Performance model for molten carbonate fuel cells

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~$1 PERFORMANCE MODEL FOR MOLTENCARBONATE FUEL CELLS

Quarter ly Report fo r5 J u l y t h r o u g h 5 October 1978

D. Bloomfidd, M. L; Finson, G. Wilemski,T. Wolf an d K. L. Wray

Prepared forU. S. DEPARTMENT OF ENERGY

b n Francisco Operations OfticeUnde r Contract No. ET=78-C-04-2(183

PHYSICAL SCIENCES INC.30 COMMERCE WAY, WOBURN, MASS. 01801


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DISCLAIMERThis report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United StatesGovernment nor any agency Thereof, nor any of their employees,makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply i ts endorsement,recommendation, or favoring by the United States Government or anyagency thereof. The views and opinions of authors expressed hereindo not necessarily state or reflect those of the United StatesGovernment or any agency thereof.


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- - - - - -DISCLAIMER I

This book war prewred esansom~ nt 1 ~ O T Xponrared by an merry of the UnitedSlaer Governmen!.NoithDl tho Unitcd Slatr, turnaammm#t u r dlly *M I Inereot.nor any of their employees. makesany~dr ran iv. express or implied. or swrumer any legal liabi lity or rewnribilily far the accuracy.amplereneu. or vmfulneu of any information, uuwratul. product. or p aiulored. orreprerents lhai i ts u r Id ng r infringe PriMICly owned rights Reference herein to any specificmmmercial product. pr aeu . or service bv trade Mme. trademark. manufa~lurer. r othenuir. do e

TR- 145

PERFORMANCE MODE L FOR MOLTENCARBONATE F U E L CELLS

Qua r te r l y Repor t fo r5 July through 5 October 1978

D. Bloomfield, M. L. Finson, G. Wilemski,T. Wolf and K. L. Wray

Phys ica l Sc iences Inc .30 Com me rc e Way

Wobur n, MA 01801

Pr e.par ed fo rU. S. DEPARTMENT O F ENERGY

San Fr anc isc o Operat ions OfficeUnder c on tr ac t No. ET-78- C-03- 20'83


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I. INTRODUCTION

TABLE O F CONTENTS

ABSTRACT

SUMMARY

11. DIFFERENTIAL CE LL MODELA. Phen omen ologic al ModelB. P re l i mi n ar y ~ r n ~ i r i c a lodel

. INTEGRAL C EL L MODEL (I)A. Cu rre nt Dens i ty Dis tr ibution

1. Without C0 U-Lilization2. With CO Util izat ion

B. Temp era ture Dis t ribut ion

REFERENCES

iii

iv.


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P ro g re ss on the development of a per forma nce model fo r moltencarbonate fuel cells i s reported. Key physical and chemical phenomena havebeen identifed fo r inclusion in the model. In some instances, mathem aticalan aly ses of thes e phenomena.have been begun. A numerical scheme to calcu-late the cel l cu rr en t density distribution has been developed. The schemeaccounts for CO utilization and nonisothermal temperature distributions inthe cell.


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SUMMARY

This re po rt is a d escr ipt ion of work performed under U. S. 'Departmentof En erg y Co nt rac t No. ET- 78- C- 03- 083 directed towards the developmentof a :performance mod el for molten carbonate fue l cells. In t h is f i r s t qua r t e rof the pr o gr am we have concentrated mainly on determin ing the key physicaland che mi cal phenomena to be included in the model.

In the different ia l cel l model, which des cr ib es the local curre ntdensity-voltage relation ship at a point in the cell, attentio n m us t be paid tothe ma ss t ran spo rt of g aseous and dissolved reacta nts and products , to theohmic resi sta nc e of electroly te within the porous electrode, and to the elec tro-lcinetics taking place a t the elec trod e surfac e. Tentatively, the thin fi lmmodel of a porous e lectrode has been assumed a s the bas is f or this invest iga-tion. While th is phenomenological mo del i s being developed, a n em piri calcurren t-voltag e relationship i s being used in computational work asso ciatedwith modeling of the int eg ral cell.

