Multicomponent Gas Diffusion in Porous Electrodes. Abstract

Transcription

1 Multcomponent Gas Dffuson n Porous Electrodes Yeqng Fu 1, Y Jang 2, Abhjt Dutta 2, Aravnd Mohanram 2, John D. Petras 2, Martn Z. Bazant 1,3 1 Department of Chemcal Engneerng, Massachusetts Insttute of Technology, Cambrdge, MA 2 Sant-Goban R&D Center, Northboro, MA 3 Department of Mathematcs, Massachusetts Insttute of Technology, Cambrdge, MA Abstract Multcomponent gas transport s nvestgated wth unprecedented precson by AC mpedance analyss of porous YSZ anode-supported sold oxde fuel cells. A fuel gas mxture of H 2 -H 2 O-N 2 s fed to the anode, and mpedance data are measured across the range of hydrogen partal pressure (10-100%) for open crcut condtons at three temperatures (800ºC, 850ºC and 900ºC) and for 300mA appled current at 800ºC. For the frst tme, analytcal formulae for the dffuson resstance (R b ) of three standard models of multcomponent gas transport (Fck, Stefan-Maxwell, and Dusty Gas) are derved and tested aganst the mpedance data. The tortuosty s the only fttng parameter snce all the dffuson cocents are known. Only the Dusty Gas model leads to a remarkable data collapse for over twenty expermental condtons, usng a constant tortuosty consstent wth permeablty measurements and the Bruggeman relaton. These results establsh the accuracy of the Dusty Gas model for multcomponent gas dffuson n porous meda and confrm the cacy of electrochemcal mpedance analyss to precsely determne transport mechansms. 1

2 I. Introducton The Sold Oxde Fuel Cell (SOFC) s currently the hghest-temperature fuel cell n development and can be operated over a wde temperature range from 600ºC-1000ºC allowng a number of fuels to be used. To operate at such hgh temperatures, the electrolyte s a thn, nonporous sold ceramc membrane that s conductve to charge carrer, O 2- ons. The operatng cency n generatng electrcty s among the hghest of the fuel cells at about 60% 1. Furthermore, the hgh operatng temperature allows cogeneraton of hgh-pressure steam that can be used n many applcatons. Combnng a hgh-temperature SOFC wth a turbne nto a hybrd fuel cell further ncreases the overall cency of generatng electrcty wth a potental of an cency of more than 70% 1. Therefore, t s a very promsng alternatve energy source that could potentally be used for home heatng or large scale electrcty producton n the future. Sold oxde fuel cell conssts of a porous cathode, an electrolyte, a porous anode and nterconnects. Two dfferent types have been explored n the development of SOFC, the electrolyte supported cell and the electrode supported cell. In the former, electrolyte s the thckest and serves as the mechancal support for the whole cell. However, due to the hgh Ohmc resstance of the relatvely thck electrolyte layer, the electrolyte supported desgn has been gradually replaced by the new electrode supported cells, n whch one of the porous electrodes s the supportng structure. Moreover, snce cathode supported cell usually gves hgher resstance, and s much harder to fabrcate due to the msmatched thermal expanson cocent of cathode support and functonal layer, the anode supported cell (ASC) s the most wdely accepted desgn n current SOFC research. The sold oxde fuel cell s operated wth fuel and oxdant beng contnuously fed from two sdes of the cell. Fuel (typcally, hydrogen and/or hydrocarbon mxture) s provded to the anode sde whle oxygen carred by ar s provded to the cathode. As the fuel and ar react, water vapor s produced and removed from anode. Fuels and oxdants have to be transported through porous electrodes before they arrve at the functonal layer, the reacton ste. At the same tme, product or water vapor has to travel through the porous anode n the opposte drecton to be taken away by the flowng stream. Therefore, gas transport through the porous electrode s an essental factor determnng the overall cell 2

