open travel thermoacoustics mean temp
TRANSCRIPT
Open cycle traveling wave thermoacoustics: Mean temperaturedifference at the regenerator interface
Nathan T. Weilanda)
School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
Ben T. ZinnSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
~Received 18 April 2003; revised 10 August 2003; accepted 8 September 2003!
In an open cycle traveling wave thermoacoustic engine, the hot heat exchanger is replaced by asteady flow of hot gas into the regenerator to provide the thermal energy input to the engine. Thesteady-state operation of such a device requires that a potentially large mean temperature differenceexist between the incoming gas and the solid material at the regenerator’s hot side, due in part toisentropic gas oscillations in the open space adjacent to the regenerator. The magnitude of thistemperature difference will have a significant effect on the efficiencies of these open cycle devices.To help assess the feasibility of such thermoacoustic engines, a numerical model is developed thatpredicts the dependence of the mean temperature difference upon the important design andoperating parameters of the open cycle thermoacoustic engine, including the acoustic pressure,mean mass flow rate, acoustic phase angles, and conductive heat loss. Using this model, it is alsoshown that the temperature difference at the regenerator interface is approximately proportional tothe sum of the acoustic power output and the conductive heat loss at this location. ©2003Acoustical Society of America.@DOI: 10.1121/1.1621859#
PACS numbers: 43.35.Ud, 44.27.1g @RR# Pages: 2791–2798
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I. INTRODUCTION
Research in thermoacoustics has been limited mostlclosed cycle designs in recent years, though there is a potial for significant improvements in efficiency by using opcycle thermoacoustic devices. In such systems, a slow mflow is superimposed on the acoustic field in order to replone of the heat exchangers with a convective heat tranprocess.1,2 This requires, however, that the mean flow veloity be an order of magnitude smaller than the acoustic veity in order to keep thermal convection effects from ovewhelming the thermoacoustic effects in the device. In mcases, an additional thermodynamic benefit is gained inthe open cycle thermoacoustic process is very efficienconverting the thermal energy in the convected fluid inacoustic energy, or vice versa. This was the major impefor the construction and testing of the first open cycle thmoacoustic device, Los Alamos’s standing warefrigerator,1–3 which cools a stream of gas as it passthrough the stack. Greater commercial potential exists, hever, for open cycle traveling wave thermoacoustic devicas they possess an efficiency advantage over their inherirreversible standing wave counterparts.4 For instance, in thenatural gas liquefier being developed jointly by Los AlamNational Lab and Praxair Inc.,5 the natural gas that is burneto power the liquefier could be routed directly through tthermoacoustic engine in an open cycle configuratithereby eliminating the hot heat exchanger and providinsimpler and possibly more efficient means of converting fenergy into acoustic energy.
A diagram of a basic open cycle traveling wave th
a!Electronic mail: [email protected]
J. Acoust. Soc. Am. 114 (5), November 2003 0001-4966/2003/114(5)/2
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moacoustic engine is shown in Fig. 1, where the slow mflow of hot gas approaches the regenerator from the rigHeat is removed at the cold heat exchanger on the left sidthe regenerator, setting up a temperature gradient throwhich an acoustic traveling wave is amplified in passifrom the cold end of the regenerator to the hot end. Howeas will be shown in the following, the joining conditions athe interface between the regenerator and the adjacentduct require that a substantial mean temperature differeexist between the incoming mean flow and the regeneratsolid material. Since this may have profound effects onacoustics, thermodynamics, and efficiency of the open cythermoacoustic engine, the remainder of this study will focon developing a model that can be used to predict the mnitude of this temperature difference.