Developments on the inte gra l cell model have been directed towardsthe determ ination of the distribut ion of cu rr en t density and tem per atur e withinthe cell. The m a ss balance scheme needed fo r the calculation of the c ur re ntdensity distribu tion in the cell has been devised. The scheme accounts forthe utiliza tion of CO and O2 in the cathode gas st re am and of H2 nd CO, which2a r e coupled by the water gas shift equilibrium, in the anode str eam . Bydefining util izations in ter m s of mo lar flow rat es , the formulation can beapplied without modiftcation to nonisothermal cell conditions. A workingcomputer code ha s been fashioned for the sim pler c ase in which CO util iza-tion i s neglected.

Calculation of-the cell tem pera ture distribution depends on the.'i.trea tme nt of heat tr an sf er within the cell . 1nitia1.l~ nly the e ffkcts o f g a s

phase t ran sp or t of heat evolved in the e lectrod e plane a r e being included.Conduction of heat in the plane of the ce ll through c el l ha rd wa re will be

- v-


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considered la ter . F o r now, then, all of the heat libera ted in the cell i sassu med to be tra nsf err ed t o the flowing gas s tr ea m s eithe r conductively o rconvectively by the flow of m a s s ac ro ss the elect rode /gas st re am boundaryari sin g fro m the =, el l alf reactions. Fo r the low Reynolds number flows ' 'typically found in anode and cathode s tr ea m s, changes in fluid pr ope rti es andthe heat addition rat e o ccur sufficien tly slowly to justify the assum ption oflocally fully developedf1 low in which velocity and temp era tur e rel ax i nstan-

taneously to fully developed profi les a t each stre amw ise station. Thisapproximation will enta il an appreciably simplified analysis , ,yet it shouldprovide the des ired accur acy for e lec trode and gas s tr ea m temper atures ;


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I. INTRODUCTION

The objective of thi s pr og ram i s the development of a molte n carbona tefue l ce l l model that descr ibe s ce l l per form ance (cu r ren t densi ty ve rsus vo ltage)a s a function of inlet gas compositions, tem per atu res, f low ra te s and pr es su re .In addition, the model should provide a cl ea r picture of the tem pera ture andcur ren t density dis t r ibut ions over the e lectrode faces . The model can beconveniently discussed in te rm s of i t s thre e main components : a differentialce ll model (DCM) and two in teg ral cell models, (ICM-I and ICM-II).

In the DCM, ce ll perfor man ce will be desc ribed on a local basi s byincorporat ing phenomenological t rea tmen ts of e lectrokinet ics , m as s t ransport ,and porous e lectrode s t ructure . . The pr im ary output of t his mo del will bethe cur r en t densi ty a s a function of lo cal ga s conditions, tem pe rat ur e andvoltage. The other princip al output of this model i s the lo cal heat r el ea serate which is easi ly determined once the local cur rent densi ty i s known.

The integral ce l l models t r ea t cef i behavior on an overal l basis ,accounting f or fuel and oxidiant uti l ization and. hea t tr an sf er w ithin the cell . .In the ICM-I the output of the DCM wi ll be com bined with tr ea tm en t of g asphase ma ss balance and convective heat t r an sf er in ord er to calculatecurr ent densi ty and tem perature dis t r ibut ions in the cel l . Heat conductionin the plane of the ce ll i s not included in the ICM-I. Thi s effect i d incor -

. porated in the ICM-11 which i s otherw ise esse ntia lly identical with ICM-I.


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11. DIFFERENTIAL CELL MODEL

A. Phenomenolopical Model. Our goal her e i s to give a phenomenological trea tme nt of the local

current density-voltage rela tionship in te rm s of molec ular, kinetic and elec-t rode paramete rs . Presently , i t a ppears most frui tful to employ an idealizedmodel of a porous electrode. In thi s analysis , the equations used to des cri bethe kinetic 'and m as s tran spo rt proc esse s occurr ' ing must be solved for a setof boundary conditions consistent with an assu me d po re geo me try and elec-trolyte distribution. Th is kind of app roac h ha s been developed in the past byseve ral groups of wor kers , 2' and we will utilize the ir methods. Ideally,a m or e rea li st ic theory of the porous str uct ur e of the electrode would be de-sirab le but analy ses start ing fr om thi s prem ise have either provided very

4over simplified tre atm ent s of kinetics and m a ss trans po rt or a r e based on5,6averaging processes that leave fundamental m as s trans po rt and kinetic,

para met er s difficult to inter pr et.Esti mat es m ade of limiting cur re nt densities support the notion that

extensive portions of the electro de sur face mu st be covered by thin fil ms of7electroly te. In Fig. 1 , a sketch of the thin fi lm mode l of a n electro de por e

i s presented. The pore i s taken to be a cylinder of radius R and i s coveredPb y z thin film of thi ckn ess 6 for a length R along the p ore wall. The axialcoordin ate along the pore i s z and r i s the radia l coordinate .

