Bdmensonal Analyss of a Termoeletr Module usng Fnte Element Tenques *Antono Arenas, Jorge Vázquez, Rafael Palaos Unversdad Pontfa Comllas Esuela Téna Superor de ngenería *Departamento de Fludos y Calor, nsttuto de nvestgaón Tenológa Alberto Agulera, 3 8015 Madrd (Span e_mal: arenas@df.a.upo.es Abstrat n ts paper an analyss s performed on a termoeletr module usng fnte element tenques takng nto aount te followng two ypotess: termal and eletral flows are present n two dmensons. Te ommeral software tool named ANSYS as been used n order to develop ts analyss. Ts software nludes te possblty to analyse ontly te Fourer and Joule effets, but not te Pelter effet. Addtonal software as been developed n order to nlude te Pelter effet. Te onsderaton of te tree effets at te same tme wll allow for a better smulaton and analyss of te performane of a termoeletr module. Te ANSYS model developed very losely represents te real onfguraton of a termoeletr module. t takes nto aount te termoelements, and ter eletral brdges and untons. All te propertes of te materals ave been onsdered onstant. Several smulatons ave been performed usng te proposed model. n all ases, te model as beome a powerful and flexble tool of analyss able to present detaled numeral and grapal results. Furtermore, te possblty to use ts model n more omplex strutures s a very attratve feature now avalable. Te omparson between te results of ts model and te performane of an elemental termoelement allows te observaton of te nfluene of every omponent n a termoeletr module. Te model and te man tests performed wll be presented n ts paper. ntroduton Numer tenques are neessary wen a detaled analyss of a termoeletr ell s requred. n ts work a fnte element model was developed usng tools probed enoug to guarantee ts relablty. Only a spef subroutne was desgned and programmed n APDL (ANSYS Parametr Desgn Language. Te study of te beavour of a termoeletr ell s basally a problem of smultaneous termal and eletral flows n a regular geometry. ANSYS, a ommeral program of fnte element metod, solves ts type of problems wt g auray as stated a lot of valdated tests. All te models n ts work were studed n two dmensons. t s a good approxmaton beause te eletral and termal flows are dstrbuted n two dmensons n te termoelements, eletral brdges, and solderng paste. Tere wll be only termal flux n 3D n te eat exange plates. So, te results of ts study are ompletely vald for te nternal elements of te ell. Struture of te program Te program allows ntrodung parametr models and makng smulatons wt a great deal of loads and boundary ondtons. Te program as a man subroutne tat ontrols two auxlary subroutnes: a data subroutne were all te values of te varables are ntrodued; and te Pelter subroutne, were Pelter effet s appled. yes no 1 3 4 5 6 7 8 9 10 11 1 13 14 1.- Start of te man program.eld..-nput of te data. Te number of ases s fxed. 3.- Te geometr model s bult. 4.- Te mes and element aratersts are defned. 5.- Te dfferent materal propertes are assgned. 6.- Mesng. 7.- Te sze of vetors and matres s fxed. 8.- Seleton of referene nodes. 9.- Te loads and te boundary ondtons orrespondng to ea ase are assgned. 10.- Resoluton of te problem wt te ntally loads.. 11.- Pelter subroutne s exeuted. 1.- Te output fles wt all te results of te ase are generated. 13.- dentfaton of te new ase. 14. End. Fgure 1. Man subroutne flow art. Man subroutne Te flow art of te man subroutne s sown n Fg. 1. ANSYS provdes two types of elements (PLANE 67 and SOLD 69 for problems of termal and eletral flows, but t does not ave any element tat takes nto aount Pelter, Fourer and Joule effets smultaneously. Te mes as te same sape n te termoeletr ell but t does not ave te same sze n all te omponents. As te