3 performance 2,3. The cacy of the gas transport through the porous electrodes often determnes the rate of electrochemcal reacton or current generaton. Furthermore, many researches have shown that the gas transport through porous electrodes s manly governed by gas dffuson wth very small convecton contrbuton 4 7. Thus, gas dffuson n porous electrodes s the man source of concentraton polarzaton (concentraton dfference between bulk gas and functonal layers) n sold oxde fuel cells. However, the dffuson process has not been well understood yet due to 1) gas phase s a multcomponent gas mxture, ncludng reactants, carrer gas and possbly products; 2) the porous electrode, through whch gas phase has to travel, could have very complcated mcrostructures. There s abundant lterature on modelng gas dffuson n porous meda usng Fck s law, Stefan-Maxwell or Dusty Gas model. It s thought that Dusty Gas model should be the most accurate, although t s also the most complcated and dffcult to valdate. Almost no analytcal results are avalable, but the Dusty Gas model has been used n a number of numercal smulatons 2 4,8,9, albet wth constant pressure approxmaton whch s nconsstent 9 (see below). Moreover, no theoretcal framework exsts to analytcally derve the dffuson resstance values from mpedance data usng these more complex dffuson models for porous meda. Instead, current researchers usually use lmtng current values from the current-voltage or I-V curves to study gas dffuson n SOFC 10. Lmtng current s usually obtaned when the reactant s nearly or completely depleted at the reacton ste. Therefore, t has often been used to derve propertes of the porous electrode that would account for slow dffuson or sluggsh mass transport However, hgh tortuosty (rato of actual dstance travelled by gas to straght lne dstance between two ponts) s commonly nvoked to explan the lmtng current values. Many prevous attempts to ft models to I- V data for SOFC have been nconclusve wth wdely varyng tortuosty values from 2 to 19 for the same system Yet, most drect measurements conducted on anode materals and reconstructon of 3D mcrostructure ndcate tortuosty values should be n the range of 1.5 to 3. At the same tme, accordng to the theores about tortuosty 21, we should expect tortuosty of porous electrode wth nce and open mcrostructures to be not 3

4 too hgh. Actually, lmtng current can have the sgnature of not only gas dffuson 22, but also dssocatve adsorpton, surface dffuson, catalytc redox reacton, or even gas transport n free channels outsde the electrode. Therefore, we studed gas dffuson n porous electrodes usng AC mpedance, whch better separates processes of dfferent tme scales and therefore provdes better assgnment of arcs n data to dfferent processes. The SOFC button cell we studed uses hydrogen as fuel, carred by ntrogen together wth 1.7% of water vapor, for anode. Oxygen n ar s used as oxdant for cathode. Therefore, the electrochemcal reacton goes as follows. Oxygen molecule dffuses through cathode bulk layer and reaches the functonal layer, where t accepts electrons and s oxdzed to oxygen on, whch s further conducted through the electrolyte layer. When t arrves at the anode functonal layer, t reacts wth hydrogen fuel, formng water and releasng electrons to the external crcut. In ths paper, we present a new theoretcal approach to predct concentraton profles and dffuson resstance usng Fck s law (Fck), Stefan-Maxwell formulaton (SM), and Dusty Gas model (DGM) and compare wth expermental data for SOFC. By usng ths approach n conjuncton wth AC mpedance, we are able to show that DGM provdes a very accurate descrpton of multcomponent gas dffuson and can be used to subtract gas dffuson response from overall data for analyzng contrbutons from other physcal processes. II. Theory 1. Models Transport of gaseous components through porous meda has been extensvely studed over the years. In general, models ncludng Fck s model (FM), the Stefan Maxwell model (SMM) and the Dusty-gas model (DGM) are wdely used to predct the concentraton overpotental. Many researches have concluded that among the above three, the dusty-gas model s the most accurate and approprate model to smulate gas transport phenomena nsde a porous electrode 4,6,8, such as SOFC electrodes. However, due to ts complexty, ths model has no analytcal solutons, and the correspondng analyss requres complcated numercal smulaton 2,3,6, In ths work, we developed a 4

