Optimal Design of Thermoelectric Devices with Dimensional Analysis

Transcription

1 Optimal Design of emoelectic Devices wit Dimensional Analysis HoSung Lee Mecanical and Aeonautical Engineeing, Westen Micigan Univesity, 93 W. Micigan Ave, Kalamazoo, Micigan , USA Office (69) Fax (69) Abstact e optimum design of temoelectic devices (temoelectic geneato and coole) in connection wit eat sins was developed using dimensional analysis. ew dimensionless goups wee popely defined to epesent impotant paametes of te temoelectic devices. Paticulaly, use of te convection conductance of a fluid in te denominatos of te dimensionless paametes was citically impotant, wic leads to a new optimum design. is allows us to detemine eite te optimal numbe of temocouples o te optimal temal conductance (te geometic atio of footpint of leg to leg lengt). t is stated fom te pesent dimensional analysis tat, if two fluid tempeatues on te eat sins ae given, an optimum design always exists and can be found wit te feasible mecanical constaints. e optimum design includes te optimum paametes suc as efficiency, powe, cuent, geomety o numbe of temocouples, and temal esistances of eat sins. Keywods: Optimal design; Dimensional analysis; emoelectic geneato; temoelectic coole; temoelectic module

2 omenclatue A coss-sectional aea of temoelement (cm ) A total fin suface aea at fluid (cm ) A total fin suface aea at fluid (cm ) A b base aea of eat sin (cm ) COP te coefficient of pefomance eat tansfe coefficient of fluid (W/m K) eat tansfe coefficient of fluid (W/m K) electic cuent (A) L lengt of temoelement (mm) temal conductivity (W/mK), p n n te numbe of temocouples n dimensionless temal conductance, A L A dimensionless convection, A A dimensionless cuent, A L V dimensionless voltage, V Vn n Q te ate of eat tansfe enteing into EG (W) Q te ate of eat tansfe leaving EG (W) P powe density (W/cm ) d R electical esistance of a temocouple () R L load esistance of a temocouple () R dimensionless esistance, R RL R junction tempeatue at fluid ( C) junction tempeatue at fluid ( C) tempeatue of fluid ( C) tempeatue of fluid ( C) maximum tempeatue of fluid ( C),max minimum tempeatue of fluid ( C),min V n W n W Z n Voltage of a module (V) powe output (W) powe input (W) te figue of meit Gee symbols Seebec coefficient (V/K), p n electical esistivity (cm), p n fin efficiencyof eat sin fin efficiencyof eat sin

3 3 t temal efficiencyof EG Subscipts p p-type element n n-type element opt optimal quantity /opt alf optimal quantity Supescipt dimensionless. ntoduction emoelectic devices (temoelectic geneato and coole) ave found compeensive applications in sola enegy convesion [], exaust enegy convesion [,3], low gade waste eat ecovey [4,5,6], powe plants [7], electonic cooling [8], veicle ai conditiones, and efigeatos [7]. e most common efigeant used in ome and automobile ai conditiones is R-34a, wic does not ave te ozone-depleting popeties of Feon, but is neveteless a teible geenouse gas and will be banned in te nea futue [9]. e petinent candidate fo te eplacement would be temoelectic cooles. Many analyses, optimizations, even manufactues pefomance cuves on temoelectic devices ave been based on te constant ig and cold junction tempeatues of te devices. Pactically, te temoelectic devices must wo wit eat sins (o eat excanges). t is ten vey difficult to ave te constant junction tempeatues unless te temal esistances of te eat sins ae zeo, wic is, of couse, impossible. A significant amount of eseac elated to te optimization of temoelectic devices in conjunction wit eat sins as been conducted as found in te liteatue [-3]. t is well noted fom te liteatue tat tee is te existence of optimal conditions in powe output o efficiency wit espect to te extenal load esistance fo a temoelectic geneato (EG) o te electical cuent fo a temoelectic coole (EC). Many eseaces attempted to combine te teoetical temoelectic equations and te eat balance equations of eat sins, and ten to optimize design paametes suc as te geomety of eat sins [], allocation of te eat tansfe aeas of eat sins [,3,8,9], temoelement lengt [4], te numbe of temocouples [5], te geometic atio of te coss-sectional aea of temoelement to te lengt [6], and slendeness atio (te geometic facto atio of n-type to tat of p-type elements) [7]. t can be seen fom te above liteatue tat te geometic optimization of temoelectic devices is impotant in design and also fomidable due to so many design paametes. e temal conductance of temoelements tat is te most impotant geometic paamete as been often addessed in analysis, wic is te poduct of tee paametes: te numbe of temocouples, te geometic atio, and te temal conductivity. n ode to educe te optimum design paametes, obviously dimensionless analyses wee pefomed in te liteatue [-6]. Yamanasi [] developed optimum design intoducing dimensionless paametes fo a temoelectic coole wit two eat sins, weein te temal conductance appeas twice in te nominatos and fout in te denominatos of te dimensionless paametes. Altoug is wo led to a new appoac in dimensionless optimum design, te analysis encounteed difficulties in optimizing te cooling

