Effect of Ambient Gradients on Sound Transmission in Narrow Permeable Rectangular Pipes with Application to Heat Exchangers

Transcription

1 Advances n Pwertrans and Auttves, 05, Vl., N., 4- Avalable nlne at Scence and Educatn Publshng DOI:0.69/apa--- Effect f Abent Gradents n Sund Transssn n Narrw Pereable Rectangular Ppes wth Applcatn t Heat Exchangers Sabry Alla * Auttve Technlgy Departent, Faculty f Industral Educatn, Helwan Unversty, Car, Egypt *Crrespndng authr: alla@kth.se Receved Aprl 0, 05; Revsed May 08, 05; Accepted May 5, 05 Abstract The effect f abent gradents n sund prpagatn n ar flled narrw tube was nvestgated. The narrw tubes were taken t be nnally straght wth very sall pres n the walls. The slutn ncludes the effect f a statc pressure, teperature and densty gradents n the presence f ean flw, whch s assued t have a unfr velcty prfle. A dspersn equatn s derved by assung the spatal varatns f the abent varables can be luped by usng ther average values. The cplex wave nuber, densty, speed f sund and the characterstc pedance f such eda were evaluated. An applcatn t fulfl narrw tubes wth rectangular crss sectn and pereable walls such as; heat exchangers s develped and presented. An accurate acustc del based n tw-prt atrx t calculate the transssn lsses n the heat exchanger (HE takng the abent gradents effects nt accunt are develped and used t study and prve the acustc perfrance f HE. The develped del s valdated wth the easured results usng nral ncdent and dffuse feld at r teperature and a gd agreeent s acheved. Based n the results presented n ths paper, the acustc perfrance f the exstng heat exchanger s bad especally at lw frequences, the peratng cndtns have se pstve effects n ts perfrance and ts acustc perfrance can be prved thrugh channel wall pedance. Extra prveents are stll needed t use t as a passve nse cntrl eleent t dap the fan nse. Keywrds: sund transssn, narrw pereable ppes, nral ncdent and dffuse feld easureent, wall pedance ptzatn and heat exchanger Cte Ths Artcle: Sabry Alla, Effect f Abent Gradents n Sund Transssn n Narrw Pereable Rectangular Ppes wth Applcatn t Heat Exchangers. Advances n Pwertrans and Auttves, vl., n. (05: 4-. d: 0.69/apa---.. Intrductn.. General Acustc wave prpagatn wthn a hgeneus ther-vscus flud-flled edu at rest, unbunded n all drectns, nvlves reactve and absrbng prcesses, whch can be characterzed, n the frequency dan, by a cplex wave nuber. The agnary part f whch s prprtnal t the shear and bulk vscsty ceffcents and the heat cnductn ceffcent. It can als nclude dsspatn prcesses due t lecular relaxatn usng the apprprate cplex specfc heat rat []. In a bunded dan (duct r cavty, the reactve and absrbng prcesses at rgd bundares arse fr nteractns between the acustc veents and bth the entrpc veent (dffusn f heat and the vrtcal veent (dffusn f shear waves, whch are created n the bundary walls, extractng energy fr the acustc wave. The entrpc and vrtcal perturbatns dffuse nt the edu n drectn nral t the bundary, whch s a cnsequence f the unfr bundary cndtns, and de ut befre reachng the ppste wall. Prvded that the lcal curvature and dstance between the walls are large enugh wth respect t the bundary layer thcknesses. In these stuatns, the absrptn f acustc waves utsde the bundary layers can be characterzed, n the frequency dan, by the agnary part f a cplex wave nuber, whch n st cases wll be prprtnal t the