Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator

Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

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Original article

Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator A. Piccolo Department of Engineering, University of Messina, Contrada di Dio, 98166 S. Agata (Messina), Italy

a r t i c l e

i n f o

Article history: Received 1 August 2016 Revised 9 October 2016 Accepted 25 October 2016 Available online xxxx Keywords: Thermoacoustics Electricity generator Heat Acoustic power

a b s t r a c t This paper is concerned with the study of low-cost low-power thermoacoustic electricity generators. Based on a target electrical output power of 100 W, a two-stage standing wave prototype integrating a commercial loudspeaker in side branch arrangement is conceived. Each stage consists of a square-pore stack sandwiched between hot and ambient heat exchangers. The working gas is air at atmospheric pressure oscillating at the operation frequency of 194.5 Hz. Design issues and optimization procedures are discussed in detail. The prototype efficiency in converting heat to electrical power is simulated by the specialized design tool DeltaEC based on the linear theory of thermoacoustics. Computations reveal that at the heating temperature of 527 °C the target electrical output power is extracted by the loudspeaker with an acoustic-to-electric efficiency of around 70% and that the overall thermal-to-electrical conversion efficiency of the engine is 5.7%. This result suggests that in applications involving the use of loudspeakers as linear alternators and air at near atmospheric pressure as working fluid, electricity generators of the standing wave type could perform, comparably to their travelling-wave counterparts. This is likely due to a simpler design of the alternator-engine coupling and a simpler and more compact configuration. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction Thermoacoustic (TA) engines has the immediate potential to be a reliable, efficient, clean, and low-cost technology. This extraordinary potential derives from their ability to directly convert heat into acoustic power without involving any moving mechanical component. In these engines, in fact, the phasing of the transformations which build up the underlying thermodynamic cycle is naturally accomplished by an acoustic wave rather than by pistons, valves and displacers. This sound wave is spontaneously excited inside a solid porous medium when a sufficiently high temperature gradient is imposed on it [1,2]. This characteristics account for the engineering simplicity of these devices and for the associated high technical reliability and low cost. Additional favorable features derive from being environmental friendly, since they use noble gases (or simply air) as working fluids and from allowing the exploitation of waste heat and renewable energy sources, since they are potentially configurable to operate with low temperature differentials [3]. The first TA engine implementations date back to the 1980 s (Los Alamos National Laboratory) and refer to the so-called standing-wave typology [4]. In these devices the working fluid undergoes an intrinsically irreversible Brayton-like cycle that limE-mail address: [email protected]

its their thermal-to-acoustic conversion efficiencies (gha) typically below 20%. At the end of the 1990 s, research switched to the development of TA engines of the travelling-wave typology. In these devices the working fluid undergoes a reversible Stirlinglike cycle that enables higher conversion efficiencies, as proved by the demonstrators developed by Backhaus and Swift [5] and Tijani and Spoelstra [6] characterized by gha = 30 and 32% respectively. A great body of research in TA technology is being recently addressed to the development of TA electricity generators (TAEG). A TAEG is essentially a TA engine coupled to an electroacoustic transducer (linear alternator, loudspeaker, piezoelectric crystal, bidirectional gas turbine, etc.) to convert a fraction of the useful acoustic power (generated from heat) into output electrical energy. Flexure bearing supported linear alternators (LA) characterized by low friction losses appear to date to be the most promising candidates for transduction in TAEGs. These devices are capable of achieving transduction efficiencies up to 90%. The first example of this new class of electricity generators integrating LAs in TA engines is the device implemented in 2004 by Backhaus et al. [7] which consists of a compact travelling wave TAEG working without a resonance tube. This prototype is able to produce up to 57 W of electrical power with 17.8% thermal-to-electric efficiency (ghe). However, when designing low-cost TAEGs the use of LAs is precluded by their high cost so a different electro-acoustic transducer

http://dx.doi.org/10.1016/j.seta.2016.10.011 2213-1388/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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has to be considered. This problem has been recently investigated by different researchers who found that standard electrodynamic loudspeakers operated in reverse mode can be conveniently used as LAs, but in low power applications (a few hundred of Watts). This is due to the fact that a loudspeaker, generally designed for high fidelity applications, is characterized by a weak and fragile cone, a short stroke, a low power handling and a low acoustic-toelectric transduction efficiency (gae). Anyway, when the particular application requires low power levels, low-cost for generated kWe, and not very high transduction efficiencies (
50%) the use of loudspeakers could be authorized. The development of TAEGs integrating conventional loudspeakers was firstly undertaken by Hartley [8] and Morrison [9] who considered vibrating air columns in standing-wave mode without stacks. More recently, Kitadani et al. [10] developed both standing- and travelling-wave TAEGs prototypes using loudspeakers as LAs and obtained a maximum electrical output power of 1.1 W with ghe  0.3%. Yu et al. [11], implemented a travellingwave TAEG of the looped-tube type with a commercially available loudspeaker placed within the loop and obtained an overall ghe efficiency of the order of 1% with around 4–5 W output power. The performance of the device was subsequently improved [12] by optimizing the impedance matching of the alternator and an output power of 11.6 W with ghe  1.5–2% was achieved. Analogous performance levels were obtained by Chen et al. [13] who developed two travelling wave TAGEs based on the same loopedtube configuration and powered by waste heat from cooking stoves. These engines are able to generate power outputs of the order of 20 W with ghe  3%. Conversion efficiencies ghe slightly greater than 1% with about 1 W electrical output power were obtained by Olivier et al. [14] by a torus-shaped TAGE coupled to a loudspeaker placed at the end of the resonator. The performance was observed to improve by about 25% when an active control mechanism was added to the device. The coaxial two-stage travelling-wave TAEG developed by the de Blok [15] and working with air at 2 bar static pressure is able to generate about 25 W electrical power with ghe  2% for temperatures of the hot heat exchanger near 350 °C. Recently, Kang et al. [16] developed a two-stage travelling-wave TAEG working with helium pressurized at 18 bar. The couple of installed loudspeakers extract 204 W of electrical power with ghe  3.4%. From the above discussion it results that research on low power TAEGs integrating standard loudspeakers has mainly focused on travelling-wave engines with looped-tube configuration. However, in applications where conversion efficiency is not a major factor, as the one here discussed, configurations based on the (intrinsically less efficient) standing-wave typology might be worthy of attention. On the other hand, the drop in performance could be compensated by a simpler design and technical implementation, a more compact structure and a lower fabrication cost. This work is specifically concerned with the study/development of low-cost TAEGs of the standing-wave type using standard loudspeakers as LAs. Based on a target electric output power of 100 W, a two-stage standingwave prototype working with air at atmospheric pressure and integrating a loudspeaker in side branch arrangement is conceived. Design issues and optimization procedures are discussed in detail and the prototype performance is simulated by standard codes based on the linear theory of thermoacoustics. Results are compared to those of analogue TAEGs integrating conventional loudspeakers found in literature.