5 new theoretcal approach whch s based on mpedance analyss to show how DGM can also been used to analytcally analyze the gas dffuson nsde the porous meda. Fck s law s the smplest dffuson model and s used n dlute or bnary systems. It assumes the net flux s proportonal to the gradent of the concentraton of the correspondng speces 26. P N = D RT dx dx (1) D n Fck s law s the ectve dffuson cocent of speces, whch takes nto account of the composton of the gas mxture. The calculaton of followng equaton 2. Where D can be carred out D s the theoretcal dffuson cocent of speces, ε P and τ P are the porosty and tortuosty of the porous electrode, respectvely. P D = D ε τ P (2) Stefan-Maxwell model s more commonly used n mult-component systems because t consders the molecular collsons among dfferent types of the gas speces by usng a more complcated left hand sde term (equaton 3) However, t s more typcally used for nonporous meda. In equaton 3, mxture, X s the mole fracton of speces n the gas N s the mole flux of speces, P s total gas pressure n Pa, R s the unversal gas constant, T s absolute temperature n K, and x s the 1 D spatal varable. j X N X N P dx = j j D, j RT dx (3) The Dusty Gas model s an extenson of the Stefan-Maxwell equaton. It further takes nto account the molecules-pore wall nteractons by ntroducng the Knudsen dffuson term (frst term n eqn. 4) 11,25. Ths model assumes the pore walls consst of large molecules that are unformly dstrbuted n space. These pseudo or dummy dust molecules also collde wth real gas molecules, brngng n the Knudsen dffuson ect. 5

6 Besdes, the vscous fluxes due to pressure gradent are also taken nto consderaton. The general form of the DGM s the followng (Equaton 4) N XN XN P dx X 1 B P dp D D RT dx RT D dx j j 0 + = 1 (4) + K, j, j K, µ where B s the permeablty of the porous medum and 0 µ s the vscosty of the gas mxture. In both SM and DGM, the bnary dffuson cocents D can be calculated by the, j Chapman-Enskog equaton (Eqn.5), where T s temperature n K, p s pressure n Pa, Ω s the collson ntegral, speces 31. σ s the collson dameter, and j M s the molecular weght of D 1/2 3 3/ T + M M, j= 2 pωσ j j (5) Knudsen dffuson cocents can be derved from the knetc theory of gases (equaton. 6) where r s the radus of the gas molecule, M s the molecular weght of speces 31. D K, 2 8RT = 3 π M 1/2 r (6) Note that ths expresson was derved from cylndrcal pore geometry that havng the mean radus, but n realty, pore geometry can devate from cylnders, therefore, ths expresson has some uncertantes n predctng Knudsen dffusvty. The ectve bnary dffusvty and Knudsen dffusvty ( D and, j D ) were defned as K, ther theoretcal counterparts ( D and, j D ) tmes a geometrc factor, whch s porosty ( K, ε P ) dvded by tortuosty ( τ P ). D D ε and D D P P, j =, j K, = K, (7) τp τp ε 6

7 Numerous studes on transport through porous meda n the absence of a chemcal reacton reveal that the DGM s superor to the Fck s law n ts ablty to predct the fluxes 32,33. In porous catalyst, the Fck s law s stll frequently used because ts smplcty allows explct and analytcal expressons to be derved. If nonunform pressure s present n a porous meda due to reactons nvolvng a change n the number of molecules, addtonal permeaton term has to been taken nto account, and therefore DGM should be adopted. Many works 34,35 showed that the DGM can successfully predct the fluxes for these reactons n varous reactng systems. For example, Daves 36 used t for the SO 2 oxdaton reacton, Blek 37 appled t to the coal gasfcaton where large pressure gradent s present, However, the pressure gradent term requres addtonal computatonal tme and cost. Therefore researchers started to use DGM wthout the permeaton term f pressure gradent can be approxmately neglected. And a comparson among dfferent dffuson models, ncludng Fck s Law, Stefan-Maxwell model and Dusty Gas Model, to predct concentraton polarzaton s presented n n Suwanwarangkul work 4. Debates on Graham s law: Interestngly, we found there s a paradox of Dusty Gas model wth constant pressure assumpton, whch has not been wdely realzed n the communty of SOFC. Eqn.4 shows the general Dusty Gas model wth an extra permeaton flux term due to the pressure varaton, f we sum over all the gas speces, the pressure gradent can be calculated as shown n Eqn.9. By takng a look at the numerator, we can fnd that the pressure gradent comes from the dfferent ectve Knudsen dffusvty D of two actve speces n equmolar counter-dffuson K, mode. N B P X dp D RT D dx = + (8) K, µ K, N RT dp DK, = dx B0 P X 1+ µ D K, (9) 7

