A Microscale Modeling Tool for the Design and Optimization of Solid Oxide Fuel Cells

Transcription

1 Energes 9,, ; do:1.339/en47 OPEN ACCESS energes ISSN Artcle A Mcroscale Modelng Tool for the Desgn and Optmzaton of Sold Oxde Fuel Cells Shxue Lu 1,,3,4, We Kong and Zjng Ln 1,, * Hefe Natonal Laboratory for Physcal Scences at the Mcroscale, Unversty of Scence and Technology of Chna, Hefe 36, Chna; E-Mal: sxlu@mal.ustc.edu.cn (S.L.) Department of Physcs, Unversty of Scence and Technology of Chna, Hefe 36, Chna; E-Mal: wkong@mal.ustc.edu.cn (W.K.) Qngdao Insttute of Boenergy and Boprocess Technology, Chnese Academy of Scences, Qngdao 6611, Chna Key Laboratory of Bofuels, Chnese Academy of Scences, Qngdao 6611, Chna * Author to whom correspondence should be addressed; E-Mal: zjln@ustc.edu.cn; Tel.: ; Fax: Receved: May 9; n revsed form: 9 June 9 / Accepted: 1 June 9 / Publshed: 3 June 9 Abstract: A two dmensonal numercal model of a sold oxde fuel cell (SOFC) wth electrode functonal layers s presented. The model ncorporates the partal dfferental equatons for mass transport, electrc conducton and electrochemcal reactons n the electrode functonal layers, the anode support layer, the cathode current collecton layer and at the electrode/electrolyte nterfaces. A dusty gas model s used n modelng the gas transport n porous electrodes. The model s capable of provdng results n good agreement wth the expermental I-V relatonshp. Numercal examples are presented to llustrate the applcatons of ths numercal model as a tool for the desgn and optmzaton of SOFCs. For a stack assembly of a ptch wdth of mm and an nterconnect-electrode contact resstance of.5 Ωcm, a typcal SOFC stack cell should consst of a rb wdth of.9 mm, a cathode current collecton layer thckness of 3 μm, a cathode functonal layer thckness of 4 μm, and an anode functonal layer thckness of 1 μm n order to acheve optmal performance.

2 Energes 9, 48 Keywords: sold oxde fuel cell; functonal layer; modelng tool; optmzaton; fnte element method 1. Introducton Sold oxde fuel cells (SOFCs) are clean and cent chemcal to electrcal energy transfer devces. A tradtonal SOFC ncludes three layers: porous anode, dense electrolyte and porous cathode [1,]. The fuel and ar are transported from the gas channels to the three phase boundary (TPB) near the electrode/electrolyte nterfaces through the porous electrodes and the electrochemcal reactons take place n the TPBs. To mprove the fuel cell performance by ncreasng the TPB length to reduce the actvaton polarzaton whle mantanng the concentraton polarzaton at an acceptable level [3-7], the new desgn of an anode-supported SOFC cell often conssts of fve layers: an anode support layer (ASL), an anode functonal layer (AFL), an electrolyte layer, a cathode functonal layer (CFL) and a cathode current collector layer (CCCL), as shown schematcally n Fgure 1. The thn functonal layers, CFL and AFL, are desgned to ncrease the TPB length wth low porosty and small partcle szes. The thck layers, CCCL and ASL, are of hgh porosty and large partcle szes to enable easy gas transport to the functonal layers. Fgure 1. Schematc dagram of an anode-supported planar SOFC stack cell unt. In order to acheve hgh performance for the fuel cells, the cell geometres should be carefully optmzed. The thckness of CFL and CCCL are usually the focus of structural optmzaton n expermental and theoretcal works due to ther strong nfluences on the cathode actvaton polarzaton and concentraton polarzaton. The thckness of ASL s also an mportant parameter for the performance optmzaton [8]. Expermentally, Zhao et al. have carred out a systematc study on the nfluence of the mcrostructures of cell components on the cell propertes [3] and provded the followng optmzed cell parameters: an ASL thckness of.5 mm and porosty of 57%, an AFL thckness of μm, a CFL thckness of μm and a CCCL thckness of 5 μm. The experments by Haanappel et al. [4] showed that a CFL thckness of at least 1 μm and a CCCL thckness of at least 45 5 μm were requred for the optmal cell performance. As experments are expensve and tme consumng, however, t s dffcult for the experments to explore a large number of desgn parameters