ell as a layered struture, t was easy to get te ondene of te nodes loated n te ontguous surfaes. n oter words, all te elements of te mes ave te same base dmenson and tey only dffer n ts egt. Data subroutne All te data regardng wt te ell (dmensons, materal propertes, loads and boundary ondtons, and some varables to ontrol te program and te presentaton of te results are nluded n ts subroutne. Pelter subroutne Solve agan 1 3 4 5 6 7 8 Contnue 1.- Start of Pelter subroutne, t s ontrolled by te program eld..- Temperature of te referene nodes s read. 3.- Te termal flow owng to Pelter effet (ftp s alulated n ea node. 4.- Te ftp are ntrodued n te model as new external termal loads. 5.- Te model s solved wt te new external termal loads. 6.- Te new temperature n te referene nodes s read. 7.- Temperature dfferene before and after applyng te ftp s alulated and ompared wt a referene value. 8.- End of te Pelter subroutne. Te man program goes on. Fgure. Flow art of Pelter subroutne. A spef subroutne was programmed n order to apply te Pelter effet n te model. Te flow art of ts subroutne s sown n Fg.. Te temperature and eletral urrent densty n all te nodes of te mes are stored. Wt tese data te termal power by Pelter effet s alulated and appled as a new external load n ea of te nodes n te orrespondng dreton. A 3 1 4 A Fgure 3. Seme of te mes. Pelter effet s produed n te separaton surfaes between two dfferent materals wen eletral urrent goes aross t. Te algortm s able to dsrmnate te nodes tat are n te separaton surfaes and ten alulates te urrent B B tat rosses t. Te program also takes nto aount te dsontnuty of te materals n te ends of te model (nlet and outlet of te eletral urrent A pee of te mes s sown n Fg. 3, t brngs togeter te four elements around a gener node, beng =1,,3,4 te elements w as te node n ommon. A node an belong to several elements (an element s ea of te small quadrlaterals n w te model s mesed. Every omponent of te termoeletr ell s dvded nto elements and a materal s assgned to every element n te model. Frstly, te followng produt s alulated for ea node as a part of an element: were: p = σ T : Pelter eat n node assoated wt te element p : ntensty n node assoated wt te element. T : Temperature at node. σ : Seebek oeffent n node assoated wt element Te sum of all te produts,,s te net eat generated p by Pelter effet n te nfluene area of node (te same nfluene area tat for te ntenstes assgned to ea node: p = (1 T σ = T σ ( = 0 (3 Te sgn of s gven by te eletral urrent. Ten, all p te are appled as new external loads and te problem s p solved agan as a termal and eletral one. A new temperature and urrent densty dstrbuton s obtaned. Ts proess wll be repeated untl te dstrbutons between two onseutve teratons wll be pratally nvarable. Te eletral urrent an be wrtten as: = (, + 1 + (, 1 (4 means te lne of separaton between te adaent elements and ± 1 around te node. Ten: σ = ( (, + 1 + (, 1 σ = (, + 1 σ + (, 1 σ (5 Were te subsrpt (, ±1 Calulatng te sum of all tese produts for all te elements tat ave te node n ommon, te eq. (6 s obtaned: ( = T ( (, + 1 ( σ σ p = T σ (6 + 1 n Fg. 3, elements 1- and 3-4 ave te same materals assgned, so σ 1 = σ and σ 3 = σ. n ts ase, te eq. (6 4 wll be: p = T ( ( 1,4 + (,3 ( σ A σ B (7