8 In the case of hydrogen molecules reactng to produce water vapor, the molar flux of all speces should add up to zero. In ths equ-molar counter dffuson mode, f the ectve Knudsen dffusvty D of hydrogen and water are the same, whch means f the force K, exerted on the pore walls by H 2 and H 2 O are exactly the same but n the opposte drecton, they wll cancel each other and no pressure wll buld up. However, the molecular weght and sze of the molecules vary among dfferent speces, therefore, Knudsen dffusvty must be dfferent, whch means total pressure has to change throughout the electrode. From another pont of vew, n the constant pressure assumpton, the summaton over all gas components wll lead to Graham s law 38, whch says the sum of molar flux ( tmes the square root of the molecular weght ( M ) should add up to zero (Eqn.10). Actually, the Graham s law s vald n the absence of chemcal reactons. But when chemcal reactons occur, the component fluxes are related through the reacton stochometry, and only somerzaton reactons wll be consstent wth Graham s law. N M = 0 (10) N ) In our case, moles of H 2 react to form equvalent number of moles of H 2 O and ths s obvously contradctory to the flux relatons mposed by the reacton. Snce the algebrac dervaton from Dusty Gas model to Graham s law s strct, ths conflct ndcates the Dusty Gas Model s ntrnscally nconsstent wth the constant pressure assumpton. Actually, Graham s law s only vald n the case of gas dffuson wthout reacton n general. In the case of SOFC, the gas dffuson n porous electrode has a boundary condton of surface reacton at the functonal layer/electrolyte nterface; therefore, the flux of actve speces (H 2 and H 2 O) cannot be captured by Graham s law. However, some current researches stll use t to study gas transport n porous SOFC electrodes 4,39. In fact, full DGM wth permeaton flux term due to pressure varaton has no problem, and s accurate enough to satsfy chemcal reacton boundary condtons. Yet wth the permeaton term, DGM s too complcated for dervng analytcal results, therefore restrct ts acceptablty n some theoretcal studes. But we wll provde a proof, n the 8

9 secton II.4 that n porous electrodes of SOFC, the pressure gradent ects on the gas transport s not sgnfcant and can be safely neglected. 2. Steady State Concentraton Profles From the governng equatons, we can derve the concentraton profles throughout the porous electrode when the bulk concentratons of dfferent gas speces are taken to be known. In Eqns , 0 X s the molar fracton of speces n the bulk gas mxture outsde the porous electrode, X ( x ) s the molar fracton of speces at poston x. I s the total current, F s the Faraday constant. R s unversal gas constant and P s the total gas pressure. 0 RTI 1 1 XN2( x) = XN2exp x (11) 2FP DH2, N2 D H2, H2O 0 H2O H2O ( D D ) D ( D D ) D ( ) ( ) 0 RTI 1 1 H2, H2O H2, N2 N2, H2O 0 RTI 1 1 XH2 = XH2 + x XN2 exp x 1 (12) 2FP DKH, 2 D H2, H2O 2FP D N2, H2O H2, N2 H2, H2O H2, N2 D N2, H2O X RTI 1 1 DH2, H2O DH2 O, N2 D N2, H2 0 RTI 1 1 = X + + x+ XN 2 exp x 1 (13) 2FP DKH, 2O D H2, H2O D 2FP D N2, H2O DH2, N2 D H2, H2O H2, N2 D N2, H2O Concentraton Polarzaton From the concentraton profle calculaton, we know the gas concentraton at the reacton surface and then concentraton overpotental can be calculated usng Nernst equaton (Eqn.14) η X X = ln (14) 0 RT H2 H2O anode _ conc 0 2F XH X 2 H2O In the case of mpedance under current, the concentraton outsde the porous electrode s very close to the bulk concentraton (the concentraton n the feed gas). However, under a non-zero current, some of the reactants need to react electrochemcally to support the current, therefore, there must be some concentraton gradent resultng from the consumpton of the reactants. We use a contnuously strred tank reactor (CSTR) 9