4 4 powe wit espect to te temal conductance because te conductance is inticately elated to te otes. Late, eseaces [,4,5] epoted optimum design using te simila dimensionless paametes used by Yamanasi [], pesenting valuable optimum design featues as Xuan [] optimized cooling powe fo a EC as a function of temoelement lengt, Pan et al. [4] sowed te optimum temal conductance fo a EC wit a given cooling powe, and Casano and Piva [5] pesented te optimum extenal load esistance atio wit eat sins fo a EG wic is geate tan unity. ee ae also some expeimental wos [7-3] compaing wit te teoetical temoelectic equations. Gou et al. [7] conducted expeiments fo lowtempeatue waste eat ecovey and demonstated tat te expeimental esults wee in fai ageement wit te solution fomulas oiginally deived by Cen et al. [5] fom te geneal teoetical temoelectic equations. Cang et al. [8] and Huang et al. [9] conducted expeiments fo a EC fom a eat souce wit ai-cooling and wate cooling eat sins, espectively. Casano and Piva [3] epoted expeimental wo on a set of nine temoelectic geneato modules wit a eat souce on one side and a eat sin on te ote side. Afte delibeately detemined te eat leaage wic tuned out to be about 3% of te supplied eat souce, tey demonstated tat te teoetical pefomance cuves of te powe output and efficiency as a function of te extenal load esistance and tempeatue diffeence wee in good ageement wit te measuements. t is ealized fom te above expeimental wos tat te teoetical temoelectic equations wit te eat balance equations of eat sins can easonably pedict te eal pefomance. Howeve, pope optimum design still emains questionable. n spite of many effots fo optimum design, its applications seem geatly callenging to system designes [,,3]. Fo example, Hsu et al. [3] in tested an exaust eat ecovey system bot expeimentally using an automobile and matematically using compute simulations. ey found a easonable ageement between te measuements and te simulations. Howeve, tey obtained te powe output of.4 W ove 4 temoelectic geneato modules wit te exaust gas tempeatue of 573 K and te ai tempeatue of 3 K. Wen te powe output was divided by te footpint of 4 temoelectic geneato modules, it gives te powe density of.3 W/cm, wic seems unusually small. Kai et al. [] in conducted a simila expeiment wit an SUV automobile. is time tey designed te exaust eat ecovey system wit an optimum coolant flow ate. ey obtained te powe output of 55 W ove 6 temoelectic geneato modules wit te exaust gas tempeatue of 686 K and te coolant tempeatue of 36 K, wic povided te powe density of.6 W/cm. is sows a significant impovement, indicating te impotance of optimum design. A ew Enegy Development Oganization (EDO) Pogam (Japan) [7] in 3 also epoted a simila expeiment wit a passenge ca, obtaining te powe output of 4 W ove 6 segmented-type modules wit te exaust gas tempeatue of 773 K and te coolant tempeatue of 98 K, wic povided te powe density of ~ W/cm. otably, te powe densities obtained ae no way to evaluate ow good it is until te bette comes because pope optimum design seems not available. Fom te eview of te above teoetical and expeimental studies including optimum design in te liteatue, it is summaized tat te pope optimum design sould be detemined basically not only by te powe output fo EG (o cooling powe fo EC) but also by te efficiency (te coefficient of pefomance) simultaneously wit espect to bot te extenal load esistance (o te electical cuent) and te geomety of temoelement wic efes to te numbe of temocouples and te geometic atio. e fome (extenal load esistance) is well attained in te liteatue but te latte (geomety) is vague. eefoe, te optimum design seems

5 5 incomplete. is is te ationale wy te pesent pape is to impove te optimum design intoducing new dimensionless paametes.. emoelectic Geneato Let us conside a simplified steady-state eat tansfe on a temoelectic geneato module (EG) wit two eat sins as sown in Figue (a). Eac eat sin faces a fluid flow at tempeatue. Subscipt and denote ot and cold quantities, espectively. We assume tat te electical and temal contact esistances in te EG ae negligible, te mateial popeties ae independent of tempeatue, and also te EG is pefectly insulated. e EG as a numbe of temocouples, of wic eac temocouple consists of p-type and n-type temoelements wit te same dimensions as sown in Figue (b). t is noted tat te temal esistance of eat sin can be expessed by te ecipocal of te convection conductance A, wee is te fin efficiency, is te convection coefficient, and A is te total suface aea in te eat sin. We eeafte use te convection conductance ate tan te temal esistance. (a) (b) Figue. (a) emoelectic geneato module (EG) wit two eat sins and (b) temocouple. e basic equations fo te EG wit two eat sins ae given by Q A () A Q n R L () A Q n R L (3)

