AND SIMULATIONS OF DEFORMATION AND TRANSPORT IN PEM FUEL CELLS

Transcription

1 Proceedngs of FuelCell008 Sxth Interntonl Fuel Cell Scence, Engneerng nd Technology Conference June 16-18, 008, Denver, Colordo, USA FuelCell MODELING AND SIMULATIONS OF DEFORMATION AND TRANSPORT IN PEM FUEL CELLS Serht Yeslyurt Sbnc Unversty, Istnbul, Turkey ABSTRACT Perfornce degrdton nd durblty of PEM fuel cells depend strongly upon trnsport nd deforton chrcterstcs of ther coponents especlly the polyer ebrne. Physcl propertes of the ebrne, such s ts onc conductvty nd Young s odulus depend on ts wter content, whch vres sgnfcntly wth opertng condtons nd durng trnsents. Recent studes ndcte tht cyclc trnsents y nduce hygrotherl ftgue tht leds to the ultte flure of the ebrne shortenng ts lfete, nd thus, hnderng the relble use PEM fuel cells for utootve pplctons. In ths work, we present two-densonl sultons nd nlyss of coupled deforton nd trnsport n PEM fuel cells. A two-densonl cross-secton of node nd cthode gs dffuson lyers, nd the ebrne sndwched between the s odeled usng Mxwell-Stefn equtons n the gs dffuson lyers, Bot s poroelstcty nd Drcy s lw for deforton nd wter trnsport n the ebrne nd Oh s lw for onc currents n the ebrne nd electrc currents n the gs dffuson electrodes. Stedy-stte deforton nd trnsport of wter n the ebrne, trnsent responses to step chnges n lod nd reltve hudty of the node nd cthode re obtned fro sulton experents, whch re conducted by ens of coercl fnte-eleent pckge, COMSOL Multphyscs. INTRODUCTION Desgn of PEM fuel cells s chllengng tsk for utootve pplctons due to durblty nd relblty constrnts, whch re prtculrly strngent s the ncubent nternlcobuston-engne technology, hs been perfected for over century. Flure of PEM fuel cells re ostly due to degrdton of terls durng trnsents whch nclude strtup, shutdown, electrc lod dends nd control nputs [1]. Ftgue echns s ong the ost suspected ones tht led to ctstrophc flure of the coponents followng ld-to-severe perfornce degrdton [1-3]. Mjor porton of PEM fuel cell odelng efforts concentrte on trnsport of rectnts nd wter, nd ts portnce on the electrochecl perfornce of fuel cells, e.g. [4-7]. Trnsport studes re hrd to extrpolte to understnd structurl spects tht led to degrdton nd flure. Moreover, there re severl structurl odelng nd nlyss of PEM fuel cells tht re not coupled to the trnsport nd electrochecl spects of PEM fuel cell operton, e.g. [,8,9]. Modelng of coupled deforton of coponents nd trnsport of speces n PEM fuel cells s recently ttrctng ttenton. Cho et l [10] use therodync odel to ddress swellng of PEM fuel cell ebrnes, typclly Nfon, by ens of checl potentl of the wter n the ebrne where the stretchng of polyers ffects the osotc presssure. In ther odel, Flory-Huggns theory, whch reltes the volue frcton to the swellng pressure (wter pressure n the ebrne), nd Young s odulus of the ebrne re used to ddress the role of echncl stresses n the polyer trx on the sorpton of wter [10]. Nzrov nd Proslow [11] developed odel, whch uses the cpllry pressure effect only t the boundres where the ebrne s n contct wth wter vpor, but uses echncl force blnce n the polyer trx nd wter pressure n the s. In ths work, we use Bot s lner poroelstcty to odel the elstc deforton of the ebrne trx wth the consderton of lqud pressure n s of the sold trx. Poroelstc theory uses the force blnce between the lqud nd the sold phse usng lner elstcty [1]. Cpllry effects only count t the nterfce where the lqud phse s exposed to gs slr to the odel presented n [11]. The lqud s llowed to ove n nd out nd wthn the trx usng Drcy s Lw [13]. Thus, the drvng forces of the trnsport re drectly coupled wth the echncl forces of the edu. Furtherore, the odel llows the consderton of dpng effect due to flow of the lqud n the porous sold durng trnsents. In the odel, whch s presented here trnsport of gs speces n gs dffuson lyers s odeled by Mxwell-Stefn equ- 1 Copyrght 008 by ASME

