Strojni{ki vestnik 48(2002)9, Journal of Mechanical Engineering 48(2002)9, ISSN ISSN UDK : :536.2 UDC 62

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1 Strojni{ki vestnik (00),-0 Journal o Mechanical Engineering (00),-0 ISSN 00-0 ISSN 00-0 UDK -:.55:5. UDC -:.55:5. Horvat Izvirni znanstveni A. - Catton ~lanek I.: Analiza (.0) vezanega prenosa toplote - An Analysis Original o Conjugate scientiic Heat paper Transer (.0) Analiza vezanega prenosa toplote v hlailniku elektronskega ~ipa An Analysis o Conjugate Heat Transer in the Heat Sink o an Electronic Chip Anrej Horvat - Ivan Catton Prispevek opisuje razvoj algoritma za izraèun vezanega prenosa toplote z namenom izbire najugonejše geometrijske oblike za hlailnik elektronskega èipa. Struktura hlailnika je bila moelirana kot homogena porozna snov z uporabo teorije prostorninskega povpreèenja (TPP - VAT). Geometrijska oblika simulacijskega obmoèja in robni pogoji so bili povzeti po eksperimentalni napravi v laboratoriju za prenos toplote Morrin- Martinelli-Gier na Univerzi Kaliornije v Los Angelesu. Primeri numeriènih simulacij so bili izveeni za izotermno testno sekcijo kakor tui za toplotno prevono testno sekcijo iz aluminija. Primerjava koeicienta upora celotne proge C kot unkcije Reynolsovega števila Re h razkriva obro ujemanje z objavljenimi rezultati, metem ko primerjava porazelitev Nusseltovega števila Nu kaže veèje razlike. Konèna toplotna prevonost trnine zmanjša koeicient prestopa toplote in Nusseltovo število Nu. Vpliv toplotne prevonosti na rezultate se zveèuje s poveèevanjem Reynolsovega števila. 00 Strojniški vestnik. Vse pravice priržane. (Kljuène besee: prenosniki toplote, hlailniki èipov, prenos toplote, razvoj algoritmov) This paper escribes the construction o an algorithm or conjugate heat-transer calculations in orer to in the most suitable orm or the heat sink o an electronic chip. Applying volume averaging theory (VAT) to a system o transport equations, a heat-sink structure was moele as a homogeneous porous meium. The geometry o the simulation omain an the bounary conitions ollowe the experimental setup use in the Morrin-Martinelli-Gier Memorial Heat Transer Laboratory at the University o Caliornia, Los Angeles. The example numerical simulations were perorme or the test section with an isothermal structure as well as or the heat-conucting aluminum pin-ins. A comparison o the whole-section rag coeicient C as a unction o Reynols number Re h reveals goo agreement with existing ata, whereas the comparison o the Nusselt number Nu istributions shows larger iscrepancies. The inite conuctivity o the soli ecreases the heat-transer coeicient an Nusselt number Nu. The inluence o conuctivity becomes larger with increasing Reynols number. 00 Journal o Mechanical Engineering. All rights reserve. (Keywors: heat exchangers, heat sinks, heat transer, algorithms) 0 UVOD Prenosniki toplote so ena o osnovnih komponent ne samo v termoenergetski in procesni inustriji ampak tui v proizvonji elektronske opreme. Navkljub pomembni vlogi, je v konstrukcijski postopek prenosnikov toplote še veno vpleteno mnogo izkustvenih spoznanj. V preteklosti sta se namreè razvoj in uporaba prenosnikov toplote razvijala loèeno na številnih, najveèkrat nepovezanih poroèjih, zlasti v avtomobilski in letalski inustriji, v kriogeni in hlailniški tehniki. Pri tem so se tehnologije, obro znane v enem poroèju, le poèasi širile na ruga poroèja []. Zarai tega lahko skupen naèin izbire in optimizacije konstrukcije prenosnikov toplote pomembno zmanjša stroške v inustriji. 0 INTRODUCTION Heat exchangers are one o the basic installations, not only in power an process inustries, but also in the prouction o electronic equipment. Despite their crucial role, there is still a great eal o empiricism involve in the esign proceure o heat exchangers. The evelopment an application o heat exchangers an their suraces has taken place in a piecemeal ashion in a number o rather unrelate areas, principally those o the automotive, aerospace, cryogenic an rerigeration sectors. A lot o etaile technology, amiliar in one sector, progresse only slowly over the bounary into another sector []. Thereore, a uniying approach to select an to optimize a heat-exchanger esign can bring signiicant cost reuction to inustry. stran