electrodynamic drivers to thermoacoustic devices has been widely discussed by Wakeland [17]. The treatment here reported, concerning the optimization of reverse-operated electrodynamic loudspeakers, is parallel and strictly refers to the Wakeland’s theory, but in the present case the load the power is delivered to is a (frequency independent) electrical resistance rather than the (frequency dependent) input acoustic impedance of a thermoacoustic device. The essence of the acoustic-to-electric transduction mechanism can be qualitatively captured observing that the difference of the acoustic pressures acting on the opposite sides of the diaphragm forces the voice coil to move in the radial magnetic field of the permanent magnets. The alternating motion of the voice coil induces then a current in it according to the Faraday’s law. This complex interaction among acousto-mecahanical and electro-magnetic physical quantities is quantitatively described by the well-known canonical equations for an antireciprocal transducer which is schematized as a two port network relating electric quantities at one port to acoustical quantities at the other. These equations are conveniently written in complex notation which allows to express any variable a(t) oscillating harmonically at angular frequency x about its mean value a0 as

aðtÞ ¼ a0 þ RefAejx t g

ð1Þ

where t is the time j the imaginary unit, Re{} signifies the real part and where the complex amplitude A accounts for both magnitude and phase of the oscillation. Neglecting hysteresis losses and considering that the electric terminals are closed on a load electric resistance RL the canonical equations assume the form of the following two linear equations (see Ref. [18] pg. 397, Eq. (14.3.11) and (14.3.12))

0 ¼ ðZ e þ RL ÞI 

ðp1  p2 Þ ¼

ðBlÞ U Ad

ðBlÞ Zm Iþ 2U Ad Ad

ð2Þ

ð3Þ

where p1 and p2 are the complex amplitudes of the acoustic pressures acting on the front and back side of the diaphragm of area Ad (see Fig. 1), I is the complex amplitude of the current flowing in the voice coil, U is the complex amplitude of the volumetric velocity due to the diaphragm motion, Bl is the ‘‘force factor” (the product of the magnetic field, B, times the length of the voice coil wire, l) and where Ze (the electrical impedance with blocked

Theoretical modelling An electrodynamic loudspeaker can be operated both in direct mode, as a source of sound (driver), or, in reverse mode, as an electric power generator driven by sound. The optimal matching of

Fig. 1. Schematic illustration of the loudspeaker. The dashpot symbolizes the frictional forces (Rm) while the spring symbolizes the stiffness (Km).

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

diaphragm) and Zm (the mechanical impedance with open electric circuit) are

Z e ¼ Re þ jX e ¼ Re þ jx Le

ð4Þ


Z m ¼ Rm þ jX m ¼ Rm þ j x m 

Km

 ð5Þ

x

Re and Le being the electrical resistance and inductance of the voice coil, Rm the mechanical resistance of the system, m the mass of the diaphragm and voice coil and Km the mechanical stiffness of the elastic suspensions. To represent by circuital analogy the overall (acoustic-to-mechanic-to-electric) transduction mechanism, the surface diaphragm is interpreted as an ideal transformer converting acoustical to mechanical quantities on the basis of the equations

F ¼ Ad ðp1  p2 Þ U ¼ Ad V

ð6Þ

This equation shows that a fraction of the absorbed acoustic power is dissipated by the mechanical resistance Rm, another fraction is consumed in the coil electrical resistance Re and the remaining fraction is extracted by the load RL as electric power. Using Eq. (7) and (8) the last contribute can be written as " # 2 2 1 2 1 jUj ðBlÞ RL W e ¼ jIj RL ¼ 2 2 A2d ðRe þ RL Þ2 þ X 2e 2

1 ðBlÞ RL 2 2 ¼ jp1  p2 j Ad 2 2 2 ½ðBlÞ  X m X e þ Rm ðRe þ RL Þ þ ½X m ðRe þ RL Þ þ Rm X e 2 ð11Þ

which shows that for delivering high electric power levels to the load high oscillation velocities of the diaphragm are required. The acoustic-to-electric conversion efficiency is 2

F being the complex amplitude of the net external force (due to the pressure difference p1-p2) acting on the diaphragm, which oscillates with complex amplitude velocity V. This leads to the circuital scheme of the generator illustrated in Fig. 2. Eq. (2) shows that the current flowing in the coil is caused by the induced voltage (Bl)(U/Ad) on the electrical impedance (Ze + RL):

I¼

ðBlÞU ðZ e þ RL ÞAd

ð7Þ

Substituting this relation into Eq. (3) and dividing both members of the resulting equation by U leads to the following expression of the acoustic impedance of the loudspeaker

" # 2 ðp1  p2 Þ 1 ðBlÞ ¼ Ra þ jX a Za ¼ ¼ 2 Zm þ U Z e þ RL Ad " # " # 2 2 1 ðBlÞ ðRe þ RL Þ 1 ðBlÞ X e ¼ 2 Rm þ X  þ j m Ad ðRe þ RL Þ2 þ X 2e A2d ðRe þ RL Þ2 þ X 2e

ð8Þ

where the term (Bl)2/(Ze + RL)Ad2 is an impedance transferred from the electrical to the mechanical side due to the electromechanical coupling. The acoustic power absorbed by the alternator can be written in the following equivalent forms

~ W a ¼ 12 Refðp1  p2 ÞUg ¼ 12 jUj2 Ra ¼ 12

jp1 p2 j2 jZ a j2

Ra ¼ 12 jp1  p2 jjUj cosðhÞ

ð9Þ

where h is the phase angle between the pressure difference (p1-p2) across the diaphragm and the volumetric velocity U. Substitution of Eq. (8) into Eq. (9) provides " # 2 2 2 1 jUj ðBlÞ Re ðBlÞ RL Wa ¼ Rm þ þ 2 A2d ðRe þ RL Þ2 þ X 2e ðRe þ RL Þ2 þ X 2e 2