2 tons, dvecton of the gs xture n the node gs dffuson lyer s odeled by Drcy s Lw subject to conservton of ss, chrge trnsport s odeled by Oh s Lw subject to conservton of chrge, nd deforton s odeled by plnestrn forulton of the lner elstcty, whch ncluded poroelstc effects only wthn the ebrne, nd the wter trnsport n the ebrne s odeled by usng the Schlogl velocty. Coupled equtons re solved usng coercl fnte-eleent code, COMSOL [14]. DESCRIPTION OF THE MODEL Two-densonl odel of the PEM fuel cell cross secton, whch ncludes gs dffuson lyers () nd the ebrne s shown n Fgure 1. The ctlyst lyers re ssued reltvely thn copred to the ebrne, nd verged out n n-plne drecton; ther effects re consdered t the ebrne- nterfces. Geoetrc preters nd vlues of the twodensonl secton of the ebrne re lsted n Tble 1. TABLE 1: GEOMETRIC PROPERTIES OF THE TWO- DIMENSIONAL PEM FUEL CELL SECTION Geoetrc property Vlue Anode nd cthode gs flow chnnel wdths, l ch Anode nd cthode shoulder wdths, l sh Anode nd cthode gs dffuson lyer thcknesses, δ Mebrne thckness, δ Anode nd cthode ctlyst lyer thcknesses, δ CL Governng equtons Trnsport of gs speces n s Mxwell-Stefn equtons re used for odelng the trnsport of H nd H O vpor n the node nd O, H O vpor nd N n the cthode s presented n [6,15]: ( ρw ) + t N p (1) ρw D x ( x w ) ρw 0 j + + = j j j u j = 1 p In (1) w s ss nd x s olr frcton of the th speces; s {H,H O} on the node sde, nd {O,H O,N } on the cthode sde; ρ s the densty of the xture; D j s the bnry dffuson coeffcent of speces nd j, whch s replced by the effectve eff 1.5 dffuson coeffcent, Dj = Dε j g for porous s hvng porosty of ε g ; p s pressure, whch s ssued to be constnt (nlet pressure) n the dffusve ter s the sll pressure grdents n s re neglgble copred to the concentrton grdents; nd lstly u s the convectve (superfcl) velocty feld, whch s clculted by Drcy s Lw n the node, nd ssued to be equl to the negtve of the dffusve velocty of the nert spece, N, n the cthode. Note tht the superfcl velocty due to the pressure grdent s consdered due to ts convectve effect n the node, but dffusve effect due to the pressure grdent s neglected. Bnry dffuson coeffcents n xtures re deterned fro [16]: D j 1/ T 1 1 M M j = p ( v + vj ) where, v s the olr volue of speces, T s teperture, nd M s the oleculr weght of speces,. y x l sh Cthode Shoulder Anode Shoulder FIGURE 1: TWO-DIMENSIONAL PEM FUEL CELL SECTION MODELED IN THIS WORK; DASHED LINES INDICATE SYMMETRY SURFACES. In the node gs dffuson lyer, gs flow s odeled s flow n the porous ed to clculte the superfcl velocty of the xture s follows: = µ Cthode Mebrne () κ u p (3) where, κ, s the pereblty of the, µ s the vscosty of the xture [16]. The velocty feld n the node gs dffuson lyer s subject to contnuty equton: κ ( ). p 0 g t ρε ρ + = µ Cthode Flow Chnnel Anode Anode Flow Chnnel δ Trnsport of wter n the ebrne Mebrne wter trnsport s due to (1) electro-osotc drg (grdent of onc potentl), () dffuson (grdent of wter concentrton), nd (3) grdent of ebrne wter pressure, ledng to n verge voluetrc flux (superfcl velocty) of the wter n the ebrne, bsed on the Schlogl velocty [13]: l ch δ δ (4) Copyrght 008 by ASME

3 κ σ v p V c D λ V n (5) F (0) = Φ w H O λ H O d SO 3 µ w where, κ s the pereblty of the ebrne; µ w s the vscosty of the lqud wter; wter; V s the olr volue of lqud H O p s the wter pressure n the s; w c (0) SO3 s the olr densty of sulfonc groups n the dry ebrne; D λ s the dffuson coeffcent of wter n the polyer ebrne; of drgged wter olecules per proton; n nuber d σ s the onc conduc- tvty of the ebrne, whch s functon of λ; F s the Frdy s constnt; nd Φ s the onc potentl; nd λ s the reltve concentrton of wter wth respect to the concentrton of sulfonc groups n the dry ebrne: λ c c (6) w (0) SO3 A slent feture of our odel, whch s bsed on Bot s poroelstcty [1], s tht the structurl deforton of the polyer trx dcttes the wter concentrton of the ebrne bsed on the ssupton tht wter olecules re clustered s s n the polyer trx nd the concentrton s, n essence, gven by the voluetrc bulk strn tself. Thus, the olr concentrton of wter n the ebrne s the rto of the voluetrc strn, ε, to the olr volue of lqud wter: kk For better ccurcy of the nuercl soluton, the followng npulton kn to the use of generlzed potentl s ntroduced: g = A p + A + A Φ (10) where, A = κ µ, p λ p w λ Φ A = V c D nda Φ = V n σ F. λ (0) H O SO 3 λ H O d Thus the velocty n (5) cn be gven n ters of the generlzed potentl s follows: v = g + vɶ (11) where the vɶ ter on the rght-hnd-sde eerges fro