The physical processes responsible for creating the tperature difference at the regenerator/open duct interfaceshown schematically for various gas parcels in Fig. 2, whcan be viewed as a magnification of the control volume srounding the regenerator interface in Fig. 1. Gas parcelsstart the acoustic cycle at various locations within the regerator are shown in Figs. 2~a!–~d!. For good gas–solid thermal contact within the regenerator, these parcels of gasmain at the regenerator’s temperature until they enteropen duct, where they undergo isentropic temperature olations in concert with the pressure oscillations of the traving acoustic wave. These oscillations cause the gas parcere-enter the regenerator at temperatures below that ofregenerator,6 where they receive heat from the regenerasolid and are rapidly heated back up to the temperature ofregenerator at the interface,Tre. At the end of the acousticcycle, fresh gas parcels enter the regenerator for the firstdue to the steady mean flow of gas toward the regenerato
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depicted in Figs. 2~e! and ~f!. These gas parcels isentropcally oscillate about their mean temperature,TH , until theyenter the regenerator, at which time they are cooled toTre bycontact with the regenerator’s solid material. In the abseof other sources of heat input or output, a heat balance ofregenerator solid at the interface shows that the heat trferred from the solid to the cold returning gas must equalheat transferred from the fresh gas to the regenerator solthe solid material is to maintain a constant mean temperaduring steady-state operation. Satisfying this criterion girise to the mean temperature difference between the re
FIG. 1. A schematic of a basic open cycle traveling wave thermoacouengine. A steady flow of gas with mean temperatureTH flows into theregenerator, whose hot end is at the temperatureTre . A cold heat exchangerrejects heat to ambient temperature at the other end of the regeneratothe resulting temperature gradient in the regenerator is used to amplifyacoustic traveling wave entering its cold end. An analysis of the convolume, which includes the interface between the regenerator and theduct, is used to determine the difference between the temperaturesTH andTre . State properties on the left and right of the control volume are labewith subscripts ‘‘L’’ and ‘‘ R,’’ respectively.
FIG. 2. A schematic of the temperature and acoustic displacement histof various gas parcels near the regenerator/open duct interface, ovecourse of an acoustic cycle. The closed squares denote the starting poof each parcel, and the open squares denote the final position. Thdisplacement between the starting and finishing positions is due to the mmass flux. Gas parcels in~a!–~d! start their motion inside the regeneratoundergo isentropic temperature oscillations outside the regenerator thduce their temperatures, and are heated to the regenerator’s tempeTre , upon re-entry. Gas parcels in~e! and~f! start their motions outside theregenerator, oscillate isentropically about the mean temperature,TH , andcool down toTre upon entering the regenerator at the end of the acoucycle.
2792 J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weila
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erator solid and the hot incoming gas, as cited earlier.The effect of a temperature difference within or betwe
thermoacoustic components has been a subject of sestudies in thermoacoustics. This topic was first broachedSwift,7 who pointed out that for net solid–gas heat transferoccur within a heat exchanger, a difference must existtween the solid temperature and the spatially and temporaveraged gas temperature within the heat exchanger. Bsteret al.8 extended this concept to include the heat transbetween a thermoacoustic stack and its heat exchangthough their analysis was limited only to standing wave thmoacoustic devices. Research that is more closely relatethe problem at hand was first performed by Smith aRomm6 for component interface losses in Stirling engineand later on by Kittel9 and Bauwens10 for similar interfacelosses in pulse tube refrigerators. Swift11 subsequently ap-plied these concepts to the general field of thermoacousand has developed relationships that are frequently useestimate losses and joining conditions between componin thermoacoustic devices. However, none of these stuconsider the effects of mean flow on the component interfdynamics, thus the development of a new theoretical framwork is required for studying the component interface coditions in open cycle thermoacoustic devices.
It should also be noted that while this study deals pmarily with open cycle, traveling wave thermoacoustic egines, the problem of determining the temperature differeat the regenerator/open duct interface is also of concerthermoacoustic refrigerators and heat pumps of the stype, as this temperature difference critically impacts theficiencies of all of these devices. However, this paper efftively demonstrates the procedure involved in the determtion of this temperature difference by focusing solely on ttraveling wave engine application.
II. MASS FLUX MODEL
The formulation used to model the open cycle thermcoustic engine follows the formulation developed by Smand Romm,6 with some modifications to account for thpresence of mean flow. The model considers conditiwithin a control volume that encloses the hot side interfaof the regenerator, as shown in Fig. 1. The control volumleft and right boundaries, designated with the subscriptLandR, are located inside the solid matrix of the regeneraand in the adjoining open duct, respectively, and are clenough together that the control volume can be assumecontain a negligible amount of mass. To simplify the anasis, it is assumed that the flow is one-dimensional, andthere is no axial conduction or mixing between the parcelsgas that move axially in and out of the control volume.
A. Mass flux
To begin, the mass flux passing through the control vume is assumed to be given by
m~ t !5m01m1 sin~vt1f!, ~1!
wherem0 andm1 are the magnitudes of the mean and osclating components of the mass flux, respectively,v is theangular frequency of the acoustic oscillations, andf is a
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phase shift used to set a reference time below. Havingsumed that the control volume contains a negligible amoof mass, it can be further assumed that the mass fluxes cing the left and right sides of the control volume are eq~i.e., mL5mR).