2 Prispevek je el širših prizaevanj za razvoj znanstvenega prostopa k problemu optimizacije geometrijske oblike prenosnikov toplote. Opisuje granjo algoritma za hiter izraèun vezanega prenosa toplote z namenom izbire najugonejše geometrijske oblike za hlailnik elektronskega èipa. Struktura hlailnika je bila moelirana kot homogen porozen meij z uporabo teorije prostorninskega povpreèenja (TPP) ([] o []) sistema prenosnih enaèb. Mesebojni vpliv tekoèine in strukture je bil opisan s koeicienti lokalnega upora C in prestopa toplote h, ki so bili prevzeti iz razpoložljive literature ([] o []) in vstavljeni v raèunalniški program. Izraèunani koeicient upora celotne proge C, toplotna uèinkovitost QW in Nusseltovo število Nu so bili primerjani z razpoložljivimi eksperimentalnimi poatki [5]. Primerjava kaže obro ujemanje s preskusi kljub poenostavitvam prestavljenega moela. MODELNI PRISTOP This paper is part o a broaer eort to evelop a scientiic proceure or optimization o heatexchanger geometries. It escribes the construction o an algorithm or ast calculations o conjugate heat transer in orer to in the most suitable orm or an electronic chip heat sink. Applying volume averaging theory (VAT) ([] to []) to a system o transport equations, a heat-sink structure was moele as a homogeneous porous meia. The interaction between the lui an the heatsink structure was escribe with local rag an heattranser coeicients, which were taken rom the available literature ([] to []) an inserte into a computer coe. The calculate whole-section rag coeicient C, thermal eectiveness QWan Nusselt number Nu were compare with available experimental ata [5]. The comparison shows a goo agreement with the experimental ata espite moel simpliications. MODEL APPROACH Zraèni tok skozi hlailnik èipa lahko opišemo z osnovnimi enaèbami prenosa snovi, gibalne kolièine in energije []. Zarai zahtev po kratkem raèunskem èasu moela, je treba prenosnim enaèbam izraèunati povpreèje po perioièni nazorni prostornini (za porobnosti glej []). To prostorninsko raèunanje povpreèja voi o problema sklenitve sistema enaèb, pri katerem je treba prenos gibalne kolièine in toplote me tekoèino in trnino opisati z empiriènimi razmerji, npr. s koeicientoma lokalnega upora C in prestopa toplote h. Da bi še naalje poenostavili simuliran sistem, smo prepostavili tok tekoèine le v vzolžni smeri z nespremenljivim znižanjem tlaka (sl. ). Zarai tega se proil hitrosti spreminja le preèno na smer toka. To pomeni, a je tlaèna sila èez celotno simulacijsko obmoèje v ravnovesju s strižnimi silami. Tako je mogoèe enaèbo prenosa gibalne kolièine zapisati v ierencialni obliki kot: æ ö a m ç è ø kjer so: a elež tekoèine, C koeicient lokalnega upora, S speciièna površina porozne snovi, Dp paec tlaka èez simulacijsko obmoèje in L olžina simulacijskega obmoèja. Temperaturno polje v tekoèini se oblikuje po vplivom ravnovesja me toplotno konvekcijo v smeri toka, toplotno iuzijo in toploto, ki se prenese s trnine na tekoèino. Iz tega izhaja ierencialna oblika energijske enaèbe za tekoèino: kjer sta: T temperatura tekoèine in T s temperatura trnine. Prenos toplote me trnino in tekoèino je u u D - p + + = y z L The airlow through a chip-cooler structure can be escribe with basic mass, momentum an heat-transport equations []. Due to the requirement or the moel to have short computing times, the