Two-Dimensional Modeling of Thermoelectric Cells

Transcription

1 udu Uivit udu -ub Itatioal Rfigatio ad i Coditioig Cofc School of Mchaical Egiig 2014 o-dimioal Modlig of hmolctic Cll Klaudio S. M. Olivia UFR, Bazil, klaudio@uol.com.b Rodigo. Cadoo UFR, Bazil, odigo.pito@ufp.b Chitia J. L. Hm UFR, Bazil, chm@ufp.b Follo thi ad additioal ok at: Olivia, Klaudio S. M.; Cadoo, Rodigo.; ad Hm, Chitia J. L., "o-dimioal Modlig of hmolctic Cll" (2014). Itatioal Rfigatio ad i Coditioig Cofc. ap hi documt ha b mad availabl though udu -ub, a vic of th udu Uivit Libai. la cotact pub@pudu.du fo additioal ifomatio. Complt pocdig ma b acquid i pit ad o CD-ROM dictl fom th Ra W. Hick Laboatoi at Hick/Evt/odlit.html

2 2108, ag 1 o-dimioal Modlig of hmolctic Cll Klaudio S. M. OLIEIR, Rodigo. CRDOSO *, Chitia J. L. HERMES Laboato of hmodamic ad hmophic, Dpatmt of Mchaical Egiig Fdal Uivit of aaá, Cuitiba-R, Bazil * Copodig autho: odigo.pito@ufp.b BSRC h pt pap i aimd at puttig foad a to-dimioal modl fo thmolctic cll. h g covatio quatio a fomulatd i od to accout fo th Foui, th homo ad th Joul ffct o th tmpatu ditibutio. h lctic fild a alo olvd to com out ith th cut ad voltag ditibutio. h govig quatio dictizd b ma of th fiit-volum mthod, ha th DM algoithm a adoptd fo olvig th t of lia quatio. plicit itativ olutio chm a mplod to add th tmpatu ifluc o th Sbck cofficit. h modl ult compad ith pimtal data, h a atifacto agmt a achivd fo both coolig capacit ad CO, ith o ithi a ±10% bad. I additio, th modl a mplod to a th ffct of th thmolctic popti ad th coupl gomt o th thmodamic pfomac of th thmolctic cll. 1. INRODUCION I th pat dcad, olid-tat coolig tchologi hav com oto paticula makt ich, pciall th applicatio latd to potabl coolig (Hm ad Baboa, 2012). h mot igificat advac hav b achivd i th alm of th thmolctic coolig, i hich a lctic cut poduc a tmpatu diffc i a pai of diimila micoducto matial. tpical thmolctic modul i maufactud ith to thi camic af ad a aa of p- ad -tp block of dopd micoducto matial adichd bt thm. pai of p- ad -tp block coctd lcticall i i ad thmall i paalll mak up a thmolctic coupl. Sval tudi hav b coductd both thoticall ad umicall to a th thmodamic pfomac of thmolctic cll. Som iflucig ok a ummaizd i ab. 1. h litatu vi poit out that mot modl a o-dimioal, big ot abl to valuat th ifluc of th coupl gomt o it pfomac. I additio, th litatu aali alo val that th f availabl multidimioal (2D/3D) modl a oft dvlopd aidd b commcial packag, hich ot ol tict th acc to th mathmatical fomulatio, but alo to th umic chm. t lat, mot modl do ot accout fo th hat taf i th ai cavit, hich alo might affct th tm pfomac. h pt pap i thfo aimd at advacig a to-dimioal modl, i th alm of o-quilibium thmodamic, hich i abl to valuat th itivit of th thmophical popti ad th cll gomt o it thmodamic pfomac. abl 1: Summa of th ct litatu o pfomac amt of thmolctic cll utho Ya ppoach homo Cavit Cavit hical opti a ffct covctio adiatio Domai ff() Huag t al altical Y Y Y 1D No amaickad Da altical Y No No 1D No L ad Kim 2006 Numical No No No 1D No Yamahita 2009 altical-epimtal Y No No 1D Y Ch t al Numical Y Y Y 3D No Mg t al Numical Y Y Y 1D Y Du ad W 2011 Numical-Epimtal Y No No 1D Sbck ol Ch t al Numical Y No No 3D Sbck ol éz-paicio t al Numical Y Y Y 3D Y 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

