Numerical study of flow and energy fields in thermoacoustic couples ofnon-zero thickness

L. Zoontjens a,∗, C.Q. Howard a, A.C. Zander a, B.S. Cazzolato a

aSchool of Mechanical Engineering, The University of Adelaide, South Australia

Abstract

A limitation in many previous numerical studies of thermoacoustic couples has been the use of stack plates which are of zerothickness. In this study, a system for modelling thermoacoustic couples of non-zero thickness is presented and implemented usinga commercial CFD code. The effect of increased drive-ratio and plate thickness upon the time-average heat transfer through thestack material is investigated. Results indicate that the plate thickness strongly controls the generation of vortices outside thestack region, perturbing the flow structure and heat flux distribution at the extremities of the plate. An increase in plate thicknessis also shown to improve the spatial integral of the total heat transfer rate but at the expense of increased entropy generation.

Key words: Thermoacoustics, Heat, Sound, Thermoacoustic couples, Heat exchangers, Acoustic streaming

1. Introduction

In a time of increased concern over environmental im-pact and operating costs of refrigeration systems, ther-moacoustic refrigeration technologies are appearing moreand more attractive for commercialisation. These de-vices possess efficiencies comparable to convential vapour-compression systems, and operate without adverse envi-ronmental impact, significant maintenance requirementsor high construction costs. Thermoacoustic refrigerationunits can be driven using heat as a direct input energysource, and as such are appealing for waste energy recoveryfrom heat sources such as hot exhaust gas streams from anexisting thermodynamic processes or solar collectors.

Detailed information regarding the design and opera-tion of thermoacoustic systems is provided by Swift [1].When optimising new thermoacoustic devices, most de-signs make use of linear order prediction tools such as thecomputer program DeltaE [2] or analytical design guides[3]. There are many publications demonstrating the useful-ness of these methods in the design and modelling of ther-moacoustic devices [1, 4, 5]. However at higher operatingstates, the accuracy of linear prediction methods deterio-

∗ Corresponding author. Tel. +61 8 8303 5460.Email address: luke.zoontjens@adelaide.edu.au (L. Zoontjens).

rates. Higher-order numerical models may therefore assistin the understanding of various loss mechanisms during op-eration. Because of the intimate interaction between pres-sure, velocity, temperature and their derivatives, numericalmodelling of thermoacoustic phenomena requires simulta-neous conservation of momentum, continuity and energyof the working fluid and surrounding solid structures in anunsteady formulation.

Ishikawa and Mee [6] used full two-dimensional Navier-Stokes equations to model the heat transport effects in whatis referred to as a ‘thermoacoustic couple’ [7]. Thermoa-coustic couples are best described as a short stack (beingmuch shorter than the acoustic wavelength) consisting ofonly a few parallel plates which can be placed at any axialposition inside the resonator duct. Modelling of a thermoa-coustic couple has typically involved consideration of onlyone side of a single, infinitely wide stack plate as a bound-ary to a two-dimensional computational domain.

Previous studies of thermoacoustic couples have consid-ered plates of non-zero thickness only. This study aims tobuild upon exiting knowledge by introducing a system ofmodelling thermoacoustic couples of finite thickness andimplementing a numerical code to investigate the effect ofincreased drive-ratio and plate thickness upon the time-average heat transfer through the stack material.

In Section 2, the numerical model, operating conditions

Preprint submitted to Elsevier 16 October 2006

Table 1Nomenclature

English letters s entropy, J/kgK

S total system entropy, J/kgK

a gas sound speed, m/s sgen rate of entropy generation per unit volume, W/m3K

A cross-sectional area, m2 Tk gas temperature, K

BR blockage ratio Tm mean temperature, K

cpk gas heat capacity, J/kgK Ts stack temperature, K

cps plate material heat capacity, J/kgK T0 mean oscillatory gas temperature, K

DR drive-ratio ts time step

ediss rate of energy dissipation, W |u1| first-order acoustic velocity amplitude, Pa

f frequency, Hz, also thermal function |〈u1〉| mean acoustic velocity amplitude, m/s

〈hhx〉t time-average heat flux over heat exchanger boundary, W/m2 |U1| first-order volumetric flow rate amplitude, m3/s

〈hx〉t time-average heat flux in the x direction, W/m2 x axial co-ordinate/dimension, m

〈hy〉t time-average heat flux in the y direction, W/m2 x′ axial distance from centre of resonator, m

k gas wavenumber, m−1 y transverse co-ordinate/dimension, m

ks plate thermal conductivity, W/mK yhxsf y co-ordinate of plate surface, m

k0 gas thermal conductivity, W/mK y0 stack plate half-spacing, m

LA axial length of subdomain ‘A’, m yx stack plate half-thickness, m

LB axial length of subdomain ‘B’, m

LCV axial length of control volume, m Greek letters

LS stack plate length, m

M Mach number γ ratio of specific heats

Ma acoustic Mach number δkm mean thermal penetration depth, m

nx number of mesh grid intervals along edge in x direction δκ thermal penetration depth, m

ny number of mesh grid intervals along edge in y direction δυ viscous penetration depth, m

pA pressure amplitude at pressure antinode, Pa ∆Tk,hx axial gas temperature difference across heat exchanger, K

pm mean operating pressure, Pa ∆x mesh interval spacing in x direction, m

|p1| first-order acoustic pressure amplitude, Pa ∆y mesh interval spacing in y direction, m

Pr Prandtl number λ wavelength, m

q surface heat flux, W/m2 ρ density, kg/m3

r internal radius, m ω angular frequency (=2πf), rad/s

Rs streaming (oscillatory) Reynolds number µ dynamic viscosity, kg/ms

Rcs critical streaming Reynolds number ρs density of the stack material, kg/m3

t time, s υ kinematic viscosity, m2/s

T0 ambient temperature, K ξc normalised stack centre position

and performance characteristics considered are introduced.By considering the basic heat transfer distributions andflow structures about finite thickness thermoacoustic cou-ples, new insight from several different aspects of this fieldis presented in Section 3. In Section 3.1 the results are pre-sented in terms of the flow field perturbed by a rigid flowimpedance, and the influence of plate thickness upon theheat transfer distribution is reported in Section 3.2.

1.1. Heat fluxes in thermoacoustic couples

Cao et al. [8] and subsequent work by Ishikawa and Mee[6] found that the heat flux in thermoacoustic stacks wasconcentrated at the edges of the stack, and that a time-averaged non-zero velocity profile existed for particles lo-cated at a quarter of the interplate spacing (i.e. 0.5y0) fromthe plate edges. They suggest that these effects are part ofa large scale, low frequency vortex just outside the plate

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edges. However, Ishikawa and Mee note that their com-putational domain is unable to fully model this phenom-enon or account for acoustic streaming at the walls, sincetheir model only considers a small region between the stackplates from a two-dimensional cross-sectional viewpoint.

Although thermoacoustic couples with zero thicknesssuch as those modelled by Ishikawa and Mee [6] offer someinsight into thermoacoustic heat transport mechanisms,they do not represent the flow impedance resulting fromthe physical presence of the finite thickness parallel platestacks, let alone other stack configurations such as pinarray or rectangular pore stacks. Furthermore, despite thework of Ishikawa and Mee [6] being a significant extensionof the work of Cao et al., there is still no experimental datato directly support the simulation results of either researchgroup.