transport equations have to be average over a perioic control volume (see [] or etails). This volumetric averaging leas to a closure problem, where an interace exchange o momentum an heat between a lui an a soli has to be escribe with aitional empirical relations, e.g. a local rag coeicient C an a local heat-transer coeicient h. To urther simpliy the simulate system, lui low was taken as uniirectional with a constant pressure rop in the streamwise irection (Fig. ). As a consequence, velocity only changes transverse to the low irection. This means that the pressure orce across the entire simulation omain is balance with shear orces. Thus, the momentum equation can be written in ierential orm as: C r u S (), where a is the lui raction, C the local rag coeicient, S the speciic surace o porous meia, Dp the pressure rop across the simulation omain an L the simulation omain s length. The temperature iel in the lui is orme as a balance between thermal convection in the streamwise irection, thermal iusion, an the heat that is transerre rom the soli to the lui. Thus, the ierential orm o the energy equation or the lui is: T T T T a r cu = a l æ + + ö h - ( T -Ts) S x ç x y z è (), ø where T an T s are the lui an soli temperatures, respectively. The heat transer between the soli an the stran

3 moeliran kot linearna ovisnost temperatur obeh az, kjer je h koeicient lokalnega prestopa toplote. V vsaki nazorni prostornini je struktura hlailnika le šibko povezana v vooravni smeri (sl. ). Zarai tega je le toplotna iuzija v navpièni smeri v ravnovesju s toploto, ki oteka skozi stièno površino kapljevine in trnine, metem ko lahko toplotno iuzijo v vooravni smeri zanemarimo. To poenostavi energijsko enaèbo trnine: 0 al Ts z kjer je a s elež trnine. Enaèbe () o (), ki so zapisane s povpreèenimi velièinami, so enaèbe ravnovesnega prenosa gibalne kolièine in toplote skozi homogeno porozno snov. Zanesljive poatke za va oatna parametra, to sta koeicienta lokalnega upora C in prestopa toplote h, smo poiskali v [] o []. lui is moele as a linear relation between both phase temperatures, where h is a local heat-transer coeicient. The chip-cooler structure in each control volume is only loosely connecte in the horizontal irections (see Fig. ). As a consequence, only the thermal iusion in a vertical irection is in balance with the heat leaving the structure through the luisoli interace, whereas the thermal iusion in horizontal irections can be neglecte. This simpliies the energy equation or the soli to: ( ) = + h T -T S s s s (), where a s is the soli raction. Equations () to (), written with the phase average variables, are equations or the steay-state transport o momentum an heat through homogeneous porous meia. The reliable empirical ata or two aitional parameters, a local rag coeicient C an heat-transer coeicient h, were oun in [] to []. SIMULACIJSKO OBMOÈJE Geometrijska oblika simulacijskega obmoèja kakor tui robni pogoji enaèb () o () sleijo geometrijski obliki eksperimentalne testne sekcije, ki je bila uporabljena v laboratoriju za prenos toplote Morrin-Martinelli-Gier na Univerzi Kaliornije v Los Angelesu za priobitev eksperimentalnih poatkov, opisanih v [5]. SIMULATION DOMAIN The geometry o the simulation omain as well as the bounary conitions or Eqs. (-) ollow the geometry o the experimental test section use in the Morrin-Martinelli-Gier Memorial Heat Transer Laboratory at the University o Caliornia, Los Angeles, where the experimental ata escribe in [5] were taken. m z x Q p=0.5" =0.5" y x Sl.. Eksperimentalna testna proga Fig.. Experimental test section stran

