswork, studied since itby better Swift fits[15], the while additive the second manufacturing one has beenprocess used for for the thepatterns.the experiments experiments The in firstin the the present configuration present work, work, since since was it itbetter studied better fits fits the by the additive Swift additive [ 15manufacturing manufacturing], while the process secondprocess for for one the the has been used for the thecores. experiments Once the ingeometry the present of the work, cell withsince parallel it better pins fits in the hexagonal additive arraymanufacturing is fixed, imposing process thefor samethe cores.cores. Once Once the the geometry geometry of of the the cell cell with with parallel parallel pins pins in in hexagonal hexagonal array array is isfixed, fixed, imposing imposing the the same same cores.thermalexperiments Once characteristic the geometry in the length of present the for cell pins with work, in parallel a square since pins array it betterin resultshexagonal fits in theequal array additive radii is fixed, of the imposing manufacturing equivalent the dashedsame process for the cores. thermalthermal characteristic characteristic length length for for pins pins in in a squarea square array array results results in in equal equal radii radii of of the the equivalent equivalent dashed dashed thermalcirclesOnce characteristicshown the geometry in Table length 1: of r1 thefor= r0 .pins cellImposing in with a square the parallel same array diameters pinsresults in in hexagonalof equal the pinsradii for of array thethe equivalenttwo is fixed,patterns dashed imposing results the same thermal circlescircles shown shown in in Table Table 1: 1:r1 r=1 r=0 .r 0Imposing. Imposing the the same same diameters diameters of of the the pins pins for for the the two two patterns patterns results results circlesin the shown same porosityin Table but1: r1 different = r0. Imposing pin spacing: the same y1 ≠ diametersy0. of the pins for the two patterns results inincharacteristic the the same same porosity porosity length but but different different for pins pin pin spacing: in spacing: a square y1 y ≠ 1 ≠ y0y. array 0. results in equal radii of the equivalent dashed circles in the same porosity but different pin spacing: y1 ≠ y0. shown in Table1: r1 = r0. Imposing the same diameters of the pins for the two patterns results in the Energies 2020, 13, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies EnergiesEnergiessame 2020 2020 porosity, 13, 13, x;, x;doi: doi: FOR but FOR PEER diPEERff REVIEWerent REVIEW pin spacing: y1 , y0. www.mdpi.com/journal/energieswww.mdpi.com/journal/energies Energies 2020, 13, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies For infinitely long slits, imposing the same thermal characteristic length as for the pins in hexagonal array provides the distance 2a between plates. Furthermore, imposing 2h = 2rp results in the same porosity as for the pins. For cores provided with circular tubes arranged in a hexagonal pattern, imposing thermal characteristic length equal to that of the pins in hexagonal array yields to radius R of the tubes equal to the characteristic length. The second condition which has been imposed, 2d = 2rp, does not provide the same porosity as for the pins. 5 A list of pore characteristics is reported in Table2, obtained by imposing rp = 9 10− m and by 5 × varying rp/b in the range [0.001–2]. The value rp = 9 10 m and the specific case rp/b = 0.29 listed in × − Energies 2020, 13, 2941 6 of 19

Table2 represent, respectively, the smallest thickness which can be manufactured by means of additive manufacturing and the characteristic ratio of the porous core with parallel pins which has been used in the experiments.

Table 2. Geometrical data of the porous cores studied and compared in this paper.

Parallel Pins Slits Circular Tubes

rp/b Λ’ [m] φ a [m] h [m] R [m] d [m] 0.001 9.05 101 1.000 4.51 101 4.50 10 5 9.02 11 4.50 10 5 × × × − × × − 0.01 9.18 10 1 1.000 4.59 10 1 4.50 10 5 9.18 10 1 4.50 10 5 × − × − × − × − × − 0.05 3.96 10 2 0.998 1.98 10 2 4.50 10 5 3.96 10 2 4.50 10 5 × − × − × − × − × − 0.1 1.08 10 2 0.992 5.40 10 3 4.50 10 5 1.08 10 2 4.50 10 5 × − × − × − × − × − 0.29 1.69 10 3 0.949 8.45 10 4 4.50 10 5 1.69 10 3 4.50 10 5 × − × − × − × − × − 0.5 7.20 10 4 0.889 3.60 10 4 4.50 10 5 7.20 10 4 4.50 10 5 × − × − × − × − × − 1 2.70 10 4 0.750 1.35 10 4 4.50 10 5 2.70 10 4 4.50 10 5 × − × − × − × − × − 1.5 1.6 10 4 0.640 8.00 10 5 4.50 10 5 1.60 10 4 4.50 10 5 × − × − × − × − × − 2 1.13 10 4 0.556 5.63 10 5 4.50 10 5 1.13 10 4 4.50 10 5 × − × − × − × − × −

2.1.4. Governing Equations For the thermoacoustic simulations of different porous cores, the spatially averaged velocity and temperature functions, fv and fk, as well as the heat capacity ratio εs of the pores to be modelled, must be used. In this section these three parameters are introduced for the geometries in focus. Analytic solutions exist for the one-dimensional momentum equation of a fluid of mean density ρm, dynamic viscosity µ, oscillating at the angular frequency ω, undergoing a spatial gradient dp1/dx of the fluctuating pressure p1: dp1 2 iωρmu = + µ u (1) 1 − dx ∇ 1

In Equation (1), u1 is the complex velocity amplitude and i is the imaginary unit. As shown by Arnot et al., this is the only differential equation which must be solved for determining the first order acoustic quantities and second order heat and work flows [11]. For all types of pores, the common boundary condition (BC) to be fulfilled by the solution of Equation (1) is the no-slip on the rigid walls, i.e., u1(wall) = 0. When the velocity field is spatially averaged in the pore section, the general solution for the problem can be written in the form:

1 dp1 u1 = (1 fv) (2) h i −iωρm dx − where < > is the symbol for the space average and fv is the spatially averaged velocity function, also known as velocity Rott’s function. Exact solutions exist for fv for infinitely wide slits and for circular tubes. For pins, the situation is more complicated because another BC must be applied, since the pins are mutually influencing. The second BC is u1, (hexagon) = 0 or u1, (square) = 0, ∇ ⊥ ∇ ⊥ where u1, is the gradient in directions perpendicular to the geometry of the pattern indicated in ∇ ⊥ brackets (see figures in Table1). It can be proved that for pins displaced in a regular, hexagonal pattern q with size y0 = r0 π/2 √3, the condition du1(r0)/dr = 0 is still a valid approximation [11]. In this case, an approximate, analytic solution exists [15]. The fv for parallel pins in hexagonal pattern can be used also for parallel pins in square pattern provided that the cells are equivalent according to the definition given in the previous subsection (r1 = r0 and same diameter of the pins). Energies 2020, 13, 2941 7 of 19