Schneider et al. [9] numerically evaluated the heat fluxthrough the surface of the heat exchangers over an acousticoscillation for a computational domain similar to Cao etal.. In addition, heat exchanger plates parallel to the stackwere included. However, other important aspects, such assignificant Mach number flows, plate thickness, variationsin plate spacing and variations in thermal and viscous pen-etration depths were not considered.

Piccolo and Pistone [10] expanded on the work ofIshikawa and Mee [6] and Mozurkewich [11] by evaluatingthe transverse heat transfer distribution in a thermoa-coustic couple, using linear ‘short-stack’ thermoacousticapproximations modified with energy conservation consid-erations within the stack region. Only fluid within the stackregion was considered, with the computational domain ex-tending horizontally from plate end to end and verticallyfrom the plate mid-section to the plate mid-spacing. As the‘short stack’ analysis assumes that the plate length is muchlonger than the acoustic wavelength and the plate is not adisturbance to the acoustic field, the axial pressure and ve-locity distribution was considered by Piccolo and Pistoneto be constant and unperturbed by the flow constrictionbetween the stack plates. It should be noted however, thatPiccolo and Pistone increased the imposed pressure andvelocity through the stack region by considering the block-age ratio of the stack and maintaining continous volumevelocity. Density was also considered constant. The platewas considered to be thermally ‘isolated’, in that axialheat transfer between the stack and fluid was prevented,and the transverse heat flux integral over the plate sur-face was zero. In effect, any heat transferred through theplate surface at one end of the plate was returned at theother end. The thermal conduction within the plate wasmodelled using a thermal conductivity of ks=10W/mK,however the basis for which ks was set is not clear.

The time-average transverse heat flux determined by Pic-colo and Pistone [10, Run 1] showed good agreement withthat presented by Cao et al. [8, Run 2] and that of Ishikawaand Mee [6, Run 7]. All three runs solved a case withnear-identical stack location and geometry using Heliumat a mean pressure of pm=10kPa, an operating frequency

of f=100Hz and a relatively low drive-ratio of DR=1.7%.However, all three studies used a different computationaldomain: the domain considered by Cao et al. extended pasteach end of the stack plates; Ishikawa and Mee extendedthe model to the nearby end wall, and the fluid region mod-elled by Piccolo and Pistone was only that within the stackplate region. Regardless of the difference in solution do-mains, the similarity in results is expected, since in all threestudies, the stack plates were considered to be of zero thick-ness thus representing zero impedance to the acoustic andflow structures and the local flow conditions at the platesurface were effectively the same.

1.2. Thermal distortion effects

‘Temperature distortion’ is a significant non-linear effectthat has been observed in numerical studies by Marx andBlanc-Benon [12, 13] at drive-ratios above 3%. Within thestack region, it was predicted that with increasing Machnumber, the ratio of oscillating gas temperature to ambienttemperature reaches a ‘pseudo saturation’ point and thetemperature temporal waveforms become more and moreinharmonic. Marx and Blanc-Benon argue that because thetemporal variations at a point distant from the stack didnot distort with higher Mach number flows, the effect mustresult from interaction with the stack plates and not non-linear acoustic effects.

The model used by Marx and Blanc-Benon [12] in 2004was similar to that of Ishikawa and Mee [6] in that thecomputational domain was one half wavelength high andextended beyond an isothermal zero-thickness plate to ahard reflective end. However, the model also considered anadditional plate located near each end of the stack, rep-resenting the hot and cold heat exchangers used in prac-tical applications. The model used air at a mean pressureof 100kPa, which prevents any immediate comparison withthe Ishikawa and Mee model which used helium at 10kPa.The 2004 Marx and Blanc-Benon model also used an oper-ating frequency of 20kHz, far higher than the 100Hz usedby Ishikawa and Mee in 2002.

Interestingly, in a later study by Marx and Blanc-Benon[13], a revised computational domain was used which issimpler and closer to that used by Ishikawa and Mee, inthat the two heat exchanger plates were omitted and thelength of the stack plate retained was set to λ/40 as perthe majority of the test cases Ishikawa and Mee considered.Regardless, in this study Marx and Blanc-Benon concludethat for high drive-ratios, temperature distortion effects aremost likely to occur throughout stack regions located nearvelocity antinodes and with stack lengths less than fourparticle displacements long.

With sufficient cause to believe that high drive-ratioinduced temperature distortions could explain the sig-nificant inaccuracies of linear thermoacoustic theory forpressure amplitudes above 1% mean pressure, Marx andBlanc-Benon [14] used the model from then-recent work

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[13] to offer comparisons in prediction accuracy with con-ventional, linear models developed from over twenty yearsof quantitative thermoacoustic research [15, 16]. The com-parison found that whilst non-linear effects could explaindiscrepancies between linear and non-linear models at highdrive-ratios or flowrates, temperature distortion effectswere present for all flow ranges of interest.

Conventional linear formulations for stacks assume a con-stant temperature gradient in the direction of gas oscil-lation: the non-linear numerical model results presentedby Marx and Blanc-Benon [14] showed a tapering effectat the stack ends. Although the temperature gradient inthe stack was essentially the same between each model,the tapering effect meant that the linear model predictedan overall stack end temperature difference greater thanthe non-linear model. Linear models also assume the stackand adjoining gas temperature gradients to be equal, how-ever Marx and Blanc-Benon [14] argue that because of thegreater heat transfer at each end of the stack, this assump-tion is not correct. To account for these differences, Marxand Blanc-Benon proposed correction terms to the linearmodels to account for such temperature effects, however itis noted that these correction terms are offered in the ab-sence of experimental findings.

1.3. Flow disturbances and streaming withinthermoacoustic stacks

Limitations in the accuracy of linear approximationsled to increased study of 2nd order flow effects in ther-moacoustic systems. A form of acoustic streaming termed‘Rayleigh streaming’ is described by Bailliet et al. [17] asa “vortex-like” streaming (non-zero mean velocity) withinthe Stokes fluid boundary layer at the pipe and stackwall surfaces. For standing wave devices where the meangas flow rate is zero, dissipative Rayleigh streaming atthese fluid-solid interfaces has a significant effect upon theperformance of the device [16, p183].

Worlikar and Knio [18] used a two-dimensional formula-tion of the Navier-Stokes equations, to model the unsteadyflow of the working gas in the vicinity of two parallel ther-moacoustic couples. Different plate thicknesses were inves-tigated and in each case were modelled as rectangular incross section. Using what was termed a ‘streamfunction’distribution (which might also be referred to as an approx-imate vorticity field distribution), Worlikar et al. showedthat the generation of vortices at each edge of the stackplates due to the gas oscillations appeared symmetric aboutthe midplane of the stacks over a full oscillation. The samestudy also considered dissipative mechanisms in regard tothe effect of varying blockage ratio BR, stack position andthe ratio of plate length LS to plate spacing 2y0. The largestaspect ratio considered by Worlikar et al. was LS/2y0=6.66(acoustic wavelength not reported), which may be consid-ered too low for comparison with realised parallel platestacks. For example, the heat pump stack used in the mod-

ified STAR device [19] equated to an LS/2y0 value of≈411.Each stack used in the “flow-through” refrigerator of Reidand Swift [20] used an LS/2y0 of ≈188, and the conserv-ative design of a parallel plate stack based upon the con-ditions simluated by Ishikawa and Mee [6] would lead toLS/2y0 in excess of 150.

