ELECTRONIC JOURNAL OF POLISH AGRICULTURAL UNIVERSITIES

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1 Elerni Jurnal f Plish Agriulural Universiies is he very firs Plish sienifi jurnal published elusively n he Inerne funded n January 998 by he fllwing agriulural universiies and higher shls f agriulure: Universiy f Tehnlgy and Agriulure f Bydgsz Agriulural Universiy f Craw Agriulural Universiy f ublin Agriulural Universiy f Pznan Universiy f Pdlasie in Siedle Agriulural Universiy f Szzein and Agriulural Universiy f Wrlaw EECTRONIC JOURNA OF POISH AGRICUTURA UNIVERSITIES 6 Vlume 9 Issue Tpi CIVI ENGINEERING Cpyrigh Wydawniw Aademii Rlnizej we Wrlawiu ISSN ŁACIŃSI Ł WOŹNIA CZ 6 ASYMPTOTIC MODES OF THE HEAT TRANSFER IN AMINATED CONDUCTORS Elerni Jurnal f Plish Agriulural Universiies Civil Engineering Vlume 9 Issue Available Online hp://wwwejpaumediapl ASYMPTOTIC MODES OF THE HEAT TRANSFER IN AMINATED CONDUCTORS Łuasz Łaińsi Czesław Wźnia Insiue f Cmpuer and Infrmain Sienes Czeshwa Universiy f Tehnlgy Pland Insiue f Mahemais and Cmpuer Siene Czeshwa Universiy f Tehnlgy Pland ABSTRACT aminaed maerials play an impran rle in ivil engineering The nribuin is fused n he mdelling f hea nduin in hese maerials The analysis is arried u in he framewr f he lerane averaging ehnique A new asympi predure fr finding sluins he speifi hea nduin prblems is prpsed General resuls are illusraed by sme numerial eamples and mpared wih hse derived frm hmgenizain ehnique ey wrds: hea ransfer laminaed rigid ndurs INTRODUCTION I is nwn ha he dire apprah he analysis f hea nduin presses in mpsies wih a dense peridi sruure leads ill-ndiined and mpliaed mpuainal prblems Tha is why sme averaged marspi mahemaial mdels fr finding sluins speial prblems have been frmulaed In ms ases averaged mdels f peridi maerials and sruures are based n he nep f hmgenizain - 4 as well as n is mdifiains lie he mirlal parameer apprah 5-7 A erain drawba f he asympi hmgenizain is ha i des n desribe he effe f he perid lenghs n he verall behaviur f a peridially inhmgeneus slid 89 Hene sme phenmena aing plae near he slid bundaries and lse is iniial sae ann be invesigaed in he framewr f hmgenizain 8 Tha is why a number f alernaive apprahes he mdeling f peridi maerials and sruures have been frmulaed The verview f hese apprahes an be fund in where fundains f wha was alled he lerane averaging ehnique have been summarized I has be emphasized ha in nras hmgenizain he lerane averaging desribes he effe f peridiiy ell size n he verall behaviur f he peridially inhmgeneus slid

2 In his nribuin we apply lerane averaging he hea nduin prblem in peridi w-mpnen laminaes The main resul is he frmulain f a erain higher-rder asympi ehnique fr finding apprimae sluin he nwn lerane hea nduin equains These equains are realled in he subsequen sein Mrever he prpsed asympi apprimain ehnique will be applied he numerial analysis f sme speial prblems The bained numerial resuls shw ha he prpsed asympi apprah an be effeively applied he analysis f prblems invlving near-bundary and near-iniial ime effes I is als shwn ha he hmgenized mdel equains an be reaed as he firs sep in he prpsed higher-rder asympi apprah Nains e he physial spae be parameerized by he rhgnal Caresian rdinae sysem and le sand fr he ime rdinae Subsrips i j and α β run ver sequenes and respeively The parial derivaives wih respe argumens are dened by and he ime derivae by he verd We als define z Fr an arbirary inegrable funin lengh parameer f f z whih an als depend n l l << we inrdue mean value f z f f seing O and and a z l f z f y dy l zl z l l If f is a peridi funin wih perid l hen f is independen f z MODE EQUATIONS The peridially laminaed rigid hea ndur under nsiderain is assumed be made f w inds f laminae and upies regin Ω in he physial spae The sheme f he ndur is shwn in Fig where l is he perid f inhmgeneiy l << and l l are laminae hinesses The speifi hea and he hea nduin ensr mpnens in perinen laminae are dened by and ij ji ij α α We assume he maerial symmery f he ndur seing Figure The rss-sein f he peridially laminaed ndur and dene ji By θ θ Ω R we dene a emperaure field a ime We assume ha in every lamina field θ saisfies he well-nwn linearized Furier hea ransfer equain θ & θ i ij j