The expressions for fv for parallel pins, inﬁnitely wide slits and circular tubes are: Y1(sr0)J1 srp J1(sr0)Y1 srp 2 − fv = fv,p rp, r0, s = (3) − 2 2 ( ) ( ) (sr ) srp Y1 sr0 J1 srp J1 sr0 Y0 srp 0 − −

tanh(s0a) fv = fv,s(a, s0) = (4) s0a 2 J1(sR) fv = fv,c(R, s) = (5) s √ i J0(sR) − J and J are the Bessel function of order 0 and 1, respectively, Y and Y are the Neumann 0 1 p 0 p 1 functions of order 0 and 1, respectively, s = iωρm/µ = (i 1)/δv and = iωρm/µ = (i + 1)/δv, − − 0 both related to the viscous penetration depth δv. The fv-s are dimensionless functions depending on s and s’, and thus on ω, ρm and µ. For circular tubes, fv,c = fv,c(R, ω, µ, ρm), but also for slits (fv,s), the same dependence occurs, provided that the plate semi-distance a replaces R. For pins (fv,s), rp replaces R, but there is also dependence on the fifth parameter, y0: fv,p = fv,p(rp, y0, ω, µ, ρm). n is the number of independent variables, and m is the number of independent physical units; n = 5 for pins and n = 4 for slits and tubes, while m = 3 (length, time and mass) in all cases. By using the Buckingham π-theorem for dimensional analysis, the number of dimensionless parameters (the π–groups) necessary to describe the functions fv is p = n m, which is − equal to 2 for pins and 1 for circles and slits. It is easy to figure out that fv,c and fv,s depend exclusively on 2 2 the dimensionless parameter µ/ρmωR , whilst fv,p depends on µ/ρmωb and rp/b. The dimensionless p parameter involving the dynamic viscosity is the Shear wavenumber Sh = R ωρm/µ. As previously mentioned, it can be shown that, under the assumption of solid material of the core having sufficient heat capacity to keep the solid-fluid interface isothermal, the solution for differential equation of the excess temperature T1 in the fluid is

dTm 2 iωρmcpT + ρmcpu = iωβTmp + k T (6) − 1 1 dz − 1 ∇ 1 which is similar to Equation (1) [11]. However, fv must be modified into fk, by replacing µ with k in the expressions of fv (k is the thermal conductivity of the fluid, cp is the isobaric heat capacity per unit mass). Equivalently, the viscous penetration length δv can be replaced by the thermal penetration length δk in the terms s and s’ of fv in Equations (3)–(5). Both fv and fk are also referred to as Rott’s functions. The use of non-dimensional analysis for fk leads to identical functions as the fv but dependent on the parameter R √ω/k, which is the dimensionless parameter related to the averaged solution of the excess temperature. Compared to the fv, the fk functions are simply “stretched” along the horizontal axis by the factor √Pr. In Figure3, a generic, spatially averaged function f is shown for different types of core. The function can be read as fv by using the bottom horizontal axis (dimensionless parameter Sh) and as fk by referring to the upper horizontal axis (dimensionless parameter Sh √Pr). Another parameter which plays an important role for the porous cores is the heat capacity ratio εs, which is defined as: εs = ρmcp fkA f luid/ρscs fk,sAsolid (7) with density, specific heat, averaged thermal function and area of the fluid in the core appearing at the numerator of the fraction and the same parameters, referred to the solid part of the stack, at the denominator. εs represents the ratio between the thermal powers required by the fluid within the core and by the solid part of the core to have a temperature increase of 1 K. It expresses the fact that the solid may not have enough heat capacity to successfully impose the boundary condition T1 = 0 on the fluid—solid interface. As shown in the next subsection, εs appears in the expression of the total power involved in the thermoacoustic conversion, within a core, subject to temperature differential. Energies 2020, 13, 2941 7 of 20

The expressions for fν for parallel pins, infinitely wide slits and circular tubes are:

2 𝑌(𝑠𝑟)𝐽𝑠𝑟−𝐽(𝑠𝑟)𝑌𝑠𝑟 𝑓 = 𝑓,𝑟,𝑟,𝑠=− (3) 𝑌 (𝑠𝑟 )𝐽 𝑠𝑟 −𝐽 (𝑠𝑟 )𝑌 𝑠𝑟 (𝑠𝑟) −𝑠𝑟

tanh(𝑠′𝑎) 𝑓 = 𝑓 (𝑎, 𝑠′) = (4) , 𝑠′𝑎

2 𝐽(𝑠𝑅) 𝑓 = 𝑓,(𝑅,) 𝑠 = (5) 𝑠√−𝑖 𝐽(𝑠𝑅)

J0 and J1 are the Bessel function of order 0 and 1, respectively, Y0 and Y1 are the Neumann functions of order 0 and 1, respectively, 𝑠=−𝑖𝜔𝜌/𝜇 = (𝑖−1)/𝛿and ′=𝑖𝜔𝜌/𝜇 = (𝑖+1)/𝛿 , both related to the viscous penetration depth 𝛿 . The fν-s are dimensionless functions depending on s and s’, and thus on ω, ρm and μ. For circular tubes, 𝑓, =𝑓,(𝑅,𝜔,𝜇,𝜌) ,but also for slits (fν,s), the same dependence occurs, provided that the plate semi-distance a replaces R. For pins (fν,s), rp replaces R, but there is also dependence on the fifth parameter, y0: 𝑓, =𝑓,(𝑟,𝑦,𝜔,𝜇,𝜌). n is the number of independent variables, and m is the number of independent physical units; n = 5 for pins and n = 4 for slits and tubes, while m = 3 (length, time and mass) in all cases. By using the Buckingham π-theorem for dimensional analysis, the number of dimensionless parameters (the π–groups) necessary to describe the functions fν is p = n−m, which is equal to 2 for pins and 1 for circles and slits. It is easy to figure out that fv,c and fv,s depend exclusively on the dimensionless parameter 𝜇/𝜌𝜔𝑅 , whilst fv,p depends on 𝜇/𝜌𝜔𝑏 and 𝑟/𝑏. The dimensionless parameter involving the dynamic viscosity is the Shear wavenumber 𝑆ℎ = 𝑅𝜔𝜌/𝜇 As previously mentioned, it can be shown that, under the assumption of solid material of the core having sufficient heat capacity to keep the solid-fluid interface isothermal, the solution for differential equation of the excess temperature T1 in the fluid is 𝑑𝑇 −𝑖𝜔𝜌 𝑐 𝑇 +𝜌 𝑐 𝑢 =−𝑖𝜔𝛽𝑇 𝑝 +𝜅∇𝑇 (6) 𝑑𝑧 which is similar to Equation (1) [11]. However, fv must be modified into fk, by replacing μ with κ in the expressions of fv (k is the thermal conductivity of the fluid, cp is the isobaric heat capacity per unit mass). Equivalently, the viscous penetration length δv can be replaced by the thermal penetration length 𝛿 in the terms s and s’ of fv in Equations (3)–(5). Both fv and fk are also referred to as Rott’s functions. The use of non-dimensional analysis for fk leads to identical functions as the fν but dependent on the parameter 𝑅𝜔/𝜅, which is the dimensionless parameter related to the averaged solution of the excess temperature. Compared to the fν, the fκ functions are simply “stretched” along the horizontal axis by the factor √𝑃𝑟. In Figure 3, a generic, spatially averaged function f is shown for different types of core. The function can be read as fν by using the bottom horizontal axis (dimensionless parameter Energies 2020, 13, 2941 8 of 19 Sh) and as fκ by referring to the upper horizontal axis (dimensionless parameter Sh√𝑃𝑟).

Sh√Pr 0123456 1.2 Parallel pins: rp/b = 0.001 Parallel pins: rp/b = 0.2913 0.8 Infinite slit Circular tubes

0.4

-0.4 imag (f) real (f) (f) real imag

-0.8 012345 Sh FigureFigure 3.3. Rott’sRott's functions for circular tube,tube, slitslit andand pinpin arrays.arrays.