square rt f the shear vscsty ceffcent as well as that f the heat cnductn ceffcent []. A sewhat dfferent apprach s needed n very sall cavtes and narrw ducts, where ne r tw f the densns are f slar agntude t the bundary layer thcknesses. In these stuatns, whch can be fund n nuerus acustc devces (re partcularly electracustc devces, the heat dffusn (entrpc veent and the shear wave dffusn (vrtcal veent have apltudes f the sae rder f agntude as the acustc wave tself (whch acts n the wall as a surce fr the entrpc and vrtcal veent. In these crcustances, the apprach ust nvlve a precse descrptn f the partcle veent nsde the bundary layers. Ths partcle veent can be expressed n ters f a superpstn f three knds f cpnents: frst, the rgnal wave (whch prvdes energy, secnd, theral and vscus dsspatn n the fr f crrectn factrs

2 Advances n Pwertrans and Auttves 5 added t the acustcal wave functn and thrd, dffusn f heat and shear waves als descrbed by eans f addtve functns []. Over the last decades, theretcal actvtes n the subject prvde a relevant glbal frulatn, whch wuld be able t satsfy the requreents entned abve fr descrbng the acustc felds n ther-vscus fluds. In publcatns whch relate t ths subject ne fnds: ( the pneerng wrks f Dkuac [], whch lead t bth a dellng f vscus bundary layers n a narrw crcular and rectangular ppes by takng n hs accunt the abent gradents effect n the case f crcular narrw ppes wth the sld wall ( the wrks f Cungs fr rectangular and square narrw ducts [5] and ( Peat et al [6], whch uses nuercal ethds t del thn layers f vscus flud trapped between parallel walls (v K.- W. JEONG etal [8] uses a nuercal ethd t study the sund transssn n capllary tubes wth ean flw. All f these papers assue that they have stheral bundary cndtn and hard wall, very few wrk such as [0], have been ntrduced t pleent the wall effect, n ther wrk an average slutn t the sund transssn n narrw ppes wth prus wall s presented and appled t desel partculate trap. Alla and Åb [] presented a splfed apprach based an equvalent flud del t study the sund transssn n HE, ths del s prtve and based n the easured flw resstvty f the HE. S, the prble f sund waves prpagatn n narrw pereable rectangular ppes carryng ean flw wth abent gradent s nt addressed yet. Generally, the heat exchanger s a specalzed devce that asssts n the transfer f heat fr ne flud t the ther. In se cases, a sld wall ay separate the fluds and prevent the fr xng. In ther desgns, the fluds ay be n drect cntact wth each ther. In the st effcent heat exchangers, the surface area f the wall between the fluds s axzed whle sultaneusly nzng the flud flw resstance. Fns r crrugatns are setes used wth the wall n rder t ncrease the surface area and t nduce turbulence. Cn applances cntanng a heat exchanger nclude ar cndtners, refrgeratrs, and space heaters. Heat exchangers are als used n checal prcessng and pwer prductn. Perhaps the st cnly knwn heat exchanger s a car radatr, whch cls the ht radatr flud by takng advantage f arflw ver the surface f the radatr []. There are als fur dfferent desgns f heat exchangers: shell and tube, plate, regeneratve, and nteredate flud r sld. The st typcal type f heat exchanger s the shell and tube desgn. Ths heat exchanger has ultple fnned tubes. One f the fluds runs thrugh the tubes whle the ther flud runs ver the, causng t t be heated r cled. In the plate heat exchanger, the flud flws thrugh