It is convenient to nondimensionalize the model eqtions in order to eliminate the dependence on the angfrequency. Introducing a dimensionless time,t[vt, and di-mensionless mass fluxes,m6 [m/m1 and m6 0[m0 /m1 , Eq.~1! can be rewritten as
m6 ~ t !5m6 01sin~ t1f!. ~2!
Note that nondimensionalizing the mass fluxes bym1 insteadof m0 is more practical for consideration of the limiting cain which there is no mean mass flux.
By choosing to letm6 50 at t50, the phase shift becomes
f5sin21~2m6 0!. ~3!
Here, Eq.~3! specifies thatum6 0u<1, though for practical pur-poses, we requireum6 0u<0.1 to satisfy the condition that thmean velocity be an order of magnitude smaller thanoscillating velocity in an open cycle thermoacoustic devic2
Further, this study will only consider the processes that ocduring one acoustic cycle, i.e., 0< t<2p, wherem6 50 at t50 and t52p, according to Eqs.~2! and ~3!.
An important parameter in this study is the time,tmid , atwhich the gas ceases to flow out of the regenerator, revedirection, and begins to flow back into the regenerator. Ning thatm6 ( tmid)50, that tmid should be close top in value,and that the sine function has odd symmetry aboutp, Eqs.~2! and ~3! can be used to show that
tmid5p22f. ~4!
B. Total mass
In later sections, it will be necessary to relate the timewhich a particular gas parcel leaves the regenerator totime at which it re-enters the regenerator. This is accoplished by tracking the total mass,m, that has passed fromthe regenerator to the open duct during the acoustic cyIntegrating Eq.~1! and nondimensionalizing yields
m~ t !5m6 0t2cos~ t1f!1cos~f!, ~5!
where the total dimensionless mass has been definedm[mv/m1 .
The mass flux and total mass displacement, as descrby Eqs.~2! and~5!, are plotted in Fig. 3, where three distintime periods within the acoustic cycle can be identified. Tfirst, denoted by the subscript ‘‘a,’’ describes the time period0< t a< tmid when the gas is flowing from left to right out othe regenerator. At the timetmid , the gas reverses directioand starts flowing back to the left, thus the total mass thatentered the open duct since the start of the acoustic cyclem,is a maximum at this time. The second time period is definas the time during which gas parcels that left the regenerreturn to the regenerator. Since a stratified flow has bassumed, the time at which a representative gas parcel lethe regenerator,t a , can be related to the time at which th
J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weiland and
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same parcel re-enters the regenerator,t b , by equating thetotal mass displacement at these times as shown in Figi.e.,m( t a)5m( t b). Using Eq.~5!, t b can thus be related tot a
by solving
m6 0t a2cos~ t a1f!5m6 0t b2cos~ t b1f!, ~6!
where t b falls betweentmid and a timet d , which marks thestart of the third time period.
In the stratified flow assumption, two parcels of gas thstart the acoustic cycle adjacent to each other, one insideregenerator and one outside, can have vastly different stthus the regenerator interface experiences a discontinuitgas properties as the gas in the open duct flows from righleft through the control volume at timet d , the ‘‘discontinu-ity’’ time. This time corresponds to the time at whichm50again, and can be found by solving
m~ t d!5m6 0t d2cos~ t d1f!1cos~f!50 ~7!
for t d , where tmid, t d,2p. This time marks the beginningof the third time period, denoted by the subscript ‘‘c,’’ t d
< t c<2p, during which fresh gas parcels enter the regenetor for the first time due to the presence of a mean flow.the end of the acoustic cycle, a net mass of
m~ t52p
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where p[p/p0 and p1[p1 /p0 . Here,p0 is the mean pressure,p1 is the amplitude of the acoustic pressure, andu is thephase angle by which the acoustic pressure leads the acomass flux, where attention is restricted to2p/2,u,p/2 inthis work, since acoustic energy flux is positive accordingthe sign conventions of this study.
A. Gas temperatures
The temperature and position histories of variousparcels near the regenerator interface are shown schecally in Fig. 2 above, although it is not necessary to calcuthese histories, as the only temperatures of interest instudy are those of the gas parcels as they enter and leavcontrol volume. To simplify matters, it is assumed that tgas properties can be related by the ideal gas equatiostate, and that there is perfect thermal contact betweengas and the solid within the regenerator~i.e., TL5Tre for allt). Since accounting for the transient heating effects inregenerator would require a finite-width control volumthermal conductivity and specific heat properties of the gand details on the regenerator’s internal geometry, neglecthese effects greatly simplifies the model at the expenseslightly diminished accuracy.
