Detailed Multi-dimensional Modeling

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1 DOI: 0.002/fuce Detaled Mult-dmensonal Modelng of Drect Internal Reformng Sold Oxde Fuel Cells K. Tserons, I.S. Fragkopoulos,I.Bons, C. Theodoropoulos * School of Chemcal Engneerng and Analytcal Scence, Unversty of Manchester, Manchester M3 9PL, UK Receved August, 205; accepted Aprl 4, 206; publshed onlne June, 206 Abstract Fuel flexblty s a sgnfcant advantage of sold oxde fuel cells (SOFCs) and can be attrbuted to ther hgh operatng temperature. Here we consder a drect nternal reformng sold oxde fuel cell setup n whch a separate fuel reformer s not requred. We construct a multdmensonal, detaled model of a planar sold oxde fuel cell, where mass transport n the fuel channel s modeled usng the Stefan-Maxwell model, whereas the mass transport wthn the porous electrodes s smulated usng the Dusty-Gas model. The resultng hghly nonlnear model s bult nto COMSOL Multphyscs, a commercal computatonal flud dynamcs software, and s valdated aganst expermental data from the lterature. A number of parametrc studes s performed to obtan nsghts on the drect nternal reformng sold oxde fuel cell system behavor and effcency, to ad the desgn procedure. It s shown that nternal reformng results n temperature drop close to the nlet and that the drect nternal reformng sold oxde fuel cell performance can be enhanced by ncreasng the operatng temperature. It s also observed that decreases n the nlet temperature result n smoother temperature profles and n the formaton of reduced thermal gradents. Furthermore, the drect nternal reformng sold oxde fuel cell performance was found to be affected by the thckness of the electrochemcally-actve anode catalyst layer, although not always substantally, due to the counter-balancng behavor of the actvaton and ohmc overpotentals. Keywords: DIR-SOFC, Dusty-Gas/Stefan-Maxwell Couplng, Multdmensonal Modelng, Parallel H 2 and CO Electro-oxdaton Introducton A method for obtanng the hydrogen requred for the sold oxde fuel cell (SOFC) operaton, s through catalytc steam reformng (CSR) of methane or natural gas. Natural gas s one of the most wdely avalable lght hydrocarbon fuels. It s a fuel mxture typcally consstng of methane and smaller amounts of hgher hydrocarbons, such as ethane, propane, butane, and of ntrogen. CSR s typcally carred out between 700 C and 900 C [], therefore, t s compatble wth the hgh SOFC operatng temperatures. Addtonally, SOFC anodes typcally comprse of metal catalysts that promote the CSR reacton. The general reformng reacton of a hydrocarbon can be wrtten as: C n H m O p þ ðn pþh 2 O f nco þ ðn p þ m=2þh 2 ; DH o > 0 () Methane, whch s the most common fuel n nternal reformng SOFC (IR-SOFC) systems, due to the hgher effcency that can be acheved from ts use [2, 3], reacts wth steam to produce hydrogen (H 2 ) and carbon monoxde (CO) n the methane-steam reformng (MSR) reacton: CH 4 þ H 2 O f CO þ 3H 2 ; DH o ð073kþ ¼ 242 kj mol (2) The MSR reacton s thermodynamcally favored at hgh temperatures and low pressures, snce the forward reacton results n volume ncrease []. At the same tme, CO further reacts wth steam (H 2 O) to generate addtonal hydrogen and carbon doxde (CO 2 ) n the water-gas shft (WGS) reacton: CO þ H 2 O $ CO 2 þ H 2 ; DH o ð073kþ ¼ 38:6 kj mol (3) As can be seen from reactons (2) and (3), MSR s a hghly endothermc reacton, whereas the WGS, as well as the electro- [ * ] Correspondng author, k.theodoropoulos@manchester.ac.uk 294 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

2 chemcal oxdaton, are exothermc reactons [4, 5]. In our model, n addton to the aforementoned reactons, the smultaneous electrochemcal oxdaton of H 2 and CO at the anode s taken nto account as can be seen n Fgure. Therefore, the electrochemcal reactons at the actve layers of the anode and cathode electrodes are as follows: Anode: H 2 þ O 2 f H 2 O þ 2e ; DS A H 2 ¼ 25 J mol K (4) Anode: CO þ O 2 f CO 2 þ 2e ; DS A CO ¼ 45 J mol K (5) Cathode: =2 O 2 þ 2e f O 2 ; DS C ¼ 55 J mol K (6) Based on these half-cell reactons, the overall electrochemcal reactons are the followng: H 2 þ 2 O 2 $ H 2 O (7) CO þ 2 O 2 $ CO 2 (8) The entropy change s evaluated separately for each electrochemcal half-cell reacton [6, 7], snce the overall reacton of a SOFC does not provde nformaton on how the heat s dstrbuted among each electrode [8]. Usng ths approach the temperature dstrbuton n the sold parts of the SOFC can be evaluated more accurately. Two operatng parameters that are commonly used to characterze reformng systems are the steam to carbon rato (S/C) and the stochometrc rato (SR) [9]. S/C s defned as the mole fracton of steam n the anode fuel to all combustble speces mole fractons and s gven by: S=C ¼ _n H 2 O _n C (9) SR s defned as the amount of oxygen n the nlet of the system over the amount of oxygen needed to acheve complete combuston and s gven by: SR ¼ Fg. Schematc representaton of the basc operaton prncples of a DIR-SOFC. _n O2 _n O2 ; stochometrc (0) A suffcent amount of steam s necessary n the reformer, n order to fulfl the reformng reacton requrements. Prevous studes on methane reformng n SOFCs have shown that a S/C rato hgher than 2 guarantees that carbon formaton does not occur n the SOFC [0]. However, a very hgh S/C rato can prove to be an neffcent choce, as t would dlute the hydrogen n the fuel channel, lowerng the electrcal effcency of the cell. Therefore, the S/C rato s usually kept below 3.5 []. Some of the benefts of CSR of hydrocarbons n SOFCs are the producton of hydrogen n stu, the hgher overall effcences and the smplcty of the resultng system n the case of nternal reformng (IR) due to the combned reformng and electrochemcal