the λ dependence of ech A ter n (10). Furtherore, snce λ s clculted fro the voluetrc strn, s n (7), one cn deterne vɶ fro the voluetrc strn: vɶ = p A + λ A + Φ A w p λ Φ A A A = p + + Φ = Substtutng (11) nd (1) n (9), one obtns: ρε w kk p λ Φ λ w ε A ε kk ε kk ε ε ε kk kk kk ( A ) ρ g = ρ ε w w ε kk (1) (13) t where the ter on the rght-hnd-sde corresponds to pseudosource ter whch s deterned fro the bulk strn. Once the generlzed potentl,g, s deterned, wter pressure n the ebrne cn be obtned fro (10). ε kk c w = (7) V HO Note tht, n (6), the voluetrc strn, ε, s deterned fro kk the poroelstc stress-strn reltonshp [1]. In (5), the dffuson coeffcent of wter nsde the polyer trx of the ebrne s gven by [17]: tnh λ λ.5 D λ = (8) The lst ter on the rght-hnd-sde n (5) corresponds to the superfcl velocty of wter due to onc currents n the ebrne. Ths ter s clculted fro the chrge blnce n the ebrne. As two ters of the Schlogl velocty n (5) cn be deterned fro lredy clculted qunttes, lst unknown, whch s wter pressure, p, cn be clculted fro the conservton w of wter ss n the ebrne: ρε w kk t ( ρ ) 0 + v = (9) whereρ w s wter s densty nd constnt. w Deforton (force blnce) In the two-densonl secton of the PEM fuel cell, deforton of sold coponents, nely s nd the ebrne s odeled usng plne-strn forulton of the lner elstcty. Only n the ebrne, the pressure due to ts lqud content (wter) s consdered nd dded to the norl stress coponents kn to Bot s poroelstcty. The generl stress-strn reltonshp used n odelng of the deforton s s follows [1]: where E ν σ = ε + ε δ γαpδ ν ( 1 + ν ) 1 j j kk j w j σ re stress coponents for, j { x, y, z} j (14) =, E nd ν re the Young s odulus nd the Posson s rto of the terl, ε re the strn coponents, whch re zero when ether or j j s z, ε = ε + ε s the voluetrc bulk strn, γ s n ndctor vrble whch s unty n the ebrne nd zero n the kk xx yy s, α s the Bot-Wlls coeffcent (unty for ncopressble fluds such s wter), nd δ s the Kronecker s delt. j 3 Copyrght 008 by ASME

4 Conservton of chrge Electrons re chrge crrers n the s. The conservton of chrge bsed on the Oh s Lw s expressed s follows: J = 0, J = σ Φ (15) e e e e where, σ s the effectve conductvty of the edu, nd e Φ s e the electrc potentl. Snce the ctlyst lyers re projected onto ebrne- nterfces, there re no chrge sources or snks n the odel geoetry. In the ebrne, the chrge crrer s the proton, for whch the conservton of chrge s gven slr to tht of electron to keep the current of the chrged speces n the se drecton: J = J = Φ (16) 0, σ + + where Φ s the onc potentl ndσ s the effectve conductvty of the ebrne, whch s wter content dependent [18]: σ 1 1 ( λ) = + exp T (17) Boundry condtons For ss frctons n the Mxwell-Stefn equton: At nterfces between the s nd gs flow chnnels, convecton ss trnsfer n lnr boundry lyer s ssued due to lnr flow n gs flow chnnels, nd pleented s follows: ρw u D x n = h j j, ch ( w w, ch ) (18) j where ρ s the densty of the xture n the,n s the surfce norl, w s the ss frcton of the th spece n the flow, ch chnnel, nd h s the ss trnsfer coeffcent deterned, ch fro the Sherwood nuber, Sh, for gs flow n squre-crosssecton, whch s constnt for lnr flow [16]: Sh h, ch ch = = D l, x.7 (19) In (19), l s the wdth of the chnnel, nd D s the dffuson coeffcent of the th spece n the gs xture, nely ch, x D, D nd D for H H -H O O -r H O-r, O nd H O vpor respectvely. At the ebrne- nterfces, we hve flux boundry condtons to c the snks nd sources n ctlyst lyers, whch re equl to rtes of consupton of the speces: M {, c} ρw u D x n = (0) j j j n F In (0), u s gven by (3) for node nd the negtve of the dffusve velocty of N n the cthode, M s the olr ss of the th spece; n s the stochoetrc coeffcent of ech speces nd negtve for H nd O nd postve for H O; nd nd c re node nd cthode exchnge currents respectvely, nd gven by Butler-Voler s expresson [19]: c ref {H,O } = {, } ( S 0 ) δ c {, c} CL ref c {H,O } (1) F F exp β η exp β η {, c} {, c} {, c} {, c} RT RT In (1), ( ref S ) s the product of the surfce re of the ctlyst nd the reference current densty for node nd cthode re- 0 {, c} spectvely; ctlyst lyers thckness, δ, s the se for both CL ref ref node nd the cthode; c nd c re reference concentrtons of H nd O t the node nd