During the first time period, gas exits the regeneratotemperatureTre and crosses the right-hand boundary of tcontrol volume, soTR,a5Tre. Upon exiting the regeneratogas parcels undergo isentropic temperature oscillationsresult of the acoustic pressure oscillations in the open ducthe right of the regenerator, as depicted in Figs. 2~a!–~d!. Foran isentropic process in an ideal gas, the pressure andperature are related by
T
Tref5S p
prefD ~g21!/g
, ~10!
whereg is the ratio of specific heats. The reference contions for a particular parcel of gas correspond to the tempture and pressure of the gas when it first encounters the itropic environment upon exiting the regenerator at timet a .The time at which this gas parcel returns to the regeneratdetermined by solving Eq.~6! for t b , and the dimensionlestemperature of the gas parcel at this time,TR,b , is then foundfrom Eq. ~10!:
TR,b5F11 p1 sin~ t b1f1u!
11 p1 sin~ t a1f1u!G~g21!/g
, ~11!
where the temperature has been nondimensionalized witspect to the regenerator temperature~i.e., T[T/Tre).
The temperature of the incoming gas during the thtime period can be similarly obtained, though the referestates for temperature and pressure must be redefined. Iopen duct, the gas temperature isentropically oscillates aa mean temperature,TH , as a result of the pressure oscilltions, as depicted in Figs. 2~e! and~f!. In the course of theseoscillations,T5TH whenp5p0 , so takingTH andp0 as thereference states in Eq.~10! for the gas parcels that begin thacoustic cycle inside the open duct, the temperature attime the gas parcels first enter the regenerator,TR,c , can beexpressed as
2794 J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weila
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TR,c5TH@11 p1 sin~ t c1f1u!#~g21!/g. ~12!
B. Thermal energy balance
Having determined the temperature of the gas crosseach of the control volume’s boundaries, the heat balawithin the control volume can now be examined. Assumithat kinetic energy and viscous effects are negligiblequasi-one-dimensional energy equation can be written a
]~re!
]t1
]~ruh!
]x1“"q50, ~13!
wherer is the gas density,e is the internal energy,u is thegas velocity in thex direction,h is the enthalpy, andq is theheat flux vector. Using the ideal gas law and the state rtionshipse5h2p/r and dh5cpdT, the first term in Eq.~13! can be written as:](re)/]t5(]p/]t)/(g21).
Integrating over the control volume then yields
V
g21
dp
dt1@mh#R2@mh#L1E “"qdV50. ~14!
Assuming that the periphery of the device is well-insulatethe heat flux vector,q, consists of two primary componentaxial heat conduction and lateral heat transfer betweengas and the regenerator’s solid material. Using the divgence theorem and Fourier’s law for the conductive heatyields
V
g21
dp
dt1F mh2Ak
dT
dxGR
2F mh2AkdT
dxGL
5Q, ~15!
whereA is the cross-sectional area occupied by the gas,k isthe thermal conductivity of the gas, andQ is genericallydefined as the heat transfer rate between the regeneratorand the gas in the control volume, where heat transfer togas is positive. This equation can be further simplifiedapplying the above assumption of an infinitely thin contvolume, which eliminates the first term on the left-hand sof Eq. ~15!. Finally, if thermal conduction effects in the gaare assumed to be negligible, Eq.~15! can be written nondi-mensionally as
Q6 5m6 ~ TR21!, ~16!
where the dimensionless heat flux is defined asQ6
[Q/m1cpTre.Although thermal conduction in the gas has been
glected, conduction in the solid of the regenerator may sbe important, as the thermal conductivities of solids are gerally orders of magnitude larger than those for gases,the solid material constitutes a significant volume fractionthe regenerator. To account for this conductive heat fluxthermal energy balance of the regenerator solid is requiAssuming that heat transfer to the gas and conducthrough the left side of the control volume are the only hfluxes in or out of the solid, this energy balance can be wten nondimensionally as
dEs
dt5Q6 k2Q6 , ~17!
nd and B. T. Zinn: Open cycle thermoacoustic temperature difference
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[2Asks(dT/dx) int /m1cpTre, where As is the cross-sectional area of the solid,ks is the thermal conductivity ofthe solid, and (dT/dx) int is temperature gradient of the solat the regenerator interface.