processes [2]. One of the man downsdes of MSR n IR-SOFCs s the formaton of steep temperature gradents wthn the cell, due to the hghly endothermc reformng reacton, whch results n thermal stresses that can lead to structural falures. Such gradents are formed manly at the nlet of the fuel channel where the hghly endothermc MSR reacton rate s large, due to the hgher methane concentraton, resultng n much lower temperatures than the feed temperature. As methane flows n the fuel channel, methane concentraton decreases, whle hydrogen concentraton respectvely ncreases, causng the exothermc electrochemcal (oxdaton) reactons to domnate the MSR reacton, resultng n a temperature ncrease downstream n the cell. Another lmtaton of MSR s that, despte ther relatvely hgh tolerance n carbon, the anode electrode catalysts can suffer from carbon deposton, whch can result n ts deactvaton n the long term. To allevate the aforementoned lmtatons some degree of upstream fuel processng, ncludng recrculaton, can be used. Natural gas can be fed n a pre-reformng reactor along wth a fracton of the outlet anode stream that s recrculated from the SOFC, n order to produce a fuel stream that s rch n H 2 and CO and dlute n hydrocarbons. The decreased methane (CH 4 ) concentraton means that the S/C rato s ncreased, and the ncreased H 2 and CO concentratons mean that a hgher current densty s observed at the nlet, whch consequently results n the producton of more steam through the electrochemcal reactons. Hgher S/C ratos and current denstes have been found to decrease the rate of carbon formaton [2]. In addton, the generated temperature gradents n the SOFC become smaller, due to the ncrease n the relatve rate FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 295

3 of the exothermc electrochemcal reactons versus the rate of the endothermc reformng reacton. In ths work, the Dusty-Gas model (DGM) s employed for the smulaton of mass transport n the porous anode and cathode electrodes of a drect nternal reformng sold oxde fuel cell (DIR-SOFC) system. Several studes have been conducted to revew and compare the mass transport approaches whch can be followed for accurate SOFC modelng and they concur that DGM can be deemed accurate and dependable [3 6]. Although the merts of DGM are hghly held of n the SOFC communty, to the best of our knowledge only a handful of works exst n the lterature employng DGM for IR-SOFC n button [7] and planar confguratons [8 20]. The closest work to the one dscussed here s Janardhanan and Deutschmann s [8] who smulate a planar Membrane Electrode Assembly (MEA) SOFC wth an teratve, fxed-pont algorthm. Here, although a MEA s not appled, the governng partal dfferental equatons (PDEs) are fully coupled and solved smultaneously usng a commercal computatonal flud dynamcs (CFD) package, yeldng a computatonally effcent and flexble model. In [8] charge transfer takes place only at the electrode-electrolyte nterface consderng the trple phase boundares (TPBs) as -D boundares, whle here the TPBs are taken nto consderaton as fnte 2-D zones n the anode electrode. Furthermore, ths study takes nto account the electrochemcal oxdaton of both H 2 and CO at the TPB of the anode. Followng the approach we ntroduced for the frst tme n [2] for anode SOFC smulatons, and then extended on entre system SOFC smulatons n [22], here the DGM s coupled wth the Stefan-Maxwell model (SMM) achevng hgher accuracy for mass transport smulatons n DIR-SOFCs. Such a detaled model can be used for the nvestgaton of the effect of varous parameters on the SOFC performance, targetng robust desgn and control of DIR-SOFC systems. Such detaled models can be used drectly n conjuncton wth model reducton technques n the context of optmsaton [23, 24] and control [25, 26], achevng both accurate and robust desgn and operaton. Ths paper s organsed as follows: the varous approaches to SOFC fuel processng are presented n Secton 2; n Secton 3, the developed model s descrbed n terms of reacton knetcs, governng PDEs and boundary condtons for the phenomena taken nto consderaton (mass, energy and charge transport) and doman consdered. In Secton 4, the constructed model s valdated aganst expermental results avalable n lterature. Furthermore, results obtaned from base case smulatons are presented and a parametrc nvestgaton s undertaken. In Secton 5, the conclusons are dscussed. 2 Approaches to SOFC Fuel Processng SOFC systems typcally obtan the hydrogen requred for ther operaton from a hydrocarbon-based fuel source. Alcohols, such as methanol or ethanol, and hydrocarbons, such as natural gas, are usually reformed nto a hydrogen-rch synthess gas by varous methods. The most representatve methods for ths purpose are CSR, partal oxdaton (POX) and autothermal reformng (ATR) [3, 27]. The fuel can be converted and reformed externally to the SOFC stack by external reformng (ER) or nternally n the SOFC anode compartment by IR. ER requres an external heat source, such as a burner or hot waste gas and a fxed bed reactor. IR utlzes the heat release from the fuel electrochemcal oxdaton reactons by provdng a convenent and effcent confguraton for energy transfer between heat source and heat snk, smultaneously lowerng the ar coolng requrements of the SOFC [3]. There are two man approaches to IR wthn a SOFC, ndrect nternal reformng (IIR) and drect nternal reformng (DIR). In the former approach, the reformer secton s physcally separated from the SOFC, but n close thermal contact wth the anode, n order to make use of the SOFC stack released heat. In the latter approach, the hydrocarbon fuel mxture s fed drectly nto the anode fuel channel and the reformng reacton occurs there [0]. The two man advantages of CSR over POX and ATR processes are that t produces a hgh composton hydrogen mxture wthout dluton of the product stream and that t operates wth a hgh fuel converson effcency (85% 95%) [3]. In contrast to the above, POX offers compactness, fast start-up and rapd dynamc response, but sacrfces fuel converson effcency. ATR, on the other hand, shares some of the features of CSR and POX technologes. The major dfference between CSR, POX and ATR processes s the mechansm for provdng the thermal energy requred for the endothermc reformng reactons [3]. In ths work, a DIR approach s gong to be used for the developed IR-SOFC model. 