3 Energes 9, 49 and dfferent materal choces. Numercal models that ncorporate the proper physcal mechansms are more versatle and flexble and are hghly desrable for the gudng analyses of a large engneerng communty. Most exstng theoretcal and modelng works are concerned wth the tradtonal SOFCs wthout the functonal layers [8-16]. For SOFCs wth the functonal layers, the electrochemcal actve areas are extended from the electrode/electrolyte nterfaces to the functonal layers and gaseous speces are consumed and produced wthn the functonal layers nstead of only at the nterfaces. The couplng of the governng equatons s not only by the boundary condtons but also by the source or snk terms and the mult-physcs theoretcal models are complcated. Consequently, only recently have there been a few theoretcal models consderng SOFCs wth the functonal layers [17-]. In the present work, a two dmensonal stack cell model s developed for SOFCs wth the electrode functonal layers. Electrochemcal reactons n the functonal layers and on the electrode/electrolyte nterfaces are consdered n detal. A dusty gas model s used to descrbe the gas transfer process n porous layers and electrc conductng equatons are used to descrbe electron and oxygen on conductng processes. In fttng the expermental data for model parameters, the ects of the nterconnect rbs and the nterconnect-electrode contact resstance on the fuel cell performance are also consdered. The model s then used to optmze several geometry parameters for stack cell.. Method The D cross-secton geometry model for the SOFC stack s shown n Fgure. Here d channel and d rb are one half of the channel wdth and one half of the nterconnect rb wdth, respectvely. The typcal geometrc, materal and operatonal parameters are shown n Table 1. The ar and the fuel are suppled to the reacton stes n the functonal layers (CFL and AFL) from the gas channels through the outsde porous layers (CCCL and ASL). Two mportant electrochemcal reactons occur n the functonal layers when cell operates wth hydrogen fuel and ar oxdant as llustrated n Fgure. At the CFL reacton ste, each half of an oxygen molecule gets two electrons conducted from the nearest cathodc nterconnect rb and s reduced to an oxygen on. The oxygen on s then transported from the CFL reacton ste to the AFL reacton ste through the electrolyte (YSZ). Fnally at the AFL reacton ste, the oxygen on reacts wth a hydrogen molecule and produces a water molecule and two electrons and the electrons are conducted to the nearest anodc nterconnect rb. The overall cell performance depends on the operatng cell potental and the output current. The operatng cell potental (V cell ) can be formally expressed as [8]: Vcell E-ASR;a -conc;a -act;a -ohm;e;a -ohm;;a -ohm;el-ohm;;c-ohm;e;c-act;c-conc;c-asr;c (1) where E s the Nernst potental, η conc;a and η conc;c are the concentraton overpotentals n the anode and cathode, respectvely, η act;a and η act;c the anode and cathode actvaton overpotentals, η ohm;e;a, η ohm;e;c the electronc ohmc overpotentals n the anode and cathode, η ohm;;a, η ohm;el and η ohm;;c the onc ohmc overpotentals n the anode, electrolyte and cathode, η ASR;a and η ASR;c the anode-rb and cathode-rb nterface overpotentals due to the contact resstance at the materal boundares.