cthode; b s 0.5 for H nd H O unty for O ; R s the unversl gs constnt, β re coeff- {, c} cents, nd η re overpotentls t the node nd cthode respectvely, nd gven {, c} by: In (), b η = Φ Φ Φ () {, c} e 0,{, c} Φ nd e Φ re electrc nd onc potentls gven by (15) nd (16), Φ s the open crcut potentl t the node, 0, whch s tken s 0, nd Φ0,c cthode nd gven by [0]: 3 = ( - 98) 0, c s the open crcut potentl t the Φ T (3) Fluxes of ll speces re set to zero t the syetry surfces of the D geoetry n the norl drectons. For node pressure n (4): The superfcl velocty t the node s clculted fro (3) by solvng (4). The boundry condtons for the node gs pressure re specfed t the gs-flow-chnnel nterfce s the node flow chnnel pressure, p, whch s very close to the nlet pressure:, ch p = p, ch (4) At the nterfce between the ebrne nd the node, the superfcl velocty of the xture cn be deterned fro the recton of H nd the electro-osotc drg of wter nto the ebrne: M H ρ v w u n = + (5) Fρ ρ 4 Copyrght 008 by ASME

5 where v s the y-coponent of the velocty of the flow n the ebrne. At the syetry surfces nd -shoulder boundry, the velocty s set to zero. For the conservton of ebrne wter: Equton (13) gves the conservton of ss for the ebrne wter by ens of vrble trnsforton tht ntroduces functon whch cobnes ll the drvng forces tht dctte the wter trnsport n the ebrne for whch the voluetrc flux (superfcl velocty) s gven by (5). In typcl dffusve odels of the ebrne wter trnsport, non-equlbru boundry condtons re deeed pproprte, nd successfully pleented by severl uthors. In ddton to the dffuson, the effect of the electro-osotc drg s lso ncluded [13]. Moreover, recently, Nzrov nd Proslow used non-equlbru boundry condtons for the checl potentl of the ebrne wter, whch ncluded ts ctvty nd pressure [11]. Bsed on the experentl results of Ge et l [1], dffusve forces contrbutng to the trnsport of the ebrne wter re pleented here. Moreover, the electro-osotc drg s ncluded due to the effect of the onc potentl of the ebrne on the trnsport of wter. Lstly, the pressure blnce between the ebrne nd the s ncluded ssung Poseulle flow through the s of the ebrne ner the nterfce where the pressure blnce between the nd the ebrne network s effectve. Cobnng these three effects tht contrbute to the voluetrc flux gven by (5), we hve the followng boundry condton for the ebrne-node- nterfce: n V u n = J n + + d H O (0) * ( ) k V c ( λ λ + λ H O SO ) 3 F ( + ) k p p p p w cp In (6), n s the surfce norl; (6) n s electro-osotc coeff- d cent, whch s functon of λ ; J s the onc flux, whch s de- + fned n (16); k λ s ss trnsfer coeffcent; concentrton for the node wter ctvty; * λ s equlbru k s the pressure p coeffcent; nd p, p nd p re ebrne wter, cpllry w cp pressure nd node pressures respectvely. The ss trnsfer coeffcent s experentlly clculted by Ge et l [1]: k λ ( ( 0 )) ( ( 0 )) f exp 416 1/ T 1/ T, λ < λ = f exp 416 1/ T 1/ T, λ > λ where f λ V / V H O ( λ V V H O ) 5 * V 5 * V (7) = + s the volue frcton of the wter n the ebrne, V s the volue frcton of the dry * ebrne nd λ s the equlbru concentrton t the boundry, whch s gven by [1]: λ * + + T = = + + = , 303K (8) , T 353 K The pressure coeffcent s clculted fro the Poseulle flow ssupton n the s ner the nterfce,.e. r v p p k p p p = = + out p w cp c 8µ l ( ) ( {, }) (9) where, r nd l re the rdus nd length respectvely nd re both bout 1 n [], thus, yeldng s-n -1. Note tht, rgubly equlbru pressure k p boundry condton s ore pproprte for wter pressure, however, kn to odels tht use checl potentl nd for consstency wth other grdent ters n (6), we ncorporte the non-equlbru odel. Moreover, reltvely lrge k cn p ensure equlbru pressure, however t cuses nuercl nstbltes n our odel. The cpllry pressure, p cp equton, e.g. [3]: p cp, s gven by the Young-Lplce σ cosθ s c = (30) r where σ s the surfce tenson, θ s the contct ngle (whch s s c bout 93 degrees for Nfon nd wter [11]), nd the rdus, r depends on the bulk strn nd the specfc surfce re of the s, r S [10]: ε kk = x, r LB S In the odel, lower bound r LB vod unphyscl results. = 0.1 n s used for r (31) At the ebrne-cthode- nterfce, slrly to the condtons t the node sde, we hve: V H O H O n V d u + J ( ) + + n = J n + F F k V c k p p p (0) * λ H O SO 3 ( λ λ ) + c p ( + w cp c ) to (3) In (3), copred to (9), we lso hve the source ter, whch corresponds to the generton of wter n the cthode * ctlyst lyer. Other specfc vrbles n (3) re, λ s equlb- c ru concentrton for the node wter ctvty; nd p s the c cthode pressure. In our odel, we hve the se node nd 5 Copyrght 008 by ASME