1 Rm ½ðRe þ RL Þ2 þ X 2e  þ ðBlÞ ðRe þ RL Þ 2 2 ¼ jp1  p2 j Ad 2 2 2 ½ðBlÞ  X m X e þ Rm ðRe þ RL Þ þ ½X m ðRe þ RL Þ þ Rm X e 2 ð10Þ

gae ¼

We jIj2 RL ðBlÞ RL ¼ ¼ Wa jUj2 Ra Rm ½ðRe þ RL Þ2 þ X 2e  þ ðBlÞ2 ðRe þ RL Þ

ð12Þ

Equations (10) and (11) and (12) allow to gain information on the conditions which optimize the loudspeaker performance in the alternator operation mode. For a given loudspeaker, the performance (expressed by the parameters gae, Wa and We) grows at increasing the force factor (Bl) and at decreasing the mechanical (Rm) and the electrical (Re) resistances (in the ideal case Rm = Re = 0 the device would have an acoustic-to-electric efficiency of 100%). The performance also increases when the electrical reactance (Xe) is decreased. An additional condenser could be added to the coil circuit to bring the alternator in favorable electrical resonant conditions (Xe = 0) at the selected operation frequency [19] (even if the inductance of the voice coil, xLe, is generally negligible at the typical working frequencies of thermoacoustic devices for standard woofers and sub-woofers). Finally, the amount of acoustic power absorbed by the alternator also grows at decreasing the mechanical reactance Xm so the performance of the device is highest when it works in both mechanical and electrical resonant conditions (Xm = 0, Xe = 0) for which the alternator acoustical impedance becomes real (p1-p2 in phase with U). The optimal value of the load resistance maximizing the efficiency can be found from Eq. (12) when written (neglecting the electrical reactance term) in the form

1

gae

¼1þ

Re 1 Rm þ ðRe þ RL Þ2 RL ðBlÞ2 RL

ð13Þ

Equating to zero the derivative of this equation we find that the optimal load resistance is

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðBlÞ RL ¼ Re 1 þ Re Rm

ð14Þ

Substituting this value in Eq. (13) we find that the maximum achievable efficiency is

Fig. 2. Equivalent circuit of the loudspeaker. The acoustical and mechanical parts are of the mobility type. The electrical part is of the impedance type. Vertical arrows point to the positive terminal.

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

gmax

pffiffiffiffiffiffiffiffiffiffiffiffi bþ11 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi bþ1þ1

ð15Þ

where b = (Bl)2/ReRm. The value provided by Eq. (14) differs from the value of the optimal load resistance that maximizes the electric output power. Taking into account Eq. (7), in fact, and indicating with |xmax| the maximum admissible diaphragm excursion (the corresponding maximum velocity being |Vmax|=x|xmax|), the maximum electrical output power is

We ¼ ¼

1 2 1 RL 2 jIj RL ¼ ðBlÞ jV max j2 2 2 ðRe þ RL Þ2 1 RL 2 ðBlÞ x2 jxmax j2 2 ðRe þ RL Þ2

ð16Þ

sure amplitudes up to 10% of the mean pressure. The code performs 1D numerical integration of the momentum, continuity and energy equations through each segment of the acoustic network. These equations, considered in their linear-approximation (low amplitude) version, are:

dp jx q0 ¼ U dx Að1  f m Þ as far as the momentum equation is concerned,

  dU Ax ðc  1Þf k ¼ j 1 þ p q0 c2 dx ð1 þ es Þ   dT 0 ðf j  f m Þ U þb dx ð1  f m Þð1  PrÞð1 þ es Þ

dT 0 ¼ dx

RL ¼ Re

with

ð17Þ

that, when substituted in Eq. (16), provides the maximum electric power obtainable at the operation frequency x

ð18Þ

Finally, we identify the extreme operative conditions which could damage the alternator. These conditions derive from the constraints on maximum stroke, maximum current and maximum pressure difference imposed by the alternator technical characteristics. The limitation imposed by the maximum stroke |xmax| is

jxj ¼

jUj

x Ad

¼

jp1  p2 j

x Ad jZ a j

6 jxmax j

ð19Þ

which allows to determine, at each frequency, the maximum oscillation velocity and pressure difference across the diaphragm for which the displacement amplitude is lower than the maximum allowable value. The limitation imposed by the maximum current, |Imax| is

jIj ¼

ðBlÞjUj ðBlÞjp1  p2 j ¼ 6 jImax j jZ e þ RL jAd Ad jZ e þ RL jjZ a j

ð20Þ

which allows to determine, at each frequency, the maximum oscillation velocity and pressure difference across the diaphragm for which the current flowing in the coil is lower than the maximum allowable value. Maximum pressure difference |p1-p2| across the diaphragm depends essentially on the strength of the material used for its fabrication. Commercial loudspeakers have cones generally made of fragile (resin impregnated) paper, polypropylene, thermoplastic sheets, etc. So, although at the typical frequencies of thermocoustic devices (<500 Hz) it is reasonable to assume they behave as rigid pistons [18], a high pressure difference across them could lead to rupture and/or to mechanical instability. In applications integrating loudspeakers as LAs, pressure differences in the kPa range are recommended [20]

jp1  p2 j ¼ jZ a jjUj 6 jp1  p2 jmax  10 kPa

n h io ~ 1  T 0 bðf j ~f m Þ H  12 Re pU ð1þes Þð1þPrÞð1~f m Þ n o ~f m Þð1þes f =f Þ q0 cP jUj2 m j Im ~f m þ ðf j ð1þ  ðKA þ K s As Þ es Þð1þPrÞ 2Axð1PrÞj1f j2

ð24Þ

m

dH_ _ ¼ qðxÞ dx

ð25Þ

as far as the energy equation is concerned. In the above equations

2

1 ðBlÞ x2 jxmax j2 8 Re

ð23Þ

as far as the continuity equation is concerned, and

which shows that for achieving high power levels, without working at increasingly values of the frequency, loudspeakers characterized by high diaphragm excursions are preferable. Equation (16) maximizes for