Fig. 1 Thin Fi lm Model of an Electrode P o r e. - - .

- 2 -


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A sbnpl if iod , but s t i l l ma the i~~ at i cs l lyoir~p licated reatrnen.tof the ma s s tran sp ort and kinetic s includes:

1, Diffusion of gas eou s reacta 'nts and pro du cts in to and out of th epore along the z axis ; because we expect R/&


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.Hero, I3 and c a r e a diffusion cocfficiont and mol ar concentration; the

subscr ip ts g and 1 lab el the ga's' and liquid phase.s, n i s the nwnb e r of molesof e lec trons trans fer red per mole of reactant or product , I(z) i s the to ta lcurr en t in the film at z and 'i i s the transf er curr ent density:

Regarding statement (3) , we may wri te fo r thin film s1

- 1where K i s the specific conductivity (ohm-' crn ) of the elec tro lyte . Finall y,the Butler-Volrner equation for the tra nsf er c urre nt density has the gen eralform: . .

. . . - .fo r assumed f i rs t o rde r fo rward and reve rse reac t ions. Here, i i s the0exchange cu rr en t density and a i s the t ransf er coeffic ient . The supersc r ip t

0 denotes open cir cui t conditions, and the concentra tions appea ring in th isexpressio n a r e to be evaluated "a tf1 he elec trode surface just outside, thecompact double layer. The actual half reactions fo r the molten carbonate c el lwill mor e than likely not be f ir st ord er in just a single reactant and product.A m or e complicated dependence i s anticipated on the ba si s of pas t cons ider a-

8tions. Thus, the Bu tl er -V o he r equation will have to be adjusted to refl ectthe actua l reaction m echanism s a s sumed to be occurring. This problem will.. . -. . ---. . . . . . .-be add ress ed in the next quar ter when we will al so begin exarniningth e.adequa cy of the f its to I'GT experim ental sm all cel l data8 with i-q relation-

. ships calcu lated fr om the above equations. We have not yet decided whethera ful l num eric al solution of the above equations i s needed o r whether th ere


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i s suff icient sepa rat ion of the physical regimes to allow sat isfactory tre at-ment fro m simp ler (probably approximate) analyt ical solutions. As' an. example of the lat te r procedure, con sider the following calculation.

At sufficiently low overpotentials (< 50.mV), the Butler-Volm er equa-tion (Eq. ( 3 ) with n=2) can be linea rized giving

Under these conditions, i t i s l ikely that m a ss t ran spo rt l imitat ions .will not aris e, . and we need only combine Eq. ( I ) , (2) and (4) for the ohmic ,drop.in the film to obtain

The solution to this equation is

~ ( z ) ~ ( 0 )osh (Kz)

w here

F r o m Eq. (2) we al so readily. find 8

I(A) = 2TrRp.bK K a n h ( ~ l ) q ( l ) . (8. . . . . .Subst ituting the par am ete rs l is ted in Table I into Eq. (7) we o btain a

- 1 .value of K = 14 cm . , 'Equation (8) ma y be solv ed.fo r the to tal por e ove r-potent ial in te rm s of the pore current . Fo r an e lec t rode curre nt dens ity of2 6 2100 mA /cm and an ele,ctrode po re density N = 4 x 10 ~ o r e s / c m Ref. (7)),P


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- 6the pore cur rent I (&)s 25 x 10 mA/pore. Equation (8) yield s a va li e of thepore overpoten.tia1 q( & ) 38 mV. .With th is value of the overpotential at . z = i ,we may solve fo r the overpoten tial at the film end z=0 usin g Eq. ( 6 ) . We find

. ,We may conclude f ro m the above ex er ci se that the 38 mV total poreoverpotential ' just i f ies the l inearizat ion and that the resist i ve lo ss in the f i lmi s about 8 mV. Finally, the slope of the pola rizat ion cu rv e ( N I (&) s ~ ( 4 ) )

3 Pi s 2. 6 r n ~ / m ~ - c m &hich i s the sam e o rd er of magnitude a s that found i n theexp erim ental IGT data. 8

. TABLE I(Ba sed on values found i n Ref. 7)