Further to this, Worlikar et al. [18] did not considerdrive-ratios above 1%. However, in a follow-up article bythe same authors [21], consideration of the drive-ratio wasextended to 2%, in addition to approximations regardingthermal conduction in the stack plates. It should be notedthat drive-ratios in excess of 3% are often associated withpractical thermoacoustic devices. As the ‘accuracy limit’for linear-order thermoacoustic approximations is also con-sidered to be DR=3% [19], DR values higher than 3% areof most interest to those considering numerical studies ofsuch devices.

Bailliet et al. [17] derived a series of analytical expres-sions for Rayleigh streaming effects in standing wave de-vices, using a series of two-dimensional Navier-Stokes equa-tions to evaluate the second-order components of pressuregradient, velocity and steady state mass-flux across a stack.While the work of Bailliet et al. is useful for calculating themagnitude of acoustic streaming for parallel plate stacks,analytical consideration of a variable cross-section as anacoustic streaming suppression measure would have beenuseful. Bailliet et al. only considered time-independent,steady-state operation within the stack; the effect of theblockages imposed by the stack wall thickness do not seemto be accounted for, and linear thermal gradients were as-sumed across the stack length [17].

Hamilton et al. [22] comments that because the work ofBailliet et al. used a Eulerian streaming velocity, it did notfully provide for mass transport effects which Hamilton etal. considered important. Further to this work, a linear an-alytical model for the average mass transport velocity gen-erated in a standing wave thermoacoustic device was de-veloped. The study considered a rectangular cross-sectionin two dimensions. However they did not consider a cylin-drical device as Bailliet et al. had.

The experimental work of Blanc-Benon et al. [23] is per-haps the most insightful of recent studies of thermoacousticstacks, since it directly compares results from both exper-imental and computational investigations. Blanc-Benon etal. used particle image velocimetry (PIV) measurementsfor comparison with low Mach number flow computationalresults for a small region at one end of a thermoacousticstack. The study showed good agreement in flow struc-ture between experimental and computational estimates. Alater study by El-Gendy et al. [24] of a miniature thermoa-coustic refrigerator design also used PIV data in their op-timisation considerations, which suggests there is increas-ing research interest in the flow structures present withinthermoacoustic devices.

Although the experimental work of Blanc-Benon et al.does not extend to significant thermoacoustic effects suchas those in pressurised, circular devices which are designed

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for effective cooling performance, it does give confidence inthe use of computational methods for investigations of flowstructure around thermoacoustic stacks and heat exchang-ers. The design of highly efficient thermoacoustic stacksmay indeed one day require use of higher order computa-tional or discrete numerical tools to minimise adverse flowstructures.

2. Numerical model

This study aims to build upon previous work by consid-ering the performance of a thermoacoustic couple for vary-ing blockage ratio BR and drive-ratio DR.

For the sake of comparison with previous studies [6, 10,8], the numerical model uses similar operating conditionsbut an expanded solution domain to account for physicalflow disturbances. The model approximates a closed-ended,half-wavelength standing wave resonator duct filled withhelium at 10kPa absolute pressure, with a parallel-platestack located in one end of the device. As the resonatorlength is fixed at 5.04m and the sound speed of the gas atambient temperature T0=300K is 1008m/s, the operatingfrequency of the device is fixed to have the first naturalfrequency at 100Hz.

Figure 1 presents the computational domain in the con-text of the theoretical half-wavelength resonator. The res-onator is shown as a two-dimensional representation of asmooth duct with rigid terminations at each end. Ignoringeffects of the resonator diameter and duct surfaces, a par-allel plate stack located inside the resonator is then for thepurposes of this study, of infinite width and plate count.

Fig. 1. Locus of the computational domain within the basic theo-retical model of half-wavelength resonator tube with parallel platestack. Acoustic source not shown.

2.1. Modelspace

The modelspace used for the current work is divided intosix subdomains which together can facilitate changes tothe fundamental characteristics of the thermoacoustic en-vironment, such as the addition of plate thickness or ductlength. Figure 2 shows a sketch of the modelspace used inthis study.

The plate and thermal reservoir comprises subdomains‘P’ and ‘H’ respectively. Subdomain ‘S’ is the region encom-passed by the plate axially within the stack. The length y0

is the half-spacing between plate centrelines. Subdomains‘A’ and ‘B’ enable consideration of flow structures whichmay develop outside the stack region due to non-zero platehalf-thickness yx, and are of sufficient lengths LA and LB

such that the pressure changes can be considered adiabatic.Subdomain ‘C’ links subdomain ‘B’ to the hard walled endof the duct at the right edge of subdomain ‘C’. Symmetricboundary conditions have been imposed on all lateral fluidboundaries and an oscillatory boundary condition on theleft side of subdomain ‘A’, has been used to represent anacoustic standing wave.

Subdomain ‘H’ is important to the solution as it pro-vides a thermal ‘reservoir’ to the model. Cao et al. [8] andIshikawa and Mee [6] used the time-averaged heat transferthrough a plate at fixed temperature to benchmark the per-formance of a thermoacoustic couple. However, Piccolo andPistone [10] demonstrated that similar results could alsobe obtained by including a finite heat capacity in the stackplate and forcing all heat fluxes to pass through the sameboundary. In following Piccolo and Pistone [10], the topand sides of subdomain ‘H’ are thermally insulated, forcingany excess heat entering the ‘H’ and ‘P’ regions to returnback through the boundaries between subdomain ‘P’ andsubdomains ‘A’, ‘S’ and ‘B’. Whilst the thermal propertiesof the stack are therefore important as unsteady tempera-ture distributions are allowed to form within the stack re-gion, the thickness of subdomain ‘H’ is minimal comparedto that of subdomain ‘P’.

To provide comparative results with Ishikawa and Mee[6] and Piccolo and Pistone [10], the subdomain ‘P’ is setto zero thickness, and subdomains ‘H’ and ‘S’ are adjacent.The number of mesh intervals in the x and y directions, nx

and ny respectively, can be easily adjusted to explore thegrid dependency of the model or match the sizings used byprevious studies. For example, in Section 2.3 where com-parisons are drawn with the results of Piccolo and Pistone[10], nxS=244 and ny=nyS=50, with LS=0.252m.

The effect of non-zero plate thickness can be accomo-dated by varying yx, indicated as the vertical size of subdo-main ‘P’. To preserve a consistent and rectangular compu-tational mesh, nyX is increased at the expense of nyS suchthat ny=nyX+nyS . The transverse dimension (thickness)of subdomain ‘H’ was held fixed at ∆y=0.16mm.

2.2. Boundary conditions

Figures 3(a) and 3(b) indicate the boundary types andvarious length scales used in this study. The conditionsimposed upon the boundaries shown in Figure 3(a) and3(b) are described by Equations 1 to 5 with u and v thecomponents of velocity in the x and y directions, and x′

being the axial distance from the centre of the duct.

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Fig. 2. Annotated sketch of the basic thermoacoustic couple modelspace. Shaded areas are regions of solid material: unshaded areas are fluid

regions. The half-thickness of the plate is modelled using finite yx, and appropriate selection of the number of mesh intervals in the x and y

directions, nxP (=nxS) and nyP .

u = 0

v = 0dTdy = 0

on axial ‘WALL’ boundaries, (1)

u = 0

v = 0dTdx = 0

on transverse ‘WALL’ boundaries, (2)

u = 0

v = 0

on ‘HX’ boundaries, (3)

v = 0dTdy = 0

on ‘SYM’ boundaries, and (4)

p = pm + Re[|p1| ej(ωt+kx′−π

2 )] }

on ‘INLET’ boundaries.