10 assumpton to approxmately calculate the gas concentraton outsde the cell n the feed tube 16, as descrbed n Eqn.20 and Eqn Dffuson mpedance (R b ) (wth and wthout dp) The above mentoned gas dffuson models, ncludng Fck s law, Stefan Maxwell and Dusty Gas model, are not new, and are wdely used to predct I-V curves and ft the lmtng current values as mentoned earler 13,20. But the SM and DGM models have rarely been used before to analytcally analyze mpedance spectra of SOFC, although they have been used to descrbe gas dffuson n porous electrodes. By takng the dervatve of the concentraton overpotental wth respect to current and evaluatng t at a specfed current, dffuson resstance (R b ) s obtaned for all three models. By takng a look at zero current R b n equatons.16, 17, and 18, we notce that compared to R b value from Fck s law, the R b of SM has an extra complcated term resultng from the consderaton of nteractons among dfferent gas speces. Also, the R b value derved from DGM further ncorporated the Knudsen ect, whch accounts for the collson of gas molecules wth the pore wall. The multcomponent gas dffuson nsde the porous electrodes was then studed by comparng these three dfferent dffuson models. In Eqns. 15 to 18, Rb _ anode( I ) s the gas dffuson resstance at current I, ηanode concentraton overpotental due to gas dffuson, 0 P and s the anode 0 X s the partal pressure and molar fracton of speces n the bulk gas mxture outsde the porous electrode. All other parameters are defned the same way as n general Dusty Gas model. R b_ anode( I= 0) dη di anode = or Rb_ anode( I) ( I = 0) dη = di anode ( I ) (15) Rb 2 RT = L Fck ( anode) a 0 0 2F PH2 O DK, H2O DH2, H2O PH2 DK, H2 DH2, H2O (16) Rb SM ( anode) PH2O DH2, H2O PH2 DH2, H2O RT = La 2F X N ef ( DH2, H2O DH2 O, N2 ) DN2, H2 ( DH2, H2O DH2, N2 ) DN2, H2O DH2, N2 D N2, H2O f 0 ( DN2, H2O DH2, N2 ) DH2, H2OPH2 O ( DN2, H2O DH N ) DH H OPH 2, 2 2, 2 2 (17) 10

11 Rb P H2 O DK, H2O D H2, H2O P RT H2 DK, H2 D H2, H2O = L F ( D D ) D ( DH2, H2O DH2, N2 ) DN2, H2O + + ( ) ( D D ) D P DGM ( anode) a H2, H2O H2 O, N2 N2, H2 X N 2 0 DH2, N2 D N2, H2O DN2, H2O DH2, N2 DH2, H2OPH2 O 0 N2, H2O H2, N2 H2, H2O H2 (18) Note that n the R b expressons, there are not too many quanttes that need to be ftted to data. Almost all the varables and parameters are determned from expermental nputs or estmatons from knetc gas theory, except for a mcrostructure factor (porosty dvded by tortuosty), whch lnks ectve dffusvty nsde porous electrode wth ts theoretcal value. When the porosty s known, the only quantty need to be determned from fttng s the tortuosty value of the electrodes. Smlarly, cathode dffuson resstance can also be estmated by dervng from a specfed dffuson model, e.g., Dusty Gas model dervaton was shown n Eqn.19 Rb (19) 2 0 RT Lc 1 XN2 DGM ( cathode ) = 0 + 4F PO2 DK, O2 DN2, O2 Comparng the theoretcal R b at anode and cathode, Fgure.1 shows the rato between cathode R b and anode R b multpled by 100%. It clearly shows that theoretcal R b of cathode s less than 0.5% of that of the anode n anode supported cells. Though the porosty and tortuosty can be slghtly dfferent n two porous electrodes, we can stll safely conclude that the dffuson resstance from anode sde domnates. Therefore, n all the followng dscusson, we treat total gas dffuson resstance to be anode gas dffuson resstance, and the low frequency arc n the mpedance data was ft wth a fnte-length Warburg element n a Randles crcut (Fg.2) to extract the anode dffuson resstance (R b ), whch was then compared to analytcal predctons from the three dffuson models (Eqns.16,17,18). 11