2. Drect Internal Reformng In DIR, the reformng reactons take place wthn the anode of the SOFC. Ths s feasble, snce the hgh operatng temperatures of SOFCs n combnaton wth the nckel content of the anode electrodes provde suffcent actvty for the MSR (2) and WGS (3) reactons [28]. In DIR-SOFC systems, there s very good system ntegraton, because the steam produced from the hydrogen oxdaton electrochemcal reacton can be used drectly n the reformng reacton. Addtonally, the heat released n the DIR-SOFC can provde the heat for the MSR reacton. The need for SOFC coolng s usually satsfed by flowng excess ar through the cathode ar channel, but n DIR-SOFCs ths s not necessary, because of the endothermc reformng reacton [0, 28]. Moreover, due to the ongong hydrogen consumpton by the electrochemcal reactons, the equlbrum of the MSR reacton (2) s further shfted to the rght, ncreasng the methane converson and leadng to a more evenly dstrbuted load of hydrogen [0]. DIR has the advantage of elmnatng the requrement for upstream processng wthout usng a separate fuel reformer, as s the case for IIR, thus smplfyng the overall SOFC system desgn, hence mnmsng ts cost, sze and complexty. Another 296 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

4 beneft of DIR over IIR s that n DIR the converson of CH 4 to H 2 s promoted to a much larger extent than n IIR [0]. A DIR related drawback, as already mentoned, s the carbon formaton on the anode electrode that not only deactvates the catalytc actve stes, but can also damage the cell through delamnaton or nckel dustng leadng to reduced SOFC performance [0, 29]. Ths problem can be allevated by ncreasng the S/C rato of the nlet fuel mxture to an amount greater than the stochometrc requrement of reacton (2). However, the amount of steam necessary for carbon-free operaton s not an easy choce, as n addton to the suppresson of coke formaton, S/C rato can also affect the equlbrum yeld of hydrogen. As the S/C rato s ncreased, the hydrogen yeld decreases. Furthermore, hgh S/C-ratos can cause nckel oxdaton, whch can lead to severe damagng of the cell through the alteraton of the lattce structure [30, 3]. Increasng extensvely the S/C rato can also have a negatve effect n the overall system effcency, snce the energy requrements for steam generaton are also ncreased. Advanced anode materals that allow DIR at low steam/carbon ratos can offer an alternatve approach wth sgnfcant benefts [28]. Another negatve aspect, mostly affectng the hgh temperature DIR-SOFCs, s the strong coolng effect caused by the hghly endothermc MSR reacton. It has been demonstrated that almost all CH 4 s reformed wthn a small dstance from the anode nlet [0, 28]. Ths can result n very steep temperature gradents n the SOFC sold structure and lead to large thermal stresses that can potentally result n system falure due to crack formatons. In order to reduce ths effect, an opton s to use partal external pre-reformng, whch reforms the hgher hydrocarbons, possbly present n the fuel mxture, and some of the methane, reducng that way cokng formaton phenomena [28]. Another approach to ths shortcomng s the lower temperature operaton, snce ths would naturally reduce the endothermc MSR reacton rate. The latter s one of the reasons behnd the ncreased research and development of ntermedate temperature IR-SOFCs [32, 33]. The overall heat producton from the electrochemcal and water gas shft reactons s about twce the amount of heat consumed by the reformng reacton [34]. In order to regulate the temperature of the SOFC, the extra heat generated s usually removed by the flow of excess ar n the cathode ar channel. An example of a commercal applcaton of the DIR- SOFC technology s the Semens-Westnghouse SOFC prototype [35]. The reformng confguratons that were dscussed n the prevous secton (IIR, DIR) can be appled to all the processes presented (CSR, POX, ATR), but ths work focuses on the study of a CSR DIR-SOFC system. The mathematcal model of the processes takng place n the SOFC fuel and ar channels, dffuson and reacton layers, and electrolyte s formulated by applyng DGM-SMM based mass transport, the momentum conservaton (Naver-Stokes equaton), the energy conservaton equaton and the charge conservaton equaton, wth the approprate subdoman and boundary source terms for the chemcal and electrochemcal reactons. Addtonally, all the correspondng boundary condtons are also determned, formng a set of coupled nonlnear PDEs that s solved wthn a suffcently fne mesh, n order to produce mesh ndependent solutons, wth the use of the fnte element method (FEM) software package, COMSOL Multphyscs [36]. The model can take as nput a set of desgn and operatng physcal and chemcal parameters and s able to generate performance polarzaton curves, operatng varables dstrbutons and other measurable output parameters. The reacton zone layers, where the anode and cathode electrochemcal reactons occur at the localsed TPB, causng fuel and oxdant to be converted to electrcal current, heat and steam, are usually treated as mathematcal boundares. In ths work, however, the electrochemcally actve reacton zones wll be treated as fnte subdomans, as can be seen n Fgure 2, n order to smulate more accurately the behavor of composte electrodes, whch are descrbed next. Another unque feature of ths model s that t takes nto account the electrochemcal oxdaton of both H 2 and CO and the contrbuton of both reactons to the current generaton. 3. Mxed Ionc-electronc Composte Electrodes Mxed onc-electronc composte electrodes have been proposed as alternatves to the smple metal cermets [37, 38]. Ther use n SOFCs has many potental benefts and can 3 DIR-SOFC Model Descrpton Fg. 2 Schematc of the computatonal doman of a cross secton of a planar SOFC. FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 297