p = T ( σ σ (8 A B Beng > 0 wen te eat s generated n te unon. p Analysed ases Two ases were studed: An elemental onstant ase (d-e: t uses a smplfed model wt sotrop materals and wt onstant propertes. Te model was also studed wt analytal equatons n order to verfy te results obtaned wt te program. A omplete onstant ase (d-: a termoeletr ell wt all te elements (eletral brdges, weldngs... was studed. Te materal propertes were onsdered onstant wt te temperature. All smulatons were done wt a onstant eletral urrent and fxng te temperature on te ot and old faes of te termoeletr ell. Normally, tese are te referene ondtons n te atalogues of te manufaturers, altoug some tmes, te eat transmsson between te ell and te ambent an ause very mportant lmtatons, beng rt te sze of te old and ot faes n te desgn and operaton of te ell []. Elemental-onstant model d-e y Y" m x T Y' Fgure 4. Seme of te model d-e. Te elemental model was only made up by te termoelements oned by a materal wtout termal and eletral resstane. Ts model s pysally nfeasble but very useful n a teoretal analyss. As sown n Fg. 4, t s possble to use only a termoelement wose propertes are te average of ea of te two termoelements (type p and n tat form te termoouple. Te dmensons were l=1.14 mm and m=1.4 mm. Materals and propertes an be seen n annexe 1. Te model worked wt te old fae temperature fxed at T=73K. Dfferent ases were studed orrespondng to temperature dfferenes (T-T from 0ºC to 70ºC wt nrements of 10ºC, and eletral urrent varyng from 1A to wt nrements of 1A (56 ases n total. All te results laked of nterest f tey were not ompared wt te analytal ones obtaned n [1, apter ], beause te teoretal equatons are exat n ts smplfed model. All te results presented n next fgures are te relatve dfferenes T l between de magntudes alulated by te numeral metod X X X. n and te analytal one a, ( n a a (n-a/a 1.0 E+0 0 5.0E-01 0.0E+00-5.0E-01-1.0E+00-1.5E+00 0A A 4A -.0E+00 Fgure 5. Relatve dfferene n.mode A.T=73K (d-e. Máx (Tn-Ta/Ta 1.0 E- 0 0.0E+00-1.0E-0 -.0E-0-3.0E-0-4.0E-0-5.0E-0 Fgure 6. Maxmum relatve dfferenes n te temperature dstrbuton nsde te termoelement. Te relatve dfferenes n termal powers are very low, beng under 0.% for and 0.0% for, exept n te proxmtes of te ange of sgn. n te ase of te termal power generated by Pelter effet te error s neglgble, and ts s te most mportant onluson beause t was te man obetve of ts analyss. Te relatve dfferene between te effenes s also very low, under 0.0% and t s always very smlar to te dfferene of te termal powers. At last, te temperature dstrbuton was evaluated n te ross seton (Y -Y. As sown n Fg. 6, te maxmum relatve dfferene s less tan 0.06%, beng te temperature alulated by te numeral metod always lower tan te analytal. Complete-onstant model d- Te model smulated s sown n Fg. 7, te dmensons and materals are spefed n annexe 1. n ts ase te model takes nto aount all te materals nvolved n a ommeral termoeletr module. Te model nludes: Te weldngs (elements bx between te termoelements (ax and eletral ondutors (x 1A 3A 5A 0A 1A A 3A 4A 5A -6.0E-0
Te dmenson n z dreton s 1mm. (D. Te materal propertes are onsdered onstant. Te sze of te mes s 0.1mm. Y Y 1 e b d 3.460.800.765.365.300 temperature dfferenes and eletral urrents as n te model d-e were studed. ANSYS allows to plot te temperature dstrbuton n all nodes, and te termal and eletral flow (Fgs. 8, 9, 10. a a 1 y x b d e 1 b 1 d 1 1.160 0.695 0.695 0.660 0 0 0.5 Y 1.9.9 Y 1 4.3 4.8 Fgure 7. Seme of te model d-, dmensons (mm. Fgure 8. Temperatures, T=73K, Fgure 9. Heat power flow (W/m, T=73K T = 40º C, =4A. T = 40º C, =4 A. n ts ase, t s possble to reprodue te way of workng of a termoeletr ell studyng a termoouple owng to te symmetry, but te model annot be redued to a unque termoelement. Two ways of workng were analysed. Te