6 cthode pressures, whch consttute reference to the ebrne wter pressure nd, thus, tken to be zero. For other boundry condtons t the syetry surfces, the velocty s set to zero. For deforton: Equton (14) s solved n the dsplceent for wth the ctul dsplceent vector, ξζ,, for whch the strn-tensor coponents re: ξ ζ 1 ξ ζ ε =, ε =, ε = + x y xy x y y x (33) The D PEM fuel secton shown n Fgure 1 s ssued to be fxed n the y-drecton t the botto but free to ove n the x-drecton,.e.: ( x ) ζ < l /,0 = 0 (34) sh Left sde s fxed n the x-drecton but cn freely defor n the y-drecton: ( y) ξ 0, = 0 (35) The rght sde cn freely ove n the y-drecton, but cn ove n the x-drecton only s uch s the botto surfce (edge) does: nd ( ) ζ l,0 = 0, (36) ξ tot (, y) = ξ (, y) l l (37) tot tot where l = ( l + l )/ s the totl wdth of the PEM fuel secton. tot sh ch Lstly, top of the secton s free to ove n the y-drecton but subject to constnt copressve lod due to clpng. Slr boundry condtons re lso used n structurl nlyss of PEM fuel cell cross-sectons [,3,9]. For conservton of chrge: The electrc potentl s set to ground t the node shoulder boundry of the node : Φ e ( x,0) = 0 (38) At the cthode shoulder, the lod current densty,, s posed: ( x,δ δ ) e cell J cell J + n = J (39) At nterfces between the ebrne nd s, electrc nd onc currents re gven by the exchnge currents, {, c}, whch re defned n (1). Thus, we hve: ( x, δ ) ( x, δ δ ) J n = (40) e c J + n = (41) e c c nd ( x, ) J δ n = (4) ( x, δ δ ) J + n = (43) c In (39) (43), n, n nd c n re surfce norls wth re- spect to the ebrne, cthode nd node respectvely. Intl Condtons In sultons, stedy-stte condtons re used s ntl condtons durng trnsents; preters tht correspond to ech sulton re explctly gven n the results secton. NUMERICAL APPROACH Governng equtons, whch re gven by (1), (4), (10), nd (13) (16) subject to boundry condtons, whch re gven n detl bove re solved usng coercl ultphyscs fnteeleent code, COMSOL [14], for stedy-stte nd trnsents. In totl of 1854 trngulr eleents wth qudrtc Lgrnge shpe functons re used; totl nuber of degrees of freedo s Nuercl soluton does not prtculrly depend on the esh selecton ore thn ssgned tolernces. A drect solver, UMFPACK, wth undped Newton nd reltve tolernce of 10-9 s used n stedy-stte nd te-dependent sultons. For the ltter, the 5 th order vrble te-step bckwrd dfferencng s lso nvoked. Slrly to the strtegy outlned n [15], stedy-stte solutons re obtned fro bootstrppng of ech ultphyscs coponent. A typcl stedy stte sulton tkes few nutes on. GHz CoreDuo, Wndows XP-SP, 3GB RAM lptop. A trnsent sulton, whch follows rpd rp tkes between 10 nutes nd 1 hour dependng on the condtons of the trnsent. RESULTS In Tble, typcl PEM fuel cell odel preters re lsted wth references where pplcble. Followng verges re used ong the perfornce etrcs tht re used n coprsons of sulton results: Cell voltge: At the cthode nterfce wth the shoulder, where the electrc lod boundry condtons re ppled the electrc potentl s verged over the surfce nd gven by: V cell l / sh = Φ dx e l sh 0 (44) Averge ebrne wter content: Nuber of wter olecules per sulfonc group n the ebrne s verged over the entre ebrne: 6 Copyrght 008 by ASME

7 λ v 1 = λdv V (45) e Ve Von Mses stress: To quntfy the stress levels n the fuel cell coponents one cn use the von Mses stress, whch s defned for plne-strn forulton s follows: where ( ) 1 x y z x y x y y z xy σ = σ + σ + σ σσ σσ σσ + σ (46) vm 3 σ re norl stress coponents nd stress on the x-y plne. σ s the sher xy In sultons, cell lod current densty, reltve hudty of the node nd cthode nlets, RH {,c}, nd the clpng force ppled to the shoulder prt of the cthode re used s sulton vrbles. Unless otherwse noted, the bse cse vlues of these vrbles re gven n Tble 3. Stedy-stte results In order to verfy tht the odel cptures the essentl fetures of typcl PEM fuel cell operton; typcl polrzton curve for the bse cse condtons s obtned fro