Synthesizing Models of Power Yield in Thermo-Electro-Chemical Systems

Transcription

1 , July 4-6, 0, Lodo, U.K. Sythsizig Modls of Powr Yild i Thrmo-Elctro-Chmical Systms S. Siiutycz Abstract This ivstigatio offrs a sythsizig thrmodyamic approach to modlig, simulatio ad powr maximizatio i various rgy covrtrs, such as thrmal, solar ad chmical gis ad ful clls. Thrmodyamic aalyss lad to fficicis i trms of propllig fluxs. Efficicy quatios ar applid to dtrmi maxima of powr itgrals (work) i dyamical systms, which work with upgradig ad dowgradig of a rsourc mdium. Whil optimizatio of static systms rquirs usig of diffrtial calculus ad Lagrag multiplirs, dyamic optimizatio ivolvs variatioal calculus ad dyamic programmig. ractig mixturs balacs of mass ad rgy ar applid to driv powr yild i trms of a activ part of chmical affiity. Powr maximizatio approach is also applid for ful clls. Maximum powr data provid powr limits for SOFC rgy grators. dx Trms gis, powr bouds, thrmodyamics, fficicy, ful clls. T. NTRODUCTON HS rsarch is a tratmt i thrmodyamic optimizatio, i.. optimizatio which applis thrmodyamic costraits ad thrmodyamic prformac critria. Thrmodyamic sythsis, prstd hr, stads for combiig various partial optimizatio modls ito a sythsizig modl from which th prformac of all th compot uits ca b prdictd. Liguistically, sythsis is a combiatio of sparat thigs, uits, idas, tc., ito a complt whol. t also may b somthig as a substac or a ida, mad by combiig various parts. Dfiitios of sythsizig approachs may vary. For xampl, i th systm s thory, wh a systm s prformac is valuatd by combiig sparat obsrvatios, sythsis mas th structur prdictio followig from th prformac obsrvatios. Th litratur of sythsizig approachs is quit limitd. Sythss ar usually achivd aftr sufficitly log rsarch tim, wh sparat compots, modls or cocpts ar wll udrstood idividually. A attmpt to sythsiz thrmal powr systms proplld by hat ad mass trasfr is prstd i our paprs [, ], Mauscript rcivd Octobr 5, 0; rvisd March, 0. This rsarch was prformd i th cotxt of a grat NN from th Polish Miistry of Scic, ad a grat 0/0/B/ST8/068from th Natioal Ctr of Scic (grats Thrmodyamics ad Optimizatio of Chmical ad Elctrochmical Ergy Grators with Applicatios to Ful Clls, ad Sythsis of Thrmodyamic Gractio of Powr ad Hat i Ful Clls ad Flow Ergy Systms.) S. Siiutycz is with th Faculty of Chmical ad Procss Egirig at Warsaw TU, SSN: (Prit); SSN: (Oli) whras a summary of rlatd computatioal aspcts is giv i [3]. Radiatio gis ar icludd [4-8]. th prst papr gralizd modls ar applid, which ivolv imprfct ful clls. Th prst papr offrs a thrmodyamic approach to modlig ad powr optimizatio i divrs rgy covrtrs, such lik thrmal, solar ad chmical gis. ts ovlty lis ot i dtaild data of powr yild but i its sythsizig mathmatical formalism or a commo thrmodyamic modl applicabl to th thrmal, radiativ, chmical, ad lctrochmical powr grators. Th iclusio of imprfct ful clls ito th modl, whil quit simpl, is bficial to th FC works. Yt, th approach limitatio maifsts whvr th FC topology diffrs sigificatly from that of thrmal or chmical systm. Thrmodyamic pricipls lad to covrtr s fficicy ad limitig gratd powr. A powr limit is a uppr (lowr) boud o powr producd (cosumd) i th systm. Efficicy quatios srv to solv problms of upgradig ad dowgradig of a rsourc mdium. Ral work yild is a cumulativ ffct obtaid i a systm of a rsourc fluid, gis, ad a ifiit bath. Whil optimizatio of stady systms rquirs usig of diffrtial