5 mprove ther performance sgnfcantly [34, 39, 40]. The composte electrodes are porous metal-loaded ceramcs, whch have the property of beng mxed electronc and onc conductors. The most commonly used materals for ths applcaton are Nckel/Yttra-Stablzed Zrcona (N/YSZ) for the anode and Lanthanum Strontum Manganate/Yttra-Stablzed Zrcona (LSM/YSZ) for the cathode. Ther prmary advantage over the tradtonal pure electronc conductng electrodes s that the reacton zone layer for the electrochemcal reacton s not lmted only on the TPB, but t extends nto the ceramc conductor some dstance from the electrode/electrolyte nterface. Subsequently, the actvaton overpotental losses are sgnfcantly reduced snce the electrochemcal reactons are better promoted [4]. Another sgnfcant advantage of composte electrodes over tradtonal ones s the reducton of thermal stress effects, as a result of the fact that the value of ther expanson coeffcent s much closer to the correspondng value of the electrolyte. Ths s mportant snce the effect of the strans s manly caused by the dfferent thermal expansons and t s a qute common problem for SOFCs [40]. 3.2 DIR-SOFC Model Knetcs Despte the fact that catalytc steam reformng has been studed by many researchers and there are many publcatons provdng nformaton related to ts knetcs, there are only a few dealng specfcally wth the reformng knetcs n SOFC anodes. In many publcatons [, ], authors clam that reformng knetcs are slow when compared wth both shftng and hydrogen electrochemcal oxdaton, thus, under certan operatng condtons, t s safe to assume that equlbrum s not reached. In the current study, for the MSR reacton (2), the followng reacton rate equaton wll be used [0, 28]: r MSR ¼ k MSR p CH4 exp EA MSR RT () where E A MSR s the MSR reacton actvaton energy and k MSR s the MSR reacton pre-exponental factor. In studes about reformng n SOFCs, several authors clam that the MSR reacton s too slow to assume that t can reach equlbrum, but on the other hand ths approach s sutable for descrbng the knetcs of the WGS reacton [0]. Therefore, the knetc equaton that s gong to be used for the WGS reacton (3) s the followng [0, 28]: r WGS ¼ k WGS p CO p CO 2 p H2 =p CO p H2 O K eq! (2) where k WGS s the WGS reacton rate constant and K eq s the WGS reacton equlbrum constant. The unts of the reacton rates r MSR and r WGS gven by Eqs. (2) and (3) are [mol s m 2 ]. The nternal reformng knetcs used n ths study are consdered typcal for N cermet anodes [0, 28], and they have also been used n many SOFC systems [42 46]. 3.3 DIR-SOFC Model Assumptons The man assumptons for ths sngle cell DIR-SOFC model are gathered below: () All the water throughout the fuel channel and the porous electrode s consdered to reman n the gas phase. Consequently, the model wll be a sngle phase model and phase change as well as two phase transport wll not be consdered. () The gas mxtures are assumed to behave deally, thus the deal gas law s appled n ths model. () The electrochemcal reactons are assumed to be nstantaneous and to take place at the electrochemcally-actve catalyst layer subdomans (see Fgure 2). (v) The porous electrodes are assumed to be sotropc and macro-homogeneous and the electrolyte s assumed to be mpermeable to mass transport [47, 48]. (v) The densty and the heat capacty of the gas mxtures are consdered constant and temperature ndependent throughout the SOFC. (v) The thermal conductvtes of the sold structures are assumed temperature ndependent. (v) The electrcal and onc conductvtes of the sold structures are descrbed by equatons that are functons of the temperature dstrbuton. (v) Heat transfer due to radaton s not taken nto account. (x) Coke formaton phenomena are not taken nto consderaton. (x) Thermal stress related phenomena are not consdered n ths work. 3.4 Mass, Energy, Momentum and Charge Transport Equatons The governng coupled equatons and ther correspondng boundary condtons of the modelng approach developed to smulate the assocated mass, energy, momentum and charge conservaton phenomena that occur n a DIR-SOFC wll be descrbed next. In Fgure 3 the conservaton prncples assocated wth each subdoman of the computatonal doman are depcted Mass Transport Equatons The equaton of contnuty for component, assumng an deal gas behavor, can be wrtten as follows [49,50]: e p RT t ¼ ÑN þ R þ R el (3) where e s the porosty (equal to unty for a non-porous gas phase), N s the total molecular flux of speces, R and R el are the volumetrc (chemcal and electrochemcal, respectvely) producton/consumpton rates of speces. Smlar to our prevous work [2, 22], for the smulaton of mass transport n the fuel and ar channels the SMM wll be employed. The multcomponent SMM s gven by the followng vector matrx equaton for deal gases [5]: 298 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