12 Fg. 1 Theoretcal comparson of gas dffuson resstance (R b ) from cathode and anode n anode supported cell at dfferent ph 2 levels. Fg.2 Fttng (top) of the low frequency arcs wth the Warburg element n a Randles crcut (bottom). Nonlnear least-squares fttng (CNLS) A fttng procedure called complex nonlnear least-squares fttng (CNLS), was mplemented, where data sets of (Z real, Z magnary ) versus frequency, or ( Z, phase angle) versus frequency ω are used. The am of the least squares fttng procedure s to fnd a set of parameters whch wll mnmze the sum of weghted devatons. 12

13 n k = 1 ' 2 ' 2 wk ( Zk, r Zk, r( ω)) + ( Zk, Zk, ( ω)) Where subscrpt k denotes the k th data pont n mpedance plot, Z s the real part of kr, the expermental mpedance data, whle ts counterpart Z s the theoretcal predcton ' kr, of the real part of the mpedance response. Smlarly, Z and k, Z ' k, are the magnary parts of the mpedance, expermentally and theoretcally, respectvely. Note that the theoretcal predcton of the mpedance s a functon of frequencyω, whch makes fttng of the Nyqust plot to be a three dmensonal curve fttng. Curves should not only match the correspondence of real and magnary parts, but also need to match ther frequency dependence as well. w s the weghtng factor, for whch we use the magntude of the k k th data pont n ths study. By mnmzng the sum usng the least square logarthm, a set of optmzed parameters wll be obtaned. We chose a Levenberg Marquardt nonlnear least-squares fttng algorthm because of ts straghtforward mplementaton. Any parameter enterng the model can n prncple be used as a free fttng parameter; however, care must be taken to stay wthn lmts of physcal sense. It should also be noted that the Levenberg Marquardt algorthm does not necessarly terate to a global optmum of the fttng parameters, nor does t gve any ndcaton for the unqueness of the optmzed parameters. Therefore t s mportant to start from realstc ntal guesses for the free parameters, and to exclude ftted results by analyzng ts orders of magntude and lookng at the ftted graphs. Or upper and lower bounds can be set n the process of nonlnear least square fttng. 4. Proof of usng sobarc assumpton n Dusty Gas model As mentoned before, Dusty Gas model s ntrnscally nconsstent wth sobarc or constant pressure assumpton. However, by comparng the gas composton profles of H 2, N 2 and H 2 O, we can see pressure varaton only leads to very small devatons of the gas composton profle (Fg.3). At the same tme, theoretcal predcton of anode gas dffuson resstance R b values derved from the full Dusty Gas model s practcally the same as that derved from the sobarc Dusty Gas model (Fg.4). Therefore, our analyss 13

14 proved that t s stll safe to neglect total pressure varaton nsde the porous electrode when usng the Dusty Gas model, although theoretcally there s some nconsstency between the model tself and the sobarc assumpton. Therefore, all the analyss and results we show n ths paper are based on sobarc assumpton, assumng total pressure nsde porous electrode does not vary n depth. Fg. 3 Comparson of anode gas composton profles under the current of I=100mA (full Dusty Gas model versus sobarc Dusty Gas model) Fg. 4 Comparson of theoretcal predcton of anode gas dffuson resstance (R b ) derved usng full Dusty Gas model versus sobarc Dusty Gas model. 14