3 On he inerfaes beween laminae we deal wih he hea flu ninuiy ndiins where i i θ θ sand fr he righ-hand side and lef-hand side derivaives respeively and i i are he bundary values f i in he perinen laminae Equains have be saisfied geher wih he apprpriae iniial and bundary ndiins The abve equains are diffiul slve sine l << and effiiens ij are unninuus highly sillaing funins Tha is why we replae he abve equains by he lerane mdel equains whih have nsan effiiens Realling resuls deailed in we inrdue he l-peridi funin g z assumed be a saw-lie shape funin; he diagram f g is shwn in Fig The lerane averaging ehnique is based n he assumpin ha he lass f emperaure fields θ is resried by ndiin θ g z 4 where and derivaives whih appear in he prblem under nsiderain T underline his fa we wrie are differeniable funins slwly varying in argumen z wih all heir SVε l SVε l 5 where ε > was referred in as a lerane parameer ε << I means ha in every inerval f he z- ais f he lengh l funins geher wih heir derivaives an be reaed as nsan wih a erain apprimain O ε Cndiins 5 are referred as he physial reliabiliy ndiins impsed n he bained sluins f speifi prblems Funin is referred as he marspi averaged emperaure and is alled he ampliude f emperaure fluuains Figure Diagram f he peridi saw-lie shape funin Define where f sands fr l l f ν f ν f ν l l f f f f f f ν ν and ν 6 7

4 Afer subsiuing he righ-hand side f 4 equain and erain manipulains we bain finally he fllwing sysem f gverning equains fr β α β α l l & & 8 Equains 8 wih perinen bundary and iniial ndiins dempsiin frmula 4 and ndiins 5 represen he lerane mdel f a peridially laminaed rigid ndur The deailed disussin f equains 8 an be fund in The hmgenized equains an be reaed as a asympi apprimain f he lerane mdel equains by he frmal negleing erms l O in 8 Under his apprimain an be eliminaed frm equain 8 and hene β α & 9 where we have dened ν ν Equains 9 wih he bundary and iniial ndiins fr represen he hmgenized mdel fr he hea ransfer in a peridially laminaed rigid ndur 5 Fllwing we shall ransfrm equains 8 a new frm seing z Hene is he new unnwn funin slwly varying in z e us subsiue he righ-hand side f equain in equain 8 and dene l z D β α where is a erain lengh dimensin and hen << we an assume prvided ha is finie Under he abve nains we bain finally D D D Equain have be nsidered geher wih he apprpriae iniial and bundary ndiins fr and as well as wih he dempsiin frmula derived frm 4 and z g θ and he physial reliabiliy ndiins l SV l SV ε ε nsiue he basis fr he subsequen analysis

5 ASYMPTOTIC MODES Due he presene f he small parameer < << in he resuling lerane mdel equains we shall apply he asympi apprah he analysis f iniial/bundary value prblems T his end independenly f argumens and we inrdue argumens and seing α α α and define differenial perar D seing D α β We shall see an asympi apprimain f sluin equains in he frm f asympi epansins ~ ~ ~ ~ where are funins f an rder Subsiuing he abve epansin in lerane mdel equain 4 and equaing effiiens f zer we ge D D The abve equains are assumed be nsidered geher wih he iniial/bundary ndiins iniding wih hse impsed n and Equains 5 represen wha will be alled he firs rder asympi apprimain f he lerane mdel equains e θ be a sluin a erain iniial/bundary value prblem in he framewr f he hmgenized mdel I means ha θ has saisfy equain 4 5 D θ θ 6 and he rrespnding nnhmgeneus iniial/bundary ndiins Hene θ θ is a sluin equain Dθ θ 7 whih saisfies he perinen hmgeneus iniial/bundary ndiins I fllws ha he firs rder apprimain is gverned by hmgeneus sysem f equains and D θ θ 8 D 9

6 wih nnhmgeneus iniial/bundary ndiins and by he nnhmgeneus equain 7 wih perinen hmgeneus iniial/bundary ndiins Bearing in mind frmula fr θ we nlude ha he iniial/bundary ndiins impsed n θ an be upled and he apprimain frmula has he frm θ θ O O I has be remembered ha he abve frmula has a physial sense nly if funins θ θ and are slwly varying in argumen z geher wih heir firs derivaive Subsiuing frmula 4 in lerane mdel equains and equaing effiiens f zer we bain where D D D θ θ The abve equains have be nsidered geher wih he apprpriae hmgeneus iniial/bundary ndiins fr Thus he send rder apprimain is deermined by frmulae θ θ O O where θ θ are fund as sluins he abve menined iniial/bundary value prblems Fllwing his predure i is pssible frmulae als higher rder asympi apprimains f he general lerane mdel equains Under he resriin f analysis he epansin 4 we arrive a he residual equain fr ~ ~ in he frm ~ ~ D ~ ~ D ~ ~ ~ D ~ Obviusly funins have saisfy he perinen hmgeneus iniial/bundary ndiins e us bserve ha if erm an be as negligibly small when mpared hen negleing hese erms we bain nly rivial sluins ~ ~ The abve saemen lses he prpsed asympi predure A jusifiain f his predure is a separae mahemaial prblem and will be sudied elsewhere NUMERICA ANAYSIS In rder verify effiieny f he prpsed asympi apprimain ehnique we shall resri urselves he sluin f w benhmar prblems Cmparing derived apprimains f he averaged emperaure and emperaure fluuain allws deermine he influene f he righ-hand sides f equains a sluin