the odel nd shown n Fgure. Actvton losses followed by Ohc nd concentrton losses re dentfble n the polrzton curve. The cell voltge s obtned fro (44). Note tht, ctlyst lodng vrbles re estted to hve relstc polrzton curve tht reflects typcl PEM fuel cell operton wthn the rnge of vlues vlble n the lterture. Averge ebrne wter content for the bse cse condtons s plotted gnst the lod densty n Fgure 3. As expected, the ebrne wter content ncreses wth ncresng wter producton t the cthode. Even though the onc currents n the ebrne re lost unfor nd only n the y-drecton, electrc currents orgntng fro the shoulders go tngentlly under the gs flow chnnels where nsulton boundry condtons re posed (Fgs. 4-d). Electrc nd onc currents pek n the porton tht lgns wth shoulders n the x-drecton for low currents (Fgs. 4-c), nd shft towrds the chnnel t hgh currents (Fg. 4d) TABLE : MATERIAL PROPERTIES AND THE BASE OPERATING CONDITIONS. Property, sybol Vlue Cell opertng teperture, T 353 K 0 Frdy s constnt, F C-ol -1 Unversl gs constnt, R 8.31 J-kg -1 ol -1 Molr volue of oxygen, v O ol -1 Molr volue of ntrogen, v N ol -1 Molr volue of wter vpor, v HO ol -1 Porosty of s, ε g 0.6 Pereblty of s, κ Anode nd cthode nlet pressures, p {, c}, n Anode nd cthode gs vscostes, µ [clc.] {, c} Pereblty of the ebrne, κ [11] Actve re of node nd cthode ref ctlyst lyers, ( 0 ) S [4, est.] {, c} Anode nd cthode coeffcents, β [4] {, c} H nd O reference concentrtons, c [5] ref {H,O } Electro-osotc drg coeffcent, n d [1] {00,00} kp {.08,3} 10-5 P-s ( λ ) {10 11,30} A- -3 {,1} 0 {56.4,3.39} ol S- Electrc conductvty of s, σ e Young s odulus of s, E 10 GP Posson s rto of s, ν 0.5 Young s odulus of ebrne, E 63 MP [11] Posson rto of the ebrne, ν 0.5 Specfc surfce of ebrne s, S [10] Surfce tenson of wter, σ s [11] 0.07 N- -1 FIGURE : POLARIZATION CURVE FOR THE BASE CASE. TABLE 3. BASE CASE CONDITIONS USED IN SIMULATIONS Property Vlue Anode nd cthode nlet reltve hudty, RH {,c} {0.5, 0.5} Clpng force 1 MP 7 Copyrght 008 by ASME

8 Effect of the clpng pressure on the verge ebrne wter concentrton (per sulfonc group) s shown n Fgure 5 for RH = RH c = 0.5 nd J cell = 5000 A- -. As the copresson ncreses ebrne s wter content decreses s f the copresson squeezes the ebrne slrly to sponge to reduce ts wter content. Slr behvor s observed by the odel developed by [11]. Cell voltge does not vry sgnfcntly wth the clpng pressure snce our odel does not nclude contct resstnces, whch re very portnt n the overll perfornce of the PEM fuel cell s ddressed elsewhere, e.g. [4]. FIGURE 3: AVERAGE MEMBRANE WATER CONCENTRA- TION PER SULFONIC GROUP AS A FUNCTION OF LOAD CURRENT FOR THE BASE CASE. Moreover, dstrbuton of the ebrne wter content becoes ore unfor s the copresson ncreses s depcted n Fgure 6, where λ dstrbuton t the node-sde of the ebrne s plotted wth respect to x-coordnte nd vryng gntudes of the clpng pressure. The squeezng effect of the copresson s ore pronounced n flttenng of λ dstrbuton (Fg. 6) thn the overll slght decrese of the wter content (Fg. 5). A B FIGURE 5: AVERAGE WATER CONCENTRATION PER SUL- FONIC GROUP IN THE MEMBRANE AS A FUNCTION OF CLAMPING PRESSURE FOR J cell =5000 A-M -. C D FIGURE 4: VON MISES STRESS DISTRIBUTION AND CUR- RENT VECTORS IN THE PEM FUEL CELL SECTION FOR J cell = 000, 5000, 10000, AND A-M - (A-D). FIGURE 6: WATER CONCENTRATION (PER SULFONIC GROUP) PROFILE AT THE ANODE-MEMBRANE INTERFACE FOR CLAMPING PRESSURES OF 1, 10, 0, 30 AND 50 MPA. 8 Copyrght 008 by ASME

9 Trnsent results Lod trnsent In Fgure 7, trnsent cell voltge s plotted gnst te n response to rpng up of the lod current densty fro 000 to 7000 A- - t t = 10 s for 0.1 seconds. The sudden drop of the cell voltge, whch s due to lod ncrese s followed by recovery yeldng n undershoot behvor, whch s observed n experentl studes, e.g. [1], nd n trnsent odels, e.g. [5]. Fgure 8 shows the vrton of λ v durng the lod trnsent. It s cler tht conductvty of the ebrne ncreses due to ncresng λ, nd leds to the recovery of the cell voltge v observed n Fgure 7. The response of λ resebles to tht of v typcl frst order syste. In Fgure 9, ebrne wter pressure t the cthode-sde nd x-postons of 0 nd 1 re plotted gnst te for the lod trnsent. At x = 0, whch lgns wth