calculus ad Lagrag multiplirs, dyamic optimizatio ivolvs variatioal calculus ad dyamic programmig [9]. Th primary rsult of th static optimizatio is th limitig valu of powr, whras that of th dyamic optimizatio is a fiit-rat coutrpart of th classical pottial of rvrsibl work (xrgy; [0]). Th gralizd pottial dpds o thrmal coordiats ad a dissipatio idx, h, i.. th Hamiltoia of th rlatd problm of miimum tropy productio. This gralizd pottial implis strogr bouds o work dlivrd or supplid tha th rvrsibl work pottial. ractig systms [, ] th chmical affiity costituts a prvailig compot of th fficicy vctor. Thrfor, i ractig mixturs flux balacs ar applid to driv powr yild i trms of a activ part of chmical affiity. As th prst papr shows, powr maximizatio approach ca also b applid for ful clls (FC) tratd as flow gis driv by fluxs of chmical ragts ad lctrochmical mchaism of currt gratio [3]. Th aalysis [3] tsts th ffct of typical dsig ad opratig paramtrs o th cll prformac. Th FC thory combis th formalism workd out for chmical machis with th Faraday s law which dtrmis th itsity of th lctric currt gratio. Stady-stat modl of a high-tmpratur SOFC is cosidrd, which rfrs to costat chmical pottials of Waryskigo Strt, Warszawa, Polad (pho: ; fax: ; -mail: siiutycz@ ichip.pw.du.pl)..

2 , July 4-6, 0, Lodo, U.K. icomig hydrog ful ad oxidat. Dyamical limits may also b likd with us of rsourcs i rgy systms. Thy rfr to situatios wh fluids ar rstrictd i thir amout or magitud of flow, ad, as such, play rol of rsourcs. A powr limit is a uppr (lowr) boud o powr producd (cosumd) i th systm. A rsourc is a valuabl substac or rgy usd i a procss; its valu ca b quatifid by spcifyig its xrgy, a maximum work dlivrd wh th rsourc rlaxs to th quilibrium. Rvrsibl rlaxatio yilds th classical xrgy; i a dissipativ rlaxatio gralizd xrgis aris, which quatify dviatios of th systm s fficicis from th prfct fficicis. Th siz limitatio of our papr dos ot allow for iclusio of all drivatios to mak th papr slf-cotaid, thus th radr may d to tur to som prvious works, [] - [5] ad [,].. LMTED RESOURCES AND POWER OPTMZATON Limitd amout or flow of a rsourc workig i a gi causs a dcras of th rsourc pottial i tim (chroological or spatial). This is why studis of th rsourc dowgradig apply th dyamical optimizatio mthods. From th optimizatio viwpoit, dyamical procss is vry o with squc of stats, dvlopig ithr i chroological tim or i (spatial) holdup tim. Th first group rfrs to ustady procsss i o-statioary systms, th scod group may ivolv stady stat systms. a procss of rgy productio two rstig rsrvoirs do itract through a rgy grator (gi). this procss powr flow is stady oly wh two rsrvoirs ar ifiit. Wh o, say, uppr, rsrvoir is fiit, its thrmal pottial must dcras i tim, which is th cosquc of th rgy balac. Ay fiit rsrvoir is thus a rsourc rsrvoir. t is th rsourc proprty that lads to th dyamical bhavior of th fluid ad its rlaxatio to th quilibrium with a ifiit lowr rsrvoir (usually th viromt). Altrativly, fluid at a stady flow ca rplac rstig uppr rsrvoir.. STEADY STATE THERMAL SYSTEMS Th grat dal of rsarch o powr limits publishd to dat dals with statioary systms, i which cas both rsrvoirs ar ifiit. To this cas rfr stady-stat aalyss of th Chambadal-Novikov-Curzo-Ahlbor gi (CNCA gi [6]), i which rgy xchag is dscribd by Nwtoia law of coolig, or th Stfa-Boltzma gi, a systm with th radiatio fluids ad th rgy xchag govrd by th Stfa-Boltzma law [7]. Du to thir statioarity (causd by th ifiitss of