4 Splošna razporeitev palènih reber hlailnika je poana na sliki. Premer palènih reber je znašal = 0,005 m (0,5"). Razmerje mepalènega razmika in premera v smeri toka je bilo p x / =,0 in v smeri preèno na tok p y / =,. Simulacijsko obmoèje je zajemalo vrst palènih reber v smeri toka in vrst preèno na smer toka. Dolžina hlailnika L kakor tui širina W sta znašali 0,5 m, metem ko je višina H znašala 0,0 m. Robni pogoji trne stene brez zrsa so bili uporabljeni za enaèbo prenosa gibalne kolièine () na vseh štirih stenah, ki so vzporene s smerjo toka: The general arrangement o the heat-sink pinins is given in Fig.. The iameter o the pin-ins was = m (0.5"). The pitch-to-iameter ratio in the streamwise irection was set to p x / =.0, an in the transverse irection to p y / =.. The simulation omain consiste o rows o pin-ins in the streamwise irection an rows o pin-ins in the transverse irection. The length L as well as the with W o the heat sink were 0.5 m, whereas the height H was 0.0 m. The no-slip bounary conitions or the momentum equation () were implemente or all our walls, which are parallel to the low irection: u ( 0,z) = 0, u ( W,z) = 0, u ( y,0 ) = 0, ( y,h) = 0 u (). Kot gonilna sila toka je bil poan tlaèni paec vzolž celotnega simulacijskega obmoèja. Absolutne vrenosti so zbrane v preglenici. Pri enaèbi prenosa energije v tekoèini () smo prepostavili izotermni vtok tekoèine kot tui izotermno sponjo steno: As a low riving orce, the whole-section pressure rop Dp was prescribe. The absolute values are summarize in Table. For the lui energy equation (), the simulation omain inlow an the bottom wall were taken as isothermal: ( 0,y,z) Tin, T ( x,y, 0 ) = Tg T = (5), metem ko so bile ruge stene aiabatne: T T ( L,y,z) = 0, ( x, 0,z) = 0, ( ) 0 x y Pri enaèbi prenosa energije v trnini () je bila sponja stena privzeta kot izotermna, metem ko je bila zgornja stena aiabatna: Prepostavka o izotermnosti sponje stene (5) in () se pomembno razlikuje o eksperimentalne postavitve [5], pri kateri so palèna rebra spojena s toplotno prevono bazno plošèo. Kljub vsemu boo rezultati pokazali, a seanji moel aje zaovoljiv približek izmerjenih vrenosti. Absolutne temperature robnih pogojev za razliène simulirne primere so zbrane v preglenici. s whereas the other walls were consiere as aiabatic: T T x,w,z =, ( x,y,h) = 0 (). y z For the soli energy equation (), the bottom wall was prescribe as isothermal, whereas the top wall was assume to be aiabatic: T s x,y,h = (). z The assumption about the isothermal bottom wall (5) an () signiicantly iers rom the experimental set-up [5], where the pin-ins were joine with a conuctive base plate. Nevertheless, as the results will show, the presente moel still gives a satisactory approximation to the measure values. The absolute temperatures in ierent simulation cases are summarize in Table. ( x,y, 0 ) Tg, ( ) 0 T = NUMERIÈNE METODE Prenosne enaèbe () o () in robni pogoji () o () so bili preoblikovani v brezimenzijsko obliko in iskretizirani, upoštevajoè naèela metoe konènih NUMERICAL METHODS The transport equations () o () an bounary conitions () to () were transorme into the imensionless orm an then iscretize ollowing Preglenica. Robni pogoji - izbrane vrenosti Table. Bounary conitions - pre-set values Št. No. D p Pa Tin o C Tg o C Št. No. D p Pa Tin o C Tg o C 5,0,00 5,0 5,,0 0,0 0,0,00, 5,,0,0 0,0,00,0,5,0,0 0,0,00,00,,5, stran 5

5 prostornin ([] in [0]). Pri vseh izveenih numeriènih simulacijah smo uporabili xx0 konènih prostornin v smereh x, y in z. Zarai robnih pogojev () o (), sta bili hitrost u in temperatura trnine T s zapisani kot voimenzionalno skalarno polje, metem ko je bila temperatura tekoèine T zapisana kot triimenzionalno skalarno polje. Zarai iskretizacijskega postopka je nastal, v primeru voimenzionalnih skalarnih polj, pet-iagonalni matrièni sistem in, v primeru triimenzionalnega skalarnega polja, seemiagonalni matrièni sistem. Za uèinkovito obraèanje matriènega sistema enaèb je bila za ta poseben primer privzeta metoa spremenjenih vezanih