baffles. Ths causes the fluds t be separated by plates wth a large surface area. Ths type f heat exchanger s typcally re effcent than the shell and tube desgn []. The regeneratve heat exchanger takes advantage f the heat fr a specfc prcess n rder t heat the flud used n the sae prcess. These heat exchangers can be ade wth the shell and tube desgn r the plate desgn. The nteredate flud r sld heat exchanger uses the fluds r slds wthn t t hld heat and ve t t the ther sde n rder t be released. Ths ethd s cnly used t cl gases whle revng purtes at the sae te. The bject f ths paper s t develp an accurate acustc del based n tw-prt atrx t study and prve the acustc perfrance f the plate heat exchangers (HE takng the abent gradents effects nt accunt, by slvng the cnvectve wave equatn n narrw rectangular ppes wth the relevant bundary cndtns... Structure f the Paper Ths paper descrbes the effect f abent gradents n the sund transssn n narrw rectangular pereable tubes carryng ean flw. An analytcal slutn t the prble s presented n sectn and suary f the easureent setup and prcedure s presented n sectn, the del s eplyed t study the acustc perfrance f HE and cpared wth easured results at r teperature n sectn 4. Fnally, cnclusns and prpsal fr Future wrk s presented n Sectn 5.. Prble Frulatn.. Wave prpagatn n Narrw tubes The gvernng equatn fr a perfect gas n a unfr narrw rectangular ppe can be wrtten usng the basc equatns presented n [4]. Assung the ean flw velcty prfle t be unfr acrss the ppe crsssectnal area and axal flw velcty, teperature, pressure gradents are taken nt accunt. The cnvectve equatns can be expressed, wth exp( ω t te dependence assued fr the fluctuatng quanttes, where ω s the radan frequency, t s the te and dentes the unt agnary nuber, as fllws. The entu equatn s; dv ρ ω + v vx + ρvx x dx ( p j µ + s vx, p p( x x The cntnuty equatn s: dv d ρ ν x ( xyz,, ω + v + ρ + vx + ρ x dx dx x ( ρ τ vτ ( x, y, z dy dz A A Here, x dentes the ppe axs, ρ A ρ ( x, y, z dydz, v x A v x( x, y, z dydz are the crss-sectnal averaged acustc densty and x cpnent f partcle velcty wth Area A s crss - sectn averaged acustc. Applcatn t Eq. ( f the dvergence there n the (x,y plane gves dvj d ρ ν x ( xyz,, ω + v + ρ + vx + ρ x dx dx x ( ρ n( yz, vτ ( xyzds,, A S

3 6 Advances n Pwertrans and Auttves where S beng the pereter f a an pre havng utward unt nral n and ds an eleent f pereter. In case f pereable walls narrw ppes equatn ( reduce t dv j d ρ ν x ( xyz,, ω+ v + ρ+ vx + ρ 0 (4 x dx dx x Fgure. The geetry f a heat exchanger tubes The energy equatn when the perfect gas beng assued and neglectng the cnvectve heat exchange wth the channel walls, s dtj dtj ρ Cp ω + v T + ρcpx v + ρcp v x dx dx (5 dp ω + vj p+ vx + kth s T. x dx The state equatn s p ρ T ρ, RT T (6 Whle p, T and ρ s the acustc pressure, teperature and average densty respectvely, µ s the shear vscsty ceffcent. k th s the theral cnductvty f the flud, R s the gas cnstant, C p s the specfc heat ceffcent at cnstant pressure. The ean flw velcty v assued t be axal and have velcty prfle, the 0 ean teperature T s assued t be a functn f the axal c-rdnate x. µ and k th are the shear vscsty ceffcent and theral cnductvty, respectvely are assued t be slwly varyng functn f T s that the gradents µ dx and κ th dx are sall t the frst rder. The Laplacan n the crss-sectn s and the dvergence f the partcle velcty.v are gven by υ y,. v x υ υ z s y z x y z (7 where, y and z dente the transverse c-rdnates as shwn n Fgure, wth the ppe crss-sectnal area lyng n, υy and υz are the cpnents f the partcle velcty n the y and z drectns. Accrdng t Eq. (, the average gas densty n the an tubes changes n te due t cpressn f the gas and as a result f ass flux ρ nxy (, vτ ( xyz,, nt the pre wall. A reasnable assuptn s that the pereable wall radus s t sall cpare t an tube and wave length» a (the duct hydraulc radus. Under ths assuptn the wall can be taken lcally t be a flat surface havng specfc acustc pedance Z w. In ths averaged sense cntnuty f the nral cpnent f at the tube wall bundary s taken t be n( ( ( xy, vτ xyz,, px Zw (8 Here p s taken t be cnstant n a gven crss sectn f a an pre. Use f the bundary cndtn, Eq. (7, n the cntnuty equatn, Eq. (, gves dvj d ρ ω+ v + ρ vx x dx + + dx S ρ. ν + ρ p AZw (9 Recall agan S s the pereter f the arbtrarly shaped an pre whch s shwn n Fgure. The axal dstrbutn f the ean teperature and pressure s assued t be lnear and are expressed n the fllwng frs ( ( + τζ ( ( + εζ T x T, p x p, ζ + xl (0 where, p dentes the abent pressure, L s the length f the ppe, the ver bar ( dentes an axal average, and the teperature change paraeters are defned as ( ( 0 ( ( 0, ε T L T p L p τ ( T p Snce the ean flw velcty n the transverse drectn s t sall cpare t axal ean flw velcty, s the axal ean can be deterned n the sae way lke the hard wall fr the cntnuty equatn fr the ean flw, ρ v cnstant. The axal varatn f the ean densty dstrbutn s then deterned by the perfect gas law p ρ RT. Hence, the abent gradents, whch ccurrng n the gvernng acustc equatns can be

4 Advances n Pwertrans and Auttves 7 expressed n ters f the teperature and pressure change paraeters by usng the fllwng relatns. dt τt dp ε p,, dx L dx L d ρ dv T τ p ε ρdx vdx T L p L ( Usng these gradents (8, (9 and (0 and drppng the ndex j t avd any dsturbance equatn (, ( and ( can be expressed t ( οτ and ( ο π as fllws ( τ ε vx vx ρω[ ( τ ε ζ] vx+ ρv + x L ( p τζ + µ + svx x Usng equatn (9, (0, and ( wth equatn (4 yeld τ T T [ ( ] { } ρ C 0 pω τ ε ζ T + v L x γτ p γτ p v x L( γ ε ωp+ v0 γ L p [ ( τ ε ζ] x τζ + kth + 0 T (4 Usng the abve-descrbed apprxatn, the slutn f equatn (, ( and ( can be searched n the fr; ( ( ( p Aexp Γ kx, υ x H yzpt,, F yz, p (5 where, A dente any arbtrary cnstants. Substtutng equatns (5 n equatns ( and (4 yelds and where where, F y H H + + β H β z y ( (, F + + σ F σ + σ H yz z Γ k ω τ Mγ β, σ M Γ+, µ κth ( γ kl p τ j γ σ, ε ( β a Φ ( ε τ ss, κth L ( γ Pr, ( σ a ( Φ ( τ s γ p ρ ω ω k, c 0, s a, c ρ µ 0 µ C v Pr P, M k th c (6 (7 (8 (9 The functn Φ s defned as M j ε Φ ( ε M Γ+ (0 kl where, a halve f the nlth wdth, M s the average ean flw Mach nuber, c s the speed f sund, γ s the rat f the specfc heat ceffcents, k s the wave nuber, s s the shear wave nuber, and Pr s the average Prandtle nuber. The slutn f the equatn (6 can be expressed n the fr f a duble Furer seres []. y nz H( yz,, n hn sn sn, a b + π, 0,,,.. ( The ndex and n. The ceffcent hn ( can be deterned by substtutng the equatn ( n equatn (6 (the strategy f the slutn s presented n the Appendx A, whch yelds 6k Γ h n ( µ n βαn ( βa By nsertng equatn ( n equatn (, H can be deterned by ntegratng and averagng the resultng ver the ppe crss-sectn area y nz H( yz,, n hn sn sn a b By ntegratng ver the ppe area and after se ath where ( Γ H J( β a (4 Φ ( ε τ ρ c J( β a 64 n, nαn ( βa a αn ( βa 4 + n ( β a b (5 Als, the slutn f the equatn (7 can be als expressed n the fr f a duble Furer seres. y nz F( yz,, n fn sn sn a b (6 The ceffcent f n can be deterned usng the sae technque and t presented n the Appendx B. 6 σ β f n σ + ( a (7 nσ α σ βαβ ( a By ntegratng ver the tube area gves σ 0 σ ( ( β J σ a F J σ a (8 σ σ β αn ( βa