6 dffuson and the Darcy s vscous flux, whch s caused due to the exstence of a total pressure gradent. The multdmensonal DGM s gven by the followng vector matrx equaton [50, 53, 54]: A DGM ðp ÞN ¼ B DGM ðp ÞÑp; ¼ ; :::; n (9) where N and Ñp are the n-dmensonal vectors of component total molar fluxes and partal pressures, respectvely, for all components, A DGM ðp Þ s a n n matrx whose elements are gven by: A DGM j ¼ p P D ;j (20) A DGM ¼ Xn j¼ j p j P D ;j þ D K (2) Fg. 3 Schematc of the conservaton equatons and models used n each subdoman of the SOFC computatonal doman. RT Ñp ¼ ASMM ðp ÞJ ; ¼ ; :::; n (4) where J s the vector of dffuson fluxes, Ñp the vector of the gradents of the partal pressures p and A SMM ðp Þ s a ðn Þ ðn Þmatrx whose elements are gven by: A SMM A SMM j 0 ¼ p B ¼ p P X n j¼ j D ;j! D ;j D ;n p j p þ D ;n C A (5) (6) The addtonal equaton needed n order to have a fully defned system of n equatons wth n unknowns s the followng [5, 52]: X n ¼ J ¼ 0 (7) The total molecular flux N for the multdmensonal SMM s gven by the followng equaton, where the convectve flux term has been added to the dffusve flux term J : N ¼ J þ c U (8) where c s the molar concentraton of speces and U s the velocty vector. For the smulaton of mass transport n the anode and cathode dffuson and electrochemcally actve catalyst layers, the DGM [53, 54] wll be used takng nto account the Knudsen B DGM ðp Þs a n n matrx whose elements are gven by: B DGM j ¼ B 0 p RT D K m B DGM ¼ RT þ B 0p D K m (22) (23) B 0 n Eq. (22) and Eq. (23), s gven by the Kozeny-Carman equaton [9]: B 0 ¼ e3 2 d P 72tð eþ 2 (24) At the anode porous electrode the MSR and the WGS reactons are takng place. Thus, n subdomans III and IV (see Fgure 4) the reacton source terms R used n Eq. (3) are as follows: R H2 ¼ ð3r MSR þ r WGS Þ=w (25) R H2 O ¼ ð r MSR r WGS Þ=w (26) R CH4 ¼ r MSR =w (27) R CO ¼ ðr MSR r WGS Þ=w (28) R CO2 ¼ r WGS =w (29) where r MSR and r WGS are the MSR and WGS reacton rates gven by Eq. () and Eq. (2), and w s the wdth of each correspondng subdoman. WGS reacton s not catalytcally actvated and t can also be consdered to take place n the fuel FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 299

7 channel. Thus, n subdoman II (see Fgure 4) the reacton source terms R used n Eq. (3) are the ones descrbed n Eqs. (25) to (29) wth r MSR equal to 0. In composte electrodes, the electrochemcal reactons take place at the anode and cathode extended TPBs, represented here by the electrochemcallyactve catalyst layers (subdomans IV and VI). It has been shown that the H 2 electrochemcal oxdaton reacton rate s about tmes and tmes hgher than the CO electrochemcal oxdaton reacton rate at,023 K and,273 K, respectvely [55]. Ths dfference s caused manly due to the larger dffuson resstance of CO than that of H 2 n the porous electrode. Therefore, n subdomans IV and VI (see Fgure 4), the electrochemcal reacton source terms, R el, of Eq. (3) are descrbed by the followng equatons: R el H 2 ¼ IA H 2 2F R el H 2 O ¼þIA H 2 2F R el CO ¼ IA CO 2F R el CO 2 ¼þ IA CO 2F w ACL ¼ 0:7 IA 2F w ACL ¼þ0:7 IA 2F w ACL ¼ 0:3 IA 2F w ACL ¼þ0:3 IA 2F w ACL (30) w ACL (3) w ACL (32) w ACL (33) R el O 2 ¼ IC (34) 4F w CCL where I A and I C are the anode and cathode current denstes computed by Eq. (52) and Eq. (55) presented next n ths work, respectvely Energy Transport Equatons The energy transport equaton s solved for every subdoman of the computatonal doman. The domnatng phenomena n the current model are heat conducton, heat convecton and thermal speces dffuson, whle energy transport by radaton s neglected. The reformng reacton s consdered to take place at the anode electrode (subdomans III and IV) and the shft reacton s assumed to take place at the anode electrode (subdomans III and IV) as well as n the fuel channel (subdoman II), whle the electrochemcal reactons occur n the anode and cathode catalyst layers (subdomans IV and VI, respectvely). The governng equaton descrbng these phenomena s the followng [5]: Pn ¼ t c H ¼ Ñ k eff ÑT þ Xn ¼ H N!þ Q h (35) where N s the total flux vector of speces gven by Eq. (8) for the gas channels and by Eq. (9) for the porous electrodes, c s the molar concentraton of speces, Q h represents the thermal sources term and k eff s the effectve thermal conductvty gven by the followng expresson: k eff ¼ ek g þ ð eþk s (36) where k g s the heat conductvty of the gas phase and k s the heat conductvty of the sold structure. For the electrolyte and the nterconnect e ¼ 0, whle for the gas channels e ¼. H s the partal molar enthalpy of speces gven by the followng equaton [8]: H ¼ D H 0 þ Z T T ref c P dt (37) where c P s the molar heat capacty of speces and DH 0 s the standard enthalpy of formaton of speces. At all the porous subdomans energy transport by convecton s neglgble. Accordngly, n the gas channels conductve energy transport s not taken nto account. Thus, for the fuel and ar channels takng nto account Eq. (8), Eq. (35) takes the followng form: c t c P T t ¼ c tc P U ÑT Xn ¼ J Ñ H þ Q h (38) Fg. 4 Schematc and numberng of the subdomans, boundares and nterfaces of the SOFC computatonal doman. where c t s the total molar concentraton of the gas mxture, c P s the molar heat capacty of the gas mxture, J s the dffusve flux vector as evaluated by the SMM n Eq. (4) and U s the velocty vector. The frst term on the rght hand sde of Eq. (38) descrbes convecton, the second one thermal speces dffuson and the thrd one s the heat sources term. 300 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