both rsrvoirs), cotrols maximizig powr ar lumpd to a fixd poit i th stat spac. fact, for th CNCA gi, th maximum powr poit may b rlatd to th optimum valu of a fr (ucostraid) cotrol variabl, which ca b fficicy η or so-calld Carot tmpratur T []. trms of th rsrvoirs tmpraturs T ad T ad th itral irrvrsibility factor Φ o fids / = ( TΦT ) SSN: (Prit); SSN: (Oli) T opt [4]. For th Stfa-Boltzma (radiatio) gi xact xprssio for th optimal poit caot b dtrmid aalytically, yt, this poit ca b foud graphically from th chart P=f(T ). Also, th mthod of Lagrag multiplirs ca succssfully b applid [8]. As thir limiatio from a st of rsultig quatios is quit asy, th problm is brok dow to th umrical solvig of a oliar quatio for th optimal cotrol T. Fially, th so-calld psudo- Nwtoia modl [4, 5], which uss stat or tmpratur dpdt hat xchag cofficit, α(t 3 ), omits, to a xtt, aalytical difficultis associatd with th Stfa- Boltzma quatio. Applyig this modl i th so-calld symmtric cas, whr both rsrvoirs ar filld up with radiatio, o shows that th optimal (powr maximizig) Carot tmpratur of th stady radiatio gi is vry clos to that for th CNCA gi [4]. V. DYNAMCAL (UNSTEADY) SYSTEMS dyamical rgy yild kowldg of a xtrmal path rathr tha a xtrmum poit is cssary. Th optimizig procdur rfrs to a powr itgral ad rquirs th applicatio of variatioal mtods to hadl fuctioal xtrma. For xampl, th us of a psudo-nwtoia modl to quatify th dyamical rgy yild from radiatio, givs ris to a xtrmum curv dscribig th radiatio rlaxatio. This curv is o-xpotial, th cosquc of th oliar proprtis of th rlaxatio dyamics. Noxpotial ar also othr curvs of th radiatio rlaxatio,.g. thos followig from xact modls usig th Stfa- Boltzma quatio (symmtric ad hybrid, [4,5]). Aalytical difficultis associatd with dyamical optimizatio of oliar systms ar svr; this is why divrs modls of powr yild ad divrs umrical approachs ar applid. Optimal (.g. powr-maximizig) rlaxatio curv T(t) is associatd with th optimal cotrol curv T (t); thy both ar compots of th dyamic optimizatio solutio to a cotiuous problm. th corrspodig discrt problm, formulatd for umrical purposs, o sarchs for optimal tmpratur squcs {T } ad {T }. Various discrt optimizatio mthods ivolv: dirct sarch, dyamic programmig, discrt maximum pricipl, ad combiatios of ths mthods. V. CHEMCAL SYSTEMS Up to ow (s,.g., th prvious paprs [-5]) w modld powr yild ad powr limits i thrmal systms. As statd abov, radiatio gis wr aalyzd as oliar systms govrd by laws of thrmodyamics ad trasport phoma. Tmpraturs T of rsourc mdia wr oly cssary variabls to dscrib ths systms. Howvr, chmical gis ad ful clls (Figs. ad 4 i Rf. []) ar mor gral systms i which both tmpraturs T ad chmical pottials µ k ar sstial. Blow w shall mak a fw basic rmarks rgardig chmical systms. chmical gis mass trasports participat i trasformatio of chmical affiitis ito mchaical powr

3 , July 4-6, 0, Lodo, U.K. []. As opposd to thrmal machis, i chmical os gralizd rsrvoirs ar prst, capabl of providig both hat ad substac. Whvr ifiit rsrvoirs assur costacy of chmical pottials, problms of xtrmum powr (maximum of powr producd ad miimum of powr cosumd) ar static optimizatio problms. For fiit rsrvoirs, howvr, amout ad chmical pottial of a activ ractat dcras i tim, ad cosidrd problms ar thos of dyamic optimizatio ad variatioal calculus. Bcaus of th divrsity ad complxity of chmical systms th ara of powr producig chmistris is xtrmly broad. Th simplst modl of powr producig chmical gi is that with a