5 8 Advances n Pwertrans and Auttves 64 n,. nαn ( σa Usng equatn (9, (0,(9 and (0 wth (8 yeld where J( σ a where S (, k I I I ρ (9 AZw c c I { Φ ( ε }, I { Φ( τ } F( yz, p T 0 0 ( ε τ and I +Γ H ( y, z kl (0 By ntegratng and take the average arund the tube crss sectn gves where ( ε k Φ { γ + CI ( σa + ρ S C I( σβ a + C I( β a} ρ AZw ( ε ( ε τ ( Φ( τ σ0 Φ( τ σ β C, 0 ργr C, ( ργr Φ ε σ Φ( ε σ β ( τ π ( Γ +Γ kl and C Φ Φ Equatn ( s slved usng Newtn-Raphsn ethd fr Γ and saple f the results s shwn n Fgure ( Γ T00 K, U0 /s Real (Γ + Real (Γ - Iag (Γ + Iag (Γ Frequency (Hz Fgure. Prpagatn cnstants versus frequency at r teperature and flw speed 0 /s. HE cell prsty σ64%. Wall prsty σ 0.5%. Cell densn, (a0.5, b.5, L8... Slt-Shaped Perfrated Wall Ipedance Fr ths type f plates that shwn n Fgure (c, Allard [4] gves an equatn fr the pedance f a slt jωt tanh( β j ηω r real( + σc β j σc ω ( and jωt tanh( β j ωb ω ag + F( e (4 σ c β j σc b where β ρωη, b s the wdth f the slts, σ s the wall prsty (nuber f slts area f slt/ slt wall area. By ntrducng an ellptcty factr e, l s the half length b f slt. e ( π dθ, and F( e l 0 ( e cs θ Whch can be expressed as a seres beces [4]: F ( e π + e +.4 e e + (5 Lsts f value f l/ b crrespndng t the ellptcty e and crrespndng F(e are shwn n Table. Table. The ass end crrectn f an ellpse fr Maa [5] l/ b e b ( ½ F(e l Sund Transssn Calculatn The sund transssn thrugh the heat exchanger s st effectvely handled usng a transfer-atrx apprach []. S, f nly plane waves exst n a syste wth just tw penngs the syste can be descrbed as an acustc twprt atrx. The st cnly used del s develped by usng acustc pressure p and velcty v t represent the nput and utput state vectr. Usng the bundary cndtns at x 0, L and the cntnuty f p and v, ples that we nly have t analyze the reflectn and prpagatn n the x drectn. A transfer atrx suted fr ths prble can be defned by: pˆ T T pˆ, vˆ T vˆ 0 T x x L where T s gven by []: cs( ΓkL sn( ΓkL T H H sn( ΓkL cs( ΓkL (6 (7

6 Advances n Pwertrans and Auttves 9 T btan the acustc transssn the bundary cndtns and the plan wave relatnshps n the nlet and utlet sdes wth the help f equatn (5 ply that: and pˆ x 0 pˆ ˆ ˆ ( + p r p + r vˆ x 0 pˆ ( r H pˆ x ˆ ˆ L p t τ p vˆ x pˆ H pˆ H L t τ (8 (9 where the apltudes f the reflected and the transtted felds are related t the ncdent wave wth a transssn ceffcent τ ( τ pˆ t / pˆ and a reflectn ceffcent r ( r pˆ r / pˆ. Substtutng equatns (8 and (9 nt equatn (6, yelds + r τ cs( Γ kl + τsn( ΓkL r τsn( Γ kl + τ cs( Γ kl (40 By evaluatng equatn (40, the transssn and reflectn ceffcents can be btaned as τ cs( Γ kl + sn( ΓkL (4 The prpagatn cnstant Γ can be btaned fr equatn (. Transssn lss TL fr a certan ncdent angle (θ, φ beces TL( n 0 lg. τ ( n (4 equally between the upstrea and dwnstrea sde as shwn n Fgure. The dstances between the ludspeakers were chsen t avd any pressure na at the surce pstn. Sx flush unted cndenser crphnes (B&K 498 were used, three upstrea and three dwnstrea f the test bject fr the plane