8 For the porous electrodes, Eq. (35) takes the followng form: c eff t c eff T P t ¼ k eff Ñ 2 T Xn N ÑH þ Q h (39) ¼ where c eff t s the effectve total molar concentraton of the porous electrode and t s evaluated n the same manner as k eff n Eq. (36). Smlarly, c eff P s the effectve molar heat capacty of the porous electrode. N s the total flux term as evaluated by the DGM n Eq. (9). In order to make these equatons compatble for use wth COMSOL Multphyscs, some of the terms need to make use of specfc propertes nstead of molar. Thus, Eq. (38) and Eq. (39) were expressed as follows: r eff C eff P T t ¼ r Xn eff Ceff P U ÑT J ÑH þ Q h (40) ¼ At the anode fuel channel, as well as at the dffuson and catalyst layers, heat assocated to the MSR and WGS reactons s released. Consequently, n subdomans II, III and IV: Q h ¼ ðr MSR DH MSR þ r WGS DH WGS Þ=w (46) where the reacton rates r and the reacton enthalpes DH are gven by Eq. () and Eq. (2), and Eq. (2) and Eq. (3), respectvely Momentum Transport Equatons The momentum transport equatons are solved for the fuel and ar channel subdomans II and VIII (see Fgure 4). The governng equatons that descrbe the momentum transport are the contnuty and the Naver Stokes equatons [36]: Ñ U ¼ 0 (47) r eff C eff P T t ¼ k eff Ñ 2 T Xn ¼ N Ñ H þ Q h (4) where C eff P s the effectve specfc heat capacty and r eff s the effectve mass densty, whch s evaluated by: r eff ¼ er g þ ð eþr s, wth r g beng the mass densty of the gas phase and r s the mass densty of the sold components. At the anode and cathode dffuson and catalyst layers heat generaton s takng place, due to the rreversble ohmc heatng from the electronc current flow. Thus, n subdomans III, IV, VI and VII: Q h ¼ s eff el ÑV el ÑV el (42) Equvalently, at the electrolyte subdoman V, heat generaton due to the resstance to the onc current flow occurs, therefore: Q h ¼ s o ÑV o ÑV o (43) At the anode and cathode catalyst layers the heat assocated to the change of entropy due to the half-cell electrochemcal reactons that cannot be utlsed as electrcal energy has to be taken nto account. Addtonally, the avalable energy from the open crcut voltage (OCV) that s not transformed nto current, but nstead s released as heat at the anode and cathode catalyst layers, has to be taken nto consderaton [56]. Thus, n subdomans IV and VI:!, Q h ¼ IA H 2 2F TDSA H 2 þ IA CO 2F TDSA CO þ IA h A act w ACL (44) Q h ¼ IC 2F TDSC þ I C h C act w CCL (45) where DS A and DS C are the entropy changes for each of the half-cell electrochemcal reactons (4), (5) and (6) [6,7]. r U t ¼ ru ð Ñ ÞU Ñp Ñ h ÑU þ ð ÑU ÞT þ F (48) where r s the densty of the flud and h ts knematc vscosty Charge Transport Equatons The charge conservaton equatons are solved for two varables, the electronc potental V el at the two electrode dffuson layers subdomans III and VII and at the two catalyst layers subdomans IV and VI, and also the onc potental V o at the electrolyte subdoman V and the two catalyst layers IV and VI (see Fgure 4). The two governng equatons that descrbe the charge conservaton are the followng [5, 57]: r el t r o t ¼ Ñ s eff el ÑV el þ Q C el (49) ¼ Ñðs o ÑV o ÞþQ C o (50) where r el and r o are the electronc and onc charge denstes respectvely, s eff el s the effectve electronc conductvty of the electrodes, s o the onc conductvty of the electrolyte, V el and V o the electronc and onc voltages, and fnally Q C el and Q C o represent the electronc and onc charge source terms, respectvely. The effectve electronc conductvty of the electrodes,, s gven by the followng expresson: s eff el s eff el ¼ es g þ ð eþs el (5) where s el s the electronc conductvty of the anode or the cathode electrode. At the anode catalyst layer the hydrogen and carbon monoxde electrochemcal oxdatons are takng place and a transfer of onc to electronc current occurs. Thus, n subdoman IV FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 30

9 the source terms of Eq. (49) and Eq. (50), respectvely, are gven by the followng equatons: 2 3 exp a A n ef, Q C el ¼ IA w ACL ¼ I0 A 6 RT ha 7 4 exp a A ne 5 w F ACL RT ha 2 3 exp a A n ef (52),w ACL Q C o ¼ IA w ACL ¼ I0 A 6 4 exp a A RT ha ne F RT ha 7 5 where w ACL s the wdth of the anode catalyst layer (subdoman IV). The sum of the anode actvaton h A act and concentraton overpotentals h A conc s gven by [5, 58]: h A ¼ h A act þ ha conc ¼ V el V o (53) Snce, n ths model both the electrochemcal oxdaton of H 2 and CO are taken nto account, the anode concentraton overpotental wll be computed based on the followng equaton: h A conc ¼ RT n e F ln pbulk H 2 p TPB H 2 p TPB H 2 O p bulk H 2 O! RT n e F ln pbulk CO ptpb CO 2 p TPB CO pbulk CO 2! (54) Accordngly, at the cathode catalyst layer, where the oxygen electrochemcal reducton takes place, the Butler-Volmer equaton s also used to obtan the rate of transfer of electronc to onc current. Hence: 2 3 exp a C n ef, Q C el ¼ IC w CCL ¼ I0 C 6 RT hc 7 4 exp a C ne 5 w F CCL RT hc 2 3 exp a C n ef (55),w CCL Q C o ¼ IC w CCL ¼ I0 C 6 4 exp a C RT hc ne F RT hc 7 5 where w CCL s the wdth of the cathode catalyst layer (subdoman VI). Correspondngly, the sum of the cathode actvaton h C act and concentraton overpotentals hc conc s gven by [5, 58]: h C ¼ h C act þ hc conc ¼ E REV ðv el V o Þ (56) where E REV s the reversble potental also known as Nernst potental. The cathode concentraton overpotental h C conc s gven by: h C conc ¼ RT ln pbulk O 2. I A=C n e F p TPB 0, used n equatons (52) O 2 and (55), s the exchange current densty for each electrode and can be expressed as a functon of the Arrhenus law and the composton of the reactant gases as: I A 0 ¼ ga p H 2 p 0 ph2 O p 0 exp EA A RT (57) I C 0 ¼ gc p O 2 p 0 0:25 exp EC A RT (58) where p s the partal pressure of the gaseous speces that takes part n an electrochemcal reacton, p 0 s the reference pressure, g A=C are the anode/cathode exchange current densty pre-exponental coeffcents and E A=C A s the anode/cathode actvaton energy. 