4 Energes 9, 43 Fgure. Schematc cross-secton model of half repeatng unt of SOFC stack. Combnng the expressons for the Nernst potental and the concentraton overpotentals, RT ph,f RT po,ar RT ph,f pho,afl E E ln( ) ln( ), conc;a ln( ) and F pho,f 4F 1atm F p HO,f p RT po,ar conc;c ln( ) H,AFL 4F p, o,cfl Equaton 1 may be rewrtten for any electrochemcally actve stes n AFL and CFL as Vcell E -ASR;a -a -act;a -ohm;e;a -ohm;;a -ohm;el-ohm;;c-ohm;e;c-act;c-c -ASR;c (1a) where subscrpts f, ar, AFL and CFL refer to the values n fuel channel, ar channel, AFL area and CFL area, respectvely. E s the Nernst potental when the partal pressure of H, H O and O are all at 1 atm ( E = 1.1 V at 973 K or 7 C), R the unversal gas constant, T the temperature at Kelvn, F the Faraday constant. p H,AFL, p HO,AFL and p O,CFL are the partal pressure of H and H O at the electrochemcally actve ste n the AFL and the partal pressure of O at the actve ste n the CFL, respectvely. a and c are respectvely the anode and cathode concentraton balance potentals and are calculated as: RT p H,AFL a ln( ) F p (a) HO,AFL RT p c ln( ) 4F 1atm o,cfl (b) The detals of computng the other overpotental terms n Equaton 1 wll be descrbed later. Notce, however, the concentraton balance overpotentals (, ), the actvaton overpotentals ( act;a, act;c ) and the electrode ohmc overpotentals ( ohm;e;a, ohm;;a, ohm;;c, ohm;e;c ) n Equaton 1a and Equaton are dependent on the gven actve stes n the AFL and CFL layers that are fned meshed to reveal ther spatal varatons. The average output current densty of the SOFC stack can be calculated by d a ptch x c ptch 1 j jydx d (3)

5 Energes 9, 431 where d ptch s the ptch wdth and d ptch = d channel + d rb. The x and y axs are along the horzontal and vertcal drectons n the D model, respectvely. j y s the y component of the current densty flux vector. Table 1. Geometrc, materal and operatonal parameters for SOFC wth functonal layers. Parameters Value Cell temperature, T ( C) 7 Inlet fuel/ar pressure, P atm (Pa) Cell output voltage, V op (V).7 ASL thckness, l ASL (μm) 1, AFL thckness, l AFL (μm) Electrolyte thckness, l YSZ (μm) 8 CFL thckness, l CSL (μm) CCCL thckness, l CCCL (μm) 5 Porosty,.48 (ASL),.3 (AFL),.6 (CFL),.45 (CCCL) N volume fracton, N.55 (ASL, AFL) LSM volume fracton, LSM.475 (CFL), 1 (CCCL) Tortuosty, 3 Angle of partcle contact, ( o ) 3 Bruggeman factor, m 1.5 Mean partcle dameter, d p (μm) 1 (ASL, CCCL),.5 (AFL, CFL) Electrcal conductvty of N, N (s m -1 6 ) T Electrcal conductvty of LSM, LSM (s m -1 7 ) / T*exp( 18.5 / T) Ionc conductvty of YSZ, YSZ (s m -1 4 ) exp( 13 / T ) Dffuson volume of H, v H (m 3 mol -1 ) Dffuson volume of H O, v HO (m 3 mol -1 ) Dffuson volume of O, v O (m 3 mol -1 ) Dffuson volume of N, v N (m 3 mol -1 ) Molar mass of H, M H (kg mol -1 ) 1-3 Molar mass of H O, M H O (kg mol -1 ) Molar mass of O, M O (kg mol -1 ) Molar mass of N, M N (kg mol -1 ) Permeablty of anode (m ) Permeablty of cathode (m ) Vscosty of fuel (Pa s) Vscosty of ar (Pa s) Knudsen dffuson cocent of H (m s -1 ) Knudsen dffuson cocent of H O (m s -1 ) Knudsen dffuson cocent of O (m s -1 ) Knudsen dffuson cocent of N (m s -1 )