Energies 2020, 13, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies The analytic expressions for the heat capacity ratio εs for cores provided with pins, slits and circular tubes, are: ! 1 2 2 kρ c 2 J0 √ iω/ksri p r r m p − 0 i s,p = fk iω/k − (8) ksρscs J √ iω/ksr − 2ri 1 − i 1 ! 2 kρmcp tan h[(1 + i)y0/δk] s,s = (9) ksρscs tan h[(1 + i)l/δs]

1 ! 2 kρmcp fk(1 + i)r0/2δk s,c = (10) ksρscs tanh[(1 + i)l/δs]

2.2. Thermoacoustic Performance

2.2.1. Power and Efficiency By following Rott’s theory, heat and work flows, as well as powers involved in thermoacoustic processes, can be expressed as functions of fv and fk, which therefore are of critical importance [7]. The time averaged acoustic power per unit area of fluid in a duct is:

. 1 1 E2 = A f luid (p1 eu1 ) = A f luid p1 u1 cos φw (11) 2 < h i 2 | ||h i| where subscript 2 indicates second order quantity (given by the product of two ﬁrst order quantities), the tilde ~ indicates the complex conjugate and φw is the phase between pressure and particle velocity. This phase is π/2 for pure standing waves and 0 for travelling waves, which would mean no power in the ﬁrst case. However, in thermoacoustic devices, the phase is neither exactly 0◦ nor 90◦, but the viscous dissipations due to the presence of the stack, accounted for in fv, make the actual φw varying from 85◦ to 95◦ in standing-wave devices, which is enough to produce power. The time averaged acoustic power per unit area converted by the unit length of a thermoacustic core is: . ! ! E2 1 d eu1 dp1 1 d eu1 = A f luid p1 h i + eu1 = A f luid p1 h i (12) dx 2 < dx h i dx 2 < dx

The right-hand side of Equation (12) is justiﬁed by the fact that the last term in the brackets contains the product eu1 u1 which is purely imaginary. Equation (12) can be rearranged in the form: h ih i . E2 rv 2 1 2 1 = u1 p1 + (gep1 u1 ) (13) A f luiddx − 2 |h i| − 2rk | | 2< h i Energies 2020, 13, 2941 9 of 19

2 In Equation (13), the terms, rv = ωρm ( fv)/ 1 fv , 1/r = ( fv)ω(γ 1)/γpm and − = | − | k −= − g = (dTm/dxTm)( f fv)/[(1 fv)(1 Pr)] represent, respectively, the viscous resistance per unit k − − − length, the thermal relaxation conductance per unit length and the complex gain/attenuation constant for volume flow rate. γ is the specific heat ratio. The term g, responsible for the energy pumping into the acoustic waves (sound generation), arises only when there is a temperature differential (dTm/dx, Tm being the temperature of the fluid) along the channels of the core. In the hypothesis of inviscid fluid, the last term of Equation (13) becomes:

. E2,g 1 1 1 dTm 1 1 dTm = (gep1 u1 ) = (ep1 u1 ) ( fk) + (ep1 u1 ) ( fk) (14) A f luid 2< h i 2 Tm dx < h i < 2 Tm dx = h i = −

In travelling-wave devices, (ep1 u1 ) is negligible, and thus the acoustic power is proportional = h i to Re(fk). Instead, in standing-wave devices, (ep1 u1 ) is negligible and the acoustic power is < h i . proportional to Im(-fk). Finally, the expression of the total power H2 (intake heat power for engines and intake acoustic power for refrigerators) per ﬂuid area is:

. " !# f fev H2 1 k− A = 2 p1 eu1 1 f luid < h i − (1+Pr) 1 fev − " # 2 e ρmcp u1 fk fv (1+εs fv/ fk) dT + h i f + − m (15) 2 2 v (1+ε )(1+Pr) dx 2ω(1 Pr ) 1 fv = s − | − | Asolid dTm k + ks − A f luid dx

In this equation, Afluid is the cross-sectional area occupied by the fluid in the solid skeleton and Asolid is the cross-sectional area of the solid skeleton. The total power is equal to the heat power delivered to the stacks by means of the heat exchangers. The first term in Equation (15) is related to the acoustic power, the second term is the second-order hydrodynamic entropy flux and the last term is simply the conduction of heat due to both fluid and solid conductivities. The heat capacity ratio εs, introduced in the previous subsection, appears in Equation (15). Notice that the last two terms of this equation must be minimized, having negative effects on the global efficiency of the thermoacoustic engine:

. E2 η = . (16) H2

In order to compare stacks provided with different geometries, Swift introduced the figure of merit M given by Equation (17): 2 ( fk) 1 fv M = √Pr= | − | (17) ( fv) = M is motivated by the fact that the heat transport and work are proportional to ( f ) in the = k standing wave inviscid limit (see Equation (14)) and the acoustic power dissipated by viscosity is 2 proportional to ( fv)/ 1 fv when dTm/dx = 0 according to the definition of viscous resistance per = − | − | unit length (in Equation (13)). Notice that lim M = 1 because lim fv/ fk = √Pr. Despite the beauty of ω ω its compactness, the figure of merit M has the→∞ following limitations:→∞

The actual dissipated and acoustic powers are not accounted for: the terms involved in M are • proportional to these powers but the terms u and p in Equations (14) affect the powers in a h 1i 1 substantial way. The total power and the effects of both entropy flux and heat conduction through the core • (see Equation (15)) are not accounted for. The power dissipation due to the thermal relaxation, included in Equation (13) is not accounted for. • No information is provided regarding the optimal size of the resonator where the core is used. • Energies 2020, 13, 2941 10 of 19

2.2.2. Simulations and Optimal Configuration Assessment In order to overcome the limitations of the function M introduced above and to provide a fairer comparison between different typologies of porous cores, a numerical procedure based on thermoacoustic simulations has been used. In this way, it is possible to assess the optimal engine configuration and maximum acoustic power for each typology. This means that, for each core and . given initial conditions (operating frequency, thermal input power H2, hot and cold temperatures TH and TC at the ends of the core), the procedure estimates the position (La) and length (Ls) of the core . which maximize the acoustic power converted (E2). The process is repeated for different operating frequencies (thus lengths Lr of the resonator), with the final goal of providing the values Lr and La which maximize the acoustic power per each core type. The thermoacoustic simulations used by the numerical procedure are run by means of a source code written in Matlab® which has been elucidated in previous works [13,14]. The code implements the Rott’s thermoacoustic equations in an iterative way:

dp1 iωρm = u1 (18) dx − (1 fv) −

iωdx ( fk fv) dTm du1 = p1(1 + (γ 1) fk) + − u1 (19) − γpm − (1 fv)(1 Pr) Tm − − . " !# Tmβ f fev H2 1 k− A 2 peu1 1 dT f luid − < − (1+εs)(1+Pr) 1 fev m = − 2 (20) dx ρmcp u1 ( f fev)(1+εs fv/ f ) 1 φ |h i| e + k− k − 2 fv (1+ε )(1+P ) k φ ks 2ω(1+Pr) 1 fv = s r − − | − | ∂p In these equations, dx is the infinitesimal length of the stack, β = ( ) /ρm is the coefficient − ∂T p of thermal expansion. Once the static pressure P0 has been fixed, the maximum pressure that can be reached in the device to guarantee the validity of the holding equations is pmax = 0.1 P0. The acoustic field is computed by means of Equations (18) and (19) and the temperature along the core by means of Equation (20). This temperature is a function of the input thermal power and of the acoustic field generated, but also of the length of the core Ls. By iterating the computations for different lengths of the core, it is possible to find the optimal length which maximizes the acoustic power converted, estimated by using Equation (13). In fact, as the length of the core increases, the gain term g in Equation (13) and the corresponding acoustic power grow. However, the viscous and thermal dissipations, rv and rk in the same equation increase too, up to the point where the acoustic power begins to decrease. Thus, a maximum is detected for the acoustic power, whilst for longer lengths the losses might equal or overcome the gain. In this case, the acoustic power is no more converted. Since the acoustic field also depends on the position La of the core within the resonator, per each La and Ls can be found which maximizes the acoustic power. This is shown, as an example, in Figure4a, where the values of Ls providing the maximum acoustic powers at different locations La are plotted as a red line. The operating frequency and input heat power are fixed. The acoustic powers so computed are plotted in Figure4b, where each La on the abscissa is still associated to the Ls of the previous graph. From Figure4b, one value of the couple ( La, Ls) can be found which globally maximizes the acoustic power of the engine for the given core type. Figure4a also reports, in black line, the temperature simulated at the cold end of the core. For stacks longer than 25 cm, the acoustic power is maximized when the cold temperature is still higher than TC. As a consequence, the acoustic conversion in these cases is strongly penalized by the fact that it is not possible to exploit all the available temperature difference. Energies 2020, 13, x FOR PEER REVIEW 11 of 20 converted. Since the acoustic field also depends on the position La of the core within the resonator, per each La and Ls can be found which maximizes the acoustic power. This is shown, as an example, in Figure 4a, where the values of Ls providing the maximum acoustic powers at different locations La are plotted as a red line. The operating frequency and input heat power are fixed. The acoustic powers so computed are plotted in Figure 4b, where each La on the abscissa is still associated to the Ls of the previous graph. From Figure 4b, one value of the couple (La, Ls) can be found which globally maximizes the acoustic power of the engine for the given core type. Figure 4a also reports, in black line, the temperature simulated at the cold end of the core. For stacks longer than 25 cm, the acoustic power is maximized when the cold temperature is still higher thanEnergies TC2020. As, 13 a, 2941consequence, the acoustic conversion in these cases is strongly penalized by the11 offact 19 that it is not possible to exploit all the available temperature difference.

TH 4 200 0.12

160 3 C) o

e ( 0.08 r 120 (m) atu

s 2 r L 80 0.04 Tempe

Acoustic power (W) 1 40 TC

0 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 La (m) La (m) (a) (b)

Figure 4. 4. ExampleExample of ofsimulations simulations for foroptimal optimal configuration conﬁguration assessment. assessment. Input Input data: data:Q = 50Q W,= f 50= 140 W,

Hz,f = 140core Hz with, core parallel with parallelpins, temperature pins, temperature at cold end at cold of the end stack of the (TC stack) = 30 (TC °C,) =and30 that◦C, and at the that hot at end the (hotTH end) = 200 (TH °C.) = (200a) Simulated◦C. (a) Simulated temperature temperature at cold atend cold and end corresponding and corresponding optimal optimal lengths lengths of the of core the forcore different for diﬀerent positions positions along along the theresonator, resonator, (b) ( bsimulated) simulated acoustic acoustic power power at at different diﬀerent positions positions for

cores provided with length Ls previously determined.

The whole whole procedure procedure above above explained explained is isrepeated repeated for for different different operating operating frequencies, frequencies, i.e., i.e.,for differentfor different lengths lengths Lr ofL ther of resonator. the resonator. As a Asfinal a result, final result, it is possible it is possible to provide to provide the resonator the resonator length andlength the and position the position of the ofcore the (and core its (and length) its length) which which maximize maximize the acoustic the acoustic power power converted converted by the by giventhe given core core type. type.

3. Results Results In this section, the the three different different typologies of porous cores are compared by means of the numerical procedure previously described. A comparison between the performance of of different different 3D printed cores cores in in a a small small thermoacoustic thermoacoustic engine engine is is also also carried carried out out experimentally. experimentally. The The main main goal goal is to is compareto compare different different solutions solutions in order in order to find to findout the out one the which one which better betterfits building fits building applications applications where thewhere available the available input inputpower power is limited. is limited. Another Another important important parameter parameter which which will will be be assessed assessed is is the ability of different different cores to work with small resonator lengths, which is often a further constraint for building applications. 3.1. Numerical Results 3.1. Numerical Results The numerical procedure described in Section 2.2.2 is used to compare the performance of the The numerical procedure described in Section 2.2.2 is used to compare the performance of the porous cores listed in Table2 in a thermoacoustic standing wave prime mover. porous cores listed in Table 2 in a thermoacoustic standing wave prime mover. For the simulations, the imposed hot temperature is similar to that reached in the experiments, For the simulations, the imposed hot temperature is similar to that reached in the experiments, TH = 200 ◦C, while the cold one is roughly equal to the environmental temperature, TC = 30 ◦C. TH = 200 °C, while the cold. one is roughly equal to the environmental temperature, TC = 30 °C. Different thermal powers Q, ranging from 50 to 150 W, are simulated. Different thermal powers 𝑄 , ranging from 50 to 150 W, are simulated. Figures5 and6 show the results, in terms of total acoustic power at the hot end of the core, Figures 5 and 6 show the results, in terms of total acoustic power at the hot end5 of the core, the the optimal position and length of the stack, for the case rp/b = 0.29 and rp = 9 10− m. As previously optimal position and length of the stack, for the case rp/b = 0.29 and rp = 9 × 10−5 m. As previously mentioned, these are the geometrical values of the cores used in the experiments. The characteristics properties of AISI 316L steel have been used for the cores in the simulations, since it is the constitutive material of the additive manufactured stacks: the thermal conductivity and specific heat are 15.5 and 19.5 W/mK at 100 ◦C, and 500 and 530 J/kg K at 400 ◦C, respectively. The results are plotted as functions of the operating frequency of the engine, which means different lengths of the resonator. For all input powers simulated, the maximum acoustic power converted by the core with parallel pins is always larger than that converted by other stack typologies. For 150 W input heat power, the maximum acoustic power reached by the core with parallel slit is 10.2% higher than that of the core with circular tubes and the maximum powered converted by the core with parallel pins is 21.1% higher Energies 2020, 13, x FOR PEER REVIEW 12 of 20 mentioned, these are the geometrical values of the cores used in the experiments. The characteristics properties of AISI 316L steel have been used for the cores in the simulations, since it is the constitutive material of the additive manufactured stacks: the thermal conductivity and specific heat are 15.5 and 19.5 W/mK at 100 °C, and 500 and 530 J/kg K at 400 °C, respectively. The results are plotted as functions of the operating frequency of the engine, which means different lengths of the resonator. For all input powers simulated, the maximum acoustic power converted by the core with parallel pins is always larger than that converted by other stack typologies. For 150 W input heat power, the Energiesmaximum2020, acoustic13, 2941 power reached by the core with parallel slit is 10.2% higher than that of the12 core of 19 with circular tubes and the maximum powered converted by the core with parallel pins is 21.1% higher than that of the core with circular tubes. The advantages in percentages for slits over tubes than that of the core with circular tubes. The advantages in percentages for slits over tubes and pins and pins over tubes become, respectively, 11.0% and 22.8% at 100 W, and 11.8% and 23.6% at 50 W. over tubes become, respectively, 11.0% and 22.8% at 100 W, and 11.8% and 23.6% at 50 W. The trends The trends also show that parallel pins provide larger acoustic powers at wider frequency ranges. also show that parallel pins provide larger acoustic powers at wider frequency ranges. Cores with slits Cores with slits and circular tubes provide the best performance at low frequency range (from 25 to and circular tubes provide the best performance at low frequency range (from 25 to 75 Hz), whilst cores 75 Hz), whilst cores with pins provide the best performance at higher frequency ranges. This is with pins provide the best performance at higher frequency ranges. This is another fundamental another fundamental advantage of pin cores, because it allows to reduce the dimensions of the device, advantage of pin cores, because it allows to reduce the dimensions of the device, which is often a which is often a crucial constraint in building installations. crucial constraint in building installations.