6 6 Q A (4) R L R (5) wee p n, p n, and p n. Equations () (5) can be solved fo and, poviding te powe output. Howeve, in ode to study te optimization of te EG, seveal dimensionless paametes ae intoduced. As mentioned in Section of ntoduction, it is eminded tat optimum design sould conside not only te powe output but also te efficiency simultaneously wit espect to bot te extenal load esistance and te geomety of temoelement wic efes to te numbe of temocouples and te geometic atio. n ode to eveal te effect of temal conductance n A L, te temal conductance is placed in te nominato wile te convection conductance A in fluid is placed in te denominato of te paamete. e convection conductance A and tempeatue at fluid ae assumed to be given. e dimensionless temal conductance, te atio of temal conductance to te convection conductance in fluid, is defined by A n A L (6) e dimensionless convection is an impotant geomety of eat sins. Since te convection conductance at fluid is given as mentioned befoe, te dimensionless convection, te atio of convection conductance in fluid to fluid, is defined by A A (7) e dimensionless electical esistance, te atio of te load esistance to te electical esistance of temocouple, is given by RL R (8) R Since te fluid tempeatue at fluid is given as mentioned befoe, te dimensionless tempeatues ae defined by (9) ()

7 7 () e dimensionless powe and eat tansfe ae defined by dividing by te poduct of te convection conductance and te tempeatue of fluid so tat te quantities depend only on te nominatos not on te denominatos since te denominato is assumed to be constant o given. e two dimensionless ates of eat tansfe and te dimensionless powe output ae defined by Q Q A () Q W n Q A Wn A (3) (4) t is noted tat te above dimensionless paametes ae based on te convection conductance in fluid, wic means tat A sould be initially povided. Also note tat te temal conductance n A L appeas only in Equation (6) among ote paametes so tat te temal conductance can be examined fo optimization. Using te dimensionless paametes defined in Equations (6) (), Equations () (5) educe to two fomulas as: Z Z R R (5) Z Z R R (6) wee Z is called te figue of meit ( Z ). Equations (5) and (6) can be solved fo and. e dimensionless tempeatues ae ten a function of five independent dimensionless paametes as f,, R,, Z (7) f,, R,, Z (8) is te input and Z is te mateial popety wit te input, and bot ae initially povided. eefoe, te optimization can be pefomed only wit te fist tee paametes (,, and R). Once te two dimensionless tempeatues ( and ) ae solved fo, te dimensionless ates of eat tansfe at bot ot and cold junctions of te EG can be obtained as:

8 8 Q (9) Q () en, we ave te dimensionless powe output as W n () Q Q Accodingly, te temal efficiency is obtained by W n Q () t Defining L A, te dimensionless cuent is obtained by Z (3) R Also, defining V V n, te dimensionless voltage is obtained by V Wn (4) Wit te inputs ( and Z ), we begin developing te optimization wit te dimensionless paametes (,, and R) iteatively until tey convege. t is found tat bot and R sow tei optimal values fo te dimensionless powe output, wile does not sow te optimal value sowing tat te dimensionless powe output monotonically inceases wit inceasing. is implies tat, if is given, te optimal combination of and R can be obtained. Howeve, te dimensionless convection actually pesents te feasible mecanical constaints. us, we fist poceed wit a typical value of fo illustation and late examine te vaiety of wit some pactical design examples. Suppose tat we ave two initial inputs of. 6 (two fluid tempeatues) and Z. (mateials) along wit. We ten detemine te optimal combination fo and R, wic may be obtained eite gapically o using a compute pogam. We fist use te gapical metod at tis moment and late te pogam fo multiple computations. e dimensionless powe output Wn and temal efficiency t ae togete plotted as a function of R, wic ae pesented in Figue (a). Bot Wn and t wit espect to R indeed sow tei optimal values tat appea close. We ae inteested pimaily in te powe output and secondly in te efficiency. Howeve, since tey ae close eac ote, we eein use te powe output fo te optimization. t sould be noted tat te dimensionless maximum powe output does not occu at

9 9 R fom Figue (a) as usually assumed fo a EG witout eat sins, but appoximately at R.7, because te dimensionless tempeatues and in Figue (b) ae no longe constant. is is often a confusing facto in optimum design wit a EG wit two eat sins. We sould not assume tat R is equal to unity fo a EG wit eat sins..5.4 W n *.5. t W n *.3.9 t (a) R * & * * W n *.4 W n * *. (b) R Figue (a) Dimensionless powe output esistance to esistance of temocouple Wn and efficiency t vesus te atio of load R and (b) dimensionless tempeatues R. ese plots wee geneated using. 3,,. 6 and Z.. and vesus Wit te dimensionless paametes obtained (, R. 7,. 6, and Z. ), we now plot te dimensionless powe output Wn as a function of te dimensionless temal