(5)

The axial length of subdomains ‘A’ and ‘B’ were set equalto LA=LB=0.15m. The axial length of the computationaldomain, LCV , was set to 1.476m. As shown in Figure 1,the origin of the computational domain is located 1.194maxially from the centre of the duct, i.e. x=x′+1.194m.

2.3. Operating conditions

Throughout the computational domain, the pressure, ve-locity and temperature were initialised at ambient condi-tions, that is

Fig. 3. Computational domains used in this study showing boundaryconditions and selected geometry. (b) provides a closer view of the

stack region shown in (a).

u = 0

v = 0

Tk = Tm

for t=0 at all x and y. (6)

By following Ishikawa and Mee [6], where the operatingfrequency f=100Hz, Tm=300K and pm=10kPa (approx-imately 0.1 atmospheres) for all runs, the mean thermalpenetration distance λkm (=

√2k/ωρcp) is ≈2.4mm for he-

lium as the working fluid.Flow and geometry parameters for each run are listed

in Table 2. Twenty-one runs were completed covering typ-ically four different drive-ratios for five different plate half-thicknesses yx. The drive-ratio DR is a defined as the ratioof the maximum oscillatory pressure amplitude divided bythe ambient pressure pm. The blockage ratio BR is definedas the ratio of open gas area to total device cross sectionalarea: the BR of a thermoacoustic couple is commonly eval-uated as simply

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Table 2Flow and geometry parameters for each simulation run used in in-

vestigating blockage ratio. Run 22 differs only from Run 1 in using

a mesh with grid mesh sizing equivalent to that used by Piccolo andPistone [10]. DR=|p1|/pm.

Run |p1| |p1|/pm yx λkm/yx BR

/ (Pa) / (%) / (mm)

1 170 1.7 0 ∞ 1

2 340 3.4 0 ∞ 1

3 510 5.1 0 ∞ 1

4 680 6.8 0 ∞ 1

5 170 1.7 0.16 15 0.98

6 340 3.4 0.16 15 0.98

7 510 5.1 0.16 15 0.98

8 680 6.8 0.16 15 0.98

9 170 1.7 0.80 3.0 0.9

10 340 3.4 0.80 3.0 0.9

11 510 5.1 0.80 3.0 0.9

12 680 6.8 0.80 3.0 0.9

13 170 1.7 1.6 1.5 0.8

14 340 3.4 1.6 1.5 0.8

15 510 5.1 1.6 1.5 0.8

16 680 6.8 1.6 1.5 0.8

17 170 1.7 2.4 1.0 0.7

18 340 3.4 2.4 1.0 0.7

19 510 5.1 2.4 1.0 0.7

20 680 6.8 2.4 1.0 0.7

21 850 8.5 2.4 1.0 0.7

22 170 1.7 0 ∞ 1

BR = 1− yx

y0(7)

In addition, Run 22 was developed to compare the cur-rent results to those published by Piccolo and Pistone [10]and hence uses the same operating conditions as Run 1,however differs in grid mesh density.

2.3.1. Turbulence criteriaAn approximation of the free stream ‘acoustic’ Mach

number Ma as defined by Equation 8 is a useful measureof the “nonlinear behaviour” of the system or an indicatorof higher order acoustic modes. Equation 9 is a modifica-tion of Equation 8 to compensate somewhat for the effect ofblockage ratio BR within stack pores: this follows from themethod used by Piccolo and Pistone [10, Equation 9] andSwift [16, Equation 61] for setting volume velocity bound-ary conditions. Poese [19] estimated that drive-ratios ofgreater than 3% would most likely result in non-linear ef-fects becoming significant, however a similar limit for Ma

is not clear. In providing Ma we aim to provide further in-

sight into a linear approximation limit based upon Ma. Thefree stream Mach number M as defined by Equation 10 willalso be used as measure of the “nonlinear” behaviour.

Ma =pA

ρma2(8)

Ma = (2−BR)|pA|ρma2

cos(kx′) (9)

M =|u1|a

(10)

In the ‘short stack’ approximation familiar to thermoa-coustic system designers, Ma may be calculated directlyand assumed constant within the stack region. In this study,where the amplitude of the local Mach number varies con-siderably through the stack region, we will numerically eval-uate M at the midspacing and the midlength of the stackregion (x = LS/2, y = 0) and compare with Ma.

Ishikawa and Mee [6] and Piccolo and Pistone [10] ne-glected turbulence effects in each of their studies on the ba-sis that the critical Reynolds number Rc

s (Equation 11) de-fined by Thompson [25], was not expected to exceed (200)2

using the results of Merkli and Thompson [26]. In the con-text of oscillatory flow, the critical Reynolds number is thecondition for which the flow transitions to a turbulent flowregime [26], that is, the Stokes layer becomes unstable [25].The study of Merkli and Thompson [26] indicated that thistransition at Rc

s is effectively localised to the boundarylayer, provided that the boundary layer thickness is signifi-cantly less than “other dimensions” such as the tube radius.In other words, the limiting Rc

s value of (200)2 was devel-oped for internal flows without small features [26]. How-ever, in typical thermoacoustic systems using parallel-platestacks, the plate thickness is much smaller than the tuberadius and could potentially be of similar thickness to theStokes layer.

Rcs =

|u1|2

υω(11)

The flow impedance created by increasing plate thick-ness 2yx leads to higher velocities and therefore higherRs values within the stack region. Decreasing the block-age ratio (thicker plates) would further increase Rs. Con-cerns that decreasing the blockage ratio would increase Rs

above (200)2 was addressed using Run 21, that had the low-est (most flow restrictive) blockage ratio (0.7) and highestdrive-ratio (8.5%) modelled.

2.4. Material properties

For the sake of comparison with the results of Ishikawaand Mee [6] and hence the results of other studies using

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Table 3Flow conditions and material properties used for all computational

runs.

Property Value Units

Operating frequency, f 100 Hz

Ambient temperature, Tm 300 K

Mean pressure, pm 10 kPa

Gas properties:

Prandtl number, Pr 0.69

Thermal conductivity, k0 0.149 W/mK

Heat capacity, cp 5200 J/kgK

Dynamic viscosity, µ 2.01 x 10−5 kg/ms

Kinematic viscosity, υ 1.24 x 10−4 m2/s

Ratio of specific heats, γ 1.665

Plate material properties:

Thermal conductivity, ks 10 W/mK

Heat capacity, cps 400 J/kgK

Density, ρs 400 kg/m3

the same operating conditions [8, 10], the properties shownin Table 3 are common to all tests in this investigation.Density was approximated using the ideal gas law: constantPrandtl number, specific heats and thermal conductivitywere assumed.

In acknowledging the increasing usage of ceramics suchas Corning CelcorTMsubstrates in thermoacoustic stacks,the heat exchanger plate material properties of materialdensity and thermal heat capacity were held constant at400kg/m3 and 400J/kgK respectively. In following Piccoloand Pistone [10], a constant thermal conductivity ks of10W/mK was applied to the stack region represented bysubdomains ‘H’ and ‘P’ (Figure 2).

2.5. Numerical implementation

The commercial CFD software FLUENT was used toconduct 2-D simulations of the system. An unsteady for-mulation was used with first order discretisation of flow.To enable sufficient resolution of each waveform, 100 timesteps per period of oscillation (1/100 seconds) were se-lected, which resulted in a time step size of one ten thou-sandth of a second (0.1ms). Convergence criteria for the so-lution at each time step was based upon the residuals for ve-locity components and continuity equal to or below 0.01%,and the residual for energy equal to or below 0.0001%.