W max ¼

ð22Þ

ð21Þ

To model the thermoacoustic engine and its coupling to the linear alternator the computer code DeltaEC (Design Environment for Low-Amplitude ThermoAcoustic Engines) [21], developed at the Los Alamos National Laboratory is used as design tool in the present work. The code integrates the linear theory of thermoacoustics, firstly formulated by Rott [22] and subsequently refined by Swift [1,2], which has demonstrated accurate precision for pres-

q0 is the mean density, c is the sound velocity, c is the ratio of isobaric to isochoric specific heats, T0 is the mean temperature, b is the thermal expansion coefficient, Pr is the Prandtl number, cP is the gas isobaric specific heat, K and Ks are the thermal conductivity of the gas and solid respectively, H is the enthalpy flux along the direction of acoustic vibration x, fj and fm are spatially averaged thermal and viscous functions depending on the geometry of the gas pores, A and As are the area of the transverse section open to gas flow and obstructed by the solid respectively, es is the ratio of specific heats per unit area of gas and solid and q is the heat exchanged with external thermal reservoirs along the direction normal to x per unit length in the heat exchangers. The solutions found for adjoining segments are then matched by imposing the continuity of pressure and volume flow rate at their junction. For further details the reader is addressed to Ref. [2]. Design strategy and choices In this section the design methodology applied for selecting the engine global requirements, the engine configuration, the loudspeaker to be used as a LA and its coupling typology to the TA engine is outlined. The target electric output power of the engine is We = 100 W. Selection of the working fluid and of the static pressure value High performance thermoacoustic engines work generally with helium (or binary gas mixtures of helium and other noble gases [1]) at high static pressures (up to 50 bar). These gases in fact, accomplish the requirements of a low Pr, that reduces viscous dissipation, a high K, that allow for larger pore dimensions of the stack/regenerator and heat exchangers (since it increases the thermal penetration depth dj , i.e., the length scale for heat diffusion along the direction normal to the particle acoustic oscillation), a high c, that increases the engine power density and a large c, that has a positive impact on the engine efficiency. A high mean pressure also increases the power density and, in general, the engine thermal and power levels, since they scale proportionally to the gas density. Considering these factors, engines operated with air at near atmospheric pressure are evidently not a good choice. However, from the perspective of working fluid cost and

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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availability and engine manufacturing requirements and size, the most straightforward choice is just the use of near atmospheric air. This choice, in fact, has the threefold advantage of avoiding the use of costly pressure vessels, eliminating the need of noble (or other exotic) gases (and associated costs) and reducing the engine size due to the relatively low sound velocity. Since this perspective fully meets the goals of the present study air at 1 bar pressure is selected as working medium. Selection of the loudspeaker According to the theory reported in Section ‘‘Theoretical modelling”, the key requirements for a loudspeaker to work at high transduction efficiencies are: (1) a low mechanical resistance Rm, (2) a low electrical resistance Re and (3) a high force factor Bl. A high stroke is also to be considered a favorable property since, for a fixed frequency, the electrical output power grows with the square of the diaphragm excursion, according to Eq. (16). On the basis of these criteria a number of commercially available loudspeakers suitable for TAEG applications have been individuated by different researchers. These devices are reported in Table 1 together with their technical specifications and the calculated value of the parameter gmax which, as previously discussed, constitutes a relevant figure of merit for loudspeakers working in reverse mode. In this work the B&C 6PS38 woofer (manufactured by B&C speakers) is selected as LA to be coupled to the thermoacoustic engine since it has the highest gmax (=72.5%), and a relatively small dimension (7 in. nominal diameter).

"

We ¼

2

1 jUj2 ðBlÞ RL 2 A2d ðRe þ RL Þ2 þ X 2e

# ð28Þ

we can substitute the first in the second for |U| and then resolve the resulting equation with respect to x. The result is

x¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 W e ðRe þ RL Þ2 t

ð29Þ

2

ðBlÞ RL jxmax j2  2 W e L2e

which allows to determine, for a given load resistance RL, the operation frequency compatible with the target output power and the maximum allowable stroke of the diaphragm (as well as the maximum admissible current flowing in the voice coil provided the condition We 6 0.5|Imax|2RL is verified). The pressure difference and volume flow rate generating the target power output are then derived by substituting this frequency in Eqs. (26) and (27). The x, |p1  p2| and |U| values, corresponding to the target output power We (=100 W), and the acoustic-to-electric conversion efficiency of the loudspeaker, gae, calculated through Eq. (12), are reported in Fig. 3 as a function of the load resistance RL. Making the choice of operating the loudspeaker at an acoustic-to-electric conversion efficiency of at least 70% the graph allows to determine the corresponding values of the operation frequency (f = 194.5 Hz), load resistance (RL = 19 X), pressure difference amplitude (|p1  p2| = 8.55 kPa) and diaphragm volume flow rate amplitude (|U| = 0.0968 m/s).

Selection of the operation frequency and of the load resistance The selection of the operation frequency is done on the basis of the target output power, We, and of the maximum diaphragm excursion |xmax|, where the LA is set to operate. This choice is motivated by the fact that we want the system to work preferably at not very high frequencies in order to limit thermoviscous dissipation (proportional to x2) and at the as low as possible amplitude of the pressure difference across the diaphragm, consistently with the guidelines of the previous section (|p1-p2|610 kPa). The last statement is justified on the basis of Eq. (19) that we rewrite in the form

jp1  p2 j ¼ x Ad jZ a ðxÞjjxmax j

ð26Þ

when taking into account that |Za(x)| is an increasing function of x at frequencies higher than its natural resonance frequency (75 Hz). Observing that at the operation frequency x the following two equations have to be verified

jUj ¼ Ad xjxmax j

ð27Þ

Fig. 3. The combinations of volume velocity, pressure difference and operation frequency values of the loudspeaker giving rise to an electrical output power of 100 W and the acoustic-to-electric conversion efficiency as a function of the load resistance.

Table 1 Specifications of loudspeakers selected for operation in thermoacoustic electricity generators as linear alternators. The parameter gmax is calculated through Eq. (15).