frst mode (Mode A n w te old fae temperature remaned onstant (T=73K, and te seond one (Mode B n w te ot fae temperature was fxed n (T=93K. Te same Fgure 10. Eletr urrent flow (A/m. T=73K, T = 40º C,=4 A. Effeny (ε f (Mode A and COP (Mode B. Te ε f and COP assoated wt mode A and mode B respetvely are sown n Tables 1 and. \ T 0ºC 10ºC 0ºC 30ºC 40ºC 50ºC 60ºC 70ºC 0 A - - - - - - - - 1 A 6.645 3.9 0.987-0.598-1.778 -.690-3.387-3.976 A 3.074.150 1.417 0.800 0.317-0.090-0.439-0.740 3 A 1.884 1.470 1.086 0.789 0.531 0.304 0.10-0.078 4 A 1.89 1.06 0.88 0.65 0.494 0.351 0.1 0.103 5 A 0.933 0.761 0.634 0.519 0.413 0.316 0.7 0.145 6 A 0.695 0.574 0.487 0.406 0.33 0.63 0.198 0.138 7 A 0.56 0.435 0.373 0.314 0.59 0.08 0.160 0.114 Table 1. Effeny ε f (%, Mode A. T =73K \ T 0ºC 10ºC 0ºC 30ºC 40ºC 50ºC 60ºC 70ºC 0 A - - - - - - - - 1 A 8.169 4.437 1.987 0.55-1.034 -.030 -.795-3.440 A 4.335 3.65.417 1.709 1.149 0.676 0.7-0.078 3 A 3.058.550.086 1.7 1.405 1.16 0.877 0.656 4 A.40.085 1.88 1.599 1.393 1.07 1.038 0.884 5 A.037 1.808 1.634 1.475 1.39 1.196 1.07 0.958 6 A 1.783 1.614 1.487 1.369 1.60 1.159 1.064 0.976 7 A 1.601 1.470 1.373 1.8 1.197 1.117 1.04 0.971 Table. Coeffent of Performane (COP,Mode B. T =93K Te values obtaned wt te numeral metod were ompared wt te orrespondng results of te elementary model (e, w represent te maxmum values for ea workng ondton. As sown n Fgs 11 and 1, te ε f was redued n a 18% ompared wt ts maxmum value. Ts drop s attrbuted to te elements n between (weldngs, eletral brdges... However, te derease n COP was less sgnfant beause te referene values are ger. Te results for ε f and COP obtaned n te omplex numeral model were n agreement wt te results derved from te analytal model ndated n [1, apter 5]. As sown n Fgs. 13 and 14, te relatve dfferenes are under 6% and 1% n ε f and COP respetvely.
(ε fn -ε fe /ε fe - -4-6 -8-10 -1-14 -16-18 -0 Fgure 11. Relatve dfferenes n ε f ompared to te elementary model. Mode A, T =73K. (COP n -COP e /COP e 0-1 - -3-4 1A A 3A 4A 5A 1A 3A 5A A 4A measure of te nfluene of te addtonal omponents tat bult a real termoeletr ell. Tese dfferenes flutuated from 3% to 17% n ε f and were lower tan 4% n te ase of COP. (COP a -COP n /COP n 1.5 1.0 0.5 0.0-0.5-1.0 Fgure 14. Relatve dfferenes n COP ompared to te analytal model. Mode B, T =93K. Termal powers Te graps of termal powers on te old fae (, mode A and on te ot fae (, mode B versus te temperature dfferene are sown n Fgs. 15 and 16. 0.0 0.15 1A A 3A 4A 5A -5 Fgure 1. Relatve dfferenes n COP ompared to te elementary model. Mode B, T =93K. (ε fa -ε fn /ε fn 8 6 4 0-1A A 3A 4A 5A Fgure 13. Relatve dfferenes n ε f ompared to te analytal model. Mode A, T =73K. Te pont n w te ell works wt maxmum effeny s onsdered te most relevant. So, te relatve dfferenes between te omplex model and te elementary model workng at ts pont onsttutes te most mportant Cn, W 0.10 0.05 0.00-0.05-0.10-0.15-0.0 0 1A A 3A 4A 5A Fgure 15.Termal power at old fae.. Mode A. T =73K. A detaled analyss of te total termal power densty n te ot and old fae of te ell was arred out. As stated n Fgs. 17 and 18, te absene of unformty s lear n bot dstrbutons. n te ot fae te maxmum value s reaed at x=1.3mm and t s 17.5% ger tan te average, wle te mnmum s at x=4.8mm and t s 39.8% lower. Te dstrbuton on te old fae s less unform, wt te maxmum loated at x=1.1mm (34.4% wle te mnmum s at x=.5mm (-39.7%. Despte of te geometr symmetry, tere s an asymmetry n te termal power densty dstrbutons. Te reason of ts asymmetry les on te dfferene n te Seebek oeffents between te termoelements and te adaent materals.