In steady state operation, the solid material of the regerator maintains a constant temperature,Tre, thus there mustbe no net energy change in the solid material of the regerator over the course of an acoustic cycle, i.e.,
E0
2p dEs
dtdt5E
0
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Q6 kdt2Etmid
tdQ6 bdtb2E
td
2p
Q6 cdtc50.
~18!
In this expression, the gas–solid heat transfer integhas been split into its contributions from the second and thtime periods,Q6 b and Q6 c , respectively, while the contribution from the first time period has been neglected sinceheat is transferred between the gas and the regeneratorduring this time. Substituting Eqs.~11! and~12! for TR,b andTR,c , respectively, into Eq.~16!, expressions for the heafluxes during the second and third time periods are obtain
Q6 b~ t b!5m6 ~ t b!S F p~ t b!
p~ t a!G~g21!/g
21D , ~19!
Q6 c~ t c!5m6 ~ t c!~ TH@ p~ t c!#~g21!/g21!. ~20!
In each of these equations, the mass fluxes and pressuregiven ~for the appropriate times! by Eqs.~2! and~9!, respec-tively, and t a in Eq. ~19! is related tot b by Eq. ~6!.
IV. THE TEMPERATURE DIFFERENCE
Substituting Eqs.~19! and ~20! into the heat balance oEq. ~18! yields, after some manipulation, an equation for tratio of the mean temperature of the incoming gas totemperature of the regenerator solid:
TH5
Et d
2p
m6 dtc1Etmid
tdm6 S 12F p~ t b!
p~ t a!G~g21!/gDdtb12pQ6 k
Et d
2p
m6 @ p~ t c!#~g21!/gdtc
.
~21!
The integrations in Eq.~21! are performed numerically onPC to determine the value ofTH that satisfies the heat baance. In an actual open cycle thermoacoustic engine,mean flow gas temperature would be a known, fixed qutity, and the steady-state heat balance of Eq.~18! would besolved to determine the temperature at the hot face ofregenerator. Regardless, Eq.~21! solves for the ratio betweethe two temperatures, so knowing one temperature allowsthe solution of the other temperature.
A plot of the calculated time history of the temperatuat the right-hand side of the control volume is shownrepresentative choices of independent parameters in4~a!. The right-side temperature,TR , is equal to the regenerator temperature during the first time period, and is lowthan the regenerator temperature during the second time
J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weiland and
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who have shown that if work is generated by the engine~i.e.,2p/2,u,p/2!, then the isentropic oscillations outside thregenerator cause the gas to re-enter the regenerator attemperatures. At the end of the second time period,t d , adiscontinuity in temperature occurs as the fresh, hotterenters the regenerator for the first time. The plot of the gasolid heat transfer rate versus time in Fig. 4~b! shows theadded effect of the mass flux, and illustrates the heat baladescribed in Eq.~18!, where the area under the curves durithe second and third time periods must equal one anothethe absence of a conductive heat flux. The various curveFigs. 4~a! and ~b! show the effects of varying mean maflow rates on the temperature and heat flux histories, whwill be discussed further in a later section.
A. Temperature difference approximation
Predicting the mean temperature difference with E~21! is fairly cumbersome, and is not very practical for quidesign calculations. Several approximations can be madthis equation to yield a simpler expression that reasonaestimates the temperature difference,TH , and provides aclearer understanding of how it is affected by various indpendent parameters. First, by expanding the pressure terthe denominator of Eq.~21! in a binomial series and assuming that the acoustic pressure amplitude is much smaller tthe mean pressure~i.e., p1!1), the denominator of Eq.~21!can be approximated by
Etd
2p
m6 @ p~ t c!#~g21!/gdtc'E
td
2p
m6 dtc52pm6 0 , ~22!
where the last integral is equal to the net mass displacemfor the entire acoustic cycle~see Fig. 3!. Substitution of Eq.~22! into Eq. ~21! for the temperature difference yields
FIG. 4. Time dependence of~a! the dimensionless temperature on the rigside of the control volume and~b! the dimensionless heat flux from thregenerator solid, during an acoustic cycle for the conditions:p150.1,g51.4, u50, andQ6 k50. The mean incoming gas temperature and relevtime periods are marked form6 0520.05 ~solid line!, which correspond tothe same time periods shown in Fig. 3. For comparison, the temperatureheat flux histories form6 0520.025 andm6 0520.1 are also shown.