Rm (kg/s) m (kg) Km (N/m) Re (X) Bl (Tm) Le (H) f0 (Hz) xmax (mm) gmax (%) Ad (m2)

B&C 6PS38(a)

B&C 10NW64(b)

B&C 8BG51(c)

Monacor SPH-170C(d)

JL-audio 6w3v3-4(e)

B&C 8NW51(f)

0.562 0.014 3099.3 5.4 10.8 0.0006 75 ±6 72.5 0.0132

3.74 0.047 5282.16 5.2 17.5 0.00047 50 ±8 61.6 0.0320

0.95 0.035 3814 5.1 11.8 0.0005 52 ±6.5 69.2 0.022

0.55 0.01643 943 6 9.44 0.0008 38 ±4 68.1 0.0127

1.198 0.05 3468.67 3.476 9.28 0.0005 41.8 ±8.89 64.6 0.0112

4.97 0.028 6242 5.2 18.9 0.0004 74 ±6 59.1 0.022

(a) Yu et al. [11,12]; (b) Abdoulla et al. [23]; (c) Chen et al. [13]; (d) Olivier et al. [14]; (e) K. de Blok [15]; (f) Kang et al. [16].

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

Selection of the engine configuration A two-stage standing-wave engine configuration is chosen in the present study since from preliminary simulations it results that a single-stage engine is unable to deliver to the load the acoustic power required to produce the target electrical power, unless to consider very large systems. In this engine the two stages are placed at either sides of a long resonator and consist of a stack sandwiched between a hot heat exchanger (HHX) and an ambient heat exchanger (AHX). The resonator is closed at both ends and constitutes therefore a half-wavelenght (k/2) type resonator. It stores the acoustic power converted by each stack from the heat delivered by the HHXs and realizes a 90° out-of-phase relationship between pressure and velocity oscillations. This condition is required for efficient operation with stacks where the thermal contact with the gas is not perfect (having pores characterized by hydraulic diameters higher than the thermal penetration depth dj ). Selection of the alternator-engine coupling typology The impedance matching between the alternator and the engine is a crucial task of the system optimization procedure. Optimal matching means that the presence of the alternator does not significantly alter the structure of the acoustic field (resonance frequency, pressure and velocity distributions and their phase) which maximizes the engine performance. Likewise, the structure of the acoustic field at the transducer location should make it to absorb useful amounts of acoustic power converting it to electrical power at high efficiency. This demand high-impedance (or noncompliant) transducers, having high masses and suspension stiffness, to be preferably installed near high impedance regions of the acoustic field and, conversely, low-impedance (or ultracompliant) transducers, having low masses and suspension stiffness, to be preferably installed near low impedance regions of the acoustic field [1]. Standard loudspeakers are generally low-impedance type transducers. For this kind of devices the above requirements on impedance matching can be effectively met by a side branch arrangement which acts as an impedance matching transformer. In this configuration the alternator is placed at the end of a relatively long duct while the other end (the branch input section) is connected to the main engine trunk in sideway. The branch duct will comprise, in general, a narrow straight segment, a divergent cone, a wide straight segment which accommodates the loudspeaker and a box enclosing the back side of the loudspeaker (see Fig. 4). Geometrical dimensions of the branch should be selected in order the following two conditions be verified at the system working frequency: - The impedance at the branch input section should be sufficiently higher than the local trunk impedance to preserve the acoustic wave in the trunk from being distorted and the branch from absorbing too great amounts of power that could lead to unwanted high onset temperatures or, worse, to a not-starting engine. At the same time, the impedance should be not too large to allow the target acoustic power enter the branch with (as far as possible) low driving pressures (which reduces termoviscous dissipation on the resonator walls proportional to |p|2). So a trade-off between these two requirements has to be found. - The impedance at the other side of the branch, where the LA is installed, should be low to match the impedance of the ultracompliant transducer. Note that a regulation of the combined loudspeaker/box impedance can be done by exploiting the gas-spring effect caused by the box which is equivalent to an increment of the suspension stiffness of the loudspeaker given by

Fig. 4. The pressure and volumetric velocity distributions along the branch to the alternator for f = 194.5 Hz and RL = 19 X.

K 0m ¼

cP0 A2d Vb

ð30Þ

Vb being the box volume. According to the statements of the previous section, this effect could be favorably used to bring the phase angle between the acoustic pressures acting on the front side of the diaphragm, p1, and its volume velocity, U, near zero in order to enhance the acoustic power absorbed by the loudspeaker. Note that in this case the acoustic power irradiated in the box (1/2Re ˜ }) is negligible since it simply corresponds to the power dissi{p2U pated on the box surfaces by thermal hysteresis. The box volume, however, also affects the input impedance of the branch and this leads again to the issues discussed at the previous point. So also in this case a trade-off between different requirements has to be found. Selecting the branch configuration in order to create a near k/4 acoustic field distribution (k being the sound wavelength) can fulfill both these conditions. This is clearly illustrated in Fig. 4 reporting the pressure and velocity distributions along the branch calculated by DeltaEC for the target operation frequency (f = 194.5 Hz) when the pressure amplitude at the input section is about 10 kPa. Geometrical dimensions are those relative to the optimal configuration resulting from the analysis of the present and next section. For this configuration the loudspeaker extracts 100 W of electric power when 153.6 W of acoustic power enter the connecting tee of the branch. The calculation indicates that the tee causes a power dissipation of about 2.4 W so the acoustic power flowing into the branch amounts to 151.2 W. About 3 W of acoustic power are then dissipated in the straight segments of the branch (including the box) while 4 W are dissipated in the conical segment (3 W as minor loss). The dissipation in the tee, determined by assuming a minor loss factor K = 1 for the branch-flow and K = 0.2 for the trunk-flow, may appear quite low. But, it must be considered that the volume velocity at the input section of the branch (a near velocity node due to the k/4 acoustic field distribution) is as low as |Ubr| = 0.041 m3/s. So, since the viscous resistance associated to the minor loss (see Ref. [2] Eq. (7.36)) amounts to 647.2 Pas/m3, we obtain that the power dissipation affecting the branch-part of the tee is equal to (1/2) Rm |Ubr|2  0.6 W. The power dissipation affecting the thrunkpart of the tee results slightly higher (
1.8 W). It must be said, anyway, that these numbers have to be regarded as a rough estimate, since the application of steady flow data for quantifying