B. Prel im inary Empir ica l Model

P a r a m e t e ri0K

6

R

%T

>

In o rd e r to proceed with the clevelopment of a nu me ric al s che me fo rLpredict ing cu rren t densi ty and tempera ture distr ibut ions in the cel l ( ICM-I). .we have decided to employ a sim plified analytical ex pre ssio n for the voltage

drop V between elec trode s a s a function of lo cal gas com position and pr es su reand local cu rren t densi ty j: .

e

Value4 x A /cm 2

- 11 (ohm-cm)- 55 x 1 0 c m- 25 x 1 0 . c m

' 2 . 5 x c m9 2 3 O ~


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ere, E :i s the open circu it potential at the local temper atur e T andgas condi-tions, X i s the mole fract ion in a gas s trea m, P i s the gas pres;ure, andzohm i s the ohmic impedance of the electrolyte. Finally, q i s the sum of theelectrode overpotentials which, for the present, we take a s linear in the-cu rr en t density based on the experim ental findings of IGT. 8

The local heat re le as e ra te i s given by the following equation (in units of2watts /c m ):

where AH i s the enthalpy change for the ov eral l cel l reaction at the loca ltempera ture T. Because the anode and cathode temp era tu res a r e not likelyto differ by m or e than a few deg ree s at any point in the cell, we a r e simplyusing an. aver age c ell tem pera ture for e ach point in the electrode plane. Wear e a lso assuming that the e lectrodes a r e perfect e le c tr ic a l conductors andthat ionic currents i n the electroly te ar e orthogonal to the electrode s. Even-tually, al l of these assumpt ions will be mor e cri tical ly examined, but for nowthey provide a physically sound ba si s for our modeling effort. Implementa-tion of , this prelim inar y DCM i s dis cuss ed in the following section. '


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IIL INTEGRAL CELL MODEL (I)

Our ultimate objective i s . the determ ination of cur ren t density and cel ltem pera ture distribution for a given se t of operating conditions (c el l voltage,inlet ga s compositions and tempe ratu res, pr e s sure). Fo r the ICM-I we haveidentified two reasonab le computation schemes. The fi rs t of these is . ite ra tiv ein which a given tem pera ture distribution i s used to compute a distribution ofcu rre nt density, which i s in turn used to update the temp erat ure distribution,etc. An alternative i s to proceed in a stepwise fashion, calculating thetem pe rat ure and cur re nt density simultaneously at ea ch point based on the

. up st re am hi stor y of the gas flows. Which scheme i s chosen wil l dependmainl y on acc ura cy and speed of execution and on the anticipated e ase 'w ithwhich in-plane heat conduction can be included at a la te r time.

Independent of the ove ral l computational scheme selected, we canfornlulate the .problem in te r m s of the per for ma nce of the two subtasks:

1. Calculation of the dist rib utio n of cu rre nt' density' asso ciat edwith a given temp erat ure distribution. . ,

2 . Calculation of the tem pera ture distribution associated with a 'given distrib ution of curr ent' density.

A. Current Density Distribution1. without CO Utilization

-,

8This c alcu la tion has p r e ~ d b s l y e en pe r fo rme d a t IGT. Here wepresen t a modification of that procedure and some prel imi nar y r esu lts of o urown. Our, modi fication con si st s of redefin ing the anode and cathode con ver -sioils A , t-' in ter m s of mo lar f low ra t es ra th er than volume flow ra t es a s in

. . _ _ - - -the IGT work: -. -- - -- -


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- %moles H consumed2 2 2anode: =, -mole s H2 in .OnH,.On - A ,moles CO consumed

cathode: p = 2 - C02 C02moles CO- in .O

where ri = mo lar flow ra te of species i. Superscript 0 denotes an inle t quantity.iThis sma ll change perm its the variables and p to provide an unambiguousm ea su re of gas utilizatio n even under noniso therm al conditions. Definitionsinvolving volume flow rat e s su ffer f ro m the fact that volume flow r at es changenonchemically because of expansion o r . ontraction ari sin g fr om temp erat urechanges. The mathematica l anal ysis needed to solve this problem under non-isothermal conditions also remains simpler when molar definitions are used,Of cour se, fo r iso the rm al conditions the two set s of definitions ar e equivalent.W e also 'assume that the anode and cathode st re am s a r e constrained to flowin the cr os s flow direction (.say by a network of para lle l channels at eachelectrode). Denoting the anode flow'd irec tion a s x; the cathode flow dir ecti ona s y, and the channel width a s w, requ irem ent of conserv ation .of ma ss l ead sto the equations

d h 1- - (anode)dx - 0 2 F (31)% . .2 ..1 . jw& = - 2 F (cathode)... . . dy -O

applicable at each point on the electrode surf aces, As mentioned ea rl ie rwe m ake the following ad ditiona l a s sumptions:


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1. The cel l potential V i s constant at al l points in the cell.2. The overpotential r\ i s l ine a r in c u rre n t. We c an then define a

total effective impedance z incorporating ohmic loss es andtotelectrode overpotentials .