A mesh spacing of ∆x equal to 0.5mm and ∆y equal to0.16mm within the stack region (subdomain ‘S’ shown inFigure 2) was used, however at increasing axial distancesfrom the stack region, that is, decreasing x for x ≤ 0 andincreasing x for x ≥ LS , ∆x was exponentially increasedto reflect the increasingly adiabatic (less ‘detailed’) oscil-lations present with increasing distance from the stack re-gion. The domains used for Runs 1 to 20 each contained ap-

Fig. 4. p1, u1 and ∆Tk,hx (Section 2.5.1) at the oscillatory boundary

(INLET) versus simulation timesteps 0 to 1000 and 1900 to 2100 for

Run 6. Positive velocity is to the right with reference to Figure 3.

proximately 33,900 nodes. Grid independency was checkedusing a finer mesh with double the number of mesh inter-vals nx, ny or four times the node density. To match themesh sizing used in Run 1 of Piccolo and Pistone [10], the x-direction mesh sizing ∆x was increased to 1.033mm whilst∆y was retained at 0.16mm within the stack region. Thisresulted in a total of approximately 16,860 nodes used forRun 22.

In transient or unsteady thermoacoustics, a key termused is the ‘limit state’ or ‘limit cycle’, which refers tothe state of operation or operating conditions in which thephase and magnitude of state variables such as p, U and Tat each phase in the cycle do not vary from one oscillationto the next. At the limit cycle, the time-averaged changein enthalpy and entropy flux is zero and the enthalpy fluxbecomes constant and uniform along the length of the de-vice [27]. Limit-cycle simplifications used in Rott’s approx-imations [15] are not capable of accurately predicting theperformance of thermoacoustic systems in transient states,although it is apparent that transient events such as stepor ramp inputs to thermal or acoustic states are currentlynot yet of specific interest to the study of thermoacousticcouples. However, such effects may be computationally pre-dicted using existing models at limit state.

For each run, a sinusoidal pressure input applied tothe inlet boundary shown in Figure 3(a) with frequencyf=100Hz and amplitude |p1| as listed in Table 2. Althoughan unsteady simulation, the model must be initialised suchthat comparisons between runs can be done under limitor steady state operation. Figure 4 shows that whilst thepressure and velocity states approach limit state withinthe first three to four cycles of simulation, Figure 5 showsthat a thermal limit state of the gas within the stack re-gion does not approach limit state until approximately thesixth or seventh cycle.

To ensure each simulation achieved limit state operation,two thousand time steps (twenty oscillations) were calcu-lated to initialise each model. An additional 100 time steps(1 full cycle) was then simulated with statistical averaging

8

Fig. 5. Temperature versus specific entropy for position ‘M1’

(0.126m,0), over the first ten cycles of Run 1. Points of interest

are marked for time steps 0017 (flow-time 0.0017s), 0076 (flow-time0.0076s) and 0176 (flow-time 0.0176s).

employed to determine properties such as the time aver-aged heat flux distribution or flow parameter.

Note that the ratio λkm/y0 is fixed at 0.3 for all runs,because in each model the plates are modelled as fixed atcentreline distances 16mm apart but increasing in thick-ness. The ratio of thermal penetration depth to plate spac-ing (=2y0), defined [3] as the normalised plate spacing δκh,is therefore also fixed at 0.15. If the design of the stackwere to follow convention in that the plate half-thicknessyx is set equal to the thermal penetration depth λkm, thenthe blockage ratio of Runs 16 to 20 would in our opinionbe to some degree structurally representative of a realisedsystem with the same operating conditions.

2.5.1. Performance scalesPerformance metrics for the model have been developed

to identify limit-state operation and for performance com-parisons between each mode of operation. ∆Tk,hx (Equa-tion 12) is a measure of the time-average difference in area-weighted average gas temperature at x=0 and x=0.252m,and could be considered the axial gas temperature differ-ence across the heat exchanging surface, hence the subscript‘hx’.

∆Tk,hx = 〈(|Tk|x=0.252m − |Tk|x=0)〉t (12)

The distribution of time-averaged heat flux density overan oscillatory cycle at a fluid-solid boundary, h is a use-ful performance measure that has been used in the major-ity of past studies of thermoacoustic couples. Since previ-ous studies considered a stack plate of zero thickness, thisquantity was also referred to as a time-average heat fluxin the y-direction, here represented by

⟨hy

⟩t. As plates of

Table 4Measurement positions of interest.

Code x y Comment

C1 0 0 ‘C’ for cold end

M1 0.5LS 0 c.f. Position ‘M’ of Ref [13]

H1 LS 0 ‘H’ for hot end

C2 0 yhxsf - 0.5δkm spaced 0.5δkm from plate surface

M2 0.5LS yhxsf - 0.5δkm ‘M’ for midlength

H2 LS yhxsf - 0.5δkm

non-zero thickness will be considered in this study, time-average axial or horizontal heat fluxes through the end tipsof the stack plates will exist and shall be denoted by

⟨hx

⟩t.

The term⟨hhx

⟩t

will refer to the time-average heat fluxthrough all fluid-solid boundaries of the stack plate.

The thermodynamic cycles experienced by the gas atfixed positions in the computational domain can also yieldinsight into the performance of the stack configuration. Ta-ble 4 indicates fixed positions (using the co-ordinate systemshown in Figure 3(b)) that have been designated as pointsof interest.

Locations C1, M1 and H1 are located transversely on theline of symmetry at mid-spacing between each plate surface(y=0) and axially within the stack region (0 ≤ x ≤ LS). Lo-cations C2, M2 and H2 are transversely located within halfthe mean thermal penetration distance of the fluid-solidinterface, and gas particles here should theoretically expe-rience a greater heat-pumping effect than those on y=0.

A final performance measure used in this study is theenergy dissipation rate due to irreversibilities associatedwith flow disturbances. With the subdomains A, B, C andS shown in Figure 3 forming the closed-system volume CV ,the entropy rate balance is described by

dS

dt=

∮CV

q

T+ sgen (13)

and

ediss = −T sgen (14)

At limit state operation, the time rate of change of entropyof the system dS/dt is considered constant, and since all ex-ternal boundaries to the model except the ‘INLET’ bound-ary are insulated, the rate of energy dissipation due to en-tropy generation ediss can be approximated using Equation14 over a full oscillatory cycle. In evaluating entropic lossesof thermoacoustic couples, Ishikawa and Mee [6] used thismethod to show that the increase in rate of entropy gener-ation sgen with drive-ratio was quadratic for a plate of zerothickness.

However, the commercial code FLUENT also allows di-rect measure of the volume-average entropy change dS/dt,and since all external boundaries to the model except the

9

‘INLET’ boundary are insulated, in this study, ¯sgen can beevaluated by

¯sgen =dS

dt−

∫INLET

¯qT

(15)

where the overbar indicates values averaged over a full os-cillation.

3. Results

In this section results are presented in terms of the com-puted flow structures and the rate of heat transportationand dissipation in each run.

3.1. Flow structures

Discussion of flow structures will be compared with linearestimates of turbulent limits, the generation of flow vorticesand velocity gradients.

3.1.1. Comparison with turbulence criteriaIt is important to first investigate whether the introduc-

tion of the plate thickness 2yx in this study leads to insta-bility of the Stokes boundary layer, since numerical mod-els specific to modelling turbulent behaviour such as theReynolds Stress Model (RSM) were not incorporated intothe computational solver for this study.