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

such minor losses under oscillating flow regime is not fully justified [2]. In the optimization procedure the box has been modelled by a compliance and the its volume adjusted to bring the resonance frequency of the composed loudspeaker-box system near the target operation frequency. For the selected box volume (equal to 0.0015 m3 after subtraction of the loudspeaker volume), the resonance frequency of the loudspeaker-box system is about 187.6 Hz and the phase angle between p1 and U is near 10°. Dotted lines are the pressure and velocity distribution inside the box obtained by modelling it as a short closed duct. The plot clearly evidences as the pressure exhibits a discontinuity (|p1|-|p2|
-4 kPa) across the transducer caused by the loudspeaker impedance (a series impedance from the circuital analogy point of view [2]) and as the volume velocity (continuous across the transducer) falls quickly down till to vanish on the rigid surface of the box. Results and discussion The TAEG model conceived in this study is depicted schematically in Fig. 5. The geometrical dimensions of its components, deriving from the optimization procedure below described, are given in Table 2 (excluding the branch whose dimensions are reported in Fig. 4). The resonator is not perfectly straight but presents a 11 cm diameter middle section connected to 13 cm diameter side sections through two conical ducts each one 16 cm long and with 7° taper angle. This shape serves to suppress the growth of non-linear effects associated to the harmonics generation phenomenon [24]. The branch with the LA is installed immediately after the AHX of the first stage. A stub (a side-branch closed pipe), whose function will be explained below, is placed in a symmetrical position just before the AHX of the second stage. The large sideducts (13 cm diameter) considered in the design of the engine are required to accommodate stacks able to match the target output power under reasonable values of the conversion efficiency and temperature span. It should be noted, however, that although the performance of the engine increases with the stack diameter (the ratio of generated to dissipated acoustic power scales with the stack radius) large cross sections unavoidably entail higher manufacturing costs. The optimization procedure involves the simulation of the TAEG performance by varying a large number of geometrical parameters such as the diameter and length of the ducts (the resonator, the

7

branch and the stub), the position (xs) and length (Ls) of the stacks, the hydraulic diameter (rh) of the stack pores, the length (LHHX) of the HHX fins, the length (LAHX) of the AHX fins etc. Specifically, with the aim of a subsequent technical implementation, the modelled stack is of the square-pore type with rh values typical of commercially available ceramic substrates for automotive applications. The modelled HXs are of the fin-and-tube type. The fin spacing, 2y0, is set equal to 2dj or to 1 mm depending on weather 2dj > 1 mm or 2dj 6 1 mm to avoid much too specific manufacturing requisites and relative high costs. The HHX temperature (TH) is assumed at maximum equal to 800 K for not exceeding the tolerances of typical materials used for its construction. The AHX is assumed to be made of copper and its temperature (TA) to be maintained near 295 K by a flow of cooling water. The total heat input of the engine (QH) is simulated to be around 2 kW so, considering that the target output power is fixed to 100 W, we are searching for operative conditions giving rise to an overall thermal-to-electric conversion efficiency of at least 5% for temperature differences between hot and ambient HXs not exceeding 505 K. In the initial phase of the optimization procedure an engine configuration without the stub is considered. The procedure starts from the selection of the dimension of the stack pores. For optimal heat-sound energy conversion operated by particles oscillating under standing-wave phasing between pressure and motion the ratio of the hydraulic radius rh to the thermal penetration dj depth should fall in the range 1 6 rh/dj 6 2 [25,26]. At the design operation frequency (194.5 Hz) and at the temperature of 547.5 K (intermediate between the design TH and TA) it results dj = 0.254 mm so rh should be comprised, according to the above rule, in the range 0.254 6 rh/ 6 0.508. Modelling by DeltaEC different square-pore type commercially available ceramic substrates, it is found (results not shown) that the substrate characterized by rh = 0.2425 mm and by a porosity of 88.1% is the best performing and, consequently, is retained in the model. The results concerning the optimization of the stack position and length are shown in Fig. 6 where the thermal-to-acoustic efficiency of the engine (gha), evaluated as the ratio of the acoustic power Wa delivered to the load (i.e. absorbed by the alternator) over the heat flow sustained by the two HHXs, is reported as a function of the stack length at selected stack locations. In each simulation TH = 800 K, TA  295 K, QH  1.9 kW, f = 194.5 Hz and the length of the HXs along the resonator axis matches the local peak-to-peak oscillation amplitude (2|n| = 2|U|/Ax) [1]. Further-

Fig. 5. Schematic of the two-stage standing-wave TAEG model.

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

Table 2 Geometrical parameters of the TAEG model and operation conditions. Operation conditions Working fluid Operation frequency Static pressure

air 194.5 Hz 1 bar

Resonator

Length

Diameter

Diameter

Taper angle

Hot duct Narrow duct Conical duct

4.5 cm 37.3 16 cm

13 cm 11 cm 11 cm

  13 cm

  7°

Stack

Length

Diameter

Porosity

Pore hydraulic radius

5 cm

13 cm

88.1%

0.2425 mm

Length

Diameter

Porosity

Fin spacing

1 cm

13 cm

58%

1 mm

Length

Diameter

Porosity

Fin spacing

1.5 cm

13 cm

58%

1 mm

Hot heat exchanger

Ambient heat exchanger

more, the diameter of the branch (the narrow duct soldered to the trunk) is adjusted in each run to bring TH to the design level (800 K). All curves exhibit a peak whose location shifts towards lower Ls values as the distance xs (of the stack centre) from the closed end of the resonator is reduced. This behavior can be explained observing that in all simulations QH is fixed so the gha trend parallels the Wa trend or, ultimately, the acoustic power produced by the stacks. This power is affected in different ways when the stack length is reduced at a given position xs. It is forced, in fact, both to increase, as an effect of the higher temperature gradient and of the lower surface affected by viscous dissipation, and to decreases, simply as a consequence of being proportional to Ls. The result of these two competing effects are the peaks we observe in Fig. 6. As for the shift of the peaks location, it reflects the circumstance that when xs is reduced the stack approaches the pressure antinode and velocity node of the standing wave. As a consequence, the stack performance increases since the produced power grows with the square of the pressure amplitude while the viscous dissipation (proportional to |U|2) decreases. The graph shows that gha maximizes to 7.52% for LS  5 cm and xS  8 cm. Consequently, these values are retained in the model. Subsequent refinements of the HXs length are performed by an analogue procedure starting from the optimal configuration (the optimal Ls and xs values) previously found. The results are shown