3.5 Mass, Energy, Momentum and Charge Transport Boundary Condtons The boundary (BC) and nterfacal (IFC) condtons are specfed for all the external boundares as well as the nternal nterfaces of the SOFC computatonal doman. As can be seen n Fgure 4 there are n total twenty eght dstnct faces, twenty of whch are external boundares and eght of whch are nternal nterfaces. In the next sectons, the BCs and IFCs for the mass, energy, momentum and charge transport equatons, respectvely, wll be presented Mass Transport Boundary and Interfacal Condtons At the nlet of the fuel and ar channels, the partal pressures p of each speces are specfed. Therefore, accordng to Fgure 4: BC5 : p ¼ p IN ; ¼ H 2 ; H 2 O; CH 4 ; CO; CO 2 (59) BC23 : p ¼ p IN ; ¼ O 2 ; N 2 (60) where p IN s the partal pressure of speces at the nlet. At the outlet of the fuel and ar channels convectve flux boundary condtons are mposed, meanng that the dffusve term of the total flux s cancelled out and only the convectve term s taken nto account. Thus: BC6 : J FC ¼ 0 ; ¼ H 2 ; H 2 O; CH 4 ; CO; CO 2 (6) BC24 : J AC ¼ 0 ; ¼ O 2 ; N 2 (62) where J FC and J AC are the dffusve fluxes of speces at the fuel and ar channels, respectvely. At the fuel channel/anode dffuson layer nterface and at the ar channel/cathode dffuson layer nterface, contnuty for the partal pressure and for the flux terms of each speces s mposed. Therefore: IFC7 : p FC IFC22 : p AC ¼ p ADL ¼ p CDL &N FC &N AC ¼ N ADL ; ¼ H 2 ; H 2 O; CH 4 ; CO; CO 2 (63) ¼ N CDL ; ¼ O 2 ; N 2 (64) 302 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

10 where p FC and p AC, p ADL and p CDL are the partal pressures of speces at the fuel and ar channels, and at the anode and cathode dffuson layers, respectvely. Correspondngly, N FC and N AC, N ADL and N CDL are the total molar fluxes of speces at the fuel and ar channels, and at the anode and cathode dffuson layers, respectvely. Smlarly, at the anode dffuson layer/anode catalyst layer nterface and at the cathode dffuson layer/cathode catalyst layer nterface, contnuty for the partal pressure and for the flux terms of each speces s mposed. Therefore: IFC0 : p ADL ¼ p ACL &N ADL ¼ N ACL ; (65) ¼ H 2 ; H 2 O; CH 4 ; CO; CO 2 IFC9 : p CDL ¼ p CCL &N CDL ¼ N CCL ; ¼ O 2 ; N 2 (66) where p ACL and p CCL are the partal pressures of speces at the anode and cathode catalyst layers, respectvely. Correspondngly, N ACL and N CCL are the total molar fluxes of speces at the anode and cathode catalyst layers, respectvely. At all the remanng boundares and nterfaces, nsulaton BCs are mposed Energy Transport Boundary and Interfacal Condtons At the nlet of the fuel and ar channels the temperature T of each gas stream s specfed. Therefore: BC5 : T II ¼ T FC IN (67) BC23 : T VIII ¼ T AC IN (68) Smlar to mass transport, at the outlet of the fuel and ar channels convectve flux boundary condtons for the energy transport are mposed, whch means that the conductve term of the total flux s cancelled out and only the convectve term s taken nto account. Thus: BC6 : n ð k II ÑT II Þ ¼ 0 (69) BC24 : n ð k VIII ÑT VIII Þ ¼ 0 (70) where h k s the convectve heat transfer coeffcent at nterface k. At the fuel channel/nterconnect nterface and at the ar channel/nterconnect nterface, convectve heat flux boundary condtons are mposed. Therefore: IFC4 : n ð k I ÑT I Þ ¼ h FC=AIC ðt II T I Þ (73) IFC25 : n ð k IX ÑT IX Þ ¼ h AC=CIC ðt VIII T IX Þ (74) At the anode dffuson layer/catalyst layer and at the cathode dffuson layer/catalyst layer nterfaces, temperature contnuty condtons are consdered. Hence: IFC0 : T III ¼ T IV & n u ð k III ÑT III Þ n d ð k IV ÑT IV Þ ¼ 0 (75) IFC9 : T VI ¼ T VII & n u ð k VI ÑT VI Þ n d ð k VII ÑT VII Þ ¼ 0 (76) where n u and n d are the unt normal vectors pontng outwards relatve to the subdomans. Accordngly, at the anode catalyst layer/electrolyte and at the cathode catalyst layer/electrolyte nterfaces, temperature contnuty condtons are appled. Hence: IFC3 : T IV ¼ T V & n u ð k IV ÑT IV Þ n d ð k V ÑT V Þ ¼ 0 (77) IFC6 : T V ¼ T VI & n u ð k V ÑT V Þ n d ð k VI ÑT VI Þ ¼ 0 (78) At the end of each of the two nterconnect subdomans, symmetry boundary condtons are mposed, assumng that the cell s placed n the mddle of a SOFC stack. In order to acheve ths, the feature of extruson couplng varables [36] ncorporated n COMSOL Multphyscs was used. Therefore, accordng to Fgure 4: IFC : T I ¼ T IX (79) where k j s the thermal conductvty of subdoman j and n s the unt normal vector pontng outwards relatve to the subdomans. At the fuel channel/anode electrode nterface and at the ar channel/cathode electrode nterface, convectve heat flux boundary condtons are mposed. Thus, accordng to Fgure 4: IFC7 : n ð k III ÑT III Þ ¼ h FC=ADL ðt II T III Þ (7) IFC22 : n ð k VII ÑT VII Þ ¼ h AC=CDL ðt VIII T VII Þ (72) IFC28 : n ð k I ÑT I Þ ¼ n ð k IX ÑT IX Þ (80) At all the remanng boundares nsulaton BCs are mposed Momentum Transport Boundary and Interfacal Condtons At the nlet of the fuel and ar channels the velocty U of each gas stream s specfed. Therefore: BC5 : U II ¼ U FC IN (8) FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 303