n, W 0.6 0.5 0.4 0.3 0. 0.1 0.0-0.1-0. Fgure 16. Termal power at ot fae,. Mode B. T =93K. f t s admtted te same number of nodes (n, a unform temperature dstrbuton ( T n = T, and a unform eletral urrent dstrbuton ( n = n, ten: P P = T σ (1 1 b However, under te same ondtons, te sum of and P1 s te same as n a dret ontat between te P termoelements wtout weldng or oter elements. P1 n 1,9 n 4,3 P1 + P = ntn t1 n n t1 n 0,5 n,9 0.0 ( σ σ + T ( σ σ n 4,3 ( σ + T ( σ σ (13 n 1,9 1 = P = ntn σ (14 t1 n n t1 n 0,5 n,9 n 1,9 n 4,3 P1 = P = ntnσ = ntnσ (15 n 0,5 n,9 0.18 e d b Termal power densty 0.16 0.14 0.1 y =,3 mm. a a 1 Fgure 17. Total termal power densty at ot fae. T =73K, T = 40º C, =4A. a a 1 0.10 Dstane x from te edge of te dsspater, mm Fgure 19. Termal power densty (W/mm by Pelter effet n te ot sde of te termoelements. T=73K, T=40ºC, =4A. 315.0 b d Fgure 18. Total termal power densty at old fae. T =73K, T = 40º C, =4A. f σ, t1 σ, and σ are te Seebek oeffents of a1, a, b and b (Fg. 16, te termal power produed by Pelter effet at te ot fae of ea termoelement s: n 4,3 P1 = ntn b t1 n,9 P n 1,9 = ntn n 0,5 Normally σ t1 = σ and ten: b e ( σ σ ( σ σ b ( σ σ b σ b t1 = + b 1 1 d 1 (9 (10 σ σ (11 Temperature, K 314.8 314.6 314.4 314. y =,3 mm. 314.0 Dstane x from te edge of te dsspater, mm Fgure 0. Temperature dstrbuton n element b. Te lak of unformty n te termal power densty (Fg. 19 n te nternal sde of te termoelements s owng to te ger eletral urrent densty n ts area and beause te drop n temperature n ts area (Fg. 0 s very small. Ts
drop of temperature s so small frstly, beause te termal ondutvty of te solderng paste s very g and seondly beause te g urrent densty produes an mportant rse n te termal power generated by Joule effet nreasng te temperatures. Ts sort of unformty an be more mportant f te materal propertes are onsdered varable wt te temperature. Termal power densty Fgure 1. Termal power densty (W/mm by Pelter effet n te ot sde of te ell. T =73K, T = 40º C, =4ª. As sown n Fg. 1, te Pelter power densty n te unons of te addtonal elements s ompletely symmetr n bot sdes of te termoelements (a1, a. As te eletral urrent rosses te elements n opposte dreton, te sgn of ts power s dfferent and te net ontrbuton to te termoouple s null. Termal power densty 0.005 0.004 0.003 0.00 0.001 0.000-0.001-0.00-0.003-0.004-0.005 Dstane x from te edge of te dsspater, mm 0.35 0.30 0.5 0.0 0.15 0.10 0.05 y = 3,46 mm. y =,3 mm. y =,765 mm. y =,3 mm. y =,365 mm. y =,8 mm. y =,365 mm. y =,765 mm. 0.00 Dstane x from te edge of te dsspater, mm Fgure. Total termal power densty (W/mm n te ot sde of te ell. T =73K, T = 40º C, =4A. Te total termal power dstrbuton per unt area n te dfferent surfaes between te termoelements and te external surfae of te ot outlet plate s sown n Fg.. Te temperature of ts plate remaned onstant owng to a boundary ondton. Tere s a dampng of te termal power peaks n te termoelements as a onsequene of te ntermedate elements. Te omponents loated on top te termoelements (eletral ondutor (, solderng paste (d and plate (e tends to unform te termal power dstrbuton not only beause tey are good sotrop termal ondutors but te termal power generated by Joule and Pelter effet s more unform. Te same study was done n te old fae and te results were very smlar. As sown n Fgs. 3 and 4. Te relatve dfferenes between te termal powers alulated wt te numeral and analytal metod sowed n [1, apter 5] are under 6% n te two ases (,. ( fa - fn / fn 6 5 4 3 1 0-1 - Fgure 3. Relatve dfferenes n ompared to te analytal model. Mode A, T =73K. Workng n te ondtons of maxmum effeny or op, te nfluene of te ntermedate elements aeves relatves values from 4% to 16% for te old fae an lower values tan 4% for te ot fae (ompared wt te results of te elementary model. ( a - n / n 3.0.5.0 1.5 1.0 0.5 0.0-0.5-1.0 0 1A A 3A 4A 5A 0 1A A 3A 4A 5A Fgure 4. Relatve dfferenes n ompared to te analytal model. Mode B, T =93K. Temperature dstrbuton. Ts study was done wt spef workng ondtons, T = 40º C, T=73K, =4. Te average temperature dstrbutons n bot termoelements for te numer, analytal and elementary model are sown n Fg. 5.