15 III. Experments Anode-supported sngle cells were fabrcated based on technology developed by Forschungszentrum Jülch. The anode was approxmately 1.5 mm thck and 1 nch n dameter, whle the cathode was 0.1 mm thck, and 0.5 nch n dameter. The anode sde conssts of an anode support layer and an anode functonal layer wth a thckness of 15~30 um, both of whch are composed of N/YSZ compostes wth dfferent loadngs and mcrostructures. The cathode sde conssts of a cathode current collecton layer made from pure LSM and a cathode functonal layer wth a thckness of 15~30 um, whch s composed of YSZ/LSM composte. A new cell was sealed at the crcumference usng LP-1071 glass from Appled technologes and dred n an oven at 120 o C for 20 mn. Then, t was placed nto a sprng loaded sngle cell testng fxture. The fxture was put nto a furnace wth N 2 (150sccm) on the anode sde and Ar (150sccm) on the cathode sde. The furnace was then heated to 800 o C at 5 o C/mn. The cell was reduced the next mornng for 3 hours by gradually swtchng the anode gas from N 2 to H 2 flowng at 300 sccm. Durng testng, a tertary gas mxture of hydrogen, ntrogen and 3% by volume water vapor was provded to the anode from a top feedng tube and ar was fed from the bottom, carryng oxygen to the cathode. Pt lead wres were connected from the current collector layers (Pt mesh on cathode and nckel on the anode) to the data collectng equpment. A 1470E Solartron Analytcal from MTechnologes and mstat program were used to control the operatng condtons and collect the data. 15

16 Fg. 5 The setup of the sprng loaded testng fxture that used for anode supported cell testng. AC mpedance data of anode-supported cell was collected at open crcut voltage (OCV) condton (Fg. 6) and 300 ma current (Fg. 10) when varyng the hydrogen partal pressure from 10% to 100% of the total pressure. OCV Impedance data were also collected at three dfferent temperatures (800ºC, 850ºC, and 900ºC) (Fg. 12). Fg. 6 OCV AC mpedance data of anode supported cell collected at varous hydrogen partal pressures. Total pressure of the anode feedng gas was fxed at 1 atm. 16

17 IV. Analyss and Results Fg. 7 Expermentally extracted anode R b and theoretcal predctons of anode R b of anode supported cell under dfferent hydrogen partal pressures at OCV. Fgure 7 shows the expermental R b values we extracted from expermental data (black curve), and the theoretcal predctons of anode R b wthout takng nto account the mcrostructure factor from three dfferent models (colored curves). From the fttng, t s noteworthy that the Dusty Gas model gves a constant structural factor (porosty dvded by tortuosty), ndependent of hydrogen partal pressure (Fg.8). Ths s consstent wth real physcs, where the mcrostructure of the porous meda does not change wth testng condtons. Moreover, wth the anode porosty known to be 46%, the tortuosty ftted from the Dusty Gas model s 2.30, whch matches both theoretcal expectatons and drect expermental measurements. After takng nto account the ftted tortuosty, the Dusty Gas model best descrbes the gas dffuson, whle the Stefan-Maxwell model shows some devatons, and Fck s law cannot capture the performance at all (Fg.9). It s also nterestng to note that Dusty Gas model wth constant N 2 composton does not gve good enough results as well, whch confrmed the necessty to calculate ntrogen concentraton wthout any assumpton, and use t to further calculate concentraton profles of other actve speces, such as H 2 and H 2 O. 17

18 Fg.8 Structural factor (porosty/tortuosty) values ftted from three dffuson models under OCV. Fg.9 Comparson between dffuson resstances (R b ) derved from models and the values extracted from expermental data after takng nto account the ftted structural factor (porosty/tortuosty). 18