10 conductance defined in Equation (6) along wit te temal efficiency t, wic is sown in Figue 3 (a). We find an optimum W appoximately at. 3. Actually, te optimal values of and R sould be iteated until te two simultaneously convege. n W n *.5. W n *.3.5 t. t (a) 4. 3 t.5 * & * *. t *.5 (b) Figue 3 (a) Dimensionless powe output temal conductance dimensionless temal conductance.6 and Z.. Fom Wn and temal efficiency t vesus dimensionless, and (b) ig and low junction tempeatues ( and. ese plots wee geneated using n A L A as sown in Equation (6), te A ) vesus, R. 7, actually detemines te numbe of temocouples n if te geometic atio A/L and ae given o vice vesa. e

11 dimensionless powe output Wn fist inceases and late deceases wit inceasing. t is impotant to ealize tat, if A is given, tee is an optimal numbe n of temocouples (o optimal temal conductance A L ) in te temoelectic module, wic is usually unnown. Pysically, te suplus numbe of temocouples vitually inceases te temal conduction moe tan te poduction of powe, wic causes te net powe output to decline. ee is anote impotant aspect of te optimal dimensionless temal conductance of. 3, wic is tat te module temal conductance n A L diectly depends on te A. n ote wods, te module temal conductance n A L must be edesigned on te basis of te A to meet.3. e infomation of te optimal temal conductance [(n)(a/l)()] is paticulaly impotant in design of micostuctued o tin-film temoelectic devices. Futemoe, tee is a potential to impove te pefomance o to povide te vaiety of te geomety by educing te temal conductivity. e dimensionless ig and low junction tempeatues ae pesented in Figue 3 (b). As deceases towads zeo, and appoac and, espectively. is indicates tat te temal esistances of two eat sins appoaces zeo, wic neve appens. t is noted tat te temal efficiency appoaces te teoetical maximum efficiency of. fo te given fluid tempeatues as appoaces zeo. Since tee is an optimal combination of and R fo a given, we can plot te optimal dimensionless powe output W opt and optimal temal efficiency opt as a function of dimensionless convection, wic is sown in Figue 4. t is vey inteesting to note in Figue 4 tat, wit inceasing, opt baely canges, wile W opt monotonically inceases. Accoding to Equation (4), te actual optimal powe output Wopt is te poduct of Wopt and A, seemingly inceasing linealy wit A. n pactice, tee is a contovesial tendency tat may decease systematically wit inceasing A if A is limited. As a esult of tis, it is needed to examine te vaiety of te A as a function of fo te optimal powe output. Figue 5 eveals te inticate elationsip between A and (o A ) along wit te optimum actual powe output (not dimensionless), wic would lead system designes to a vaiety of possible allocations ( A and ) fo tei optimal design. ow we loo into te actual optimal design wit te actual values. Fo example, using Figues 4 and 5, we develop an optimal design fo automobile exaust gas waste eat ecovey. A temoelectic geneato module wit a 5-cm 5-cm base aea is subject to exaust gases at 5 C in fluid and ai at 5 C in fluid. We estimate an available maximum convection conductance in fluid (exaust gas) wit. 8, W m K 6, and A cm and also an available maximum convection conductance in fluid (ai) wit.8, W m K 6, and A cm, wic gives A A 4. 8W K and. ote tat and ae typical fin efficiencies, and and ae te easonable convection coefficients fo exaust gas eating and ai cooling, of wic te convection

12 coefficients wit exaust gas o ai typically ave values anging between and W/m K depending on te flow ate and te type of fin. e middle value of 6 W/m K was used in te pesent wo. Eac aea of A and A is based on fins ( sides) wit a fin eigt of 5 cm on a 5-cm 5-cm base aea of te module, wic gives an total fin aea (5cm 5 cm sides fins = cm ). e typical temoelectic mateial popeties ae assumed to be p = n = V/K, p = n =. -3 cm, and p = n =.4 - W/cmK. e above data appoximately detemines tee dimensionless paametes as,. 6 and Z opt W* opt opt.4.. W* opt.9.8. Figue 4. Optimal dimensionless powe output convection W opt and efficiency opt vesus dimensionless. is plot was geneated using. 6 and Z.. As mentioned befoe, te pesent dimensional analysis enables te tee dimensionless paametes (,. 6 and Z. ) to detemine te est two optimal paametes, wic ae found to be. 3 and R. 7 as sown befoe. is leads to a statement tat, if two individual fluid tempeatues on eat sins connected to a temoelectic geneato module ae given, an optimum design always exists wit te feasible mecanical constaints tat pesent. is optimal design is indicated appoximately at Point in Figue 5. ote tat tee ae ways to impove te optimal powe output, inceasing eite A o o bot, wic obviously depends on te feasible mecanical constaints, wiceve is available. e inputs and optimum esults at Point in Figue 5 ae summaized in able. e inputs ae te geomety of temocouple, te mateial popeties, two fluid tempeatues, and te available convection conductance in fluid. e dimensionless esults ae conveted to te actual quantities as sown. e maximum powe output is found to be 65. W fo te 5cm 5cm base aea of te module. e powe density is calculated to be.6 W/cm, wic appeas significantly ig compaed to an available powe density of ~ W/cm wit te simila opeating conditions by EDO pogam (Japan) [7].