6 pins - 50W slits - 50W tubes - 50W pins - 100W slits - 100W tubes - 100W pins - 150W slits - 150W tubes - 150W 5

2 cousticpower (W) A 1

0 0 50 100 150 200 250 Frequency (Hz)

Figure 5. Acoustic power delivered by different diﬀerent porous cores for different diﬀerent input heat powers (50, 100 and 150 W). Other numerical results are shown in Figure6, where curves are plotted in logarithmic scale for Other numerical results are shown in Figure 6, where curves are plotted in logarithmic scale for readability reasons. Figure6a indicates that, with high input powers (100 and 150 W) at low frequencies, readability reasons. Figure 6a indicates that, with high input powers (100 and 150 W) at low the optimal lengths of the cores are several tens of centimeters, which is unpractical in most cases. frequencies, the optimal lengths of the cores are several tens of centimeters, which is unpractical in At higher frequencies, the general trend shows a decrease in length at increasing frequencies, for all most cases. At higher frequencies, the general trend shows a decrease in length at increasing stack typologies. frequencies, for all stack typologies. Figure6b shows the optimal position of the stack within the resonator in terms of distance La Figure 6b shows the optimal position of the stack within the resonator in terms of distance La between the hot end of the core and the closed end of the resonator (see sketch in Figure1). The optimal between the hot end of the core and the closed end of the resonator (see sketch in Figure 1). The LEnergiesa decreases 2020, 13 at, x increasingFOR PEER REVIEW frequencies, but it is always smaller in case of core with parallel pins.13 of 20 optimal La decreases at increasing frequencies, but it is always smaller in case of core with parallel pins. pins - 50W slits - 50W tubes - 50W pins - 50W slits - 50W tubes - 50W pins - 100W slits - 100W tubes - 100W pins - 100W slits - 100W tubes - 100W pins - 150W slits - 150W tubes - 150W pins - 150W slits - 150W tubes - 150W 1 4 (m) k

1 stackthe (m) f 0.1 Optimal length o length Optimal

Optimal position of the stac 0.01 0.1 0 50 100 150 200 250 0 50 100 150 200 250 Frequency (Hz) Frequency (Hz) (a) (b)

Figure 6. Optimization resultsresults for for di differentﬀerent input input powers powers (50, (50, 100 100 and and 150 150 W). W). (a) ( Optimala) Optimal length length of the of core.the core. (b) Optimal(b) Optimal position position of the of the core. core.

3.2. Experimental Results Experiments have been carried out by using the setup shown in Figure 7, which emulates a small‐scale solar driven standing wave prime mover. The experimental results are not directly related to those obtained by numerical simulations and the comparison between the two is beyond the scope of this paper. The main goal here is to prove that also in “real” conditions, porous cores with parallel pins exhibit better performance in applications with low input powers and small resonator lengths. Obviously, the ideal conditions used in numerical simulations (e.g., perfectly adjustable length of the resonator or changeable working frequency of the device, precise temperature values along the stack, perfectly zero heat conduction along the walls of the resonator, ideal heat exchangers, precise value of the input thermal power) typically result in optimistic estimations of the sound level and acoustic power converted. The experimental setup is equivalent to that sketched in Figure 1 with the exception represented by the heat exchangers [28]. The hot heat exchanger is replaced by a light parabolic dish collector, which focusses at one end of the porous core the light radiation generated by a 200 W lamp located at 4.5 cm distance. This focalization is possible because of the good transparency of the wall of the resonator to the light radiation. The cold heat exchanger is provided with a copper coil, accommodated around the resonator and carrying cold water at 12 °C, to reduce the heat transfer from the dish to the resonator wall. The length of the resonator is 40 cm and the diameter is 2.8 cm. The acoustic pressure has been measured in two locations at 3.5 and 6.5 cm away from the resonator open end by using two ¼” pre‐amplified G.R.A.S pressure‐filled microphones, type 46BD (±0.06 dB uncertainty at 250 Hz). The temperature in proximity of the hot and cold end of the stacks has been measured by using two type‐K thermocouples (±2.2 °C uncertainty). The tests have been carried out for different positions of the porous cores along the resonator, which has been possible because both the dish collector and the heat exchanger could slide along the glass tube, while the distance between the dish and the lamp remains constant. Two different additive manufactured cores have been tested: one is constituted by predominantly parallel pins and the another one by parallel plates. Both samples have length 3.8 cm. By following the indications provided in Section 2.1.3, the geometrical parameters have been chosen to result in “equivalent” pores for the two stacks: the pins had rp = 9 × 10−5 m, ratio rp/b = 0.291, supported by a certain number of thin obliquus struts. The parallel plates has semi‐thickness h = 9 × 10−5 m and semi‐distance a = 1.98 × 10−5 m (see Table 1). Also in this case the plates are supported by means thin struts. The constituting material is AISI 316L.

Energies 2020, 13, 2941 13 of 19

3.2. Experimental Results Experiments have been carried out by using the setup shown in Figure7, which emulates a small-scale solar driven standing wave prime mover. The experimental results are not directly related to those obtained by numerical simulations and the comparison between the two is beyond the scope of this paper. The main goal here is to prove that also in “real” conditions, porous cores with parallel pins exhibit better performance in applications with low input powers and small resonator lengths. Obviously, the ideal conditions used in numerical simulations (e.g., perfectly adjustable length of the resonator or changeable working frequency of the device, precise temperature values along the stack, perfectly zero heat conduction along the walls of the resonator, ideal heat exchangers, precise value of the input thermal power) typically result in optimistic estimations of the sound level and acoustic power converted. The experimental setup is equivalent to that sketched in Figure1 with the exception represented by the heat exchangers [28]. The hot heat exchanger is replaced by a light parabolic dish collector, which focusses at one end of the porous core the light radiation generated by a 200 W lamp located at 4.5 cm distance. This focalization is possible because of the good transparency of the wall of the resonator to the light radiation. The cold heat exchanger is provided with a copper coil, accommodated around the resonator and carrying cold water at 12 ◦C, to reduce the heat transfer from the dish to the resonator wall. The length of the resonator is 40 cm and the diameter is 2.8 cm. The acoustic pressure has been measured in two locations at 3.5 and 6.5 cm away from the 1 resonator open end by using two 4 ” pre-amplified G.R.A.S pressure-filled microphones, type 46BD ( 0.06 dB uncertainty at 250 Hz). The temperature in proximity of the hot and cold end of the stacks ± has been measured by using two type-K thermocouples ( 2.2 C uncertainty). ± ◦ The tests have been carried out for different positions of the porous cores along the resonator, which has been possible because both the dish collector and the heat exchanger could slide along the glassEnergies tube, 2020 while, 13, x FOR the PEER distance REVIEW between the dish and the lamp remains constant. 14 of 20

FigureFigure 7. 7.Experimental Experimental setupsetup utilizedutilized to emulate a small small-scale‐scale sun sun-powered‐powered thermoacostic thermoacostic prime prime movermover for for building building applications. applications.