Figure 6 indicates the estimated distribution of mean andRMS velocity magnitude along the mid-plate line (y=0) forRun 20. As expected, the RMS velocity is greater throughthe stack region due to the reduced cross section with con-tinuous mass flow rate at the stack extremities. However,Figure 6 shows that whilst the mean velocity is effectivelyzero at sufficent distance from the stack (which is to be ex-pected at these locations where the flow is theoretically anadiabatic standing wave), there is non-zero mean gas flowat the extremities of the stack. These non-zero mean flowsacknowledge that the flow is perturbed by the plate struc-ture and suggest flow recirculation (streaming) effects atthese locations.

Table 5 presents the estimated flow parameters for theassessment of turbulence used here. The free stream Machnumber M as defined by Equation 10 and the ‘acoustic’Mach number Ma as defined by Equation 8 are also pro-vided in Table 5.

The leftmost plate edge at x=0 (which is closer to thevelocity antinode in the theoretical half-wavelength res-onator) has greater velocities than at x=0.126m, howeverthe changes in flow disturbance with changes in plate thick-ness do not allow such straightforward comparison as perTable 5. It can be seen that in the midsection of the stackwhere x is approximately 0.126m, the critical Reynoldsnumber criterion for turbulence of Rc

s = (200)2 as definedby Thompson [25] was exceeded for Run 21. As expected,

Fig. 6. Estimated distribution of mean and RMS velocity magnitudealong the mid-plate spacing (y=0), for the time period 0.2000s to

0.2100s. Data shown is for Run 20, with BR=0.7 and DR=6.8%.

the conditions used in Run 21 led to the highest reported|Rs| value of (239)2, or 140% of the Rc

s criterion. The causeof this Rc

s exceedance is primarily assigned to the platethickness 2yx. Runs 1 to 20 and Run 22 will therefore beused here in the absence of a numerical model specific toflows with |Rs| ≥ Rc

s.Table 5 also shows that the numerical result M exceeds

the calculated Ma result by around 10 to 15% for BR=1. AsBR increases, the ratio M/Ma also increases. It is impor-tant therefore that whilst the volume velocity through thestack plates is preserved, the transverse distribution in u1

is perturbed significantly by non-zero yx, such that exces-sive errors may arise from use of Ma within stack regions.

3.1.2. Vortex generationFigures 7 and 8 illustrate the evolution of flow vortices

at the left end of the stack for Run 20, over a full oscil-latory period beginning at 0.2028s in 1ms steps. In thistimeframe, the flow of gas increases to the left with a ve-locity maximum at around 0.2052s which retards to zeroat around 0.2077s. The velocity of the prevailing flow thenincreases to the right, to again retard to zero at approx-imately 0.2128s. Pathlines are generated at selected fluidnodes along x=0 (the left end of the plate) and at distancesof 5mm and 10mm in each direction from the y-axis. Path-lines could be considered as massless ‘strings’ which areuseful in visualising the flow structure, and follow stream-lines generated at each time step. The velocity directionat the inlet is indicated to the left of each figure to showthe prevailing velocity in the centre of the device. A scaleis provided at the top of each figure to indicate the spatialscales of the vortices generated.

In Figure 7(a), the fluid is starting to move predomi-nantly left, with fluid closest to the stack plate leading tothe left. As the flow shifts to the left, entrained flow over

10

Fig. 7. Flow pathlines for Run 20, at (a) t=0.2028s, (b) t=0.2038s, (c) t=0.2048s, (d) t=0.2058s, (e) t=0.2068s. The prevailing flow direction

and velocity magnitude is indicated to the left of each figure.

the lip of the plate edge (Figure 7(b)) results in a smallrecirculation zone forming (Figure 7(c)) and growing (Fig-ure 7(d)) despite the fact that the prevailing flow velocityis now decreasing. This recirculation increases in size un-til it dominates the flow structure (Figure 7(e)) and as theflow moves to the right, fluid entrained by the recircula-tion breaks out to the right and with high velocity streamsaround the edge of the plate (Figure 8(a)).

Figures 8(a) and 8(b) show that the flow of gas enteringthe domain from upper left becomes increasingly dominant,pushing the recirculation down further. By 0.2098s (Figure8(c)), the recirculation has completely collapsed, and flowupstream of the plate edge begins to approach uniformity.Downstream of the plate, a small recirculation zone is seento develop (Figure 8(d)) and the boundary layer thicknessincreases (Figure 8(e)). At this timeframe, the prevailing

11

Fig. 8. Flow pathlines for Run 20, at (a) t=0.2078s, (b) t=0.2088s, (c) t=0.2098s, (d) t=0.2108s, (e) t=0.2118s. The prevailing flow direction

and velocity magnitude is indicated to the left of each figure.

flow is decelerating, and the flow reverses direction as shownby Figure 7(a) to again repeat the cycle.

It can be seen that the structure shown in Figure 7(c) issimilar to the result presented by Blanc-Benon et al. [23].Although Run 20 uses helium for a working gas comparedwith air in the Blanc-Benon et al. result, and the two fig-ures are produced for different time values two hundredthsof a phase apart, the shapes of each vortex generated are

proportionately similar. The formation of separation zoneson the plate inside edge in Figures 8(c) to 8(e)was alsodemonstrated by Worlikar and Knio [18] for much lowerdrive-ratios (DR ≤1%).

3.1.3. Entropy generationFigure 9 shows the increase in volume average system

entropy S versus timestep for Runs 1 and 20. As expected,

12

Table 5Predicted |u1|, Ma and M values at Location M1 (Table 4) for each

run.∣∣Ma

∣∣0.126m

is found from Equation 9:∣∣M∣∣

0.126mis found from

Equation 10, i.e.∣∣M∣∣

0.126m=|u1|0.126m/a.

Run BR |u1|0.126m

∣∣Ma

∣∣0.126m

∣∣M∣∣0.126m

|Rs|0.126m

/ (ms−1)

1 1 8.16 0.007 0.0081 (29.3)2

2 1 16.1 0.014 0.0160 (57.7)2

3 1 24.1 0.021 0.0239 (86.3)2

4 1 32.2 0.028 0.0319 (115)2

5 0.98 8.38 0.007 0.0083 (30.0)2

6 0.98 16.5 0.014 0.0164 (59.1)2

7 0.98 24.7 0.021 0.0245 (88.5)2

8 0.98 33.1 0.029 0.0328 (119)2

9 0.90 9.38 0.008 0.0093 (33.6)2

10 0.90 18.6 0.015 0.0185 (66.6)2

11 0.90 27.8 0.023 0.0276 (99.6)2

12 0.90 37.1 0.031 0.0368 (133)2

13 0.80 11.1 0.008 0.0110 (39.8)2

14 0.80 22.0 0.017 0.0218 (78.8)2

15 0.80 32.9 0.025 0.0326 (117)2

16 0.80 43.9 0.034 0.0436 (157)2

17 0.70 13.5 0.009 0.0134 (48.4)2

18 0.70 26.9 0.018 0.0267 (96.4)2

19 0.70 40.7 0.027 0.0404 (146)2

20 0.70 53.4 0.036 0.0530 (191)2

21 0.70 66.7 0.046 0.0662 (239)2

22 1 8.04 0.007 0.0080 (28.8)2

the rate of entropy generation in Run 20 is higher than thatin Run 1, with higher flow velocities and increased vortexgeneration from increased plate half-thickness yx.

Figure 10 presents the rate of entropy generation sgen

with respect to drive-ratio DR for Runs 1 to 21, grouped byblockage ratio. This figure shows the increase in irreversiblelosses with increasing plate thickness 2yx and drive-ratioDR, against previous results from Ishikawa and Mee [6]and the analytical result was obtained using an expressionfrom Swift [16, Equation 89] with BR set to 0.7.