2795B. T. Zinn: Open cycle thermoacoustic temperature difference
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~23!
where Eq.~22! has also been applied to the first term in tnumerator of Eq.~21!.
To simplify the pressure term within the integral in E~23!, the pressure ratio can be expanded in a Taylor sewith the use of Eq.~9!, and the resulting expression canfurther expanded in a binomial series to account for the psure ratio’s exponent. Extracting theu dependence from thisresult and using Eq.~6! yields
12F p~ t b!
p~ t a!G~g21!/g
'g21
gp1m6 0~ t a2 t b!sinu
1g21
gp1 cosu@sin~ t a1f!
2sin~ t b1f!#, ~24!
where all terms that are second order and higher inacoustic pressure amplitude have been neglected. Thispression can be further simplified by assuming that the mmass flux is much smaller than the acoustic mass flux,m6 0!1, which eliminates the first term on the right-hand siof Eq. ~24!. Substituting Eqs.~2! and ~24! into Eq. ~23! andeliminating terms of the order ofm6 0 from the integral yields
TH'11Q6 k
m6 01
g21
g
p1 cosu
2pm6 0E
tmid
tdsin~ t b1f!
3@sin~ t a1f!2sin~ t b1f!#dtb , ~25!
where the integral now contains only variables that arfunction ofm6 0 . In the limiting case in whichm6 0 approacheszero, it can be shown that:f→0, t a→2p2 t b , tmid→p, andt d→2p.
Using these limits to evaluate the integral in Eq.~25!yields a compact expression for the estimated temperadifference:
TH'12g21
g
p1 cosu
2m6 01
Q6 k
m6 0. ~26!
Note that in this estimate,TH.1, sincem6 0 and Q6 k are al-ways negative by the sign conventions used in this study,2p/2<u<p/2.
Given the number of approximations made in arrivingthis result, Eq.~26! works remarkably well as an estimate fthe true temperature difference given in Eq.~21!, with errorsof less than 5% forp1<0.1, g<1.67, 2p/2<u<p/2, 20.1<m6 0<20.0001, andQ6 k<0. Even for higher acoustic pressure amplitudes~e.g., p1<0.3), Eq. ~26! approximates thetemperature difference to within about 10% of its true val
To take the estimation of the temperature differencstep further, note that the acoustic energy at the hot sidthe regenerator can be approximated by
Ere'p1m1 cosu
2r re. ~27!
2796 J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weila
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By using Eq.~27!, the ideal gas law, and the definitionsthe dimensionless variables, Eq.~26! can be rearranged toproduce the following interesting result:
m0cp~Tre2TH!'Ere1AsksS dT
dxDint
. ~28!
Equation ~28! indicates that a substantial temperatudifference must exist across the regenerator interface ifengine is going to produce a significant amount of acoupower. A more broad interpretation of Eq.~28! states that ofthe thermal energy convected to the regenerator interfactemperatureTH , some of it is convected into the regeneratat temperatureTre , some of it is conducted through the regenerator solid, and the remainder is used to createacoustic energy exiting the regenerator. This acoustic enis not generated by the processes occurring at the regeneinterface, but rather, this term represents a ‘‘thermoacouheat flux’’ into the regenerator. Note that in an ideal regeerator of a traveling wave thermoacoustic device, thatacoustic energy flux traveling in one direction is accompnied by a ‘‘thermoacoustic heat flux’’ of equal magnitudtraveling in the opposite direction.11 The thermal energy carried by this thermoacoustic ‘‘heat-pumping’’ effect, as italso called, is converted into work in the form of acousenergy within the regenerator. Therefore, in accordance wEq. ~28!, a larger temperature difference at the regenerainterface results in a larger ‘‘thermoacoustic heat flux’’ inthe regenerator, consequently increasing the acoustic pothat is generated with this thermal energy.