in Fig. 7 where the gha efficiency is plotted against the length of one HX when the length of the other is fixed to an arbitrary value. The optimal lengths LHHX  1 cm and LAHX  1.5 cm, corresponding to the maxima locations, are approximately equal to 60% of the 2|n| local values. It must be said that this result could be affected by an error of about a factor of 2 [21] since DeltaEC estimates the heat transfer coefficients (h) at the HXS on the basis of a simple boundary layer model (h
K/dj) [27]. Nevertheless, the above lengths are retained in the model since investigations performed by 2D computational models [28,29] indicate that the optimal length of the HXs could be lower than the peak-to-peak oscillation amplitude. After correcting the HXs length by the above values the gha efficiency is found to increases from 7.52 to 8.48%. Simulations reveal that the branch slightly perturbs the acoustic field inside the resonator, although its impedance (
105 Pa s/ m3) is around one order of magnitude higher than the local impedance of the resonator (
104 Pa s/m3). This makes the second stage on the right to produce less acoustic power than the left-one as a consequence of absorbing less heat under a slightly slower temperature span. In the last configuration discussed, for example, setting TH(1st stg.) = TH(2nd stg.) = 800 K, an output power of 100 W is obtained with gha = 8.48% for QH(1st stg.) = 890.7 W, QH(2nd stg.) = 801.5 W, TA(1st stg.) = 295 K and TA(2nd stg.) = 297.6 K. While, setting TA(1st stg.) = TA(2nd stg.) = 295 K, the target output power is achieved (results not shown) with gha = 7.91% for QH(1st

Fig. 6. The effect of the stack length and position on the thermal-to-acoustic efficiency.

Fig. 7. The effect of the heat exchangers length on the thermal-to-acoustic efficiency.

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

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stg.) = 901.3 W, QH(2nd stg.) = 914.2 W, TH(1st stg.) = 717 K and TH(2nd stg.) = 800 K. To restore a perfect symmetry between the two stages a stub (a side-branch closed pipe of adjustable length) is introduced just before the AHX of the second stage. The stub technique is often practiced in thermoacoustic technology [12,16] to induce changes in the local acoustic impedance of an acoustic network according to the continuity law

1 1 1 ¼ þ Z 1 Z stub Z 2

ð31Þ

(Z1 and Z2 being the acoustic impedance immediately before and after the stub). Simulations reveal that, by a proper selection of the stub length (
74 cm) and diameter (
4.5 cm), Zstub can be tuned to compensate for the perturbation caused by the branch in order to bring the two stages in the same operative conditions (same temperature differences and same energy fluxes.). After introducing the stab, in fact, the target output power is achieved with gha  8.2% for QH(1st stg.) = QH(2nd stg.) = 877.9 W, TH(1st stg.) = TH(2nd stg.) = 800 K and TA(1st stg.) = TA(2nd stg.) = 295 K. These data suggest that, although the stub allows to enhance the acoustic power produced by the second stage to the level of the firstone, it leaves the overall engine efficiency practically unaffected since it causes additional losses which cancel out the acoustic power gain of the second stage. There could be therefore no benefit in connecting a stub because it introduces extra costs without increasing the engine performance. Neverthless, this components is hereafter included in the model for the following two reasons:  it allows an active control of the performance of the second stage;  it allows to (slightly) increase the performance level of the engine configured to work with TA(1st stg.) = TA(2nd stg.) = 295 K (a setting more complying to usual experimental conditions). The simulation results that will be discussed below include therefore the stub component whose length is systematically adjusted to makes the two stages to perform identically. Fig. 8 shows the acoustic field distribution along the optimized device when the system is driven by a heat flow of 1756 W with TH = 800 K and TA = 295 K to produce the target (maximum) electrical output power (100 W). The distribution is that typical of a k/2 resonator, with pressure antinodes at the solid terminations and a velocity maxima at the centre. The pressure amplitude is continuous along the resonator while the volumetric velocity and the impedance exhibit sharp variations in correspondence of the branch to the LA and of the stub. It is evident as the stub is able to ‘‘correct” the impedance drop caused by the branch. The branch also introduces a sharp increase of the phase angle between pressure and velocity from / = 86.2° on the left of the branch to / = 95.4° on its right. This phase shift simply reflects the fact that the acoustic power entering the branch is composed by the fraction which comes from the left stage and flows in the positive x direction (for which / < 90°) and by the fraction which comes from the right stage and flows in the negative x direction (for which 90° < / < 180°). This circumstance is more clearly evidenced by the acoustic power flow (Ea) distribution shown in the same figure where the sudden change of Ea at the branch location equals the power entering the branch. From the graph it can be deduced that a net acoustic power of around 222 W is generated by the two stacks, so each one works at a thermal-to-acoustic efficiency of 12.6%. Of this amount about 42 W are dissipated in the HXs, about 22 W are dissipated on the resonator and stub walls and about 7.4 W are dissipated as minor losses in the conical segments and in the tees. The remaining 151 W flow into the branch. Here the LA absorbs about 144 W and delivers 100 W to the electrical load with gae  0.7. In this operative condition the gha and overall ghe efficiencies of the engine are therefore 8.2 and 5.7% respectively.

Fig. 8. The pressure amplitude |p|, volumetric flow rate |U|, acoustic impedance |Za|, phase angle h and acoustic power flow Ea spatial distributions along the TAEG model for f = 194.5 Hz, RL = 19 X, QH = 1724 W and TH = 800 K.

Fig. 9. The electric power output and the overall thermal-to-electric conversion efficiency as a function of the heating temperature.