13 3 W opt (W) =. η A (W/K) = =... Point Point Figue 5. Optimal powe output W vesus convection conductance A in fluid as a opt function of dimensionless convection and Z... is plot was geneated using = 5 C,. 6 able. nputs and Results fom te Dimensional Analysis fo a EG nputs Dimensionle ss ( W, ) n opt Actual (Wn,opt) = 5 C, = 5 C, = 475 C =.3 n = 54 A = mm, L = mm = A = 4.8 W/K =.8, = 6 W/m K, A = cm R =.7 RL =.7 n R = 4.3 A= 4.8 W/K. 6 = 5 C Base aea Ab of module = 5 cm 5 cm Z. Z. p = -n = V/K. 7 = 374 C p = n =. -3 cm. 367 = 37 C p = n =.4 - W/cmK W n. 45 Wn = 65. W (Z = K - ) t. 8 t. 8 (R =. pe temocouple) =.36 = 3.9 A ( 8 K 6, A cm ) V =.5 V = 6.7 V (Powe Density Pd = Wn/Ab) - Pd =.6 W/cm Ai was used in fluid so fa. Howeve, we want to see te effect of A o by canging fluid fom ai to liquid coolant. Otewise te same conditions wee applied to as te pevious example. We ten estimate an available convection conductance in fluid (exaust gas) wit te same one of. 8, W m K 6, and A cm, but an available convection

14 4 conductance in fluid (liquid coolant) wit. 8, W m K 3, and A cm, wic gives A 4. 8W K and A 4W K, espectively, wic yields.. e aea of A is based on fins ( sides) wit a fin eigt of 5 cm fo te 5-cm 5-cm base aea of te module and A is estimated to be one tent of A (liquid coolant does not equie a lage eat tansfe aea). ese inputs and optimum esults give all te five dimensionless paametes as.7,., R. 5,. 6 and Z., fo wic te optimum at A 4W K is indicated at Point in Figue 5. e effect of on te ig and cold junction tempeatues was also pesented in Figue 6. t is inteesting to see tat, altoug a small vaiation in te optimal powe outputs between Point ( ) and Point (. ) appeas in Figue 5, a significant tempeatue vaiation between and. appeas in Figue 6. is may be an impotant facto paticulaly wen temoelectic mateials ae consideed in te optimal design. e poximity of te powe outputs between Points and is an example sowing te vaiety of te mecanical constaints ( A and A ) even wit te same powe outputs. t is impotant to ealize tat, wen A is limited, simply inceasing A invoes deceasing, wic esults in deceasing not only te ig and cold junction tempeatues but also sligtly te tempeatue diffeence as sown in Figue 6. e coexistence tat te Seebec coefficient deceases wit deceasing te tempeatue and educing te tempeatue diffeence diminises te pefomance will cause te powe output to decline. Howeve, inceasing A will diectly incease te powe output as mentioned ealie. e net powe output of loss and gain by inceasing A may be a ole of system designe. Anyow tee will be a small cange in te efficiency opt ( C). opt... opt Figue 6. Hot and cold junction tempeatues and optimal efficiency vesus dimensionless convection. is plot was geneated wit = 5 C,. 6 and Z..

15 5 3. emoelectic Coole Let us conside a simplified steady-state eat tansfe on a temoelectic coole module (EC) wit two eat sins as sown in Figue 7. Eac eat sin faces a fluid flow at tempeatue. Subscipt and denote te entities of fluid and, espectively. Conside tat an electic cuent is diected in a way tat te cooling powe Q entes eat sin. We assume tat te electical and temal contact esistances in te EC ae negligible, te mateial popeties ae independent of tempeatue, and also te EC is pefectly insulated. e EC as a numbe of temocouples, of wic eac temocouple consists of p-type and n-type temoelements wit te same dimensions. Figue 7. emoelectic coole module (EC). e basic equations fo te EC wit two eat sins ae given by Q A (5) A Q n R L (6) A Q n R L (7) Q A (8) wee p n, p n, and p n. n ode to study te optimization of te EC, seveal dimensionless paametes ae intoduced. e dimensionless temal conductance, wic is te atio of temal conductance to te convection conductance in fluid, is

16 6 A n A L (9) e dimensionless convection, wic is te atio of convection conductance in fluid to fluid, is A A (3) e dimensionless cuent is given by (3) A L e dimensionless tempeatues ae defined by (3) (33) (34) e dimensionless cooling powe, ate of eat libeated and electical powe input ae defined by Q Q W n Q A Q A Wn A (35) (36) (37) t is noted tat te above dimensionless paametes ae based on te convection conductance in fluid, wic means tat A sould be initially povided. Using te dimensionless paametes defined in Equations (9) (34), Equations (5) (8) educe to two fomulas as: Z (38)