3. The composition i s uni form over the channel depth. Th is :isapproximately valid fo r sufficiently nar row channels, . wherespe cie s .difuse ac ro ss the chanriel within a few ch annel widthsdownstream of their appearance a t the electrode surface.

The mole fract ion s X. in Eq. ( 9 ) can now be writ ten in te r m s of conversions1

and 1-1. Solving Eq. ( 9 ) fo r curren t j , where we have substituted-T + jzoh - jztot, give s

RT.i(x,p) = ztotwhere

. ., . - -. ... - , --.. - , --Eqs. ( l l ) , (12) and (13) can be solved si mu ltaneo usly for the unknowns j, A, y.A "marching forward" ,..,-.umerical in tegration scheme is appropria te , us ing


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/' jdy 1~ ( x , Y ) = 0 ' o2Ffico 2with the boundary conditions:

We have so fa r wri tten and debugged a computer code that compu testhe cu rr en t density distribution fro m the above equations fo r a given celltel-rlperitture distri butio n. Results fro m an i l lustra t ive computation, ca rr ie dout for an isoth erm al cel l , a r e pre se nt ed in Tab le 111, f or the followingconditions:.

TABLE 11 ,CE LL CONDITIONS FOR

CURRENT DENSITY DISTRIBUTION CALCULATIONT

Inlet Ga s Comp osition

Inlet Molar F ~ O W ate'$..-...(moles sec)

0 2C e ll V o lt a ge = 0 ,7 5 V , T = 9 2 0 K , P = a t m , z = 1 . 0 2 5 o h . m - c m . , ' . .tot -

Anode Stream'64% H2, 10% H 206% GO2, 20% GO .

- 45 x 10

Cathode Stream20% GO 2, 20% 0 2 ,

. 60% N2

1 . 5 ~o m 3

f


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TABLE IIIILLUSTRATIVE CURRENT DENSITY DISTRIBUTION

. . .. . . ..! .. . CURRENT DENSITY, AW/CM**2

0.331 0.301 0.281 0.2660 . 3 2 9 0.300 0.210 0.2650.327 0.298 0.279 0.2648 . 3 2 5 . 0.237 0.277 0.2620.323 '0.295 0.275 0.2610.320 0.293 0.274 0.2590.318 0.291 0.272. 6.2580.315 0.280 8.270' 0.2568.312 0.286 0.263 0.25,,10.305 0.333 0.265. 0.252LGCRL ANODE CONVERSIONS

, 0.809 0.054 0.102 0.1486.86(1 6.055 0.1C2 0.14'0.058 . 6.853 0.10! 0.1469.000 6 .053 8.1E1 0.146a.ee0 c . 052 o . 1 ~ .1450.800 C.C52 o . R F 4 0.1440.000 6.L51 6.095 0.1439.600 0 . < 5 1 0.035 0.141U.7JCb j j . ( j5 ; . ' 0 . fc:a0.663 E .C50 0.896 0.135

L C i CG i CGTr-OLil- : CfjI4VEF':S Itil.ISD.266 O . U3 0.E183 %.BE03 .C57 O . 552 B.043 0.0460.114 C.184 0 . 0 3 7 9.092e . 171 G . 155 u. 145. E!.137p.22; o . - i j? . u. 132 o . 182'0 .283 0.241 0 . 2 2 80,338 . C.28Z. 0 . 2 7 20,393 e .250 C . 335 0.3170.447 0.488 2 . 3 s : e . 3 6 10 . 50 1 6. ;"j ;.;.;.?.:: G, . :;IS

0 . 2 5 40 . 2 5 20.2510 . 2 5 00.2430.2470.246[: . 3 -10 ::'jC.241

0.1310 . 19130.1690.188l).G7B.lE6? I . l t lJ3.1838.181t j . lag0.0068.244E . 0579.131i;. 743 . 2 ; 7!, -: , -r: . .,a3r4 "* . 2 Z0 .2440. I(26