3 2108, ag 2 2. MHEMICL FORMULION thmolctic cll i compid of val pai of p ad micoducto coctd lcticall i i ad thmall i paalll, ad paatd fom ach oth b a cavit filld ith ai. h phical modl i tictd to a thmolctic pai, a illutatd i Fig. 1, hich i tu i ubdividd ito t domai, a ummaizd i ab. 2. h dimio i ab. 2 f to th thmolctic dvic (llu, 2007), hich ha b tak a fc fo th pt tud. h mathmatical modl i bad o th folloig k aumptio: (i) tad-tat to-dimioal modl, (ii) th thmophical popti of ach matial a fuctio of th tmpatu ol, (iii) th ital cotact itac (both thmal ad lctic) a gligibl, (iv) both ad p lmt hav th am Sbck cofficit, but ith difft ig, ad (v) th hat taf b both advctio ad adiatio a digadd, o that Nu1 (pu hat coductio) i th cavit. Hc, a local g balac ild, q q& (1) h q& i th at of hat gatio, ad th hat flu, q, i calculatd fom th folloig latio obtaid fom th ivibl thmodamic (Rold, 1968): q k αj (2) h th fit tm o th ight-had id tad fo th hat coductio (fd haft a Foui ffct), h k i th thmal coductivit, ad th cod tm i aociatd ith th Sbck ffct, big α th Sbck cofficit. h divgt of q. (2) ild, ( k ) j ( α) α( j) q (3) h j 0 at tad-tat coditio to u th cotiuit of th lcto flu. I additio, th dfiitio of lctic fild ild, ρj α (4) Figu 1: Schmatic ptatio of th phical modl 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

4 2108, ag 3 abl 2: Summa of th phical domai Subdomai Dciptio Matial Dimio [mm] 1 & 6 Bottom (1) ad top (6) lctic iulato l 2 O & 9 Lft (2) ad ight (9) lctic coducto at th bottom Cu & 7 -tp (3) ad p-tp (7) micoducto Bi Elctic coducto at th top Cu & 10 Lft (5) ad ight (10) id ai caviti i Ctal ai cavit i hfo, th at of hat gatio q& i calculatd a follo: q& j (5) ( ) ρ( j j) αj h th fit tm o th ight-had id tad fo th Joul hatig, ha th cod tm i gadd ith th ok poducd b th lctic cut agait th Sbck ffct. Ivokig th 2 d thmolctic latio, τd dα (6) ad placig q. (3) ad (5) ito q. (1), th folloig quatio fo th tmpatu ditibutio i a thmolctic matial ca b divd, ( k ) τj ρ( j j) 0 h th fit tm f to th Foui coductio, th cod o to th homo (thmolctic) ffct, ad th thid to th Joul hatig. Witig q. (4) fo th lctic cut, ad callig that j 0 at tad-tat coditio, o ca div th folloig pio fo th voltag ditibutio alog th domai, ( γ ) ( γα ) 0 h γρ -1 i th lctical coductivit. h fit tm tad fo th lctic coductio, ha th cod o f to th ditotio o th lctic fild iducd b th thmolctic ffct. Equatio (7) ad (8) ul th thmolctic phoma, big both pd fo 2-D Catia domai a follo: (7) (8) k γ 2 2 ( j j ) 0 k τ j j ρ γ γα γα 0 (9) (10) h j ad j a th ad compot of th lctic cut dit, pctivl, i [/m 2 ]. Equatio (9) ad (10) qui to bouda coditio ach. Fo th latt, pcibd ilt ( i ) ad outlt ( out ) voltag adoptd. I additio, baig i mid that th i o lcto flu though ubdomai 1 ad 6, d/d0 bouda coditio hav alo b adoptd. I ca of q. (9), pcibd tmpatu ud fo both hot ( h ) ad cold ( c h -) d. Zo hat flu bouda coditio (d/d0) alo mplod fo th cll mmt. Figu 2 dpict th coditio ud fo ach bouda of th phical domai. h thmophical popti of th Bi lmt calculatd fom 2 d -od polomial fit obtaid fom data povidd b Ro (1995), ρ (11) k (12) α (13) 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