17 7 Z (39) Equations (38) and (39) can be solved fo and. e dimensionless tempeatues ae ten a function of five independent dimensionless paametes as f,,,, Z (4) f,,,, Z (4) is te input and Z is te mateial popety wit te input, and bot ae initially povided. eefoe, te optimization can be pefomed only wit te fist tee paametes (,, and ). Once te two dimensionless tempeatues ( and ) ae solved fo, te dimensionless ates of eat tansfe at bot junctions of te EC can be obtained as: Q (4) Q (43) as Q is called te dimensionless cooling powe. en, we ave te dimensionless powe input W n (44) Q Q Accodingly, te coefficient of pefomance is obtained by Q COP W n (45) Defining V V n, te dimensionless voltage is obtained by V Wn (46) Wit te inputs ( and Z ), we ty to find te optimal combination fo te dimensionless paametes (,, and ) iteatively until tey convege. t is found tat bot and sow te optimal values fo te dimensionless cooling powe Q, wile does not sow te optimal Q monotonically inceases wit inceasing value sowing tat te dimensionless cooling powe. is implies tat, if any is given, te optimal combination of and can be obtained.

18 8 Howeve, te dimensionless convection actually pesents te feasible mecanical constaints. us, we poceed wit a typical value of fo illustation and late examine te vaiety of wit a pactical design example. Suppose tat we ave. 967 (two abitay fluid tempeatues) and Z. (mateials) along wit as inputs. en, we can detemine te optimal combination fo and, wic may be obtained eite gapically o using a compute pogam. We fist use te gapical metod at tis moment and late te pogam fo multiple computations (a Matematical softwae Matcad was used). e optimal combination of and fo eac maximum cooling powe ae found to be =.5 and =.3, espectively, wic ae sown in Figues 8 and 9. e maximum cooling powe of Q. 37 in bot figues is actually te optimal dimensionless cooling poweq,opt. Howeve, te COP also sows an optimal value at.74, wic gives Q. 6. e optimal COP usually gives a vey small cooling powe o sometimes even no exists, wic seems impactical, albeit te ig COP. eefoe, it is needed to ave a pactical point fo te optimal COP, wic is detemined in te pesent wo to be te midpoints of te optimum and. Fo example, te pactical optimal COP in tis case occus simultaneously at. 5 and. 5, wic leads to Q. 9 tat may be seen afte e-plotting wit te two values...8 W n *.5 Q * & W n *.6.4 COP Q *.5 COP Figue 8. Dimensionless cooling powe Q, powe input W n and COP vesus dimensionless cuent. is plot was geneated wit. 3,,. 967 and Z..

19 Q *.8 Q *.3.6 COP..4. COP Figue 9. Dimensionless cooling powe Q and COP vesus dimensionless temal conductance. is plot was geneated wit,. 5,. 967 and Z.. e existence of an optimum cooling powe as a function of cuent is a well nown caacteistic of ECs. Howeve, te existence of te optimum in ECs as not been found in te liteatue to te auto s nowledge. Wit Equation (9) tat is optimum of =.3 implies tat te module temal conductance A L n A L A n is at optimum since te A is given, wic leads to te optimum n (te numbe of temocouples) if A L is given o vice vesa. is is one of te most impotant optimum pocesses in design of a temoelectic coole module. t is good to now in Figue tat te dimensionless tempeatue becomes lowest at te optimal dimensionless cooling powe Q, not at te optimum COP., te Q * *.4 * & *...3. Q * * Figue. Dimensionless tempeatues vesus dimensionless temal conductance. is plot was geneated wit,. 5,. 967 and Z..

20 We also conside two eat sins as a unit witout a EC to examine te limitation of use of te EC, wic is sown in Figue. e geomety of te unit is te same as te one sown in Figue 7 except tat tee is no EC between te eat sins. ee sould be a cooling ate wit given fluid tempeatues, wic is Q. We want to compae te cooling powe Q wit tis cooling ate Q to detemine te limit of use of te EC. Figue. Heat sins witout a EC. e basic equations fo te unit can be expessed as Q A (47) Q A (48) e dimensionless goups fo te unit ae (49) Q Q A (5) Using Equations (3), (34), (49) and (5), Equations (47) and (48) educe to a fomula as (5) e dimensionless cooling ate witout eat sins can be obtained by Q (5)