The cur ren t densi ty is seen to be larg est a t anode and cathode inlet2(331 rnA/cm ) , , wh ere the g as s t re am s a r e r iche st in fuel and oxidant con-

2tent, and s ma lle st at outlet (202 rnA/cm ) due to depletion of the g as st re am s.Note that cu rr en t density falls off mo re sharp ly in the cathode flow directio nalong the anode inlet (770)han along the anode outlet YO), due to the tendencyof the fr es h anode gas' to deplete the oxidant m or e readily than the pa rtiallyutilized anode ga s at outlet. The low util izations fo r this c as e (A = 37%,avg= 39%) could be inc rea se d by reducing the gas flow rat es , the reb y'avg,boosting ov eral l cel l efficiency.


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2. With CO UtilizationIr1 order to account for the simul taneou s util izat ion of CO and H we ,2

note the fo llowing points. Under conditions typical of oper atin g mo ltencarbonate cells , dir ect electr ochem ical rea ction of CO 'seems to be quiteslow. On the other hand, wa ter ga s shif t equilibr ium

does see m to be pre ser ved throughout the anode gas str eam , probably viarapid heterogeneous rea'ctions on exposed nickel anode surfaces. Thus COutilization occ urs indirectly via the shift reaction as HZ i s consumed e lectro-chemically. Whether this sequence of events actually does occur i s im-m at er ia l for the pre sent calculation provided that the shift equilibrium issomehow m aintained.

We use ou r e ar li er definition of 1-12 utilization and 'provide a sim ila rdefinition for the net CO utilizati on p:

The total fuel utilization, on a mol ar ba sis , is the sum of X and p.Let v be the frac tio n of cu rr en t contributed by CO utilization.. Then,

from m as s balance considera tions , we f ind

By adding thes e two equations together and substituting fo r fi in t e r ms ofp and X we obtain . ..


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This equation and the e ar l i er equ atio nfo r the oxidant uti l izat ion p must besimultaneously integrated to d etermine p , X and p at al l points on the ele ctrod esurfa ce. As befo re, th is integration re qu ire s knowledge of ' the dependence ofj ,on the local gas com position and, hence, on p, and p. In the remainder ofthi s s ectio n we simplify the pro ble m posed by E.q. (15) by relating p andthrough the water gas shift equilibrium.

F o r that equilibrium, in te rm s of mole frac tions X, we have accordingto Eq. (14) :

Fr om stoichiometric considerat ions we also have

and

3 'Now we introduce the volume flow rate rp (c m /sec) , the to ta l (cons tant)Ap r e s s u r e P, tempera ture T and gas constant R in the form'

Then, the mole frac tion s Xi call 'be exp res sed a s


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where

.When Eqs. (17) ar e. substituted into Eq. (16), a l i t t le 'algebra p roduces

0 . .where K i s evaluated a t the inlet temperatu-re. Equation (18) can 'be udid toSeliminate p fr o m Eq. (15), leaving i t in a fo rm suitable for integration.

In the ver y near future, we plan to determine c urr en t density dis tri-bution on th eb as is of Eqs. ( 6 ) and (15) in ord er to asc er tain the effects ofshift equil ibrium on ce ll performance.

B. Temperature DistributionThe in itial weeks of this effort have. been spent identifying the p re-...

dominant heat transfe r me chanism s that occur in fuel cells, with part icul arattention given to modeling the heat tra ns fe r problem without undue complexity.. .. . . . - *-- ---


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We ass um e that a ll of the heat 4 s e e Eq. (10)) l iberated in the cell i sca rr ie d off ' by ther ma l convection in the gas str eam s, that the gas st re am s. ,a r e constraine d to flow in na rro w channels, that the anode and cathode ar elocally at the same tem pera ture , and that in-plane heat conduction can beneglected (fo r the presen t). Never theless, a prec ise calculation of the celltem pera ture rem ain s a formidable task, due pr im ari ly to the following effects:

1. Gas ~ o ~ ~ o s i t i o nnd tempe rature , hence fluid pr ope rt i is , varyboth a cr o ss the depth of the channel and in the st rea mw isedirection.