wave decpstn, the crphne separatns are chsen t fulfll the frequency ranges f nterest up t KHz. All easureents are perfred usng the surce swtchng technque [7] and the flw speed was easured upstrea f the test sectn usng a sall ptttube fxed at a dstance f 000 fr the upstrea ludspeakers sectn and cnnected t an electrnc aneter at SWEMA AIR 00. The pressure drp acrss each saple s als easured n usng the sae electrnc aneter. Three dfferent errrs can be ccurred ( the errrs n the easured nput data; ( the errr senstvty f the plane wave decpstn and ( the errr senstvty f the atrx equatn. Pnt ( and ( have been dscussed n [8] and cncluded that t btan gd easureent results, the plane wave decpstn ust be restrcted t the frequency range ( M ks π ( M 0.π < < 0.8 (45 Where k s the wave nuber and s s the crphne separatn. Regardng pnt ( t wll anly depend n the nversn f the atrx. When ths atrx s sngular; r alst sngular, large errrs can be expected. T avd ths, the tw test states fr the tw-prt ust be sgnfcantly dfferent. All these precautns have been taken nt accunt t avd any easureent errrs. where n s the unt vectr alng the ncdence drectn. By averagng equatn (4 ver all pssble ncdent angles, the transssn lss n a dffuse feld s btaned [4] TL 0 lg τ. (4 where τ ( n all angles π / π (, τθφ sn θ d θ d φ. π θ 0 φ 0 (44. Suary f Experental Prcedure.. -Prt Experental Prcedure Experents were carred ut at r teperature usng the flw acustc test faclty at the Marcus Wallenberg Labratry (MWL fr Sund and Vbratn research at KTH. The test duct used durng the experents cnssted f a standard steel-ppe wth a wall thckness f, duct nner daeter d 9 and verall length f arund 7 eters. Fur ludspeakers were used as external acustc surces, and they were dvded Fgure. Measureent set up used durng the - Prt Experental prcedure

7 0 Advances n Pwertrans and Auttves.. ISO 586 Experental Prcedure In ths prcedure, the bject s treated as a wall eleent and the sund surce ettng whte nse was unted n the reverberatn r and the sund reductn ndex can be easured wth ISO 586-:000 [9]: TL L Sp 6 L SI + 0lg( A / A (46 where L Sp s the sund pressure level easured by a rtatng crphne n the reverberatn r and L SI s the sund ntensty level btaned by scannng the surface f the heat exchanger wth an ntensty prbe (L.D. 800 n the recevng anechc r. The whle setup can be seen n Fgure 4. Here, A s the ttal area f the easureent surface and A s the area f the test specen under test, and n these easureents the scannng s dne verall the test specens... Cparsn between Results fr Tw Measureent Methds Due t the dfference between the nral ncdence that s used wth -prt easureent and blque ncdence plane wave, whch s used wth ISO 586, ne can ntce that there s deference between the easured results as shwn n Fgure 5. Ths dfference s fund t be between (-.5 db n st f the easured cases. 4. Results and Dscussn Fgure 4. Measureent setup used wth ISO standard (586 -:000 prcedure Fgure 5. Dfference n easured TL usng ISO and TMM. HE densns ( , cell prsty σ64% Fgure 6. Transssn lss n ne thrd ctave band versus frequency. HE Densns ( , cell prsty σ64%. Wall prsty σ 0.5%, T 00 K, τ0. The develped del presented n sectn, s used t calculate the sund transssn lss n heat exchangers fr bth nral ncdent sund feld (where θ 0, and φ0 and dffuse sund feld. Fgure 6 shws the transssn lss fr a saple f an auttve heat exchanger at tw dfferent flw speeds (U0, and 