These efficiencies may appear quite low but are typical for near atmospheric air operated thermoacoustic devices [15]. They are comparable, anyway, to the theoretical performances evaluated for the travelling wave TAEGs designed in [11,12] (ghe
2.4%), [23] (ghe
6%), [30] (ghe
3.5%) and about half the one of the TAEG designed in [16] (ghe
10.9%), but in the last case helium at 18 bar static pressure was simulated and the temperature of the HHX was 850 K (compared to 800 K of the present study). The effect of the heat input on the engine performance is shown in Fig. 9 where the electrical output power, We, and the overall ghe conversion efficiency are reported as a function of the heating temperature TH (which varies as a consequence of the heat input change). In Fig. 10 the corresponding variations of the square of the pressure difference across the diaphragm, |p1  p2|2, and of its stroke, 2|x|, are also shown. In these simulations the input heat varies from 100 to 1700 W by steps of 100 W. From the graphs it can be observed as We increases linearly with TH as |p1  p2|2 (proportional to QH) does according to Eq. (11) and that the onset temperature of the engine falls around 350 °C. The behavior of ghe vs TH simply reflects the growth of the stacks conversion efficiency with TH. When the temperature of the HHX reaches 710 K the LA extracts 50 W of electrical power with the design gae efficiency (70%) while the overall ghe efficiency amounts to around 5.4%. At

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

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A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

Fig. 10. The square of the pressure difference across the LA and its stroke as a function of the heating temperature. Fig. 12. The thermal-to-acoustic conversion efficiency gha as a function of the load resistance RL at selected values of the input heat flux QH.

this operative point the amplitude of the pressure difference across the diagram and its stroke are 5.9 kPa and 8.7 mm respectively. When TH = 800 K the engine performance maximizes to the target (maximum) power and efficiency levels previously reported and the stroke of the diaphragm matches exactly the maximum allowable value (12 mm). In Fig. 11 the effect of the load resistance RL on the electrical power output is shown at selected values of the input heat flux QH. Each We curve exhibits a peak which corresponds to the optimal load resistance that maximizes the electric output power obtainable from a given heat input. These optimal load resistance values are higher than the one that maximizes the power output of the linear alternator (RL = Re = 5.4 X) and result a decreasing function of the input heat, ranging from around 19 X at QH = 200 W to 14.5 X at QH = 700 W. This behavior can be explained considering that the power output of the engine can be expressed in the form We = gheQH = ghagaeQH that evidences that since QH is fixed the dependence of We (as well as of ghe) on RL is given by the combined factor ghagae. The dependence of gae on RL, expressed by Eq. (12) and shown in Fig. 3, is determined uniquely by the alternator parameters while the dependence of gha on RL (shown in Fig. 12 at selected QH values) is influenced both by the alternator parameters and by the thermoacoustic engine. It must be considered, in fact, that the load RL modifies the input acoustic impedance of the alternator-box system (Zab) which, in

Fig. 11. The electrical output power We as a function of the load resistance RL at selected values of the input heat flux QH.

turn, modifies the input impedance of the branch Zbr. For an alternator installed at the end of a straight branch of length L, for example, this dependence is expressed analytically by the relation:

Z br ¼ Z c

Z ab ðRL Þ=Z c þ j tanðkLÞ 1 þ jZ ab ðRL Þ=Z c tanðkLÞ

ð32Þ

where k = 2pk and Zc = q0c/Abr (Abr being the cross section area of the duct) in the case of inviscid flow and where

Z ab ¼

1

"

A2d þj

2

Rm þ (

1

A2d

"

ðBlÞ ðRe þ RL Þ

#

ðRe þ RL Þ2 þ X 2e

Xm 

2

ðBlÞ X e ðRe þ RL Þ2 þ X 2e

# 

1 x Cb

) ð33Þ

and Cb = Vb/cP0. For the operative conditions considered in the present study, it is found that a decrease of RL is accompanied by a decrease of the branch impedance so that more power enters the branch for a given heat input. This also causes an increase of the heating temperature and, in turn, of the stack efficiency in converting heat to acoustic power. This effect can be observed in Fig. 13 (a), (b) and (c) where the acoustic power Wbr entering the branch, the

Fig. 13. The acoustic power Wbr entering the branch, the pressure amplitude |p1| in front of the alternator, the volume velocity amplitude |U| of the alternator diaphragm and the phase angle between p1 and U as a function of RL for QH = 600 W.

Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011

A. Piccolo / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

pressure amplitude |p1| in front of the alternator, the volume velocity amplitude |U| at the alternator diaphragm and the phase angle between p1 and U are reported as a function of RL for QH = 600 W. The plots show how Wbr and |p1| increase monotonically when RL is decreased. As a consequence, more power is available to the linear alternator for being absorbed. This explains the monotonic increase of gha when RL is decreased. At relatively high RL values this increase combines with the analogue of gae enforcing the observed increase in We and ghe. However, a decrease of RL also produces an increase of the input impedance of the LA/box system given by Eq. (33). This increased impedance reduces the oscillating volume flow rate of the diaphragm surface as shown in Fig. 13 (b). This effect, which becomes relevant at low RL values, counteracts evidently the increase of Wa associated to lower Zbr values. This explain the decrease (or lower growth) of the gha curves when RL is reduced at low values. In any case, at sufficiently low RL values (near the maximum) the sharp drop of gae dominates over gha giving rise to a decrease of the extracted electrical power and of the overall thermal-to-electrical conversion efficiency. This complex behavior, where acoustical, electrical and thermal physical quantities interacts together, is further complicated by the slight increase of the resonance frequency with RL (result not shown).

Conclusions This paper describes the design methodology of a thermoacoustic electricity generator using a standard loudspeaker as linear alternator. Based on a target electrical output power of 100 W, a two-stage standing wave prototype working in half-wavelenght mode and coupled to the LA by a side-branch arrangement is conceived. The TAEG model, intended for low-power and low-cost applications, uses air at atmospheric pressure as working fluid and operates at a frequency of 194.5 Hz. Simulations of the model-prototype performance by the specialized design tool DeltaEC reveal that at the heating temperature of 527 °C the target electrical output power is extracted by the loudspeaker with an acoustic-to-electric efficiency of around 70% while the overall thermal-to-electrical conversion efficiency of the engine amounts to 5.7%. These values are comparable to the ones theorized for analogue TAEGs of the travelling wave type working with air at near atmospheric pressure. This result suggests that in applications involving the use of loudspeakers as linear alternators and air at near atmospheric pressure as working fluid, electricity generators of the standing wave type could perform comparably to their travelling-wave counterparts. This is likely due to a simpler design of the alternator-engine coupling and a simpler and more compact configuration. The construction of the TAEG model conceived in this work is actually in progress to clarify some issues which needs the integration of experimental investigation. These issues concern the use of HXs shorter than the gas displacement amplitude, the mechanical stability of loudspeakers working at pressure differences near 10 kPa and the effectiveness of the stub technique in improving the global performance of the engine.

Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Please cite this article in press as: Piccolo A. Design issues and performance analysis of a two-stage standing wave thermoacoustic electricity generator. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.10.011