isothrmal ad isomric ractio, A -A =0 []. Powr xprssio ad fficicy formula for th chmical systm follow from th tropy cosrvatio ad rgy balac i th powr-producig zo of th systm (activ part). a dorvrsibl chmical gis total tropy flux is cotiuous through th activ zo. a isothrmal cas p = ( µ ' µ ' ) () th Hamiltoia H for Eq. (4). Howvr, gttig a complt solutio for th maximum of th itgral (4) rquirs th us of umrical approachs which usually apply Bllma s mthod of discrt dyamic programmig [9]. Th optimality coditio for a discrt dyamic path is rprstd by Bllma s rcurrc quatio V ( x ) = max{ D ( x, u, θ ) u, θ (5) ( ( + V x f x, u, θ ) θ )} with cotrol u=dx /dτ ad stat x. Th o stag profit D is th discrt rprstatio of th itgrad i Eq. (4). Low dimsioality assurs xcllt accuracy of umrical rsults. Numrical issus ar discussd i [3]. Th cascad schm, which illustrats th pricipl of calculatios for th optimal powr output of a gi, is prstd i Fig.. Multi-ractio xtsios of Eq. (3) ar availabl [].. whr is a ivariat molar flux of ragts. Procss fficicy ζ is dfid as powr yild pr molar flux,, i.. ζ = p / = µ ' µ ' (). For a stady gi th followig fuctio dfis th chmical fficicy i trms of ad mol fractio x x g ζ = ζ + 0 RT l (3) g + x Figur 3 of Rf. [] illustrats th implicatios of th abov quatio which prdicts th dcrasig atur of th fficicy of powr productio ζ with ful flux. Equatio (36) shows that a ffctiv coctratio of th ractat i uppr rsrvoir x ff = x g - is dcrasd, whras a ffctiv coctratio of th product i lowr rsrvoir x ff = x + g - is icrasd du to th fiit mass flux. Thrfor th fficicy ζ dcrass with ful flux. Wh ffct of rsistacs is igorabl or flux is vry small, rvrsibl fficicy, ζ C, is attaid. Th powr output, dscribd by th product ζ(), xhibits a maximum for a fiit valu of th ful flux,. Applicatio of Eq. (3) to a ustady systm lads to th fuctioal of a itgral work [] W f τ = τ i X /( + X ) + dx / dτ dx ζ + RT l dτ 0 x jdx / dτ dτ (X=x/(-x).) Som particular proprtis of this fuctioal ca b dductd from th costacy of th Hamiltoia fuctio. For low rats ad larg coctratios X (mol fractios x clos to th uity) optimal rlaxatio rat is approximatly costat. Yt, i a arbitrary situatio optimal rats ar stat dpdt so as to prsrv th costacy of SSN: (Prit); SSN: (Oli) (4) Fig.. A cascad schm for th calculatio of th dyamical gis by Bllma s mthod of dyamic programmig. V. ELECTROCHEMCAL ENGNES: FUEL CELLS Ful clls (FC) ar lctrochmical gis proplld by fluxs of both rgy ad substacs. Th mai advatag of ful clls i compariso to othr gis is that thir fficicy is ot a major fuctio of dvic siz. A ful cll cotiuously trasforms a part of chmical rgy ito lctrical rgy by cosumig ful ad oxidat. Th rol of ful clls for viromtal protctio is quit sstial. Basic structur of ful clls icluds lctrolyt layr i cotact with a porous aod ad cathod o ithr sid. Gasous fuls ar fd cotiuously to th aod (gativ lctrod) compartmt ad a oxidat (i.., oxyg from air) is fd cotiuously to th cathod (positiv lctrod) compartmt. Elctrochmical ractios tak plac at th lctrods to produc lctric currt. Basic ractio is th lctrochmical oxidatio of ful, usually hydrog, ad th rductio of th oxidat, usually oxyg. a FC procss th itractio of ful ad oxidat is proplld by diffusiv ad/or covctiv fluxs of hat ad mass, trasfrrd through som coductacs or layrs. Th rgy flux (powr) is cratd i th cll grator which xploits ful stram cotactig with th aod ad th oxidat stram cotactig with th cathod. Both lctrods ar sparatd by th lctrolyt. As i th hat ad radiatio gis [4-8,3] both trasfr mchaisms ad proprtis of