TwoTwo di examplesfferent additive of the manufacturedtime data acquired cores during have been the tests tested: are one depicted is constituted in Figure by 8a, predominantly b. The two parallelfigures pinsare andrelated the to another the tests one which by parallel provided plates. the Both highest samples and have the smallest length 3.8 acoustic cm. By pressure following the indications provided in Section 2.1.3, the geometrical parameters have been chosen to result in measured for the core with parallel pins. In the first case, the core has been positioned at distance La 5 “equivalent” pores for the two stacks: the pins had rp = 9 10 m, ratio rp/b = 0.291, supported by = 9.5 cm, in the second case at La = 12.0 cm. After a warm‐up stage× − of ~80 s, the temperature differential a certain number of thin obliquus struts. The parallel plates has semi-thickness h = 9 10 5 m and reaches ~110–130 °C and acoustic instabilities occur, which result in an abrupt increase in× the acoustic− semi-distance a = 1.98 10 5 m (see Table1). Also in this case the plates are supported by means thin pressure measured by× the− microphone M1 at 3.5 cm from the open end. Both the acoustic pressure struts.and the The temperature constituting differential material isincrease AISI 316L. during the entire test. In case of Figure 8b, the temperature differentialTwo examples between of the the hot time and data cold acquired ends remains during smaller the tests during are the depicted whole test. in Figure 8a,b. The two figuresBy are using related the to two the‐microphone tests which technique, provided the it is highest possible and to thereconstruct smallest the acoustic acoustic pressure field (acoustic measured forpressure the core and with particle parallel velocity) pins. In[24]. the In first this case,way, thefor the core core has with been pins positioned located at at L distancea = 9.5 cm,L a~1500= 9.5 Pa cm, inacoustic the second pressure case at andLa =~412.0 m/s cm. particle After velocity a warm-up have stage been of estimated ~80 s, the in temperature proximity of di fftheerential closed reaches and open end of the resonator, respectively.

200 Pressure 200 200 Pressure 200 T_cold T_cold T_hot T_hot 160 160 Temperature ( 100 ( Temperature 100

120 120 e at M1 (Pa) e e at M1 (Pa) r r 0 0 essu essu r 80 r 80 o o C) C) -100 -100 40 40 coustic p coustic coustic p A A

-200 0 -200 0 0 50 100 150 200 250 0 50 100 150 200 250 Time (s) Time (s) (a) (b)

Figure 8. Time domain values of acoustic pressure measured by microphone M1 and temperatures

measured in proximity of the core ends. (a) Optimal position of the core, La = 9.5 cm. (b) Non‐optimal

position of the core, La = 12.0 cm.

In Figure 9, the amplitude of the acoustic pressure measured by the microphone M1 and the temperature differential along the core are plotted for tests at different positions of the cores. The values are obtained by averaging the measurements taken during the last 10 seconds of each test. Figure 9a, b are, respectively, related to tests with cores with parallel pins and with slits. In both cases, the general trend shows a certain increase in temperature differential at the locations where the emissions increase. It is also confirmed what has been shown by the simulations, i.e. that the optimal position for cores with parallel pins has a small La than the optimal positions for cores with slits. Great differences are noticeable between the performances of the two equivalent cores, with 160 Pa

200 Pressure 200 200 Pressure 200 T_cold T_cold T_hot T_hot 160 160 Temperature ( 100 ( Temperature 100

120 120 e at M1 M1 e at (Pa) e at M1 (Pa) r r 0 0 essu essu r 80 r 80 o o C) C) -100 -100 40 40 coustic p coustic coustic p A A

-200 0 -200 0 0 50 100 150 200 250 0 50 100 150 200 250 Time (s) Time (s) (a) (b)

200 Pressure 200 200 Pressure 200 T_cold T_cold T_hot T_hot 160 160 Temperature ( 100 ( Temperature 100

120 120 e at M1 M1 e at (Pa) e at M1 (Pa) r r 0 0 essu essu r Energies 2020, 13, 2941 80 r 8014 of 19 o o C) C) -100 -100 40 40 coustic p coustic coustic p A ~110–130 ◦C and acoustic instabilities occur, which resultA in an abrupt increase in the acoustic pressure measured-200 by the microphone M1 at 3.5 cm0 from the-200 open end. Both the acoustic pressure and0 the 0 50 100 150 200 250 temperature differential increase during the entire test. In case0 of Figure 508b, 100 the temperature 150 200 di ff 250erential Time (s) Time (s) between the hot and cold ends remains smaller during the whole test. By using the two-microphone(a) technique, it is possible to reconstruct (b) the acoustic field (acoustic pressure and particle velocity) [24]. In this way, for the core with pins located at La = 9.5 cm, ~1500 Pa acoustic pressure and ~4 m/s particle velocity have been estimated in proximity of the closed and open end of the resonator, respectively.

Figure 8. Time domain values of acoustic pressure measured by microphone M1 and temperatures measured in proximity of the core ends. (a) Optimal position of the core, La = 9.5 cm. (b) Non-optimal position of the core, La = 12.0 cm.

In Figure9, the amplitude of the acoustic pressure measured by the microphone M1 and the temperature differential along the core are plotted for tests at different positions of the cores. The values are obtained by averaging the measurements taken1 during the last 10 s of each test. Figure9a,b are, respectively, related to tests with cores with parallel pins and with slits. In both cases, the general trend shows a certain increase in temperature differential at the locations where the emissions increase. It is also confirmed what has been shown by the simulations, i.e., that the optimal position for cores withEnergies parallel 2020, 13, pins x FOR has PEER a small REVIEWLa than the optimal positions for cores with slits. Great differences15 of are 20 noticeable between the performances of the two equivalent cores, with 160 Pa maximum acoustic pressuremaximum measured acoustic bypressureM1 for measured the core with by M1 pinsand for the 0.007 core Pa with maximum pinsand acoustic 0.007 Pa pressure maximum for acoustic the core withpressure slits. for the core with slits.

200 200 0.008 160 Pressure - M1 Pressure - M1 DeltaT - core DeltaT - core 160 180 0.006 150 DeltaT ( DeltaT ( 160

e at M1(Pa) 120 e at M1(Pa) at e r r 0.004 140 o o essu essu C) C) r r 80 140

0.002 130 40 120 Acousticp Acoustic p Acoustic

0 100 0 120 0.22 0.24 0.26 0.28 0.3 0.32 0.24 0.26 0.28 0.3 0.32 La/Lres La/Lres (a) (b)

Figure 9. Ten second averaged acoustic pressure (at microphone M1) and pressure differential diﬀerential along the core for didifferentﬀerent positions.positions. (a)) Core with parallelparallel pins.pins. (b) Core with slits.

This circumstance is better represented in Figure 10, where the contour plots of the spectral components, expressed in sound pressure level, are carried out at each second of the tests for the two cores, once located at the optimal positions (La = 9.5 cm for core with pins and La = 11.5 cm for core with slits) and once at the worst positions (La = 12.0 cm and La = 10.0 cm, respectively). The acoustic emission for tests involving core with pins is always remarkable (Figure 10a, b), up to ~140 dB at 218 Hz, measured by microphone M1, when the core is in the optimal position. The situation radically changes for the stack with parallel plates because, as visible in Figure 10c, d, which show that, at best, ~80 dB of rather fragmented sound is produced around 216 Hz. In other words, no effective thermoacoustic conversion takes place when this core is used. Noticeably, the same core with parallel plates has been used in a different setup in the paper [19], consisting of a longer resonator (60 cm length) and larger input heat power delivered by a blowtorch. In that case, ~150 dB have been measured at 132 Hz by a microphone located at 10 cm from the open end, suggesting a good thermoacoustic conversion.

(a) (b)

Energies 2020, 13, x FOR PEER REVIEW 15 of 20 maximum acoustic pressure measured by M1 for the core with pinsand 0.007 Pa maximum acoustic pressure for the core with slits.