6 Energes 9, Electrochemcal Reactons n AFL and CFL In the present model, emprcal cathodc (anodc) Butler Volmer equatons are used to descrbe the electrochemcal reacton rate functons n CFL (AFL) and on the CFL/electrolyte (AFL/electrolyte) nterface. The transfer current denstes per TPB length (A/m 1 ) n the AFL ( TPB,a expressed [17] as: TPB,a act, a 4 act, a ph p HO exp(11 / T) act, a 4 act, a ph 14, exp(11 / T) act, c act, c TPB, c.5 4F.136 po exp(1741/ T) p p HO HO 14,Pa 14,Pa ) and CFL ( TPB,c ) are (4a) RT exp( F / RT ) exp( F / RT ) (4b) where η s the actvaton polarzaton, p the partal pressure of speces. For the reacton occurs n the functonal layers, the current source (A/m 3 ) can be calculated as V V TPB TPB. Here TPB s the volume specfc TPB length (mm -3 ) and can be expressed [17,1] V TPB 6dn p tel opp el o sn (5a) 3 where d p s the partcle dameter, n t ( 6(1 ) / d p ) the number densty of all partcles, j the volume fracton of partcle type j (el: electron conductng partcle, o: onc conductng partcle or.5 YSZ), P j (=( 1 (( ) /.47). 4 j ) the probablty of partcle type j to form percolated or globally contnuous clusters, the angle of partcle contact. For the reacton occurs on the electrode/electrolyte nterface, the current flow (A/m ) can be A calculated as. Here s the area specfc TPB length (mm - ) and may be expressed [17] TPB TPB A TPB dn Psn (5b) A TPB p t el el The actvaton polarzatons, η act,a and η act,c, are related to the electrc and balance potentals by [8] V V (6a) V V (6b) act,a e,afl,afl a act,c,cfl e,cfl c where V e,afl and V,AFL are respectvely the electronc and onc potentals at the electrochemcally actve ste n the AFL, V,CFL and V e,cfl the onc and electronc potentals at the electrochemcally actve ste n the CFL.

7 Energes 9, Gas Transport n Porous Electrode..1. Dusty gas model The dusty gas model s used here as t s more accurate than the Darcy s law and Maxwell Stefen model to descrbe the mass transport n porous electrodes. The orgnal dusty gas model can be expressed as [1,14,,3] n N xn j xn j 1 kp px xpxp (7) DK j1 Dj RT DK where N s the total molar flux of speces, x ( c / c ) the molar fracton, k the permeablty, µ the vscosty, p the total gas pressure, D K (= D K / ) the ectve Knudsen dffuson cocents and D (= / ) the ectve bnary dffuson cocents, D j j D j T 1 1 1/ 1/3 1/3 pv ( vj ) M M j (= [ ] j j ) the bnary dffuson cocent, ε and τ the porosty and tortuosty, c, ν and M the molar concentraton, dffuson volume and molar mass of speces, respectvely [4,5]. The requred parameters are shown n Table 1. For bnary gas, Equaton 7 may be reduced to where the ectve dffuson cocents of the speces, D, are gven by N Dc cu (8) D D D 1 1K 1 D 1 xd 1 K x D 1K D D D D xd x D 1 K 1 1 K 1K (9a) (9b) The ectve molar flow velocty u s gven by D1KDK k u p RTctot D1 x1d K xd1 K (1)... Governng equatons The mass transport processes n porous electrodes are governed by the mass dffuson and convectve equatons. For speces, the transport equaton can be expressed by [6] N R (11) where R s the reacton rate of speces (mol m -3 s -1 ) n the functonal layers and R n ASL and CCCL because there s no reacton, R V F for the hydrogen consumpton and H TPB,a TPB / R V /F for the water steam producton n AFL and R V HO TPB,a TPB O /4F TPB,c TPB for the oxygen consumpton n CFL.