the center of the shoulder, wter pressure ncreses fro bout 6.5 MP to 13 MP durng the trnsent nd stblzes. However t x = 1, where lgns wth the center of the flow chnnel, fnl pressure s sller thn tht of t x = 0. Wter pressure ncrese s expected due to ncresng producton of wter. Lstly, the Mses stresses t the node nd cthode sdes of the ebrne nd t the dplne of the shoulder (x = 0) re shown n Fgure 10 for the lod trnsent. Increse of stress on the cthode-sde of the ebrne s ore pronounced thn the one on the node-sde due to the producton of wter on the cthode. Anode hudty trnsent For J = 000 A- -, RH cell c = 0.5 nd F = 1 MP, node reltve hudty s ncresed to 1 fro 0.1 t t = 10 s wth clp rp n 0.1 seconds. Mebrne wter content ncreses s expected followng the ncrese of the reltve hudty of the node nlet s shown n Fgure 11. Response of the cell voltge s shown n Fgure 1. As the ebrnes conductvty ncreses, so does the cell voltge lbet sll. Mses stresses t the node nd cthode sde of the ebrne t x = 0 (shoulder d-plne) re shown n Fgure 13. Cthode sde stress s lrger throughout the trnsent. However despte tht the stress ncreses t the node sde t the begnnng of the trnsent, stress t the cthode sde hs the opposte tendency, whch recovers only fter the trnsent progresses. J = 000 A- -, nd F = 1 MP. The response of cell clp theλ v resebles to tht of frst order response (nerly exponentl) s shown n Fgure 14. Cell voltge does not chnge sgnfcntly (not shown here). σ t the node nd cthode vm sdes of the ebrne nd the syetry surfce of the shoulders ncrese wth the trnsent (Fg. 15). Althoughσ vm t the cthode sde s lrger thn the one t the node sde before the trnsent, node sde stress becoes lrger fter the trnsent. However the dfference between the two s sll. CONCLUSIONS A two-densonl te-dependent odel of PEM fuel cell cross-secton tht contns ebrne nd gs dffuson lyers s developed here. Trnsport of gs speces n gs dffuson lyers, conservton of chrge, lner elstc deforton of s, poroelstc deforton of the ebrne, nd wter trnsport n the ebrne re consdered n the odel, whch s, then, used to nlyze the coupled effect of copressve clpng pressure nd trnsport of wter n the ebrne. The odel cptures the typcl chrcterstcs of PEM fuel cell nd ebrne wter trnsport, nd elucdtes the effect of clpng pressure on the perfornce of the fuel cell nd the effect of opertng condtons on the stresses tht develop n the ebrne nd. In prtculr, ncresng lod current densty results n hgher stresses n the ebrne ore thn 10 tes of the pressure due to clpng forces for lod densty of A- -. Slrly, ncresng clpng pressure squeezes the ebrne resultng n consderble wter loss nd fltter wter dstrbuton wthn the ebrne. Durng lod trnsents, cell voltge deonstrtes the typcl undershoot (for ncresng lod) behvor; nd ebrne stresses ncrese n the porton tht lgns wth the shoulders syetry plne (ddle). RH trnsents of the node nd cthode do not cuse sgnfcnt vrton n the cell voltge, but ffect the ount of wter n the ebrne, nd result n ncresng ebrne stresses. Cthode hudty trnsent Cthode reltve hudty s ncresed fro 0.1 to 1.0 t t = 10 seconds wth rp tht lsts 0.1 seconds for RH = 0.5, 9 Copyrght 008 by ASME

10 FIGURE 7: CELL VOLTAGE AS A FUNCTION OF TIME IN RESPONSE TO A FAST RAMP IN LOAD CURRENT DENSITY FROM 000 TO 7000 A-M - AT T=10 S WITH A DURATION OF 0.1 S. FIGURE 10: MISES STRESS AT THE ANODE (BLUE LINE SQUARE MARKERS) AND CATHODE (GREEN LINE TRIAN- GLE MARKERS) SIDES OF THE MEMBRANE DURING THE LOAD TRANSIENT. FIGURE 8: AVERAGE WATER CONCENTRATION PER SUL- FONIC GROUP IN THE MEMBRANE AS A FUNCTION OF TIME IN RESPONSE TO RAMP LOAD CURRENT DENSITY FROM 000 TO 7000 A-M - AT T=10 WITH THE DURATION OF 0.1 S. FIGURE 11: AVERAGE WATER CONCENTRATION PER SULFONIC GROUP IN THE MEMBRANE AS A FUNCTION OF TIME IN THE ANODE-RH TRANSIENT. FIGURE 9: WATER PRESSURE IN THE MEMBRANE AT THE MEMBRANE-CATHODE INTERFACE AND X=0 (UNDER THE SHOULDER), AND X=1 MM (UNDER THE CHANNEL) DUR- ING THE LOAD TRANSIENT. FIGURE 1: CELL VOLTAGE AS A FUNCTION OF TIME IN RESPONSE TO THE RAMP CHANGE OF ANODE RH FROM 0.1 TO 1.0 AT T=10 S WITH THE DURATION OF 0.1 S. 10 Copyrght 008 by ASME