2. The ma ss and volume flow ra te s vary along the st ream wis edirection for ea ch electrode.

: . The rat e of heat addition va ri es along'the st ream wis e direction.4. Complex flow geom etri e s ar e possible, e. g. baffling and chan nels

of varying cr os s- section may be present. . .Under the se circu mst ance s, a detailed calculation of the tem per atu re

profiles ac ro ss the gas st re am s would req uire a finite-difference solution ofthe governing differential equations, involving signific ant computational com-plexity. . Fortunately, we a r e only interes ted in the e lectrode temp erat ures ' 'and mean gas tem pera ture s, which suggests the use of a "heat exchanger"type anal ysis . Thi s off ers consider able simplification, and can be expectedto afford the desi red degr ee of accu racy for the quantities of interest. Thi stype of an alys is i s actually exact in the limiting ca se of co nstant propert y,fully developed flow with a constant ra te of heat addition. F or the pr ese ntcas e of low Reynolds number flow, changes in fluid pr op er ti es and the ra te ofheat addition occ ur sufficiently slowly to justify the a s surnption of lllocallyfully developed" flow 5n .which veloc ity and tempera ture re lax in s ' ta n t an eo ~ s l ~to fully 'developed profiles at each stream wise station.


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W e mu st a lso account for :hca t f luxes associated with m as s addit ion(anode) o r subtract ion (cathode) a t 'the ga s /e lectrode interfa ce, a s speciesen te r ing or leav ing the ga s s t r ea m s c ar ry with them a char ac te r i s t ic en thalpy .

I t is im portant to re al ize that the interface i s taken to l ie in inf ini te-, s im al d i s tance ins ide the gas s t ream , so that the "electrode" encom passesthe chemical react ion zone (ga s phase react ions a r e presume d not to occurin the bulk str ea m ). The model .need not be sensi t ive to whether heat re le as e/absorpt ion occ urs a t the anode or cathode separately; because of suffic ientlyhigh ti l e th er m al conductivity account i s taken only of the total heat r el ea sepe r unit a re a evolved within the ele ctrod e-ti le sandwich.

To summarize, i f we neglect in-pla ne. heat conduction, the lo ca l heatre leas e a t each poin t in the ce l l i s t ransfe r red to the gas s t r ea m s as .a con-ductive flux by vir tue of a tem per atu re gra die nt. at the gas/e ' lectrode in ter-face, or a s a convective f lux due to m a s s t ran sf er a t the boundaries . Thema them atica l formulation of these concepts, the quantitative justification ofour working assum ption s, and the development of a com puter code to imple-ment them a r e scheduled to receiv e im mediate a t tent ion in the next s tage ofour inodeling efforts.


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REFERENCES

S. Srin ivas an and H. D. Hurwitz, l lT he or y of a Th in F il m Model ofPo ro us Gas-Diffusion Elec t rod est1 , Elect roc himica Acta 12, 495 (1967).-S. Srinivasan, H. D. Hu rwitz and J. OfM. Bock ris , "FundamentalEquations of Ele ctr oc he mi ca l Kinetics of Poro us. Gas-DiffusionElect rodesI1, J. Chem. Phys. -6, 3108 (1967).'D. T; Wasan, T. Schmidt and B. S. Baker , "Ma ss Tr an sf er in Fu elC e l l s : P a r t I. Models for Po ro us Elect ro desH, Chem. Eng. Pro g.Symp. Ser. -3(77), 16 (1967).R, Chas. Burshtein, V. S. Ma rkin , A. G. Psh enic hni kov , V. A.Chismadgev and Y. G. Chirkov, "The Relat ionship Between Stru ctur eand Elec t rochemica l ~ r o ~ e r t i e sf Por ou s Gas E lec t rodes l l , E lec t ro-chim.' Acta 9, 773 (1964).-K , M i c k a , l l ~ h e o r yf Po la r i za t ion of Poro us E lec trodes1I, i n "Fue lCel l Systems", Advances in Che mis t ry-7, 73, h e r i c a n C he mic alSociety, Washington, D. C. (1965).J. Newman and W. Tiedemann, l lPorous Elec t rode Theory wi thB a tt e ry A p p l i c a t i o i ~ s ~ ~ ,IChE Journal-1, 25 ( 1975).Inst itute of Gas Technology, Fue l Cel l Res ea rc h on Second-Generat ionMolten- Carbonate Systems, Pr oj ec t 8984 Quarter ly Status Report ,Janua ry- Marc h 1977.Inst i tute of Gas. Technology, Fu el Ce ll Re se ar ch on Second Ge nerat ionMolten-Carbonate System s, Vol. 11, Pr oje c t 8984 F ina l S tatus Repor t ,July- Septem ber 1977.Institute of Gas Technology, Fue l Cel l Res ear ch on Second-Genera t ionMol ten-Carbonate Sys tems, p ro jec t 9 105 Quarter ly s t a tu s ~ e ~ o r t ,October-December 1977. .