graientov (MSVG - PCGM), ki je porobneje opisana v []. REZULTATI IN RAZPRAVA the principles o the inite-volume methos ([] an [0]). In all the perorme numerical simulations, xx0 inite volumes were use in x, y an z irections, respectively. Due to the bounary conitions () to (), the velocity u as well as the soli temperature T s were escribe as two-imensional scalar iels, whereas the lui temperature T was escribe as a threeimensional scalar iel. This resulte in a nonsymmetric ive-iagonal matrix system or the twoimensional scalar iels an a seven-iagonal matrix system or the three-imensional scalar iel. In orer to invert the matrix systems eiciently, the preconitione conjugate graient metho (PCGM), as escribe in [], was aopte or this speciic problem. RESULTS AND DISCUSSION Rezultati izraèunov za primer aluminijastega (Al) hlailnika so prestavljeni na slikah in. Tlaèni paec Dp =, Pa povzroèi zraèni tok z Reynolsovim številom Re h = 0, pri èemer je Reynolsovo število einirano na polagi hiravliènega premera h hipotetiènega kanala porozne snovi: u h n Slika prikazuje temperaturno polje v Al strukturi v Celzijevi skali, metem ko slika razkriva temperaturno polje v zraènem toku. Kakor prikazuje slika, ima Al struktura najvišjo temperaturo blizu izotermne sponje stene in najnižjo temperaturo ob levem zgornjem robu, kjer je struktura izpostavljena vtoku zraka z nizko temperaturo. Slika kaže, kako se zrak postopoma segreva o vtoka na levi o iztoka na esni strani. Sponji el temperaturnega polja prav tako razkriva intenzivno segrevanje s sponje izotermne meje, kar ima za posleico vooravno toplotno razslojenost prehajajoèega zraka. Re = h = ç The results o an example calculation or an aluminum (Al) heat sink are presente in Figs. an. The impose pressure rop Dp =. Pa causes airlow o the Reynols number Re h = 0, where the einition o the Reynols number is base on a hyraulic iameter h o a hypothetical porous meia channel: u æ a ö (). n è S ø Fig. shows the temperature iel in the Al structure in egrees Celsius, whereas Fig. reveals the temperature iel in the airlow. In Fig., the Al structure has its highest temperature close to the isothermal bottom, an the lowest close to the let upper ege, where the structure is expose to a low-temperature inlow. Fig. shows how the air is graually heate rom the inlet on the let sie to the outlet on the right sie. The lower part o the temperature iel also shows intensive heating rom the isothermal bottom bounary, which results in horizontal thermal stratiication o the passing air z[m] x[m] Sl.. Temperaturno polje v trnini pri Re h = 0, T in =,5 o C, T g =, o C Fig.. Temperature iels in the soli at Re h = 0, T in =.5 o C, T g =. o C stran

6 z[m] x[m] Sl.. Temperaturno polje v zraku pri Re h = 0, T in =,5 o C, T g =, o C Fig.. Temperature iels in the air at Re h = 0, T in =.5 o C, T g =. o C Poleg vzorènega izraèuna sta bili opravljeni še ve seriji izraèunov pri osmih razliènih tlaènih pacih. Pri izraèunih uporabljeni robni pogoji so zbrani v preglenici. V obeh serijah smo za tok hlaiva vzeli snovske lastnosti zraka. Pri prvi seriji izraèunov smo za hlailnik vzeli snovske lastnosti aluminija, metem ko je bila v rugi seriji prepostavljena izotermiènost strukture. Kakor je obièajno pri tovrstnih simuliranjih, sta bila izraèunana koeicient upora C () in Nusseltovo število Nu (0) za celotno sekcijo, obljene vrenosti pa primerjane z eksperimentalnimi rezultati [5]. kjer sta: A g = L W površina gretega na in DT = T g -T in temperaturna razlika me greto sponjo steno in vtokom zraka. Pri einiciji Nusseltovega števila Nu je konvektivni toplotni tok einiran kot: C Nu Besies the example calculation, two other series o calculations with eight ierent pressure rops were perorme. The bounary conitions or these calculations are summarize in Table. In both series