It can be seen that the entropy generation rate sgen us-ing a thermoacoustic couple modelled with zero thicknessobtained in the present study is approximately 40% higherthan that calculated by Ishikawa and Mee, however this re-sult is attributed in part to use here of a mesh density fivetimes greater in the axial direction than that used previ-ously by Ishikawa and Mee. Whilst the axial dimension ofsubdomain ‘A’ (Figure 2) is also noted to be approximately75% greater than the equivalent region in the domain usedby Ishikawa [6], the zones in which the rate of entropy gen-eration is significant are well within regions common toboth models.

Fig. 9. Volume-average system entropy S versus timestep for Runs

1 and 20. ‘Moving average’ trendlines based upon the previous 100

time steps are presented for each curve as a solid black line.

Fig. 10. Rate of entropy generation sgen versus drive-ratio DR, withselected runs grouped by blockage ratio, as follows: BR=1.0 (Runs

1 to 4), BR=0.98 (Runs 5 to 8), BR=0.9 (Runs 9 to 12), BR=0.8

(Runs 13 to 16), BR=0.7 (Runs 17 to 21). The analytical result ofSwift was generated from use of Equation 89 of Ref. [16].

The analytical result of sgen for BR=0.7 utilising Equa-tion 89 of Swift [16] is shown to exceed the numerical re-sults of Ishikawa and Mee by typically 13 to 15%. Thisis expected since the viscous shear dissipation componentwas calculated with the gas velocity u1 increased using theblockage ratio as in Equation 9 which in turn increases sgen.

3.2. Heat transportation

3.2.1. Comparison with other modelsFigure 11 indicates the time average heat flux across the

plate surface in the y-direction hy versus axial position xfor Run 22 of the current study and Run 1 of Piccolo andPistone [10]. In forming good agreement with Run 1 of Pic-colo and Pistone [10], and hence Run 7 of Ishikawa and

13

Mee [6] and Run 2 of Cao et al. [8], the numerical methodused is considered appropriate for the simulation of ther-moacoustic couples with zero thickness. Slight differencesare expected as the temperature at the surface of the plateis not fixed at T0 in the present study.

Fig. 11. Distribution of time-averaged heat flux in the y-direction⟨hy

⟩t

along the horizontal plate surface according to axial position

x. Open circles are approximate numerical data from Run 1 of Ref[10]. Positive hy indicates net heat transfer from the plate to the

working fluid.

The sign conventions used in this paper are such thatpositive hy indicates net heat transfer from the plate to theworking fluid and negative hy indicates net heat transferfrom the working fluid to the plate. The time-average heatflux across the plate surface in a thermoacoustic couple isshown to be concentrated to the extremities or edges ofthe plate surface, and appears symmetric across the platemidpoint. In fact, the distribution of hy is assymetric ina thermoacoustic couple, for reasons such as the variancein velocity amplitude between each of the stack. Figure 11appears symmetric since the thickness of the plate is notmodelled. Run 1 indicates that the point at which

⟨hy

⟩t=0

is not x=0.5LS , but rather x ≈0.2LS . We attribute this toa positive bias in hy across the plate surface.

3.2.2. Thermophysical cyclesIn Figure 12, temperature-entropy graphs are plotted for

selected locations of Run 1 at limit state. It can be seenthat the oscillations of the gas at location M1 are effectivelyadiabatic, however the cycles experienced at locations C1and H1 are noticeably different. Locations C1 and H1 arelocated along y=0, at more than three times the mean ther-mal penetration distance from the plate surface. However,the T-s curve at these two locations are perturbed: the T-s curve is expanded at locations C1 and H1 for approxi-mately 270◦ of the limit cycle. The perturbances of C1 andH1 also occur at opposite phases in the cycle, for predomi-nantly negative p1 at location C1 and predominantly pos-

Fig. 12. Temperature versus entropy cycles for selected fixed positions

defined in Table 4 at limit state for Run 1. The position and directionof the cycle at ts 2000 (flow time 0.2s) is indicated with a solid

arrow. Phases of maximum and minimum pressure are indicated for

locations C2, M2 and H2.

itive p1 at location H1. This suggests that effective heattransportation is occurring at these locations despite theirdistance to the stack plate being significantly greater thanthe thermal penetration depth. It is feasible that flow re-circulations at the plate extremities (Figure 6, Figure 8(a))may lead to fluid at distances greater than δkm participat-ing in heat transfers with the plate.

Figures 13 and 14 plot temperature versus specific en-tropy for selected locations of both Run 4 and Run 20during limit state operation. The major difference betweenRuns 4 and 20 is the value of yx: Run 4 considers the plateto be of zero thickness, whereas Run 20 models the plate as2.4mm thick. The perturbation to the temperature-specificentropy curve is significant, not only at locations at theedges of the plate but also at the mid-plate locations ‘M1’and ‘M2’.

In Figure 13, indicators ‘A’ and ‘B’ designate features ofthe curve specific to flow disturbances over each plate edge.The value indicated by ‘A’ is obtained at a flow time of0.2082s, between the timeframes used in Figures 8(a) and8(b). As can be seen from these two frames, the specificentropy at Location C1 during that short time period isboosted from excess energy dissipation from the collapsingvortex upstream of the flow. The same effect, occuring withopposite phase at the other end of the plate, is indicatedby ‘B’ in Figure 13.

Indicators ‘C’ and ‘D’ in Figure 13 point to phases inthe cycle for which the flow velocity into the stack region(rightward for Location ‘C1’, leftward for Location ‘H1’) isgreatest. This phase in the cycle for Location ‘C1’ occursbetween the timeframes used in Figures 8(c) and 8(d).

A heat pumping effect can be seen from comparison ofFigures 13 and 14 in the temperature bias to the ‘H1’ and‘H2’ ends. The ‘folded’ or ‘figure of 8’ curves at Locations‘C2’ and ‘H2’ indicate that there is an effective transfer

14

Fig. 13. Limit state behaviour of temperature versus specific entropy

at Locations C1, M1 and H1 for Runs 4 and 20.

Fig. 14. Limit state behaviour of temperature versus specific entropyat Locations C2, M2 and H2 for Runs 4 and 20.

of heat, whereas the open, elliptical shape of the curve forLocation ‘M2’ indicates a storage of potential energy in-stead. This result is expected since whilst Locations ‘C2’‘M2’ and ‘H2’ are all located δκ/2 transversely from theplate, Figure 11 shows that the time-average heat transferto the plate occurs only at the plate edges (Location ‘M2’is at the midpoint).

The effect of plate thickness is also seen to reduce theperformance of the plates, in that the average temperaturesin Run 20 (compared to Run 4) are lower at the ‘hot’ end(Locations ‘H1’ and ‘H2’) and higher at the ‘cold’ end (Lo-cations ‘C1’ and ‘C2’).