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D i s tr i bu t io n L i s t f o r Q u a r t e r l y R e p o r tUnder Contr act #ET-78-C-03-2083

D r . Jo h n A c k e r m a n M r . E. Gi l l i sAr gonne Nat ional Labor a to r y E l e c t r i c P o w er R e s e a r c h I ns t i tu t eArgonne, I L 60439 , 3412 Hillview Avenue

P. 0 . Box 10412Dr . A. Boruc ka Pa lo Al to , CA 94304Bor ucka Re se a r ch Company60 Chestnut S t r eet Dr . Theod ore R. Bec kLivingston , NJ 07039 Ele ct r och emi cal Technology Cor p., 3 9 3 5 Le a r y W a y , N .W .P r o f e s s o r E. Yeag er Seatt le, WA 98107C a s e W e s t e r n R e s e r v eE l e c t r o c h e m i s t r y La b o r a t o r y D r . B e r n a r d S. B a k e rUniver s i ty Ci r c l e E n e r g y R e s e a r c h C o r po r at i o nCleveland, OH 44106 3 Gre at Pasture RoadDanbury , CT 06810Mr . J. HuffDRDME- E C Mr . J. W. Ha rr is onD e p a r t me n t of t h e A r m y G e n e r a l E l e c t r i c C o mp a ny D E C PU. S. A rm y Mobi l ity Equ ipment 5 0 F o r d h a m R o adRe se ar ch and Development C ommand Wilmington, MA 01887F or t Belv oir , VA 22060 Dr . D. Chat ter j i , ManagerM r . G a r y E. V oe l ke r E l e c t r o c h e m i s t r y B r a n c hDepar tment of Ene rgy Gener a1 E l e c t ri c R e sear ch &20 Massachu se t t s Ave . , NW Development CenterWashington, D. C. 20545 P. 0. Box 8

Bldg. K-1, Ro om 2A54Mr. Ma r t in Zlotn ick Schenectady, NY 12301Depar tment of En er gy20 Massach use t t s Ave . , NW Dr. J. G i n e r , P r e s i d e n tWashington , D. C. 20545 Gi ne r , Inc.14 Spr ing St re etM r . Arnold P. Fic ket t Waltham, MA 02154El e c t r i c P o w e r R e se a r c h I n s t i tu t e34 12 Hillview Airenue ~r . Leona rd M ar anow sk iP. 0. Box 10412 Inst i tu te of G as TechnologyPa lo Alto, CA 94304 .. 3424 S. State St re et..\._. Chicago, IL 60616


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Pag e 2 .of Dis t r ibu t ion Lis t for Qua r te r l y Repor t

Mr . Ron G.uidotti . Mr . Jo sep h M. KingMontana En erg y and MHD R'esear ch P r o g r a m M a n ag e rand Development Insti tute , . Inc. Uni ted Technologies Corpora t ionP. 0. Box 3809 P. 0. Box 109Butte , MT 59701 South Wind sor, CT .06074Dr. J. J. R a s m u s s e nMontana Ene rgy and MHD R e se ar ch

and Development Ins ti tute , Inc.P. 0. Box 3809Butte, MT 59701Dr. D. L. JohnsonTechnologica l Ins t i tuteDepar tment of Mate r ia l s Sc ienceNorthwe s t er n Univer s i tyEvanston, IL 60201M.r. Les l ie M. F e r r i sOak Ridge Nat ional La bo ra t or yP. 0. Box XOak Ridge, T N 37830Dr . Kur t WrayPhy s ica l Sc iences Inc.30 Co mm erc e WayWoburn, M A 01801Mr. Leona rd Nani sS tanford Re sea rc h Ins t i tu te333 Ravenswood Avenu eMenlo P ar k, CA 94025M.r. R. D. WeaverStanford Re sea rch Ins t i tu te .333 Ravenswood Avenue 'Menlo P ar k , CA 94205\.,..


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