4 , July 4-6, 0, Lodo, U.K. coductig layrs dtrmi rat of powr yild. Powr maximizatio is applid hr for th purpos of dtrmiig limits of imprfct clls, whr powr output dcrass with lctric currt for sufficitly larg currts bcaus of prvailig ffct of loss phoma. V. POWER YELD N FUEL CELLS Kowldg of opratioal voltag is rquird to dfi a cll fficicy as th ratio χ = V/E 0, whr E 0 is th rvrsibl cll voltag or th quilibrium cll pottial. For th powr dsity i trms of χ o has p=ie 0 χ or p=χp rv, which mas that this fficicy is qual to th ratio of th actual powr to th maximum rvrsibl powr. This dfiitio liks th ful cll fficicy with th scod law, ad strsss substatial rol of th opratioal voltag. Assum that all icomig strams (thos with highr Gibbs flux G i = G ) rprst a commo phas of substrats (all systm s compots i th stat bfor th chmical trasformatio, idx ). All outgoig strams (thos with lowr Gibbs flux G out = G ) rprst th commo phas of products (all systm compots i th stat aftr th trasformatio, idx ). Powr xprssio follows from tropy cosrvatio ad rgy balac i th rvrsibl subsystm. For a isothrmal ractor p = µ & µ & ' ' ' ' + µ & ' ' µ & ' ' +... µ & i' i'... µ & i' i'... + µ & m' m'.. µ & m' m' This formula shows that, i a stady ad isothrmal procss, powr yild of a gi systm is th diffrc btw th iput ad output flux of th Gibb s fuctio [,, 4, 5]. W ca trasform Eq, (6) to a prooucig form of Eq. (7) blow, spcific to th cas of th complt covrsio. this cas th compots ar umbrd such that spcis, i ar substrats ad spcis i+, i+ m ar products. Total powr yild of a isothrmal multi-ractio procss is R R p = { p j} = { µ ν j µ ν j... µ i ν ' + ' + ' ij j= j= + µ i+ νi j. +. µ m νm j.. + µ m νmj)}& ' + ' ' j (6) (7) Quatitis & j ar molar chmical fluxs of ragts, i.. products of th lctrod surfac ara F ad htrogous rats, r j. th cas of complt covrsio, powr yild from th uit lctrod ara quals th sum of products of th affiity drivig forcs ad th ractio rats R R p = { & } = F { A r } (8) A j j j= j= Yt, th assumptio about th complt trasformatio of substrats ito products ca b rlaxd, ad th prst papr shows how this ca b do for ful clls. By cosidrig th chmistry of systms with powr productio ad trasport phoma o ca quatitativly stimat SSN: (Prit); SSN: (Oli) j j ffcts of icomplt covrsios. Th rlatd formula rsmbls th o which dscribs ffct of th itral tropy productio withi ths systms []. For a sigl isothrmal chmical ractio th powr formula which gralizs Eq. (8) to iclud ffct of icomplt covrsios ca b writt i th form p= Π ΞΠ ) & = ( ' ' ' ia ff ff g ( T, p) = G& F whr primd quatitis rfr to th iputs ad outputs of th chmically activ zo. Π is o-way chmical affiity attributd to ractats with kow chmical pottials [,4], & ' is th (positiv) chmical flux dfid as th product of th htrogous ractio rat ad th lctrod ara. tral imprfctio fuctios, Φ ad Ξ, ar rspctivly rlatd to itral tropy productio ad icomplt covrsio. Th fractio Ξ is th rciprocity of cofficit Ψ itroducd i []; thy both charactriz dtrimtal icras of chmical pottials of products causd by thir dilutio by rmaiig ractats. Hat ffcts ar rprstd by a total hat flux (ivolvig th ssibl hat flux, q, ad th sum of products of partial tropis ad fluxs of spcis multiplid by th tmpratur T), Powr formula of Eq. (9) gralizs th idalizd powr of a dorvrsibl systm (with Ξ =) i which cas diffrc Π Π is th chmical affiity or g. This is th chmical compot of powr, which dscribs powr