B. Effects on the temperature difference
The above-presented analysis shows that the tempture difference between the gas and regenerator depupon the following five dimensionless parameters: the raof mean to oscillating mass fluxes,m6 0 , the ratio of acousticto mean pressures,p1 , the phase angle by which oscillatinpressure leads oscillating mass flux,u, the ratio of specificheats,g, and the thermal conduction loss in the regeneraQ6 k . The parameter with the largest effect on the temperadifference ism6 0 , as shown in Fig. 5. As the magnitude omean mass flux decreases relative to the magnitude of
FIG. 5. The dependence of the temperature ratio of the incoming gas anregenerator solid upon the mean to oscillating mass flux ratio and theof specific heats,g, for the conditions:p150.1, u50, andQ6 k50. Note thatthe negative mean mass flux indicates gas flow toward the regenerato
nd and B. T. Zinn: Open cycle thermoacoustic temperature difference
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acoustic mass flux,TH becomes fairly large. The functionarelationship between the temperature difference andmean mass flux is shown in Eq.~26!, where the temperaturdifference is a function of the inverse of the mean mass fl
This dependence on the mean mass flux can beplained by considering its effect on the discontinuity timt d , given in Eq.~7!. A reduction inm6 0 increasest d , thusdecreasing the fraction of the acoustic cycle during whichfresh gas enters the regenerator. This effect is depicted inheat flux plot of Fig. 4~b!, where the time at which the discontinuity occurs form6 0520.1 is earlier than the discontinuity time for m6 0520.05. As shown in the figure, the totaheat transferred to the gas during the second time period~thearea under the curve! is only slightly increased by the increased limits of integration. The primary effect of decreing the mean mass flux and increasing the discontinuity tis to decrease the time allotted for heat transfer fromfresh gas to the regenerator solid. To compensate, the mtemperature of the incoming gas must increase so that theheat input from the gas~the area under the curve for the thitime period! equals that of the heat output from the solid,expressed in the heat balance in Eq.~18!.
As Fig. 5 illustrates, the mean temperature differenbetween the gas in the regenerator and the open duct isnecessarily small, and can be much larger than the otypes of temperature differences noted in the thermoactics literature to date.6–11 This is primarily a result of themean flow’s role as the heat source, whereas other anahave only investigated temperature differences associwith the use of heat exchangers. Temperature differencethe magnitudes seen here can have a profound effect onefficiency of the engine, although Eq.~28! would suggestthat it may not necessarily be advantageous to minimizetemperature difference, as it is directly linked to the acoupower output of the engine.
Figure 5 also describes the effect of the ratio of specheats,g, on the magnitude of the temperature differenceexpressed in Eq.~21!. These effects are the result of thinfluence ofg on the isentropic relationship between temperature and pressure outside the regenerator, as describEq. ~10!. As Eq.~26! shows, the ratio of specific heats hassmaller effect upon the temperature difference than the oindependent parameters, however, it is the only one offive independent parameters that depends upon the gaserties within the engine. As such, monatomic gases~g'1.67!result in higher temperature differences than nonmonatogases. For nonmonatomic gases the mean gas temperplays a small role, as gaseous combustion productsT'2000 K), for whichg'1.33, will yield slightly lower tem-perature differences according to Fig. 5, than will ambietemperature nonmonatomic gases, for whichg'1.4.
The dependence of the temperature difference uponacoustic phase lag,u, and the acoustic pressure magnitudp1 , is shown in Fig. 6. Consistent with the cosu dependencepredicted in Eq.~26!, the acoustic phase shift produces tlargest temperature differences nearu50, corresponding totraveling wave acoustic phasing. This condition maximizthe difference in a gas parcel’s temperature from exit toturn to the regenerator, which increasesQ6 b , the heat flux
J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weiland and
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from the regenerator to the incoming gas, and requiresQ6 c andTH increase correspondingly. The temperature diffence decreases away fromu50, until no temperature differ-ence exists for standing wave acoustic phasing atu56p/2.Since the acoustic displacement, pressure, and temperare all in phase in a standing wave, a gas parcel will undeisentropic oscillations outside of the regenerator but willturn to the regenerator at the same temperature and preas when it left.11 The addition of mean flow introduces slighphase differences between temperature and displaceover the course of an acoustic cycle, but the resulting hfluxes cancel one another foru56p/2.