8 Energes 9, Electrcal Conducton The electronc charge transfer n the electrodes (ASL, AFL, CFL and CCCL) are governed by: ( V ) Q (1a) e e e where e s the ectve electronc conductvty of the electrode, V e the electrc potental n the electrode. e Ve s the flux vector of the electronc current densty. Q e s the electronc current source (A/m 3 ) and Q e n ASL and CCCL because there s no electrochemcal reacton, V V Q n AFL and Q n CFL. The electronc potental dfferences along the e TPB,a TPB,a e TPB,c TPB,c electrcal current flux paths yeld the ohmc overpotentals, η ohm;e;a and η ohm;e;c. The onc charge transfer n the functonal layers (AFL, CFL) and electrolyte are governed by ( V) Q (1b) where s the ectve onc conductvty of the functonal layers or the electrolyte, V the electrc potental. V s the flux vector of the onc current densty. Q s the oxygen onc current source (A/m 3 V ) and Q n ASL and CCCL as there s no electrochemcal reacton, Q TPB,aTPB,a n AFL V and Q n CFL. The onc potental dfferences along the onc current flux paths yeld TPB,c TPB,c the ohmc overpotentals, η ohm;;a, η ohm;;c and η ohm;el. The ectve conductvty can be expressed [17]: m j j((1 ) jpj) (13) where σ j s the conductvty of speces j, m Bruggeman factor consderng the ects of tortuous conducton paths and constrcted contact areas between partcles. The electrc potental loss nsde the nterconnect plate s assumed to be neglgble due to the hgh conductvty of the metallc materal. The local current denstes cross the nterconnect/anode ( j I a ) and the cathode/nterconnect ( jc I) nterfaces are determned by the assocated electrc potental changes, or the nterface overpotentals: V V e,i/a e,a/i ASR;a Ia (14a) ASRcontact ASRcontact j j V V e,c/i e,i/c ASR;c ci (14b) ASRcontact ASRcontact where V e,i/a and V e,a/i are respectvely the nterconnect and ASL electrc potentals at the anodenterconnect boundary, V e,c/i and V e,i/c the CCCL and nterconnect electrc potentals at the cathodenterconnect boundary, ASR contact the area specfc contact resstance..4. Boundary Condtons (BCs) The boundary settngs for the mass transport equatons are shown n Table. The molar concentratons at the channel/electrode nterface are related to the molar fractons by the deal gas equaton of state, and can be calculated by c x P RT. atm /

9 Energes 9, 435 The boundary settngs for the electronc and onc charge transfer equatons are shown n Table 3. For the electronc charge transfer equaton the current flow on the electrode/electrolyte nterface s nward/outward current boundary, whle for the onc charge transfer equaton the current flow s nteror current source. The contact resstance s set on the nterface between nterconnect rbs and the electrodes. Table. Boundary settngs for mass transports n electrodes. Equatons Boundary ASL/channel nterface AFL/Electrolyte nterface All others Fuel transfer H molar BC Type concentraton c BC H H O molar concentraton c H O H nward molar flux A F / A / TPB,a TPB TPB,a TPB H O nward Insulaton/Symmetry molar flux F Ar transfer Boundary CCCL/channel nterface CFL/Electrolyte nterface All others O molar BC Type concentraton c BC O N molar concentraton c N O nward molar N nward Insulaton/Symmetry flux molar flux A /4F TPB,c TPB Table 3. Boundary settngs for the electronc and onc charge transfer equatons. Equatons Boundary Rb/CCCL nterface Rb/ASL nterface CFL/Electrolyt e nterface AFL/Electrolyt e nterface All others Electronc transfer BC Type Reference potental Reference potental BC V cell - E Inward current flow TPB,c A TPB Inward current flow TPB,a A TPB Electrc nsulaton Ionc transfer BC Type BC Interor current source A TPB,c TPB Interor current source A TPB,a TPB Electrc nsulaton.5. Numercal Method The fnte element commercal software COMSOL MULTIPHSICS Verson 3.4 [6] was used n the present study to solve the requred partal dfferental equatons (PDEs) such as the mass dffuson and convectve equatons, the electronc and onc charge transfer equatons. The Butler Volmer equatons are ntegrated nto the PDEs as sources or boundary settngs. The COMSOL statonary nonlnear solver uses an affne nvarant form of the damped Newton method [6] to solve the dscretzed PDEs wth a relatve tolerance of Free trangle meshes wth a maxmum element sze of m for electrolyte and AFL areas were used. It s worth mentonng a subtle feature n the numercal soluton process. Accordng to Equaton 4b, some current may be generated by the electrochemcal reacton even when P O s very small and may result n some small negatve value for P O due to the oxygen consumpton by the current producton. Ths n turn may cause numercal falure for Equaton 4b. To overcome the numercal problem, oxygen