yild causd by chmical flux & '. Elctrochmical gratio of powr occurs with a o-idal chmical fficicy ξ =Π ' ΞΠ. Effctivly, i th gi mod whr Ξ =<, a imprfct systm bhalvs as it would oprat with a dcrasd affiity of a ffctiv valu Π - Ξ Π '. Of cours, powr is dcrasd by this imprfctio [6, 7]. t should b udrlid that th quatios cotaid i th prst sctio (or othr os rlatd to it) ar ot stadard quatios of th classical thrmodyamics. Ev appartly classical Eqs. (6) ad (7) prtai to flow systms, hc thy do t blog i stadard thrmodyamics. Similarly, Eq. (8) is a affiity quatio i a flow systm, ad dscribs a spac distributd affiity rathr tha th classical or stadard chmical affiity which is localizd i a sigl poit of th phas spac. V. EFFECT OF TRANSPORT PHENOMENA ON POWER LMTS N THERMO-ELECTRO-CHEMCAL SYSTEMS For brvity w limit this sctio to th cas of a simpl isomrisatio ractio A -A = 0. Trasportd rgy ad compots driv th procss of powr gratio i ful clls. trstigly, thr xists a formal lik btw th mathmatics of thrmal gis ad ful clls. This lik has origially b show for chmical gis, ad w shall ow show how this approach ca b xtdd to lctrochmical grators ad ful clls. Lt us focus o th powr grators dscribd by th formalism of irt compots [8, 9] rathr tha th ioic (9)

5 , July 4-6, 0, Lodo, U.K. dscriptio [0]. Withi this formalism a powr xprssio ca b formulatd for th cas of coupld hat, mass ad charg trasfr i all th dissipativ coductors of th systm. This will lad us to a gral rsult for powr limits i liar thrmo-lctrochmical systms. Lt us assum that, i th lctrochmical cas, th activ (powr producig) drivig forcs ivolv: o tmpratur diffrc, sigl chmical affiity ad a opratig voltag φ - φ. Th rlatd powr xprssio is P = ( T = ( T - T ) ss s s ' s T ) ' s ' ' ' s + ( µ µ ) + ( φ φ R s + ( µ µ ) s + ( φ φ ) ) ' (0) Aftr itroducig th largd vctor of all drivig pottials µ = (T, µ, V), th flux vctor of all currts = ( s,, ), ad th ovrall rsistac tsor R, Eq. (0) ca b writt i a cocis matrix-vctor form p = ( µ µ ). Ι : () Maximum powr corrspods with vaishig of all partial drivativs of powr with rspct to all currts or th vctor coditio p / Ι = 0. This lads to a coditio which stats that at th maximum poit th optimal (powr-maximizig) currt vctor is qual to o half of th purly dissipativ currt vctor of th Fourir-Osagr poit,. Th corrspodig limit of maximum powr is pmp F = R : F F () 4 Sic th powr lost at th Fourir-Osagr poit is pf = R : (3) F compariso of Eqs. () ad (3) provs that, i liar thrmo-lctro-chmical systms, oly at most 5% of powr dissipatd i th atural trasfr procss ca b trasformd ito th valuabl form of th mchaical powr. This is a gral rsult which, probably, caot b asily gralizd to oliar trasfr systms whr sigificat dviatios from Eq. () may appar dpdig o th atur of divrs oliaritis. fact, th forms of Eqs. (0) ad () ar sufficit to claim that th thrmal forc formula ad th powr formula for a thrmal gi ar similar to th voltag ad powr formula of a ful cll systm. This provs that a lik xists btw th mathmatics of thrmal gis ad ful clls, ad that a uifyig thory xists for both systms. X. SOME EXPERMENTAL DATA FOR FUEL CELLS Voltag lowrig i ful clls blow th rvrsibl voltag SSN: (Prit); SSN: (Oli) F is a good masur of thir imprfctio. Yt w d to distiguish btw Nrst idal voltag E 0 or ad idl ru voltag, E 0. t is th lattr quatity from which all rat dpdt losss of voltag should b subtractd. A umbr of approachs for calculatig ths polarizatio losss hav b rviwd i litratur by Zhao, Ou ad Ch [6]. Th dtails of calculatios of th idl ru voltag E 0 ar thoroughly discussd by Wirzbicki [7] who has implmtd th Asp Plus TM softwar to ivstigat th bhavior of SOFC basd rgy systm usig his ow thortical modl of powr yild kitics. His calculatios wr compard with th xprimtal fidigs of th voltag ad powr i a laboratory FC systm. som situatios diffrc btw E 0 ad E 0 is a currt idpdt loss which may b dscribd by a fractio Ξ which charactrizs th dtrimtal icras of chmical pottials of products causd by thir dilutio by u-ractd substrats. With ffctiv oliar rsistacs, opratig voltag ad powr ca b rprstd i trms of th dpartur from th idl ru voltag E 0 p=vi = (E 0 - V it )i = (E 0 -V act -V co - V ohm ) i = E 0 i i (R act + R coc + R ohm ) (4) Not th aalogy btw this quatio (which is a quatio of th classical ful cll thory) ad Eq (0) abov. Th losss, calld polarizatio, iclud thr mai sourcs: activatio polarizatio (V act ), ohmic polarizatio (V ohm ), ad coctratio polarizatio (V coc ). Thy rfr to th rlatd rsistacs: activatio rsistac (R act ), ohmic rsistac (R ohm ), ad coctratio rsistac (R coc ). Activatio ad coctratio polarizatio occur at both aod ad cathod locatios, whil rsistiv polarizatio rprsts ohmic losss throughout th cll. V act is glctd i th modl of rf. [7], othlss th powr curv is typical. As th voltag losss icras with currt, th iitially icrasig powr bgis fially to dcras for sufficitly larg currts, so that maxima of powr vrsus currt ar obsrvd. Voltag-currt dsity ad powr-currt dsity charactristics of th SOFC for various tmpraturs wr obtaid i Wirzbicki s MsD thsis, [7]. Ths data ar show i Fig. whr th cotiuous lis rprst th Asp Plus TM calculatios tstig th modl vrsus th xprimts, whras poits rfr to xprimts of Wirzbicki ad Jwulski i Warsaw stitut of Ergtics (Wirzbicki [7], ad his rf. 8). Ths data show that th limitig powr icrass with tmpratur ad ful flux. Błszowski [3] has also coductd xprimtal ivstigatios of SOFC s dirctd towards powr maxima. H obtaid, i particular, lis of powr dsity for various cotts of hydrog i th ful. Ths rsults ar prstd i his fficicy chart which shows a xampl of th rlatio btw th powr dsity of a SOFC ad fficicy of lctrochmical ractio. A sigl powr xtrmum is typical for all xprimts. A xcssiv icras of fficicy causs a powr dcras of a cll. Th iclusio of imprfct ful clls ito th sythsizig modl, is bficial to th FC works. Usig that modl all

6 , July 4-6, 0, Lodo, U.K. optimizatio opratios ar prformd i a uifid way, ad th lost powr data rfrrig to th atural trasfr procss ca b applid to assss th maximum of th availabl powr. Fig.. Exampl of Wirzbicki s data dscribig voltag ad powr dsity of a SOFC i trms of th currt for various tmpraturs [7]. X. CONCLUSONS Th uciatio of th mathmatical aalogy btw xprssios for powr productio i thrmal, radiativ, chmical ad lctrochmical grators allows th costructio of a sythsizig thrmodyamic modl (thrmodyamic sythsis). For this modl all optimizatio opratios ar prformd i a uifid, simpl way. Th iclusio of imprfct ful clls ito th sythsizig modl, is simpl ad bficial to th FC works. fact, a lik xists btw th mathmatics of th thrmal gis ad ful clls, so that th thory of ful clls ca b sythsizd with th thory of othr gis. Th ffct of this sythsis is th commo mathmatical modl applicabl to thrmal, radiativ, chmical, ad lctrochmical powr yild systms (ful clls). Th modls dvlopd i this papr dscrib physical ad chmical prformac of thrmal machis ad irrvrsibl ful clls at various opratig coditios. Lowrig of