21 Q * & Q * Q * COP Q * COP Figue. Dimensionless cooling powe, cooling ate of te unit, and COP vesus dimensionless fluid tempeatue. is plot was geneated wit. 3,,. 5 and Z Q*,opt.8.3 Q* COP.6. COP opt.4... Figue 3. Optimal (cooling powe optimized) dimensionless cooling powe and COP vesus dimensionless convection. is plot was geneated wit. 967 and Z.. e dimensionless cooling powe Q and cooling ate of te unit Q vesus te dimensionless fluid tempeatue along wit te COP ae pesented in Figue. e coss point in te figue is found to be =., wic is a design point as te limit of use of te EC. f te dimensionless fluid tempeatue is ige tan te coss point, tee is no justification fo use of te EC altoug te EC still functions. is coss point is defined as te maximum

22 dimensionless tempeatue. ee is also a minimum point at. 83, wee Q,, max wic is called te minimum dimensionless tempeatue, min effective cooling witin a ange fom, min. 83 to, max =... ote tat te EC can pefom Since tee is an optimal combination of and fo a given, we can plot te optimal dimensionless cooling powe wic is sown in Figue 3. t is seen tat bot inceasing Q,opt Q,opt and COPopt as a function of dimensionless convection, Q,opt and COPopt incease monotonically wit. Accoding to Equation (35), te actual optimal cooling powe Q, opt is te poduct of and A, seemingly inceasing linealy wit A. n pactice, tee is a contovesial tendency tat may decease systematically wit inceasing A. As a esult of tis, it is needed to examine te vaiety of te A as a function of fo te optimal cooling powe. Figue 4 eveals te inticate elationsip between A and (o A ) along wit te optimum actual cooling powe (not dimensionless), wic would lead system designes to a vaiety of possible allocations ( A and ) fo tei optimal design. ote tat te analysis so fa is entiely based on te dimensionless paametes. ow we loo into te actual optimal design wit te actual values Q,opt (W) 5 Point 4 3 =.. = η A (W/K) =. Figue 4. Optimal cooling powe vesus convection conductance in fluid as a function of dimensionless convection. is plot was geneated wit = 3 C,. 967 and Z.. Fo example, using Figues 3 and 4, we develop an optimal design fo an automobile ai conditione. A temoelectic coole module wit a 5-cm 5-cm base aea is subject to cabin ai at C in fluid and ambient ai at 3 C in fluid. We estimate an available maximum convection conductance in fluid (cabin ai) wit. 8, W m K 6, and A cm and also an available maximum convection conductance in fluid (ambient ai) wit. 8,

23 3 6W m K, and A cm, wic gives A A 4. 8W K and. ote tat and ae te fin efficiencies, and and ae te easonable convection coefficients fo te cabin ai cooling and te ambient ai cooling, espectively. Eac aea of A and A is based on fins ( sides) wit a fin eigt of 5 cm on a 5-cm 5-cm base aea of te module. e typical temoelectic mateial popeties ae assumed to be p = n = V/K, p = n =. -3 cm, and p = n =.4 - W/cmK. e above data appoximately detemines tee dimensionless paametes as,. 967 and Z.. As mentioned befoe, te pesent dimensional analysis enables te tee dimensionless paametes (,. 967 and Z. ) to detemine te est optimal paametes, wic ae found to be. 3 and. 5 as sown befoe. is leads to a statement tat, if two individual fluid tempeatues on eat sins connected to a temoelectic coole module ae given, an optimum design always exists wit te feasible mecanical constaints tat pesent. is optimal design is indicated appoximately at Point in Figue 4. ote tat tee ae seveal ways to impove te optimal powe output by inceasing eite A o o bot, wic appaently depends on te feasible mecanical constaints, wiceve is available. 8,opt 6 8 ( C) 4 Q,opt 6 Q (W),opt 4. Figue 5. wo junction tempeatues and cooling powe vesus convection conductance in fluid. is plot was geneated wit, A 4. 8W K = 3 C,. 967 and Z.. e inputs and optimum esults at Point in Figue 4 ae summaized in te fist two columns of able. e inputs ae te geomety of temocouple, te mateial popeties, two fluid tempeatues, and te available convection conductance in fluid. e dimensionless esults ae conveted to te actual quantities. e optimal cooling powe is found to be 54.4 W fo te 5cm 5cm base aea of te module. e cooling powe density is calculated to be.8 W/cm. Wen A is limited, simply deceasing A invoes deceasing, wic is sown in Figue 5. is attibutes to te eat balance tat te ot and cold junction tempeatues must decease wen moe eat is extacted fom te limited input, wic is a