10 Energes 9, 436 molar fracton was set as x x 6 O max( O,1 ). Smlarly, the actvaton polarzatons were set to be non-negatve, act,a max( act,a,) and act,c max( act,c,), to comply wth the physcal requrement. 3. Results and Dscusson 3.1. Model Fttng of the Expermental I-V Curve As shown n [3], slver meshes were pressed aganst CCCL and ASL for sngle cell testng. The meshes may be regarded as nterconnect rbs wth small ptch wdth. Here a ptch wdth of.3 mm and a rb wdth of.5 mm correspondng to the expermental descrpton were used for the model fttng. When the contact resstance s.8 Ωcm and the boundary hydrogen/oxygen molar fracton s.97/.1, the theoretcal curve shown n Fgure 3 agrees well wth the expermental data. It should be noted here that when the ptch wdth s small, the current through the electrode/rb nterface j I a and jc I are unform [8], then the ectve contact resstance s approxmately ASR ASR d / d. The ectve contact resstance s an ectve value assumng that the contact contact ptch rb contact resstance was dstrbuted over the entre ptch wdth nstead of the rb wdth. Accordngly, ASR s found here to be about.96 Ωcm, n good agreement wth the expermental estmaton contact that the total ohmc loss s.19 Ωcm at 7 C and half the total ohmc loss s due to the contact resstance [3]. The agreement between the theoretcal and expermental results valdates the numercal model descrbed here. Moreover, t s strkng to see that the nterconnect-electrode contact resstance s magnfed by a factor of d ptch /d rb and even a very small contact resstance may substantally affect the fuel cell testng result. Fgure 3. Comparson of theoretcal and expermental I-V curves. The onc current densty dstrbuton s qute unform along the x drecton and may be vewed as one-dmensonal along the y drecton n the model for the testng sngle cell. Fgure 4 shows the onc current densty n the CFL, electrolyte and AFL at x = when operatng cell potental s.7 V. The maxmum current densty across the electrolyte s.64 A/cm. The current denstes generated by the functonal layer body reacton and the functonal layer/electrolyte boundary reacton are respectvely.591 A/cm and.49 A/cm n the cathode sde, and.439 A/cm and.1 A/cm n the anode sde.

11 Energes 9, 437 Ths means that the body reacton s the domnant contrbutor for the fuel cell current generaton and the model pcture wth only boundary reactons may be too smplfed. Fgure 4. The onc current densty dstrbuton n the CFL, electrolyte and AFL at x =. As shown n Fgure 4, the onc current densty n AFL decreases rapdly wth the dstance from the AFL/electrolyte nterface, and almost all current s generated n the 4 μm regon around the nterface. The electrochemcal reacton occurs n the whole area of CFL, but a layer thckness of 11 μm near the nterface contrbutes about 9% of the total current densty. Ths ndcates that the ectve layer thcknesses for the AFL and CFL electrochemstry reactons are qute small. As the cathode actvaton overpotental for a gven current densty s hgher than the anode counterpart, the cathode sde requres an ectve area larger than the anode sde n order to generate the same amount of the total current. Therefore, the thckness of the ectve cathode actve layer s hgher than that of the anode sde. 3.. Dstrbutons of Physcal Quanttes n Stack Cell The output current densty s affected by the cathode layer thckness and a thcker layer benefts to the oxygen transport and the stack performance [8]. Two CCCL thckness l CCCL, 5 μm and μm, are used to show the physcal quanttes n stack cell. The ptch wdth s mm and the rb wdth s.8 mm. The boundary hydrogen and oxygen molar fracton are.7 and.1 respectvely. The area specfc contact resstance on the anode/nterconnect and the cathode/nterconnect boundares are unform (.5 Ωcm ). Fgure 5 shows the physcal quantty dstrbutons for l CCCL of 5 μm. The average output current densty s.374 A/cm. Fgure 6 shows the dstrbutons for l CCCL of μm and the average output current densty s.49 A/cm.

12 Energes 9, 438 Fgure 5. The dstrbutons of physcal quanttes when the CCCL layer thckness s 5 μm: (a) hydrogen and oxygen molar fractons; (b) total electronc current densty and current flux vector; (c) the y component of the current flux vector. (a) (b) (c)

13 Energes 9, 439 Fgure 6. The dstrbutons of physcal quanttes when the CCCL layer thckness s μm: (a) hydrogen and oxygen molar fractons; (b) total current densty and current flux vector; (c) the y component of the current flux vector. (a) (b) (c)