the air material properties were taken or coolant low. For the irst set o calculations the Al material properties were taken or the heat sink, whereas in the secon set the heat sink was consiere as isothermal. As is usually the case in such calculations, the whole-section rag coeicient C () an Nusselt number Nu (0) were calculate an compare with the experimental results [5]. Dp = (), r ul S Q h = DTA l (0), g where A g = L W is the area o the heate bottom an DT = T g -T in is the temperature ierence between the heate bottom an the inlow air. In the Nusselt number Nu (0) einition, the convective heat-low rate is eine as: ( ) Q = a r c u T -T A out in ^ (). kjer je A^=H W. Primerjava na sliki kaže koeicient upora celotne proge C kot unkcijo Reynolsovega števila Re h. Slika kaže obro ujemanje z že objavljenimi rezultati. Kljub temu pa se pri veèji vrenosti Reynolsovega števila Re h zarai narašèajoèe turbulence, ki ni bila zajeta v moel, pokaže razlika v velikosti nekaj ostotkov. Primerjava porazelitev Nusseltovega števila celotne proge Nu na sliki 5 prikazuje veèje ostopanje. Zarai razlike v toplotnih robnih pogojih kažejo izraèuni za 0 ostotkov veèji toplotni tok o eksperimentalnih vrenosti [5]. Naalje je razvino, a konèna toplotna prevonost aluminijaste strukture (na slikah 5 in oznaèeno z Al) znižuje koeicient prestopa toplote in Nusseltovo število Nu v primerjavi s strukturo z neskonèno toplotno where A^=H W. The comparison in Fig. shows the wholesection rag coeicient C as a unction o Reynols number Re h. It reveals goo agreement with alreay publishe ata. Nevertheless, at a higher Reynols number a ierence o a ew percent appears ue to increasing turbulence, which was not taken into account in the moel. The comparison o the Nusselt number Nu istributions in Fig. 5 shows larger iscrepancies. Due to the ierence in the thermal bounary conitions, the calculate ata reveal up to 0 percent higher heat-transer rate than the measure values [5]. Furthermore, it is evient that the inite thermal conuctivity o the Al structure (in Figs. 5 an marke with Al) ecreases the heat-transer coeicient an the Nusselt number Nu in comparison stran

7 Simulacije Rizzi in soelavci (00) Simulations Rizzi et al. (00) Simulacije (Al) Simulacije (Nesk) Rizzi in soelavci (00) Simulations (Al) Simulations (In) Rizzi et al. (00) 0 C Nu 00 Re Sl.. Koeicient upora celotne sekcije kot unkcija Reynolsovega števila Fig.. Whole-section rag coeicient as a unction o Reynols number prevonostjo (na slikah 5 in oznaèeno z Nesk). Ta vpliv konène toplotne prevonosti trnine se z veèanjem vrenosti Reynolsovega števila še poveèuje. Konstrukcija prenosnikov toplote mora upoštevati tako vrenosti toplotnega toka kakor tui mehanskega ela, ki je potrebno za premagovanje trenja tekoèine in za premikanje le-te skozi samo strukturo. V tem pogleu je glavni cilj konstrukcije poveèati toplotni tok Q () pri najmanjši moèi èrpanja: Sl. 5. Nusseltovo število celotne sekcije kot unkcija Reynolsovega števila Fig. 5. Whole-section Nusselt number as a unction o Reynols number with the ininite thermal conuctivity case (in Figs. 5 an marke with In). This inluence o inite thermal conuctivity o the soli becomes larger with increasing Reynols number. The esign o a heat sink involves consieration o the heat-transer rate an the mechanical pumping power expene to overcome lui riction an move the lui through a structure. Thus, the main esign goal is to maximize the heattranser rate Q () or the minimum pumping power: W = a Dp A u ^ (). Slika prikazuje toplotno uèinkovitost prenosa toplote, ki je einirana kot razmerje me toplotnim tokom Q in mehansko moèjo W. Razvino je, a so eksperimentalni in numerièni rezultati blizu skupaj. Pri paajoèi toplotni uèinkovitosti Q W se vpliv toplotne prevonosti poveèa in povzroèi 5 ostotkov razlike pri