Temperature, K Fgure 5. Average temperature dstrbuton n te termoelements, T =73K, T = 40º C, =4A. Temperature, K 30 310 300 90 80 70 315.0 314.5 314.0 313.5 313.0 1.0 1. 1.4 1.6 1.8.0..4 Dstane y from te old edge of te ell, mm y =,365 mm. y =,8 mm. y =,3 mm. y =,765 mm. y = 3,46 mm. modnum modana_ modana_e 31.5 Dstane y from te edge of te dsspater, mm Fgure 6. Temperature dstrbuton n te unons between te ntermedate elements at te ot sde, T =73K, T = 40º C, =4A. Te results between te numer and te analytal model were very smlar. But te results found n te omplex model were sgnfantly dfferent from te results n te elementary model. Te temperature dfferene s 1.97% ger tan n te elementary model. Ts dsrepany s attrbuted to te ntermedate elements and t s more pronouned n te ot sde. At last, te temperature dstrbuton n te ons of te elements of te ot sde s sown n Fg. 6. Takng nto aount tat te temperature s onstant n te external fae of te eat exanger, te flattenng of te temperature dstrbuton from te respetve fae of te temoelement s lear. t s also possble to observe te asymmetres n te nfluene area of ea termoelement owng to te asymmetry n te termal powers generated by Pelter effet. Conlusons A study of a real termoeletr ell usng fnte element tenques as been developed, onsderng tat termal and eletral flows were produed n two dmensons. Te metodology s based on te use of a ommeral program w study Fourer and Joule effets n a oupled way, ts program as been ompleted makng a subroutne n order to norporate te Pelter effet. Several models were analysed. Frstly an elementary model, ust onsderng termoeletr elements, was studed. Te great advantage of ts analyss t tat a matematal soluton exsts and te numer metod an be valdated. Te errors were under 0.% for and 0.0% for, smlar values were obtaned for te effeny and te COP. Te dfferenes between temperature dstrbutons were smaller. At last a omplex model of a ell was analysed, n ts ase all te elements and materals of a ommeral termoeletr module are onsdered. Te enormous possbltes of ts tool ave been stated. t s very easy to obtan numeral and grapal results of all te varables n all parts of te ell. Te nfluene of te ntermedate elements (weldng, eletral brdges, dsspaters an be evaluated ontrastng te results between te omplex and te elementary model. Nomenlature COP: Coeffent of performane, W. : Amount of eat absorbed at old fae. : Amount of eat gven away at ot fae. T : Temperature of te old fae durng operaton. T : Temperature of te ot fae durng operaton. W :Eletral power. f e ε : Effeny, Anexxe 1 Elements W e Materal Fg. 7 e Poston Dmensons (mm Fg. 7 Hegt (y Wdt (x Lengt Termoelement (p BTe3 a1 1.14 1.4 1 Termoelement (n BTe3 a 1.14 1.4 1 Weldng Sn-B bf,b 0.065 1.9 1 Eletr brdge Cu f, 0.4 1.9 1 Weldng Sn-B df, d 0.035 1.9 1 Dsspaters AlO3 ef,e 0.66..4 1 λ W/(mK Propertes of materals BTe3 (p BTe3 (n AlO3 Sn Cu 1.600 1.600 4.90E+01 6.000E+01 4.030E+0 ρ Ωm 9.000E-06 9.000E-06 1.000E+1 1.150E-07 1.440E-08 σ V/K.000E-04 -.000E-04 3.000E-05 1.000E-05 1.400E-05 τ V/K 1.339E-04-1.339E-04 ------- ------- ------- Referenes 1. Arenas, A. Determnaón de nuevos rteros que permtan la optmzaón de parámetros de dseño de una bomba de alor por efeto Pelter. (PD Tess. Unversdad Pontfa Comllas. Madrd, 1999.. Palaos, R., Sanz-Bob, M.A., et al, Prototype of eat pump based on Pelter effet. Result of performane tests, Pro. nd European Worksop on Termoeletrs, Nany, Frane, November, 1995, pp. 93-97.