0 /s at r teperature. It can be seen fr the results, the del gve a gd agreeent wth the easured results n bth nral ncdent and dffuse sund felds. It can be als nted that sund transssn lss f the Heat exchanger s very sall especally at lw frequency, then keywrds t prve the heat exchanger acustc perfrance thugh the ppes wall pereablty.e. thrugh wall pedance and heat exchanger wall thckness. The wall pedance can be prved usng dfferent wall prsty, ateral and wall thckness. But, these paraeters are crtcal because they are affect the theral perfrance f the heat exchangers and the pressure drp, n ther wrds

8 Advances n Pwertrans and Auttves se knd f nnlnear ult-bject ptzatn s needed t prve the heat exchanger perfrance and use t t as acustc eleent as well as theral eleent. The effect f flw speed n HE pressure drp s very sall as shwn n Fgure 7, ths due t ts shrt length and cell penng prsty, whch s σ64% but t can be reduced r at least kept cnstant durng redesgnng and ptzatn prcess. The effect f desgnng paraeters such as wall pedance, cell densn and prsty can be used as exaples t prve the acustc perfrance f HE. The effect f wall prsty n the sund transssn lss s presented n Fgure 9, t can be seen that TL drect prprtnal t the wall prsty, but t avd ts effect n the theral perfrance a full ptzatn prcess shuld be dne. Transssn lss (db Car Heat Exchanger σ 0.09 σ σ 0.0 σ Measured usng TMM at U0 /s Frequecy (Hz Fgure 9. Effect f wall prsty n transssn lss. Cell prsty, σ64%, U0 /s, T 00 K, τ0. 5. Cnclusn and Future wrk Transssn Lss (db Fgure 7. Pressure drp versus flw speed at r teperature τ 0.0 τ 0.07 τ HE, V 5 /s Frequency (Hz Fgure 8. Transssn lss n ne thrd ctave band versus frequency. HE Densns ( , cell prsty σ64%. Wall prsty σ 0.5%. T 0 K. Cparng Fgure 8 and Fgure 7, t can ntce that the peratng cndtns have bg effects f acustc perfrance f HE. Ths s bascally ces fr the effect f peratng cndtns n the ar densty and speed f sund. It can be als, ntce fr Fgure 8 that flw drectn has se effect. It can be entned that flw drectn depends n type f unt; vable (plus drectn r statnary (negatve drectn. In ths paper, an apprxate slutn t sund transssn n narrw tubes carryng ean flw wth abent gradents s presented. The develped del s eplyed t study the acustc perfrance f the heat exchanger ang at ntrducng t as an acustc eleent fr fan passve nse cntrl and t has been exaned bth experentally and theretcally. The experental assessent has been perfred usng tw dfferent ethds; dfed versn f ISO 586- :000 and the acustc Tw-Mcrphne Methd (TMM. Theretcally, the basc cnfguratn s assued t be a atrx f parallel and rectangular narrw pereable channels. The develped del s based n quas D wave prpagatn n narrw ppe wth pereable bundary. Dfferent results t characterze the acustc perfrance f the heat exchanger are presented. Fr these results t s clear that the sund reductn n the exstng HE s qute pr typcally less than 5 db up t - khz and t can be prved usng the develped del. The peratng cndtns have se pstve sgn effects n ts perfrance. Stll an nterestng questn s therefre f, wth unchanged theral effcency and pressure drp, ne can prve the acustc perfrance by ntrducng wall perfratns wth ptu pedance. Acknwledgeent Part f ths wrk has been fnanced by EU cssn Grant Agreeent n: SCP8-GA ECOQUEST. The techncal and scentfc supprt fr Mats Åb fr KTH, Sweden and Manuel Henner, fr Vale, France are beynd f y estatn.