3.2.3. Evolution of thermoacoustic couple heat fluxdistribution

The evolution of the distribution of transverse heat fluxhy through the plate surface for Run 1 is shown in Figure15. The surface plot is given for the 0.2s to 0.21s timeframe

of interest, and negative heat flux as shown indicates heattransfer from fluid to the plate (i.e. cooling of the surround-ing gas). Figure 16 shows the same result for Run 20, whichdiffers to Run 1 in blockage ratio (0.7 in Run 20 to 1) anddrive ratio (6.8% in Run 20 to 1.7%). It is important tonote that whilst Figures 15 and 16 do not include axial heatflux hx via the edges of the stack plate, this component ofthe overall heat flux through the solid-fluid boundary hhx

is significant (Section 3.2.4).The results shown in Figures 15 and 16 demonstrate the

phasing of heat transfer in such detail that the cycle can bebroken down into four distinct phases of compression, heat-ing, expansion and cooling of 90◦ phase duration, in similarfashion to most studies discussing the operation of standingwave thermoacoustic devices [16, 28]. At 0.2s, the positive(rightward) velocity maximum should denote the midpointof the compression phase, since at this point in time, thepressure is increasing and the velocity is decreasing froma maximum (Figures 8(c) and 8(d)). However, thermal de-lays in the system are present and so there exists a phasebias in the commencement of each cycle.

Fig. 15. Surface plot of transverse heat flux hy through the horizontal

plate surface for the limit state cycle of Run 1.

At around 0.2012s for Run 1, the heating phase is appar-ent, with heat transfer to the fluid occuring at both endsof the thermoacoustic couple. The cycle then shifts to anexpansion phase which begins at around 0.2037s, and leadsto the cooling phase denoted by the significant decrease inhy near x/LS=0. The flow structure about x/LS=0 duringthe cooling phase for Run 20 is captured in Figures 8(a) to8(c).

In both Figures 15 and 16, hy is shown to be oscil-latory at both ends, with a superimposed heat flux pro-portional to the oscillatory pressure. At around t=0.2037snear x/LS=1, an additional increase in hy is noted, witha decrease around t=0.2087s near x/LS=0 also noticeable.

15

Fig. 16. Surface plot of transverse heat flux hy through the horizontal

plate surface for the limit state cycle of Run 20. Note the difference

z-axis scaling to Figure 15.

These two variations or ‘spikes’ from the prevailing hy dis-tribution in Run 1 are captured by the time averaged mea-sure

⟨hy

⟩tsuch as that presented in Figure 11. These spikes

are more apparent in Run 20 (Figure 16).

3.2.4. Plate edge heat transferThe results so far presented have demonstrated a ther-

moacoustic heat-pumping effect from the left side (x=0)to the right side (x=0.252m) of the plate. Time averagedtemperatures of gas in the vicinity of the left side have de-creased, and temperatures on the right side have increased.

Figure 17 shows the relative magnitude of axial heat fluxthrough the plate ends in comparison to transverse heatflux through the plate facing for Run 20. The time-averagedheat flux via each of the left (‘cold’) and right (‘hot’) plateends 〈hx〉t|cold can be compared to the time-averaged heatfluxes 〈hy〉t|cold and 〈hx〉t|hot through the plate horizontalsurface. For Run 20, the point at which the time-averagetransverse heat flux is zero was found to be slightly closerto the cold end of the plate (4% of LS). The tranverse heatflux through surfaces forming the physical ends of the plateis shown by Figure 17 to be significant fractions of the totalheat flux through the plate. Also of interest is the higherproportion of heat leaving the plate than that entering: thisdemonstrates a time-averaged net loss in thermal energywithin the plate material.

3.2.5. Effect of blockage ratio upon heat transfer rateFigure 18 shows that increases in yx and hence decreases

in blockage ratio BR lead to a larger integral of 〈hhx〉t andhence the performance of the thermoacoustic couple. Thisperformance increase comes at the expense of increasedentropy losses, as discussed in Section 3.1.3.

Fig. 17. Sketch showing the proportion of time-averaged heat flux

through the plate surface for limit-state operation of Run 20. Per-

centage values quoted are referenced to the total heat flux enteringthe plate over an oscillatory cycle, 〈hhx〉t|cold.

Fig. 18. Comparison of time average transverse heat flux for

DR=6.8% and increasing BR. Data shown is for Run 4 (yx = 0), Run

8 (yx = 0.16mm), Run 12 (yx = 0.8mm), Run 16 (yx = 1.6mm),Run 20 (yx = 2.4mm),

Figure 19 presents the performance criterion ∆Tk,hx nor-malised by the result obtained using Runs 1 to 4 witha blockage ratio of 1, versus the drive-ratio. This figurecompares the increase in temperature difference across thestack with increasing yx and DR. For a drive-ratio of 1.7%,it can be seen that the performance of the stack is re-duced with increasing plate thickness 2yx. However, for alldrive-ratios considered above 3%, the performance of thethermoacoustic couple was actually better than the zero-thickness condition (BR=1). Regardless of drive-ratio, itis evident that for thermoacoustic couples of rectangularcross section, maximising BR will enable an increase in theperformance scale ∆Tk,hx. However the influence of subdo-main ‘H’ (Figure 2) may be significant at such a small platethickness and a source of error here.

As noted in previous studies [19], drive-ratios above 3%can not be expected to correspond to a linear increase in theperformance of the stack, and this is shown by Figure 19:the normalised ∆Tk,hx does not increase linearly with DRat drive-ratios above 3%, and higher order effects such as

16

Fig. 19. ∆Tk,hx normalised by same result obtained using a blockage

ratio BR of 1 (Runs 1 to 4) versus drive-ratio DR. Data is presentedfrom Runs 5 to 8 (BR=0.98), Runs 9 to 12 (BR=0.9), Runs 13 to

16 (BR=0.8) and Runs 17 to 20 (BR=0.7).

the increased rate of entropy generation and viscous lossesare considered contributors to this result. Interestingly, theperformance increase from DR=5.1% to DR=6.8% is not-icably greater than DR=3.4% to DR=5.1%, although aspecific cause for this result is as yet unidentified.

4. Conclusions

This study concludes that flow impedances from non-zero thickness stack plates and other fixed objects in theacoustic field will introduce flow and heat transportationcharacteristics new to previous studies of thermoacousticcouples.

Results indicate that the parameter yx strongly controlsthe generation of vortices outside the stack region and per-turbs the flow structure and heat flux distribution at theextremities of the plate. Increases in yx is also shown to im-prove the integral of the total heat transfer rate but at theexpense of increased entropy generation.

In practice, typical parallel or rectangular section stacksdo not have perfectly square edges. With existing litera-ture considering only rectangular or zero-thickness (1-D)plates, it would be interesting to see if gains in perfor-mance could be achieved using non-rectangular cross sec-tions, such as rounded or elliptical shaped edges. The modelsystem shown here is currently being modified to provide forthe investigation of non-rectangular shapes such as roundedor aerofoil cross sections.

References

[1] G. Swift, Thermoacoustics: A unifying perspective forsome engines and refrigerators, Acoustical Society ofAmerica, 2002.

[2] B. Ward, G. Swift, Design Environment for Low-Amplitude ThermoAcoustic Engines (DeltaE) Tutor-

ial and Users Guide (Version 5.1), Los Alamos Nation-alLaboratory (June 2001).

[3] M. Wetzel, C. Herman, Design optimisation of ther-moacoustic refrigerators, Int. J. Refrig. 20 (1) (1997)3-21.

[4] S. Backhaus, G. Swift, A thermoacoustic-Stirling heatengine: Detailed study, J. Acoust. Soc. Am. 107 (2000)3148-3166.

[5] D. Gardner, G. Swift, A cascade thermoacoustic en-gine, J. Acoust. Soc. Am. 114 (4) (2003) 1905-1919.

[6] H. Ishikawa, D. Mee, Numerical investigations of flowand energy fields near a thermoacoustic couple, J.Acous. Soc. Am. 111 (2) (2002) 831-839.

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