5 2108, ag 4 h i i [K], ρ i [m/], k i [W/mK], ad α i [/K]. h homo cofficit, τ, a calculatd fom q. (6) ad (13). Both th ai ad th l 2 O 3 -lmt aumd to b pfct lctic iulato, ith thmal coductiviti (at 300 K) of 30 ad W/mK, pctivl. Fo th copp, a thmal coductivit of 400 W/mK ad a lctical itivit of m/ hav b adoptd. h hat taf iid th ai cavit a modld aumig a uita Nult umb, o th ffct of f covctio ad adiatio glctd. Figu 2: Schmatic ptatio of th bouda coditio 3. NUMERICL SCHEME Bcau of th o-liaiti, q. (9) ad (10) hav to b olvd itativl to com out ith th tmpatu ad voltag ditibutio alog th domai. computatioal cod a itt bad o th o-calld fiit-volum mthod (ataka, 1980). h mthod coit of dividig th phical domai ito o-ovlapig cotol volum i hich th ma, momtum ad g quatiti a covd. h ctoid of ach cotol volum, a illutatd i Fig. 3.a, copod to a itgatio cll of th dictizd domai. h popti (, ) a valuatd at th ctoid, ha th flu (q, j) a valuatd at th cotol ufac. o-uifom Catia mh a gatd b ma of th quatio itoducd b Wood (1996). Mh idpdt olutio hav b foud fo computatioal gid ith 3000 cotol volum. aphot of a computatioal domai ith 3120 cotol volum i dpictd i Fig. 3.b. (a) Figu 3: Computatioal mh: (a) tpical cotol volum, (b) o-uifom gid ith 3120 cotol volum (b) 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

6 2108, ag 5 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014 Equatio (9) ad (10) dictizd uig a 2 d od ctal-diffcig chm. h o-lia tm hav b icopoatd ito th ouc tm. h ultig algbaic quatio fo tmpatu ad voltag a a follo: ( ) ( ) ( ) ( ) τ ρ j j j j B k k k k B 2 2 N W E (14) ( ) α α α α α α α α γ γ γ γ S N W E N W E B B (15) h th upcipt atik tad fo th popt availabl fom th pviou itatio. h t of lia quatio hav b olvd itativl though th DM algoithm. h popti at th itfac bt difft matial hav b calculatd i od to guaat th cotiuit of th lcto ad hat flu. Mo dtaild ifomatio o th umical chm ca b foud i Olivia (2014). 4. MODEL LIDION h cod pdictio validatd agait pimtal data obtaid fom th maufactu of a paticula thmolctic modul. ll imulatio caid out fo h 323 K, but vaig th bt th hot ad th cold d fom 0 to 60 K, ad th applid to th hol thmolctic modul fom 11 to 16. Figu 4 ho a compaio bt th calculatd ad maud lctic cut, h o ca th maimum diffc achivd (fo 0 K ad 16 ) a blo th 10% thhold. I all ca, o ca th modl i abl to follo th pimtal td clol. (a) (b) Figu 4: Compaio bt calculatd ad pimtal lctic cut: (a) 0 K, (b) 60 K

7 dditioal validatio paamt a th coolig capacit h 2108, ag 6 Q & ad th CO, calculatd pctivl a follo: c m c Q & c NLz k i 1 m i (16) Q& c CO Q& Q& (17) h c Q & i th hat lad at th hot d, calculatd a follo: h h 1 Q & h NLz k (18) i 1 1 i h L z i th cll dimio i th z dictio (ppdicula to th pap ht), N i th umb of thmolctic pai i th cll, ad ad m a th umb of cotol volum i th ad dictio, pctivl. Figu 5 ho th calculatd ad th pimtal coolig capaciti agd to ith o ithi th 10% thhold. h high diffc i obvd fo lo voltag ad 0 K. imila bhavio i obvd i Fig. 6 fo th CO. I all ca, th pimtal td a ll poducd b th modl. Figu 7 plo th tmpatu ditibutio obtaid fo fou difft ca: (a) o thmolctic ffct (Joul hatig ol) ad 0 K, (b) thmolctic ffct ad 0 K, (c) thmolctic ffct ad 30 K, ad (d) thmolctic ffct ad 60 K. I all ca, 16. h tmpatu pofil alog th - cut (at 1.1 mm) a alo dpictd i Fig. 7. Fo ca (a), h o thmolctic ffct tak plac, o ca that th Joul hatig i mmticall diipatd b Foui coductio i uch a a th maimum tmpatu tak plac at th ct of th thmolctic lmt. hi i o a 0 K. I ca th thmolctic ffct i o, th locu of th maimum tmpatu migat fom th ct to th bottom iamuch th ica. Figu 5: Compaio bt calculatd ad pimtal coolig capacit Figu 6: Compaio bt calculatd ad pimtal CO 5. SENSIIIY NLYSIS 5.1 Sitivit to hmophical opti h itivit aali a caid out coidig a po vaiabl th coolig capacit ad th CO, ha th thmophical popti (i.. thmal coductivit, Sbck cofficit, ad lctic coductivit) tak a idpdt paamt. 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