11 ACKNOWLEDGMENTS Ths work s prtlly supported by The Scentfc nd Technologcl Reserch Councl of Turkey (TUBITAK). Author s currently on sbbtcl leve n The Unversty of Mchgn n Ann Arbor, nd expresses hs grttude for resources provded by Professor Ann Stefnopoulou nd the Mechncl Engneerng Deprtent. FIGURE 13: MISES STRESS AT THE ANODE (BLUE LINE SQUARE MARKERS) AND CATHODE (GREEN LINE TRIANGLE MARKERS) SIDES OF THE MEMBRANE DURING THE ANODE-RH TRANSIENT. FIGURE 14: AVERAGE WATER CONCENTRATION PER SULFONIC GROUP IN THE MEMBRANE AS A FUNCTION OF TIME IN THE CATHODE-RH TRANSIENT. FIGURE 15: MISES STRESS AT THE ANODE (BLUE LINE SQUARE MARKERS) AND CATHODE (GREEN LINE TRIANGLE MARKERS) SIDES OF THE MEMBRANE DURING THE CATHODE-RH TRANSIENT. REFERENCES [1] Benzger, J., Ch, E., Moxley, J.F., Kevrekds, I.G., 005, The dync response of PEM fuel cells to chnges n lod, Che. Eng. Sc., 60, pp [] Kusoglu, A., Krlsson, A.M, Sntre, M., Cleghorn, S., Johnson, W.B., 007, Mechncl behvor of fuel cell ebrnes under hudty cycles nd effect of swellng nsotropy on the ftgue stresses, J. Power Sources, 170, pp [33] Sols, R., Zou, Y., Hung, X., Refsnder, K., Condt, D., 007, On echncl behvor nd n-plne odelng of constrned PEM fuel cell ebrnes subjected to hydrton nd teperture cycles, J. Power Sources, 167, pp [4] Bernng, T., Djl, N., 003, A 3D, ultphse, ultcoponent odel of the cthode nd node of PEM fuel cell, J. Electoche. Soc., 150(1), pp. A1589-A1598. [5] Psogullr, U., Wng, C.-Y., 005, Two-phse odelng nd floodng predcton of polyer electrolyte fuel cells, J. Electroche. Soc., 15(), pp. A380-A390. [6] Serncn, M.F., Yeslyurt, S., 007, Trnsent nlyss of Proton Electrolyte Mebrne Fuel Cells (PEMFC) t strtup nd shutdown, Fuel Cells, 7(), pp [7] Shh, A.A., K, G.S., Su, P.C., Hrvey, D. 007, Trnsent non-sotherl odel of polyer electrolyte fuel cell, J. Power Sources, 163, pp [8] Zhou, P., Wu, C.W., M, G.J., 007, Influence of clpng force on the perfornce of PEMFCs, J. Power Sources, 163, pp [9] Tng, Y., Sntre, M.H., Krlsson, A.M., Cleghorn, S., Johnson, W.B., 006, Stresses n Proton Exchnge Mebrnes due to hygro-therl lodng, J. Fuel Cell Sc. Technol., 3, pp [10] Cho, P., Jln, N.H., Dtt, R., 005, Therodyncs nd proton trnsport n nfon I: Mebrne swellng, sorpton, nd on-exchnge equlbru. J. Electroche. Soc., 15(3), pp. E84-E89. [11] Nzrov, I., Proslow, K., 007, The pct of ebrne constrnt on PEM fuel cell wter ngeent, J. Electroche. Soc., 154(7), B63-B630. [1] Wng, H.F., 000, Theory of Lner Poroelstcty wth Applctons to Geoechncs nd Hydrogeology, Prnceton Unversty Press, Prnceton, New Jersey. [13] Ekerlng, M., Khrkts, Y.I, Kornyshev, A.A., Volfkovch, Y.M., 1998, Phenoenologcl theory of electro-osotc effect nd wter ngeent n polyer electrolyte proton- 11 Copyrght 008 by ASME

12 conductng ebrnes, J. Electroche. Soc., 145(8), pp [14] COMSOL, 008, COMSOL Multphyscs User Gude, COMSOL A.B., Stockhol. [15] Yeslyurt, S, 007, Three-densonl sultons of trnsent response of PEM fuel cells, n Proc. of IMECE007, Settle, Wshngton. [16] Brd, R.B., Stewrt, W.E., Lghtfoot, E.N., 00, Trnsport Phenoen, nd ed, Wley, New York. [17] Kulkovsky, A.A., 003, Qus-3D odelng of wter trnsport n polyer electrolyte fuel cells,, J. Electroche. Soc., 150(11), A143-A1439. [18] Sprnger, T.E., Zwodznsk, T.A., nd Gottesfeld, S., 1991, Polyer electrolyte fuel cell odel, J. Electroche. Soc., 138(8), [19] Brbr, F., 005, PEM Fuel Cells: Theory nd Prctce, Acdec Press. [0] Prthsrthy, A., Srnvsn, S. Appleby, A.J, 199, Teperture dependence of the electrode knetcs of oxygen reducton t the Pltnu/Nfon nterfce A croelectrode nvestgton, J. Electroche. Soc., 139(9), pp [1] Ge, S., L, X., Y, B., Hsng, I-M., 005, Absorpton, desorpton, nd trnsport of wter n polyer electrolyte ebrnes for fuel cells,, J. Electroche. Soc., 15(6), A1149- A1157. [] Ekerlng, M., Kornyshev, A.A., Stng, U., 1997, Electrophyscl propertes of polyer electrolyte ebrnes: A rndo network odel, J. Phys. Che., 101, pp [3] Jln, N.H., Cho, P., Dtt, R., 004, Phenoenologcl ethnol sorpton odel for Nfon 117, Sold Stte Ioncs, 175, pp [4] Lee, W-K., Ho, C-H., Vn Zee, J.W., Murthy, M., 1999, The effects of copresson nd gs dffuson lyers on the perfornce of PEM fuel cell, J. Power Sources, 84, pp [5] Wng, Y., Wng, C.-Y., 005, Trnsent nlyss of polyer electrolyte fuel cells, Electrochc Act, 50, pp Copyrght 008 by ASME