24 4 caacteistic of te temoelectic coole wit eat sins. e ot and cold junction tempeatues ae sometimes a design facto, noting tat te optimal cold junction tempeatue,opt eaces zeo Celsius at. 4, wic may cause icing and educing te eat tansfe. able. nputs and Results fom te Dimensional Analysis fo a emoelectic Coole Module nput Q,opt Q,opt COP / opt COP / opt (dimensionless) (Actual) (dimensionless) (Actual) = C, = 3 C =.3 n = 57 =.5 n = 8 A = mm, L = mm = A = 4.8 W/K = A = 4.8 W/K =.8, =6 W/m K, =.5 = 6.36 A =.5 = 3.8 A A = cm. 967 = C. 967 = C Base aea Ab = 5 cm 5 Z. Z. Z. Z. cm p = -n = V/K. 93 = 8.7 C. 949 = 4.4 C p = n =. -3 cm. 45 = 73.9 C. 3 = 39.4 C p = n =.4 - W/cmK Q. 37 Q = 54.4 W Q. 9 Q = 6.9 W (Z = K - ) COP =.35 COP =.49 (R =.pe temocouple) V =.75 Vn = 4.5 V V =.33 Vn = 5.7 V, max.,max =, max.6,max = 9.3 C 49.7 C.83,min =.84,min = (Cooling Powe Density Pd = Q/Ab), min.8 C - Pd =.8 W/cm, min 9. C - Pd =.8 W/cm As mentioned befoe, te optimal COP is sometimes in demand in addition to te optimal cooling powe. Howeve, te eal optimal COP usually gives a vey small value of te cooling powe, albeit te ig COP. eefoe, in te pesent wo, te midpoints of te optimal and ae used to povide appoximately a alf of te optimal cooling powe and at least fou folds of te cooling-powe-optimized COP. is modified optimal COP is called a alf optimal coefficient of pefomance COP/opt. ese esults ae also tabulated in te last two columns of able, so tat designes could detemine wic optimum is bette depending on te application. e optimum cooling powe is usually selected wen te esouces (electical powe o capacity of coolant) ae abundant o inexpensive o te efficacy is not impotant as in micopocesso cooling, wile te alf optimum COP is selected wen te esouces ae limited o expensive o te efficacy is impotant as in automotive ai conditiones. 4. Conclusions e pesent pape pesents te optimal design of temoelectic devices in conjunction wit eat sins intoducing new dimensionless paametes. e pesent optimum design includes te

25 5 powe output (o cooling powe) and te efficiency (o COP) simultaneously wit espect to te extenal load esistance (o electical cuent) and te geomety of temoelements. e optimal design povides optimal dimensionless paametes suc as te temal conduction atio, te convection conduction atio, and te load esistance atio as well as te cooling powe, efficiency and ig and low junction tempeatues. e load esistance atio (o te electical cuent) is a well nown caacteistic of optimum design. Howeve, it is found tat te load esistance atio would be geate tan unity (.7 in te pesent case). is is a confusing facto in optimum design wit a EG. One sould not assume tat te load esistance atio RL/R is equal to unity fo a EG wit eat sin(s). e optimal temal conductance [(n)(a/l)()] consists of te numbe of temocouples, te geometic atio, and te temal conductivity. t is impotant tat tee is an optimum numbe of temocouples n fo a given te convection conductance A if te optimal temal conductance A/L is constant o vice vesa. ese ae te optimum geomety of temoelectic devices. e infomation of te optimal temal conductance is paticulaly impotant in design of micostuctued o tin-film temoelectic devices. Futemoe, tee is a potential to impove te pefomance o to povide te vaiety of te geomety by educing te temal conductivity. Finally, it is stated fom te pesent dimensional analysis tat, if two individual fluid tempeatues on eat sins connected to a temoelectic geneato o coole ae given, an optimum design always exists and can be found wit te feasible mecanical constaints. Refeences [] Kaeme D., McEnaney K., Ciesa M., Cen G., Modeling and optimization of sola temoelectic geneatos fo teestial applications, Sola Enegy, 86, ,. [] Kai M.A., ace E.F., Helenboo B.., Exaust enegy convesion by temoelectic geneato: wo case studies, Enegy Convesion and Management, 5, 596-6,. [3] Hsu C.., Huang G.Y., Cu H.S., Yu B., Yao D.J., Expeiments and Simulations on lowtempeatue waste eat avesting system by temoelectic powe geneatos, Applied Enegy, 88, 9-97,. [4] Hendeson J., Analysis of a eat excange-temoelectic geneato system, 4 t ntesociety Enegy Convesion Engineeing Confeence, Boston, Massacusetts, August 5-, 979. [5] Stevens J.W., Optimal design of small temoelectic geneation systems, Enegy Convesion and Management, 4, 79-7,. [6] Cane D.., Jacson G.S., Optimization of coss flow eat excanges fo temoelectic waste eat ecovey, Enegy Convesion and Management, 45, , 4. [7] Rowe D.M., emoelectics andboo: mico to nano, aylo & Fancis, ew Yo, 6. [8] Cein R., Huang G., emoelectic coole application in electonic cooling, Applied emal Engineeing, 4, 7-7, 4. [9] Vining C.B., An inconvenient tut about temoelectics, atue Mateials, 8, 9. [] Wang C.C., Hung C.., Cen W.H., Design of eat sin fo impoving te pefomance of temoelectic geneato using two-stage optimization, Enegy, 39, 36-45,.

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