8 2108, ag 7 (a) (b) (c) (d) Figu 7: mpatu ditibutio alog th domai fo 16 ad difft 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

9 2108, ag 8 2-lvl, 3-facto factoial dig a th plad totalizig u. h lvl t a ±5% pa takig th figu povidd b q. (11) to (13) a fc. h imulatio u caid out fo 0 K ad 16. h gio modl adoptd i thi ok i a follo: Ψ ˆ λ λ kˆ λ α ˆ λ γ ˆ λ kˆ α ˆ λ kˆˆ γ λ αγ ˆ ˆ λ kˆ αγ ˆ ˆ (19) h γ1/ρ i th lctic coductivit, Ψˆ i th dimiol po vaiabl, λ a th cofficit calculatd fom th lat-qua mthod, ad ϕˆ a th dimiol valu of ϕ, calculatd fom: 2( ϕ ϕ ) ( ϕ ϕ ) 1 ϕ ˆ (20) mi ma mi Figu 8 ho th coolig capacit i mail affctd b th lctic coductivit ad th Sbck cofficit, ad magiall affctd b th thmal coductivit, hich pla a gativ ol o th coolig capacit. h highod itactio hav ot plad a matial ffct o th coolig capacit. Figu 8 alo ho th ffct of th thmophical popti ov th CO, h o ca th Sbck cofficit pla a domiat ol, follod b th thmal ad lctic coductiviti, hich plad a magial ol. h td a cofimd b th dfiitio of th figu-of-mit of th thmolctic matial, Z α 2 γ/k, hich i taightfoadl latd to th CO. Figu 8: Rult of th itivit aali: coolig capacit ad CO 5.2 Sitivit to pct Ratio o a th ifluc of th gomt, th apct atio a vaid b icaig th hight of th thmolctic cll, L, i to fahio: (a) cotaid ba aa (i.. fid L, Fig. 9), ad (b) cotaid volum of thmolctic matial ( Fig. 10). I all ca, 0 K ad th voltag a vaid fom 14 to 20. Figu 11 ho th CO i akl affctd b L. Idd, a light ica ca b obvd. hi i o a th coolig capacit dplt iamuch th lctic cut dca, hich dimiih th po coumptio at th am at. th CO i th atio bt th coolig capacit ad th po coumptio, o ca pct th CO figu i ot igificatl chagd fom o ca to th oth. (a) (b) Figu 9: Sampl of gomti aalzd i ca of cotaid ba aa: (a) apct atio ½, (b) apct atio 2 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

10 2108, ag 9 (a) (b) Figu 10: Sampl of gomti aalzd i ca of cotaid volum: (a) apct atio ½, (b) apct atio 2 Figu 11: Ifluc of th apct atio o th CO i ca of cotaid aa (olid bullt) ad cotaid volum (op bullt) 6. CONCLUDING REMRKS fit-picipl to-dimioal tad-tat modl a put foad to valuat th thmodamic pfomac of thmolctic cll i th alm of th o-quilibium (ivibl) thmodamic. h modl tak ito accout th Foui coductio, th Joul hatig, ad th homo ffct, big abl to pdict th coolig capacit, th po coumptio, ad th CO i ca of pcibd voltag uppl ad pcibd tmpatu at th hot ad cold d. h govig quatio dictizd b ma of th fiit-volum mthod uig a ctal-diffcig chm. h o-liaiti tpical of th thmolctic phoma mbddd ito th ouc tm, ad th ultig t of algbaic quatio olvd itativl b th DM algoithm. h tailo-mad modl a codd i-hou ad it pdictio fo lctic cut, coolig capacit ad CO compad agait pimtal data obtaid fom th maufactu of a paticula thmolctic cll. It a obvd th umical pdictio ad pimtal data ot ol agd to ithi 10% thhold, but alo th modl i abl to follo th pimtal td v clol. h ifluc of th thmophical popti o th po vaiabl (coolig capacit ad CO) a ad b ma of a 2 3 factoial dig, hich ha poitd out that th Sbck cofficit ad th thmal coductivit pla majo ol o th coolig capacit, ha th CO i mo ibl to th Sbck cofficit. h ifluc of th gomt a alo ad b vaig th apct atio accodig to to difft a: cotaid ba aa ad cotaid volum. It a obvd that both th coolig capacit ad th po coumptio va at th am at, i uch a a th CO, hich lat th coolig capacit ad th po coumptio, ha hod a imila bhavio fo cotaid ba aa ad cotaid volum of thmolctic matial. 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014