vrenosti Reynolsovega števila Re h = 0. Fig. shows the thermal eectiveness o the heat-transer process, which is eine as the ratio between the heat-transer rate Q an the mechanical power W. It is evient that the experimental an numerical results are close. With ecreasing thermal eectiveness QW, the inluence o the structure s thermal conuctivity increases an causes a 5 percent ierence at the Reynols number Re h = Simulacije (Al) Simulacije (Nesk) Rizzi in soelavci (00) Simulations (Al) Simulations (In) Rizzi et al. (00) Q/W Re Re Sl.. Toplotna uèinkovitost hlailnika v ovisnosti o Reynolsovega števila Fig.. Heat-sink eectiveness as a unction o Reynols number stran

8 Kakor je prikazano, se toplotna uèinkovitost QW hlailnika zmanjšuje z veèanjem vrenosti Reynolsovega števila Re h (). Kljub temu, a manjše vrenosti Reynolsovega števila Re h prinašajo veèjo toplotno uèinkovitost, pa morajo biti rezultirajoèi majhni toplotni tokovi naomešèeni z veèjo površino in zarai tega z veèjimi imenzijami samega hlailnika. V nekaterih primerih to ni mogoèe zarai gospoarnosti in omejitev velikosti. 5 SKLEPI Prispevek opisuje elo, ki je bilo vloženo v razvoj raèunsko hitrega numeriènega algoritma za izraèun prenosnikov toplote. Namen naloge je bil posveèen numerièni raziskavi ovoa toplote iz elektronskega èipa. Pri tem je bila notranja struktura hlailnika, v obliki palièastih reber s premaknjeno postavitvijo, obravnavana kot homogena porozna snov. Vrenosti koeicientov lokalnega upora C in prestopa toplote h, ki so bile potrebne za zaprtje prenosnih enaèb, smo prevzeli iz [] o []. Razvite parcialne ierencialne enaèbe so bile iskretizirane z upoštevanjem ohranitvenih lastnosti metoe nazornih prostornin. Sistem pol-linearnih enaèb je bil nato rešen z metoo spremenjenih vezanih graientov. Za preveritev postopka izraèuna smo uporabili eksperimentalne poatke laboratorija za prenos toplote Morrin-Martinelli-Gier. Izveeni sta bili ve seriji izraèunov za hlailnik èipa, ki je bil hlajen z zraènim tokom. Izraèunane vrenosti koeicienta upora celotne proge C kažejo obro ujemanje z že objavljenimi poatki, metem ko izraèunane vrenosti Nusseltovega števila Nu celotne proge razkrivajo ostopanje zarai razlik v toplotnih robnih pogojih. Prav tako smo raziskali vpliv konène toplotne prevonosti trne strukture. Izkazalo se je, a konèna toplotna prevonost aluminija zmanjša toplotno uèinkovitost QW za 5 ostotkov pri Re h = 0. Prav tako je prièakovati veèji vpliv toplotne prevonosti pri veèjih vrenostih Reynolsovega števila. Prikazani rezultati potrjujejo ustreznost izbranega naèina za preraèun hlailnika, kjer je treba upoštevati toplotno prevonost trne strukture. Vzorèni izraèuni prav tako potrjujejo, a razvit numerièni program aje rezultate z zaostno natanènostjo za razširitev njegove uporabe na ruge bolj zahtevne geometrijske oblike. Zahvala Prvi avtor se želi zahvaliti za inanèno poporo sklau Kerze-Cheyovich in Ministrstvu za šolstvo, znanost in šport Republike Slovenije. The thermal eectiveness QW o the examine heat sink is reuce with increasing Reynols number Re h. Although the lower Reynols numbers Re h bring higher eectiveness, the resulting low-heat-transer rates have to be compensate with a larger heat-transer surace an consequently with a larger heat sink. In some cases this is not possible ue to economics an size limitations. 