19 Fg. 10 AC mpedance data of anode supported cells for anode supported cell (303-03) at OCV (left) and under a current of 300mA, when hydrogen partal pressure s vared. We also derved the dffuson resstance R b for a non-zero current mpedance. The mpedance was measured at the current of 300 ma/cm 2 (Fg.10), and the correspondng R b values for all three models were numercally evaluated usng Maple software. It s worth notng that at a non-zero current, the bulk gas concentraton (concentraton at the nterface between porous electrode and gas feedng tube) can devate from the feedng concentraton due to the concentraton polarzaton resultng from consumpton of reactants by electrochemcal reactons. And, the concentraton gradents n the gas feedng tube can be approxmated usng a contnuous strred tank reactor (CSTR) model. Therefore a CSTR correcton (Eqn.20, 21) s ntroduced for calculatng the concentraton boundary condton of the bulk gas concentraton. 0 P s the deal bulk concentraton of speces, and 0* P s the corrected bulk concentraton of speces after the CSTR formulaton. In these equatons, s the electrode area n m 2, N s the molecular flux of speces n m s the total flux of feedng gas n T mol / s mol m s 2 /( ), A N 0* 0 H A 2 PH = P 2 H P 2 m N 0* 0 HOA 2 PHO= P 2 HO+ P 2 m T T (20) (21) 19

20 Wth the CSTR correcton, the ftted porosty/tortuosty s almost ndependent of feedng gas composton and s practcally the same as the zero current mpedance analyss, whch s another valdaton of the method of analyzng R b to extract tortuosty. Fg. 11 Expermentally extracted anode R b and theoretcal predctons for anode R b of anode supported cell under dfferent hydrogen partal pressures at both zero and 300 ma/cm 2 current, usng Dusty Gas model (left). Comparson of the ftted structure factors at two dfferent currents (rght). We further appled ths gas dffuson analyss to zero-current mpedance measured at dfferent temperatures (800ºC, 850ºC and 900ºC) (Fg. 12), and the tortuosty ftted at these temperatures only vares a lttle, from 3.1 to 3.3 (Fg. 13). Ths also shows the applcablty of the proposed anode gas dffuson resstance to AC mpedance measured at dfferent temperatures. 20

21 Fg. 12 OCV AC mpedance data of anode supported cell collected at varous temperatures. Hydrogen partal pressure s fxed at 100%, and total pressure of the anode feedng gas was fxed at 1 atm. Fg. 13 Comparson of the ftted structure factor at three dfferent temperatures. (800ºC, 850ºC and 900ºC) 21

22 V. Dscusson and Concluson In ths work, we nvestgated the mult-component gas transport n porous electrodes usng anode-supported sold oxde fuel cells, and proposed a new theoretcal approach to predct gas dffuson resstance (R b ). Explct analytcal expressons for gas dffuson resstance (R b ) were derved at zero current condtons, and values of gas dffuson resstance (R b ) were evaluated numercally at non-zero current condtons. Comparson of cathode and anode gas dffuson resstance shows that n anode supported cell, anode s the major contrbutor to gas dffuson resstance. Expermental R b values were determned by fttng the low frequency arc of the anode supported cell to the fnte length Warburg mpedance n a Randles crcut. Then, they were compared to predctons from three analytcal models, ncludng Fck s Law, Stefan-Maxwell model, and Dusty Gas model, to determne the structure factor (porosty dvded by tortuosty) or tortuosty when porosty s known. An nconsstency between sobarc assumpton and the Dusty Gas model was dentfed, but numercal smulaton confrmed that total pressure varaton only has very small ects on gas composton profles and the predcted gas dffuson resstance (R b ). Therefore, we can stll safely use sobarc assumptons wth Dusty Gas model. By ncorporatng nteractons between dfferent gas molecules and collsons between gas molecules and pore walls (Knudsen ect), Dusty Gas model works best and gves a more or less constant tortuosty value over a wde range of operatng condtons (10% to ~100% of hydrogen partal pressure, zero and non-zero currents, and three dfferent temperatures), and the ftted tortuosty value matches well wth drect expermental measurements. In summary, ths work developed a new theoretcal approach to utlze AC mpedance data and varous analytcal models to nvestgate multcomponent gas dffuson n porous meda. The remarkable data collapse of the measured gas dffuson resstance for a wde range of hydrogen partal pressures, currents and temperatures wth sngle, reasonable tortuosty establshes DGM as the best model for gas dffuson n porous meda (at least under these condtons). Therefore, ths approach can be used to estmate tortuosty for porous meda or to estmate gas dffuson resstance for further nvestgatng other physcal processes occurrng nsde the porous electrodes. 22

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