14 Energes 9, 44 The hydrogen and oxygen dstrbutons for a CCCL layer of 5 μm are shown n Fgure 5a and are smlar to the dstrbutons n a stack cell wthout the functonal layers [8]. The oxygen dstrbuton n the area under the rb s easly exhausted and the area s hardly actve. Wth a thcker CCCL layer of μm (Fgure 6a), the oxygen transfer s more ectve and can supply oxygen to the area under the rb, leadng to less concentraton polarzaton and more actve area. The exstence of thn functonal layers almost has no ect on the gas transfer. The total electronc current denstes and the current flux vectors are shown Fgures 5b and 6b. The total current densty n the CFL s obvously less than that n the CCCL. The reason s that the electronc conductvty of CCCL ( s m -1 ) s much largely than that of CFL (537 s m -1 ), and the current flux n CCCL has a much larger x-component than that n CFL to carry the current to the nterconnect rb. Understandably, the current densty dstrbuton s more unform and wth a less ohmc polarzaton for a l CCCL of μm than that for a l CCCL of 5 μm. Fgures 5c and 6c show the y component of the electronc current densty flux. The values for AFL and CFL other than at the AFL/electrolyte or CFL/electrolyte nterfaces are approxmately the local current denstes produced by the bulk and boundary electrochemcal reactons. At the ASL/AFL nterface, the value decreases substantally from the area under gas channel to the area under the rb for l CCCL = 5 μm (Fgure 5c) whle t s qute unform for l CCCL = μm (Fgure 6c), also ndcatng that the latter s more favorable Optmzaton of Geometry Parameters Optmzaton of the CCCL thckness As descrbed n Secton 3., dfferent CCCL layer thcknesses for otherwse dentcal cells may have dfferent performances. Fgure 7 shows the varatons of the average output current densty wth l CCCL. The current densty ncreases by 9.4% when l CCCL ncreases from 5 to μm, but the dfference s less than.4% when the l CCCL changes between and 4 μm. Consderng the performance beneft and materal cost, the optmzed CCCL layer thckness should be 3 μm. Unless stated otherwse, the layer thckness of μm s used for optmzng other parameters descrbed below. Fgure 7. Varaton of the average output current densty wth the CCCL layer thckness.

15 Energes 9, Optmzaton of the rb wdth The optmal rb wdth s the result of balancng the polarzatons of the current collecton and the gas transport. The most mportant factor affectng the optmzed rb wdth s the contact resstance [8]. Fgure 8 shows the optmzed rb wdth and the optmal output current densty versus the area specfc contact resstance. Larger contact resstance requres larger optmzed rb wdth to balance the ncreased ohmc loss, η ASR;a and η ASR;c. The optmal current densty decreases rapdly wth the ncrease of the contact resstance for two man reasons: a larger contact resstance leads to a hgher ohmc loss and the ncreased optmal rb wdth leads to a smaller actve area. For the reference contact resstance of.5 Ωcm, the optmzed rb wdth s.9 mm. Fgure 8. The optmzed rb wdth and the optmal output current densty vs the area specfc contact resstance Optmzaton of the CFL thckness Lmted by the catalytc ablty of the cathode materals, the cathode actvaton polarzaton s often the most mportant component of the total fuel cell polarzaton and a CFL layer wth small partcle sze and small porosty s desrable for enlargng the TPB length and reducng the actvaton polarzaton. For a gven mcrostructure, the CFL thckness s also an mportant factor affectng the cell performance. Fgure 9 shows the varaton of the average output current densty wth the CFL thckness (l CFL ). Sgnfcant mprovement s observed when l CFL s ncreased from 5 μm to μm and may be attrbuted to the ncrease of CFL actve area as ndcated n Fgure 4. The output current densty s qute smlar when l CFL s ncreased from μm to 4 μm as nether the actve CFL thckness s extended nor the gas transport s affected apprecably. As shown n Fgure 9, further ncrease of l CFL may cause reduced output current densty due to the ncreased concentraton polarzaton. Therefore, the optmal l CFL s between μm and 4 μm.