Figure 6 also shows the approximately linear relatioship between the acoustic pressure magnitude and theperature difference of Eq.~21!, with higher acoustic pressures resulting in higher temperature differences. Trelationship is predicted by Eq.~26! and can be linked to theTaylor series expansion in Eq.~24!, where the temperature othe gas returning to the regenerator during the second pathe acoustic cycle,TR,b , is shown to be approximately proportional top1 for small values of the acoustic pressure manitude. Therefore, the total heat flux from the solid to the gduring the second part of the acoustic cycle is proportionap1 , which is in turn approximately proportional to the temperature difference required to maintain the heat balanRecognizing the link between the temperature differencethe heat flux across the regenerator’s interface, the resulFig. 6 for both the pressure and phase angle are in qualitaagreement with the results of Smith and Romm.6
Finally, Fig. 7 shows the combined effects of the coductive loss term,Q6 k , and the mean mass flux on the temperature difference. While the mean mass flux effect issame as that shown in Fig. 5, the addition of a conducheat loss increases the required temperature difference aregenerator’s interface. For a given mean mass flux, thispendence is shown to be approximately linear, as predicby Eq. ~26! for the estimated temperature difference. Thbehavior is best explained in conjunction with Eq.~18!,which states that the total energy change in the regenersolid over the course of an acoustic cycle is equal to the htransfer from the gas to the solid during the third time periominus the heat transferred from the solid to the gas dur
FIG. 6. The dependence of the temperature ratio of the incoming gas anregenerator solid upon the phase angle by which oscillating pressurethe oscillating mass flux,u, and the acoustic to mean pressure ratio,p1 , forthe conditions:m6 0520.05,g51.4, andQ6 k50.
2797B. T. Zinn: Open cycle thermoacoustic temperature difference
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the second time period, minus the heat leaving the convolume via thermal conduction. Therefore, for increasheat conduction losses, an increasing heat input fromthird time period is required~which is proportional to thetemperature difference!, in order to enforce the condition thathere be zero net energy change in the regenerator’s sfrom one acoustic cycle to the next. As such, minimizingconductive heat leak increases the portion of the thermalergy input available for conversion to acoustic energy.
V. CONCLUSIONS
The theoretical analysis developed in this study allothe determination of the temperature difference that occurthe regenerator interface in open cycle traveling wave thmoacoustic engines, when mean flow replaces the hotexchanger as a means for supplying heat to the engine.temperature difference is shown to be a function of fivedependent dimensionless parameters, and is well appmated by Eq.~26! above. With slight modifications, the theoretical framework developed in this study could alsoused to evaluate open cycle designs for traveling wave tmoacoustic refrigerators and heat pumps. The temperadifferences predicted by these analyses could significaaffect the performance of these open cycle devices, andpositively or negatively affect their feasibility as alternativto their closed cycle counterparts. Previous studies on tmoacoustic temperature differences typically associate twith loss mechanisms, where minimizing them generally iproves the performance of the thermoacoustic device. Th
FIG. 7. The dependence of the temperature ratio of the incoming gas anregenerator solid upon the mean to oscillating mass flux ratio and themensionless conductive heat loss,Q6 k , for the conditions:p150.1,u50, andg51.4.
2798 J. Acoust. Soc. Am., Vol. 114, No. 5, November 2003 N. T. Weila
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not found to be the case in the open cycle thermoacouengine, however, since reducing the temperature differehas the direct effect of reducing the acoustic power outputhe engine, according to Eq.~28!.
Predicting the existence of this temperature differencthe first step in the evaluation of the feasibility of open cyctraveling wave thermoacoustic devices, and suggests sesubsequent avenues of research. For instance, the losseresult from the heat transferred across the potentially latemperature difference could be analyzed and comparethe losses in the hot heat exchanger of a closed cycle tmoacoustic engine. Also, the effects of this temperatureference on the theoretical efficiencies of open cycle devicould be studied and compared to those of closed cyclevices. Finally, given the potential for increasing the efficiecies of thermoacoustic devices by employing open cyconfigurations, these analyses need to be verified in the laratory to determine the accuracy of the assumptions anddictions that are made in these studies.
ACKNOWLEDGMENT
The authors would like to thank Greg Swift for his helful discussion of this topic.
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2R. S. Reid and G. W. Swift, ‘‘Experiments with a flow-through thermocoustic refrigerator,’’ J. Acoust. Soc. Am.108, 2835–2842~2000!.
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6J. L. Smith and M. Romm, ‘‘Thermodynamic loss at component interfain Stirling cycles,’’ Proceedings of the 27th Intersociety Energy Convsion Engineering Conference, San Diego, CA, Society of Automotivegineers, 1992, pp. 5.529–5.532.
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9P. Kittel, ‘‘The temperature profile within pulse tubes,’’ Adv. Cryog. En43, 1927–1932~1998!.
10L. Bauwens, ‘‘Interface loss in the small amplitude orifice pulse tumodel,’’ Adv. Cryog. Eng.43, 1933–1940~1998!.
11G. W. Swift, Thermoacoustics: A Unifying Perspective for Some Enginand Refrigerators~Acoustical Society of America, Melville, NY, 2002!.
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nd and B. T. Zinn: Open cycle thermoacoustic temperature difference