5 CONCLUSIONS The present paper escribes an eort to evelop a ast-running numerical algorithm or heatexchanger calculations. The purpose o the task was to numerically investigate heat removal rom an electronic chip. The heat sink s internal structure, in the orm o a staggere arrangement o pin-ins, was treate as a homogenous porous meia. The local values o rag coeicient C an heat-transer coeicient h, which were neee to close the transport equations, were taken rom [] to []. The resulting partial ierential equations were iscretize using conservation properties o the inite-volume metho. The system o semi-linear equations was solve with the preconitione conjugate graient metho. To test the calculation proceure, experimental ata obtaine in the Morrin-Martinelli-Gier Memorial Heat Transer Laboratory were use or the comparison. Two series o calculations were perorme or the heat sink coole with airlow. The calculate values o the whole-section rag coeicient C show a goo agreement with alreay publishe ata, whereas the calculate whole-section values o the Nusselt number Nu reveal some iscrepancies ue to ierences in the thermal bounary conitions. Also, the inluence o the inite thermal conuctivity o the soli structure was examine. It was shown that the inite thermal conuctivity o aluminum ecreases the thermal eectiveness QW by 5 percent at Re h = 0. Furthermore, it is expecte that at a higher Reynols number this thermal conuctivity eect woul increase. The presente results emonstrate that this approach is appropriate or heat-sink calculations where the thermal conuctivity o a soli structure has to be taken into account. The example calculations also veriy that the evelope numerical coe yiels suiciently accurate results to be also applicable or other more emaning geometries. Acknowlegements The irst author s inancial support by the Kerze-Cheyovich scholarship an the Ministry o Eucation, Science an Sport o the Republic o Slovenia is grateully acknowlege. stran

9 LITERATURA REFERENCES [] Hesselgreaves, J.E. (00) Compact heat exchangers selection, Design an Operation, Pergamon Press. [] Whitaker, S. () Diusion an ispersion in porous meia, AIChE Journal, Vol., No., 0-. [] Travkin, V.S., I. Catton (5) A two temperature moel or lui low an heat transer in a porous layer, J. Flui Engineering, Vol., -. [] Travkin, V.S., I. Catton () Transport phenomena in heterogeneous meia base on volume averaging theory, Avans. Heat Traser, Vol., -. [5] Rizzi, M., Canino, M., Hu, K., Jones, S., Travkin, V., Catton, I. (00) Experimental investigation o pin in heat sink eectiveness, Procs. o the 5th National Heat Transer Conerence, Anaheim, Caliornia. [] Horvat, A., I. Catton (00) Development o an integral computer coe or simulation o heat exchangers, Procs. o the Con. Nuclear Energy in Central Europe 00, Portorož, Slovenia, Sept. 0-, No.. [] Launer, B.E., T.H. Massey () The numerical preiction o viscous low an heat transer in tube bank. Trans, ASME J. Heat Transer, Vol. 00, [] Kays, W.M., A.L. Lonon () Compact heat exchangers, r E. Krieger Publishing Company, Malabar, Floria, -. [] Žukauskas, A.A., R. Ulinskas (5) Eiciency parameters or heat transer in tube banks, J. Heat Transer Engineering, Vol.5, No., -5. [0] Versteeg, H.K., W. Malalasekera (5) An introuction to computational lui ynamics, The Finite Volume Metho, Longman Scientiic & Technical: Englan, 0-5. [] Ferziger, J.H., M. Periæ () Computational metho or lui mechanics, Chapter 5: Solution o linear equation systems, Springer Verlag: Berlin, 5-. Naslova avtorjev: r. Anrej Horvat Osek za reaktorsko tehniko Institut Jože Stean Jamova Ljubljana pro.r. Ivan Catton Morrin-Martinelli-Gier Memorial Heat Transer Laboratory Department o Mechanical an Aerospace Engineering School o Engineering an Applie Science University o Caliornia, Los Angeles 0 Westwoo Plaza, Eng. IV, 005 Los Angeles, Caliornia, USA Author s Aresses: Dr. Anrej Horvat Reactor Engineering Division Institute Jože Stean Jamova SI- Ljubljana, Slovenia Pro.Dr. Ivan Catton Morrin-Martinelli-Gier Memorial Heat Transer Laboratory Department o Mechanical an Aerospace Engineering School o Engineering an Applie Science University o Caliornia, Los Angeles 0 Westwoo Plaza, Eng. IV, 005 Los Angeles, Caliornia, USA Prejeto: Receive:.5.00 Sprejeto: Accepte:..00 stran 0