11 2108, ag 10 NOMENCLURE Roma CO cofficit of pfomac [W/W] j lctic cut dit [ m -2 ] k thmal coductivit [W m -1 K -1 ] L idth [m] L hight [m] L z lgth [m] m umb of itgatig cll (-dictio) umb of itgatig cll (-dictio) N umb of thmolctic pai i th cll q& hat gatio [W m -3 ] q hat flu [W m -2 ] Q & hat taf at [W] tmpatu [K] voltag [] Z figu-of-mit [K -1 ] Gk α Sbck cofficit [ K -1 ] γ lctical coductivit [ -1 m -1 ] ϕ gic vaiabl ρ lctical itivit [ m -1 ] τ homo cofficit [ K -1 ] Subcipt c cold d,,, cotol ufac, E, W, N, S cotol volum h hot d REFERENCES Ch M, Rodahl L, Coda, th-dimioal umical modl of thmolctic gato i fluid po tm, It. J. Hat Ma af 54 (2011) Ch W, Liao C, Hug C, umical tud o th pfomac of miiatu thmolctic cool affctd b homo ffct, pplid Eg 89 (2012) Rold WC, 1968, hmodamic, McGa-Hill, N Yok, US Du C, W C, Epimtal ivtigatio ad umical aali fo o-tag thmolctic cool coidig homo ffct, It. J. Hat Ma af 54 (2011) Hm CJL, Baboa JR, hmodamic compaio of lti, Stilig, ad vapo compio potabl cool, pplid Eg, pplid Eg 91 (2012) Huag M, Y R, Wag, h ifluc of th homo ffct o th pfomac of a thmolctic cool, It. J. Hat Ma af 48 (2005) L KH, Kim OJ, ali o th coolig pfomac of th thmolctic mico-cool, It. J. Hat Ma af 50 (2006) Mg F, Ch L, Su F, umical modl ad compaativ ivtigatio of a thmolctic gato ith multiivibiliti, Eg 36 (2011) Olivia KMS, 2014, Numical mt of th hmodamic Bhavio of hmolctic Cll, M.Eg. thi, Fdal Uivit of aaá, Cuitiba-R, Bazil, 140p. (i otugu) ataka S, 1980, Numical Hat af ad Fluid Flo, Hmiph ublihig Co., N Yok, US éz-paicio JL, alma R, alo RL, Fiit lmt aali ad matial itivit of lti thmolctic cll cool, It. J. Hat Ma af 55 (2012) amaick K, Da K, Cotuctal dig of a thmolctic dvic, It. J. Hat Ma af 49 (2006) Ro D, 1995, CRC Hadbook of hmolctic, CRC, Boca Rato-FL, US llu, 2007, Modul Spcificatio Z-ma C , av Cit-MI, US Wood W, 1996, Multigid ppoach to Icompibl icou Cavit Flo, chical Mmoadum, NS Lagl Rach Ct, Hampto-, US Yamahita O, Effct of lia ad o-lia compot i th tmpatu dpdc of thmolctic popti o th coolig pfomac, pplid Eg 86 (2009) CKNOWLEDGEMENS Fiacial uppot fom th Bazilia fudig agci CES ad CNq i dul ackoldgd. 15 th Itatioal Rfigatio ad i Coditioig Cofc at udu, Jul 14-17, 2014