9 Advances n Pwertrans and Auttves References [] R. Bssart,N. Jly and M. Bruneau Hybrd nuercal and analytcal slutns fr acustc bundary prbles n thervscus fluds Jurnal f Sund and Vbratn 6 ( [] Alland Perce Acustc Mc Graw-Hll, Inc. cpyrght 98. [] E. Dkuac On Transssn f Sund n Crcular and Rectangular Narrw Ppes wth Superpsed Mean Flw jurnal f sund and vbratn (998 0 (, [4] E. Dkuac AN APPROXIMATE DISPERSION EQUATION FOR SOUND WAVES IN A NARROW PIPE WITH AMBIENT GRADIENTS Jurnal f Sund and vbratn (00 40(4, [5] R. J. ASTLEY and CUMMINGS Wave prpagatn n catalytc cnverter: Frulatn f the prble and fnte eleent schee. Jurnal f Sund and vbratn (995 88(5, [6] K. Peat A frst apprxatn t the effects f ean flw n sund prpagatn tn capllary tubes Jurnal f Sund and Vbratn ( [7] K. Peat Cnvected acustc wave tn alng A Capllary Duct wth an axal teperature gradent Jurnal f Sund and Vbratn (997 0 ( [8] K.-W JEONG and J.-G IH A nuercal study f the prpagatn f sund thrugh capllary tubes wth ean flw Jurnal f Sund and Vbratn ( [9] N. Dckey, A. Selaet, J. Nvak Mult-pass perfrated tube slencers: A cputatnal apprach Jurnal f Sund and Vbratn (998 (, [0] Sabry Alla and Mats Åb Sund Prpagatn n An Array f Narrw Prus Channels wth Applcatn t Desel Partculate Flters Jurnal f Sund and Vbratn Vl. 9, (006, [] Sabry Alla and Mats Åb Acustc Mdellng and Characterzatn f Plate Heat Exchangers. SAE Paper [] Raesh K. Shah and Dusan P. Sekulc; 00 by Jhn Wley & Sns, Inc. Fundaentals f Heat Exchanger Desgn. [] L.S. Han Hydrdynac Entrance Lengths fr Incpressble Lanar flw n Rectangular Duct Jurnal f Appled Matheatcs, Transactns f ASME, Septeber, 960, [4] Allard, J. F., 99, Prpagatn f Sund n Prus Meda, Mdellng Sund Absrbng Materals, Elsever Appled Scence, Lndn. [5] D. Y. Maa, Ptental f Mcr-perfrated panel absrber, J. Acust. Sc. A. 04, (998. [6] L.R.Kval 976. The Jurnal f the Acustcal Scety f Aerca 59, Effect f ar flw, panel curvature and nternal pressurzatn n feld-ncdence transssn lss. [7] Mats Åb. Measureent f the Scatterng-Matrc f Acustcal Tw-Prts. Mechancal Systes and Sgnal Prcessng (99 5(, [8] Hans Bdén and Mats Åb Influence f errrs n the twcrphne ethd fr easurng acustc prpertes n ducts. J. Acust. Sc. A. 79, 54 (986. [9] ISO 586-:000 Acustcs - Measureent f sund nsulatn n buldngs and f buldng eleents usng sund ntensty - Part : Labratry easureents. Appendx A The ceffcent hn can be deterned by substtutng the equatn ( n equatn (6 and the slutn can dscuss as fllw y z n h sn sn, n n a a b π y z n n + h sn sn, n n b a b y z n k Γ + h ( β sn sn, n n a b µ A Splfyng equatn (A yelds β j y z k, h sn sn n Γ n n A a a b µ n b Integratng and splfyng π a, n h nβ 4 + n ( β a b y z k sn sn n Γ a b µ A π By puttng a αn ( βa 4 + n ( β a b n equatn (A yelds, ( sn y z k h a sn n Γ n nβ α β a b µ A4 By slvng equatn (A5 and usng the advantage f the f the rthgnalty accrdng t [] whch yelds; k Γ 6ab h nβ α( βa ab µ n 6k Γ h n µ n β αn β Appendx B ( a A5 A6 Als, the slutn f the equatn (7 can als be expressed n the fr f a duble Furer seres, and the ceffcent b can be deterned by substtutng the n equatn (4, and (5 n equatn (6, usng the sae technque and the fnal result can be expressed; and y nz F( yz,, n fn sn sn, a b n,,,5,,... B 6σ σ h f n n + B n n n α α 4a 4b 4a 4b Splfyng equatn B yelds

10 6 σ f β n σ + nσ α σ β α β ( a ( a Advances n Pwertrans and Auttves B By nsertng equatn (B and equatn (A7, F can be deterned by ntegratng and averagng the resultng ver the ppe crss-sectn area; as fllw; and F sn ( y, z 6 n, σ α σ σ β { σ } + y z n sn a b n ( a βα( βa ( σ a ( β a B4 σ0 σ β J F J( σ a B5 σ σ β αn