thrmal fficicis is attributd to diffrcs btw th tmpraturs ad chmical pottials i th bulks ad thir coutrparts i th circulatig fluid. Similarly, lowrig of th SOFC fficicy is likd with polarizatios. Optimal powr data diffr for powr gratd ad cosumd, ad dpd o paramtrs of th systm,.g., currt itsity, umbr of mass trasfr uits, polarizatios, uit surfac ara, avrag chmical rat, tc.. Ths data provid bouds for th rgy grators, which ar mor xact ad iformativ tha th classical bouds for th rvrsibl trasformatio. REFERENCES [] S. Siiutycz, Maximizig powr gratio i chmical systms ad ful clls, Procdigs of WCE: Th 0 tratioal Cofrc of Applid ad Egirig Mathmatics, Ed. A. Korsusky t al., Lodo, 6 8 July 0, [] S. Siiutycz, Carot cotrols to uify traditioal ad work-assistd opratios with hat & mass trasfr, tratioal J. of Applid Thrmodyamics, 6, 59-67, 003. [3] S. Siiutycz, Dyamic programmig ad Lagrag multiplirs for activ rlaxatio of fluids i o-quilibrium systms, Applid Mathmatical Modlig, 33, , 009. [4] S. Siiutycz ad P. Kura, Noliar modls for mchaical rgy productio i imprfct grators driv by thrmal or solar rgy, tr. J. Hat Mass Trasfr, 48, , 005. [5] S. Siiutycz ad P. Kura., Modlig thrmal bhavior ad work flux i fiit-rat systms with radiatio, tr. J. Hat ad Mass Trasfr 49, , 006. [6] F.L. Curzo ad B. Ahlbor, Efficicy of Carot gi at maximum powr output, Amrica J. Phys. 43, -4, 975 [7] A. D Vos, Edorvrsibl Thrmodyamics of Solar Ergy Covrsio, Oxford: Uivrsity Prss, 994, pp [8] P. Kura, Noliar Modls of Productio of Mchaical Ergy i No-dal Grators Driv by Thrmal or Solar Ergy, PhD thsis, Warsaw Uivrsity of Tchology, 006. [9] R. E. Bllma, Adaptiv Cotrol Procsss: a Guidd Tour, Pricto, Uivrsity Prss, 96, pp.-35. [0] J. Jtr, Maximum covrsio fficicy for th utilizatio of dirct solar radiatio, Solar Ergy 6, 3-36, 98. [] S. Siiutycz, Aalysis of powr ad tropy gratio i a chmical gi, tr. J. of Hat ad Mass Trasfr 5, , 008 [] S. Siiutycz, Complx chmical systms with powr productio driv by mass trasfr, tr. J. of Hat ad Mass Trasfr, 5, , 009 [3] S. Siiutycz, Z. Szwast, P. Kura, A. Poświata, M. Zalwski, R. Przkop ad M. Błszowski, Thrmodyamics ad optimizatio of chmical ad lctrochmical rgy grators with applicatios to ful clls.rsarch rport NN for th priod :warsaw TU: Faculty of Chmical ad Procss Egirig. [4] S. Siiutycz ad J..Jżowski,, Ergy Optimizatio i Procss Systms. Oxford: Elsvir, 009 [5] S. Siiutycz, Fiit-rat thrmodyamics of powr productio i thrmal, chmical ad lctro-chmical systms, tr. Joural of Hat ad Mass Trasfr, 53, , 00. [6] Y. Zhao, C.Ou ad J. Ch, 008. A w aalytical approach to modl ad valuat th prformac of a class of irrvrsibl ful clls. tratioal Joural of Hydrog Ergy, 33, , 008 [7] M. Wirzbicki, Optimizatio of SOFC basd rgy systm usig Asp Plus TM, Thsis (MsD) suprvisd by S. Siiutycz (Faculty of Chmical ad Procss Egirig, Warsaw Uivrsity of Tchology) ad J. Jwulski (Laboratory of Ful Clls, stitut of Ergtics, Warsaw), Warsaw TU, 009. [8] B.R. Sudhim, Trasport proprtis of liquid lctrolyts, p.p i B.R. Sudhim, d., Fusd Salts, Mc Graw Hill, Nw York, 964. [9] A. Ekma, S. Liukko ad K. Kotturi, Diffusio ad lctric coductio i multicompot lctrolyt systms, Elctrochmica Acta 3, 43-50, 978 [0] J. Nwma, Elctrochmical Systms, Prtic Hall, Eglwood Cliffs, 973. ACKNOWLEDGMENT Rsults i Fig. ar foud with M. Wirzbicki durig th prst author s suprvisig of his MsD thsis. Rsarch coopratio with dr J. Jwulski of th Warsaw stitut of Ergtics is also apprciatd. SSN: (Prit); SSN: (Oli)