16 Energes 9, 44 Fgure 9. The varaton of the average output current densty wth the CFL thckness Optmzaton of the AFL thckness The ect of AFL thckness on the output current densty s also studed and shown n Fgure 1. As shown n Fgure 4, the actve AFL thckness s only 4 μm. It s not surprsng that the output current densty n Fgure 1 decreases wth l AFL. Fgure 1. The varaton of the average output current densty wth the AFL thckness. However, the decrease of the output current densty s qute small and s less than.68% when l AFL vares from 5 μm to μm as the concentraton polarzaton s not sgnfcant for such thn AFL layer. An AFL thckness of less than μm may be consdered to be optmal. 4. Conclusons A D fnte element model for anode-supported planar SOFC stack cells wth functonal layers s demonstrated. The model consders the mcrostructures of electrode materal, the gas transfer process n the porous electrode layers, the electronc conductng process n the electrode layers, the onc conductng process n the functonal layers and electrolyte, the bulk electrochemcal reactons n the functonal layers and the boundary electrochemcal reactons at the functonal layer/electrolyte nterfaces. The model was valdated by comparson wth expermental results on button cell I-V data. Dstrbutons of physcal quanttes n SOFC stack cells wth functonal layers are llustrated. The

17 Energes 9, 443 numercal model s also used to optmze geometry parameters of representatve stack cells, demonstratng ts capabltes as a desgn and modelng tool for the development of SOFC technology. Acknowledgements We gratefully acknowledge the fnancal support of the Knowledge Innovaton Program and the Key Program of the Chnese Academy of Scences (KJCX1.YW.7), the Natonal Hgh-tech R&D Program of Chna (7AA5Z156) and the Natonal Scence Foundaton of Chna ( ). References and Notes 1. Km, J.W.; Vrkar, A.V.; Fung, K.Z.; Mehta, K.; Snghal, S.C. Polarzaton ect n ntermedate temperature, anode-supported sold oxde fuel cell. J. Electrochem. Soc. 1999, 146, EG&G Techncal Servces, Inc., Fuel cell handbook, 7th ed., U.S. Department of Energy: Morgantown, WV, USA, Zhao, F.; Vrkar, A.V. Dependence of polarzaton n anode-supported sold oxde fuel cells on varous cell parameters. J. Power Sources 5, 141, Haanappel, V.A.C.; Mertens, J.; Rutenbeck, D.; Tropartz, C.; Herzhof, W.; Sebold, D.; Tetz, F. Optmsaton of processng and mcrostructural parameters of LSM cathodes to mprove the electrochemcal performance of anode-supported SOFCs. J. Power Sources 5, 141, Basu, R.N.; Blass, G.; Buchkremer, H.P.; Stover, D.; Tetz, F.; Wessel, E.; Vnke, I.C. Smplfed processng of anode-supported thn flm planar sold oxde fuel cells. J. Eur. Ceram. Soc. 5, 5, Moon, H.; Km, S.D.; Park, E.W.; Hyun, S.H.; Km, H.S. Characterstcs of SOFC sngle cells wth anode actve layer va tape castng and co-frng. Int. J. Hydrogen Energy 8, 33, Leone, P.; Santarell, M.; Asnar, P.; Calì, M.; Borchelln, R. Expermental nvestgatons of the mcroscopc features and polarzaton lmtng factors of planar SOFCs wth LSM and LSCF cathodes. J. Power Sources 8, 177, Lu, S.; Song, C.; Ln, Z. The ects of the nterconnect rb contact resstance on the performance of planar sold oxde fuel cell stack and the rb desgn optmzaton. J. Power Sources 8, 183, Ln, Z.; Stevenson, J.W.; Khaleel, M.A. The ect of nterconnect rb sze on the fuel cell concentraton polarzaton n planar SOFCs. J. Power Sources 3, 117, Suwanwarangkul, R.; Croset, E.; Fowler, M.W.; Douglas, P.L.; Entchev, E.; Douglas, M.A. Performance comparson of Fck s, dusty-gas and Stefan-Maxwell models to predct the concentraton overpotental of a SOFC anode. J. Power Sources 3, 1, Khaleel, M.A.; Ln, Z.; Sngh, P.; Surdoval, W.; Collns, D. A fnte element analyss modelng tool for sold oxde fuel cell development: coupled electrochemstry, thermal and flow analyss n MARC. J. Power Sources 4, 13,

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