200 200 0.008 160 Pressure - M1 Pressure - M1 DeltaT - core DeltaT - core 160 180 0.006 150 DeltaT ( DeltaT ( 160

e at M1(Pa) 120 e at M1(Pa) at e r r 0.004 140 o o essu essu C) C) r r 80 140

0.002 130 40 120 Acousticp Acoustic p Acoustic

0 100 0 120 0.22 0.24 0.26 0.28 0.3 0.32 0.24 0.26 0.28 0.3 0.32 La/Lres La/Lres (a) (b)

EnergiesFigure2020, 139. ,Ten 2941 second averaged acoustic pressure (at microphone M1) and pressure differential along15 of 19 the core for different positions. (a) Core with parallel pins. (b) Core with slits.

This circumstance is better represented in Figure 10,10, where the contour plots of the spectral components, expressed in sound pressure level, are carried out at each second of the tests for the two cores, once located at thethe optimaloptimal positionspositions ((LLaa = 9.59.5 cm for core with pins andand LLaa = 11.511.5 cm cm for core with slits) and onceonce atat thethe worstworst positionspositions ( (LLaa == 12.0 cm and LLaa = 10.010.0 cm, cm, respectively). respectively). The acoustic emission forfor teststests involvinginvolving core core with with pins pins is is always always remarkable remarkable (Figure (Figure 10 a,b),10a, upb), up to ~140 to ~140 dB atdB 218 at 218 Hz, Hz,measured measured by microphone by microphone M1, M1, when when the core the core is in is the in optimal the optimal position. position. The situation radically changeschanges for for the the stack stack with with parallel parallel plates plates because, because, as as visible visible in in Figure Figure 10 10c,c,d, whichd, which show show that, that, at at best, best, ~80 ~80 dB dB of of rather rather fragmented fragmented sound sound is is produced produced aroundaround 216216 Hz. In other words, nono eeffectiveﬀective thermoacoustic thermoacoustic conversion conversion takes takes place place when when this this core core is used. is used. Noticeably, Noticeably, the same the samecore with core parallelwith parallel plates plates has been has usedbeen inused a di inﬀerent a different setup insetup the paperin the [paper19], consisting [19], consisting of a longer of a longerresonator resonator (60 cm length)(60 cm length) and larger and input larger heat input power heat delivered power delivered by a blowtorch. by a blowtorch. In that case, In that ~150 case, dB ~150have beendB have measured been atmeasured 132 Hz by at a 132 microphone Hz by a located microphone at 10 cm located from the at open10 cm end, from suggesting the open a goodend, suggestingthermoacoustic a good conversion. thermoacoustic conversion.

(a) (b)

Figure 10. Spectral components in time, expressed in sound pressure level, of the sound measured by microphone M1. Results for porous core with parallel pins: (a) La = 9.5 cm; (b) La = 11.5 cm. Results for porous cores with slits: (c) La = 12.0 cm; (d) La = 10.0 cm.

Although the geometrical characteristics of the two cores have been chosen to provide equivalent thermoacoustc behaviors, the experimental results conﬁrm what has been indicated by thermoacoustic simulations: cores with parallel pins work properly also in shorted resonators and with small input heat powers.

4. Discussion Small thermoacoustic devices can have, in principle, numerous applications ranging from harvesting the heat released by biomass-burning cook stoves to cooling electronic devices in space applications. Nevertheless, there is still a wide margin of development for small thermoacoustic devices for building applications. Different destinations in this area can be foreseen, for instance thermoacoustic generators to supply the huge number of sensors in modern buildings, thus to reduce the amount of Energies 2020, 13, 2941 16 of 19 necessary cables. Thermoacoustic devices can also be used for air cooling and exploiting sun radiation as input thermal power. At the present date, the main obstacles for the use of small thermoacoustic devices in buildings are represented by small input heat powers available and size constraints which limit the length of the resonator. Since the porous core is responsible for the heat-to-acoustic power conversion, it represents the most important part of the engine. For this reason, the study presented here has assessed the potentialities of different cores typologies and shown which one is more suitable for building applications. Although cores with parallel pins are not new, manufacturing issues have limited their diffusion and the number of related studies is quite limited. Nowadays, with the advent of modern techniques for additive manufacturing, the production issues have been largely overcome. Cores with parallel pins and cores provided with slits and circular tubes have been thoroughly studied in this paper. The method used for the assessment of the thermoacoustic performance of the cores overcomes the rather qualitative approach provided by previous studies. It consists of a detailed comparison of the best performance exhibited by each core typology and it shows that cores with parallel pins represent the best solution for the problem tackled here. Future investigations will involve the use of additive manufactured lattice cells with different pin orientations, the optimization of pin-based core geometries and the use of alumina as manufacturing material.

5. Conclusions In this paper a “fair” comparison between the thermoacoustic performance of diﬀerent core typologies, including cores with circular tubes, with slits and with parallel pins, has been provided. Thus, the best solution for small thermoacoustic devices in building applications has been suggested. The following steps have been undertaken:

The geometrical characteristics of different cores (diameter of the inner tubes, width of the • slits, pin diameter and porosities of the cores) have been chosen to result in “equivalent” thermo-viscous behavior. The behavior of each core has been modelled by means of one dimensional thermoacoustic • simulations which have been included in a numerical optimization procedure. The optimization procedure has been able to find, per each core analyzed, the engine configuration • (length of the resonator, operating frequency and position of the core) which maximizes the acoustic power converted from a fixed small input thermal power. The input power used in the simulations is limited to the range 50–150 W.

The numerical procedure has shown that cores with parallel pins provide the maximum acoustic power by using the smallest length of the resonator. As a consequence, these cores are the best candidates to meet the power and size constraints in buildings. To strengthen this conclusion, a further step has been carried out:

An experimental setup has been used to emulate a small-scale solar-driven standing wave prime • mover. The setup is equipped with a relatively short resonator (40 cm length) and the input thermal power is delivered by the focussed light of a small projector. An additive manufactured core with parallel pins has been tested at diﬀerent positions in the resonator and compared with an equivalent core provided with rectangular slits.

The experiments conﬁrmed the trend delineated by the numerical procedure. In fact, for the small thermal power used in the tests, the acoustic emission of the core with parallel pins is remarkable (up to 140 dB close to the open end) and steady over time, while that of the core with slits is negligible (~80 dB) and unsteady. Energies 2020, 13, 2941 17 of 19

Author Contributions: Conceptualization, F.A.; methodology, R.D.; numerical simulations, E.D.G., validation and formal analysis, M.N.; writing—original draft preparation, F.A.; writing—review and editing, R.D., E.D.G., M.N. All authors have read and agreed to the published version of the manuscript. Funding: This research has been funded by: PoC grant EAG28, Smart micro-perforated silencer for heat, ventilation and air conditioning systems financed by Estonian Research Council. Estonian Centre of Excellence in Zero Energy and Resource Efficient Smart Buildings and Districts, ZEBE, grant 2014- 2020.4.01.15-0016 funded by the European Regional Development Fund. Italian Ministry of University and Research (MIUR): Project number PRIN 2017JP8PHK. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