7 Fr he sae f simpliiy we assume ha every lamina is hmgeneus and isrpi Hene e us nsider firsly he ne-dimensinal sainary prblem wih bundary ndiins We l fr sluin in he frm θ θ θ θ z π H z θ s z A z π s π π z sin z The mpuains have been arried u fr differen values f he inhmgeneiy effiien κ defined by and under assumpin ns κ Sluins bained bh in he framewr f he lerane mdel using he prpsed asympi apprimain ehnique and hmgenizain are presened in Fig and Fig 4 fr H l The firs apprimain inrdues an insignifianly small rrein he zerh apprimain Figure Averaged emperaure fr he sainary prblem versus spaial rdinae parallel layering Differene beween he firs and send apprimain as well as beween he differen mdels fr a given κ are insignifian

8 Figure 4 Ampliude f inrinsi emperaure fluuain fr he sainary prblem e us nsider in urn a nn-sainary prblem We inrdue iniial ndiins and we see a sluin in he frm π sin z A π s π sin z 4 The bained resuls are presened in Fig 5 and Fig 6 Figure 5 Averaged emperaure fr he nn-sainary prblem versus ime rdinae Differene beween he firs and send apprimain is n nieable The sluin bained in he framewr f he hmgenizain mdel is inadequae θ

9 Figure 6 Ampliude f inrinsi emperaure fluuain fr he nn-sainary prblem In rder verify hese resuls we shall slve he prblems in he framewr f he Furier mdel The abve prblems will be slved numerially using he finie differene mehd FDM We apprimae he emperaure field θ FDM using frmulae 4 and r 4 where and are reaed as unnwn apprimain parameers The apprimain will be realized by he leas square mehd T his end we shall l fr he values f funins minimizing he leas square errr θ FDM θ d where θ is a emperaure field in he frm 4 The resuls f mpuains are shwn in Fig -6 CONCUDING REMARS e us summarize new resuls and infrmain n he hea nduin in a laminaed rigid ndur whih have been bained in his nribuin º I was shwn ha he prpsed asympi ehnique allws find sluins lerane mdel wih suffiien auray Thereby he nsidered mdel in general frm wihu a simplifiain by miing he righhand sides f equain an be used º An influene f righ-hand sides f he lerane mdel equains sluin is f an rder O Tw firs apprimains seem be suffiien in ms praial ases º Sluins he seleed prblems bained in his nribuin are mpared wih hse derived frm hmgenizain The main nlusin is ha he lerane mdel in nras he nwn hmgenizain mdel desribes he bundary layer effe n he hea nduin in a laminaed rigid ndur Mrever differenes beween sluins bained in he framewr f he Furier equain and hse relaed he lerane mdel are negligible in he bundary layer Ouside his layer sluins bained by using he lerane mdel hse derived frm he hmgenizain mdel as well as resuls alulaed in he framewr f he Furier hery nearly inide REFERENCES Wznia C Wierzbii E Averaging ehniques in hermmehanis f mpsie slids Czeshwa Tehnlgial Uniwersiy Press Czeshwa Bensussan A ins J Papanilau G Asympi analysis fr peridi sruures Nrh-Hlland Amserdam 978

10 Sanhez-Palenia E Zaui A Hmgenizain ehniques fr mpsies media eure Nes in Physiss Fish J Chen W A dispersive mdel fr wave prpagain in peridi heergeneus media based n hmgenizain wih muliple spaial and empral sales J Appl Meh Wznia C A nnsandard mehd f mdelling f hermelasi peridi mpsies In J Engng Si Maysia SJ Nagr W Mirlal parameers in he mdelling f mirperidi plaes Ing Arh Maysia SJ Uhansa ON On hea nduin prblem in peridi mpsies In Cmm Hea Mass Transfer Wierzbii E Wznia C ainsa Bundary and iniial fluuain effe n dynami behaviur f laminaed slid Arh App Meh Wierzbii E Siedlea U A nribuin he mdelling f hea nduin in peridially mulilayered laminaes EJPAU 5 8 #5 ainsi Numerial verifiain f w mahemaial mdels fr he hea ransfer in a laminaed rigid ndur J Ther Appl Meh ainsi Wznia C Bundary-ayer Phenmena in a aminaed Rigid Hea Cndur J Thermal Sresses 6 in he urse f publiain Łuasz Łaińsi Insiue f Cmpuer and Infrmain Sienes Czeshwa Universiy f Tehnlgy Czeshwa Pland ul Dabrwsieg 7 4- Czeshwa luaszlainsi@iispzpl Czesław Wźnia Insiue f Mahemais and Cmpuer Siene Czeshwa Universiy f Tehnlgy Czeshwa Pland ul Dabrwsieg 7 4- Czeshwa