Symbol Description a Semi width of the slit A Cross section area of the pore Afluid Fluid area of the core cross section Asolid Solid area of the core cross section Awet Wet area of the core Ak Thermal diffusion area within the pore Av Viscous diffusion area within the pore b = y rp 0 − c Speed of sound cp Isobaric heat capacity per unit mass of the fluid cs Isobaric heat capacity per unit mass of the solid frame CX Cold heat exchanger d Semi-distance between two adiacent tubes Dr Diameter of the resonator and of the core . E2 Time averaged acoustic power per unit are of the fluid fk Spatially averaged temperature function within the pore fk,s Spatially averaged temperature function of the solid part of the pore fv Spatially averaged viscosity function within the pore g Gain/attenuation constant h Semi thickness of the plate . H2 Total thermal input power HX Hot heat exchanger HR Hydraulic radius HVAC Heat, ventilation and air conditioning Imaginary part = k Thermal conductivity La Distance between the hot end of the stack and the closed end of the resonator Lr Length of the resonator M Figure of merit M1 and M2 Microphone 1 and 2 p and p1 Pressure and fluctuating pressure pm Mean pressure Pr Prandtl number rp Radius of the pin R Diameter of a tube Re Real part r0 Radius of the circle inscribed in the hexagonal pattern of pins r1 Radius of the circle inscribed in the square pattern of pins rv Viscous resistance per unit length rk Thermal relaxation conductance per unit length Energies 2020, 13, 2941 18 of 19

Symbol Description

S (i 1)/δv − Sh Shear wavenumber s’ (i+1)/δv T and T1 Temperature and fluctuating temperature T Period of the acoustic oscillation TAHE Thermo acoustic heat engine TC1, TC2 Thermocouple 1, 2 u and u1 Particle velocity, fluctuating particle velocity Vfluid Volume of fluid in the core Vsolid Volume of solid in the core Vtotal Total volume of the core Vk Thermal diffusion volume Vv Viscous diffusion volume y0 Semi distance between adjacent pins in hexagonal pattern y1 Semi distance between adjacent pins in square pattern γ Specific heat ratio ∆Tc Critical temperature differential Greek letters εs Heat capacity ratio εs,c, εs,s and εs,p Heat capacity ratios of circular pore, slit and pin array pore δk Thermal boundary layer δv Viscous boundary layer k Thermal diffusivity Λ’ Thermal characteristic length µ Dynamic viscosity η Global efficiency Π Perimeter of the cross section of the pore ρ and ρm Fluid density and mean fluid density φ Porosity φw Phase between acoustic pressure and particle velocity ω Angular frequency Subscripts c Circle p Pin s Slit m Mean 1 First order 2 Second order

References

1. Kurnitski, J. Technical definition of nearly zero energy buildings. REHVAJ. 2013, 2013, 22–28. 2. Piccolo, A. Study of standing-wave thermoacoustic electricity generators for low-power applications. Appl. Sci. 2018, 8, 287. [CrossRef] 3. Hong, B.-S.; Lin, T.-Y. System identification and resonant control of thermoacoustic engines for robust solar power. Energies 2015, 8, 4138–4159. [CrossRef] 4. Hariharan, N.M.; Shvashanmugam, P.; Kasthurirengam, S. Experimental investigation of a thermoacoustic refrigerator driven by a standing wave twin thermoacoustic prime mover. Int. J. Refrig. 2013, 36, 2420–2425. [CrossRef] 5. Timmer, M.A.G.; de Blok, K.; van der Meer, T.H. Review on the conversion of thermoacoustic power into electricity. J. Acoust. Soc. Am. 2018, 143, 841–857. [CrossRef] 6. Olivier, C.; Penelet, G.; Poignand, G.; Lotton, P. Active control of thermoacoustic amplification in a thermo-acoustic-electric engine. J. Appl. Phys. 2014, 115, 174905-1-6. [CrossRef] Energies 2020, 13, 2941 19 of 19

7. Swift, G.W. Thermoacoustics—A Unifying Perspective for Some Engines and Refrigerators, 2nd ed.; ASA Press Springer: Berlin/Heidelberg, Germany, 2002. 8. Kabral, R.; Rämmal, H.; Lavrentjev, J. Acoustic studies of micro-perforates for small engine silencers. SAE Tech. Pap. 2012, 32, 107. [CrossRef] 9. Lavrentjev, J.; Rämmal, H. Acoustic study on motorcycle helmets with application of novel porous materials. SAE Tech. Pap. 2019, 32, 0531. 10. Rott, N. Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys. 1969, 20, 230–243. [CrossRef] 11. Arnott, W.P.; Bass, H.E.; Raspet, R. General formulation of thermoacoustics for stacks having arbitrarily shaped pore cross sections. J. Acoust. Soc. Am. 1991, 90, 3228–3237. [CrossRef] 12. Hofler, T.J. Thermoacoustic Refrigerator Design and Performance. Ph.D. Thesis, Physics Department, University of California, San Diego, CA, USA, 1986. 13. Dragonetti, R.; Napolitano, M.; Di Filippo, S.; Romano, R. Modeling energy conversion in a tortuous stack for thermoacoustic applications. Appl. Therm. Eng. 2016, 103, 233–242. [CrossRef] 14. Napolitano, M.; Romano, R.; Dragonetti, R. Open-cell foams for thermoacoustic applications. Energy 2017, 138, 147–156. [CrossRef] 15. Swift, G.W.; Keolian, R.M. Thermoacoustics in pin-array stacks. J. Acoust. Soc. Am. 1993, 94, 941–943. [CrossRef] 16. Tu, Q.; Zhang, X.; Chen, Z.; Guo, F. Design of miniature thermoacoustic refrigerator with pin-array stack. Acta Acust. United Acust. 2006, 92, 16–23. 17. Matveev, K. Thermoacoustic energy analysis of transverse-pin and tortuous stack at large acoustic displacements. Int. J. Therm. Sci. 2010, 49, 1019–1025. [CrossRef] 18. Asgharian, B.; Matveev, K.I. Influence of finite heat capacity of solid pins and their spacing on thermoacoustic performance of transverse-pin stacks. Appl. Therm. Eng. 2014, 62, 593–598. [CrossRef] 19. Auriemma, F.; Holovenko, Y. Performance of additive manufactured stacks in a small scale thermoacous-tic engine. SAE Tech. Pap. 2019, 1, 1534. [CrossRef] 20. Auriemma, F.; Holovenko, Y. Use of selective laser melting for manufacturing the porous stack of a ther-moacoustic engine. Key Eng. Mater. 2019, 799, 246–251. [CrossRef] 21. Babaei, H.; Siddiqui, K. Design and optimization of thermoacoustic devices. Energy Convers. Manag. 2008, 49, 3585–3598. [CrossRef] 22. Hariharan, N.M.; Shvashanmugam, P.; Kasthurirengam, S. Influence of stack geometry and resonator length on the performance of thermoacoustic engine. Appl. Acoust. 2012, 73, 1052–1058. [CrossRef] 23. Majak, J.; Kuttner, F.; Pohlak, M.; Eerme, M.; Karjust, K. Application of evolutionary methods for solving optimization problems in engineering. In Proceedings of the NordDesign, NordDesign Conference, Tallinn, Estonia, 21–23 August 2008; pp. 39–48. 24. Majak, J.; Pohlak, M.; Karjust, K.; Eerme, M.; Kurnitski, J.; Shvartsman, B.S. New high order Haar wavelet method: Application to FGM structures. Compos. Struct. 2018, 201, 72–78. [CrossRef] 25. Tiikoja, H.; Auriemma, F.; Lavrentjev, J. Damping of acoustic waves in straight ducts and turbulent flow conditions. SAE Tech. Pap. 2016, 1, 1816. [CrossRef] 26. Champoux, Y.; Allard, J.F. Dynamic tortuosity and bulk modulus in air-saturated porous media. J. Appl. Phys. 1991, 70, 1975–1979. [CrossRef] 27. Johnson, D.L.; Koplik, J.; Dashen, R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J. Fluid Mech. 1987, 176, 379–402. [CrossRef] 28. Piccolo, A.; Sapienza, A.; Guglielmino, C. Convection heat transfer coefficients in thermoacoustic heat exchangers: An experimental investigation. Energies 2019, 12, 4525. [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).