11 BC23 : U VIII ¼ U AC IN (82) At the outlet of the fuel and ar channel pressure boundary condtons are mposed, whch are typcal for ths type of problem. Thus: BC6 : p II ¼ p A (83) BC24 : p VIII ¼ p C (84) At all the remanng boundares no slp BCs are mposed Charge Transport Boundary and Interfacal Condtons At the fuel channel/anode dffuson layer nterface, the electronc voltage, V el, s arbtrarly set to zero [47, 59, 60]. Therefore: IFC7 : V el ¼ 0 (85) At the ar channel/cathode dffuson layer nterface, the electronc voltage, V el, s set equal to the operatng voltage V C of the SOFC [47, 59, 60]. Thus: IFC22 : V el ¼ V C (86) At the anode dffuson layer/catalyst layer and at the cathode dffuson layer/catalyst layer nterfaces, electronc voltage, V el, contnuty condtons are appled. Thus: IFC0 : n IFC9 : n s eff el ÑVIII el s eff el ÑVVI el s eff el ÑVIV el s eff el ÑVVII el ¼ 0 (87) ¼ 0 (88) Accordngly, at the anode catalyst layer/electrolyte and at the cathode catalyst layer/electrolyte nterfaces, onc voltage, V o, contnuty condtons are appled. Hence: IFC3 : n IFC6 : n s eff o ÑVIV o s eff o ÑVV o seff o ÑVV o seff o ÑVVI o ¼ 0 (89) ¼ 0 (90) At all the remanng boundares and nterfaces, nsulaton BCs are mposed. 4 Results and Dscusson The DIR-SOFC model developed here s based on fundamental equatons wrtten n a partal dfferental form, makng t flexble n terms of the geometres that can be used. Thus, planar, tubular and monolthc DIR-SOFC confguratons can be smulated usng ths model, provded that the correspondng BCs are modfed accordngly. The computatonal doman was dscretzed n 29,02 doman and 3,473 boundary trangular elements, whch were suffcent to produce mesh-ndependent solutons. In the current model the operatng voltage V C s consdered constant throughout the SOFC, therefore t wll be a degree of freedom for the model. On the other hand, the current densty I C s consdered a dependent varable, thus t wll vary throughout the sold structure of the SOFC. The preexponental coeffcents, actvaton energes and equlbrum constants, used n the MSR and WGS reactons rate Eqs. () and (2), depend on the physcal and chemcal propertes of the DIR-SOFC materals. In ths work data from the lterature have been used. In Table the DIR-SOFC model parameters Table DIR-SOFC model energy and charge transport parameters. Parameter / Unts Symbol Value Anode electrode thermal conductvty k A 0 [6] /Wm K s Fuel mxture thermal conductvty /Wm K k A g 0.32 [62] Cathode electrode thermal conductvty /Wm K k C s [6] Ar mxture thermal conductvty /Wm K k C g 0.58 [62] Electrolyte thermal conductvty k E 2[6] /Wm K Interconnects thermal conductvty k IC 6[6] /Wm K Anode specfc heat capacty /Jkg K C A p 0.65 [63] Cathode specfc heat capacty / J kg K C C p 0.9 [63] Electrolyte specfc heat capacty / J kg K C E p 0.3 [63] Interconnects specfc heat capacty /Jkg K C IC p 0.8 [63] Anode electrode densty / kg m 3 r A s 6,200 [63] Cathode electrode densty / kg m 3 r C s 6,000 [63] Electrolyte densty / kg m 3 r E s 5,560 [63] Interconnects densty / kg m 3 r IC s 7,700 [63] Convectve heat transfer coeffcent h 5[64] /Wm 2 K Anode electrode electrcal conductvty / W m s el ADL Cathode electrode electrcal conductvty / W m s el CDL 9:5 0 7 T 4:2 0 7 T exp 50 [65] T exp 200 [65] T [65] T Electrolyte onc conductvty / W m s o E 33400exp 0300 Anode transfer coeffcent / a A 0.5 Cathode transfer coeffcent / a C 0.5 Anode pre-exponental coeffcent / A m 2 g A [66] Cathode pre-exponental coeffcent / A m 2 g C [66] Anode actvaton energy / J gmol E A A 20,000 [66] Cathode actvaton energy / J gmol E C A 0,000 [66] 304 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

12 are lsted along wth the correspondng sources, they were extracted from. In order to study the effect of dfferent operatng and desgn parameters on the SOFC performance, base case parameters have to be defned. The base case operatng and desgn parameters used n ths model, consderng steadystate condtons, are lsted n Table 2. The bnary dffuson coeffcents of the speces used are temperature and pressure dependent and they are calculated usng the Fuller s method [68]: D A;B ¼ PM =2 AB 0:0043T :75 h 2 (9) ðs u Þ =3 ð S uþ A B The Knudsen dffuson coeffcents of the partcpatng speces are calculated by[69]: sffffffffffffffffff D K ¼ 4 3 K 8RT 0 pm 4. Model Valdaton (92) Our model has been valdated aganst expermental data avalable n lterature [70]. The geometrcal aspects of the expermental setup have been reported n [7]. The geometry of the model defned n Tables and 2 has been adapted to match the expermental one. Table 3 outlnes the geometrcal aspects of the model used for the valdaton, as well as the materal parameters, as reported n [7]. All parameters not mentoned n Table 3, have the same value as n the base case (Tables and 2). The feed reported n [7] s ls, Table 2 Base case operatng condtons and knetc parameters for the DIR-SOFC model. Parameter / Unts Symbol Value Fuel nlet temperature / K TIN A,73 Ar nlet temperature / K TIN C,73 Fuel channel nlet velocty / m s U FC IN 0.2 Ar channel nlet velocty / m s U AC IN 5.0 Operatng pressure / atm P Gas law constant / m 3 atm gmol K R Vscosty / N s m 2 m Permeablty / m 2 B Faraday s constant / A s gmol F Porosty / e 0.46 Tortuosty / t 4.5 Average pore radus / m r Fuel cell length / m L 0.05 Fuel channel wdth / m w FC Ar channel wdth / m w AC Anode dffuson layer wdth / m w ADL Cathode dffuson layer wdth / m w CDL Anode catalyst layer wdth / m w ACL Cathode catalyst layer wdth / m w CCL Electrolyte wdth / m w E Anode nterconnect wdth / m w AIC 0.00 Cathode nterconnect wdth / m w CIC 0.00 Operatng Voltage / V V C 0.6 MSR reacton actvaton energy / J mol E A MSR 82,000 [0] MSR reacton pre-exponental constant / mol s m 2 atm k MSR 4,33 [0] WGS reacton equlbrum constant / K eq.44 [67] WGS reacton pre-exponental factor k WGS [67] / mol s m 2 atm Table 3 Parameters for the DIR-SOFC model used for model valdaton [7]. Parameter / Unts Symbol Value Fuel nlet temperature / K TIN A,073 Ar nlet temperature / K TIN C,073 Fuel channel nlet velocty / m s U FC IN 0.7 Ar channel nlet velocty / m s U AC IN 0.7 Operatng pressure / atm P Gas law constant / m 3 atm gmol K R Vscosty / N s m 2 m Permeablty / m 2 B Porosty / e 0.35 Tortuosty / t 3.8 Fuel channel wdth / m w FC Ar channel wdth / m w AC Anode dffuson layer wdth / mm w ADL 400 Cathode dffuson layer wdth / mm w CDL 5 Anode catalyst layer wdth / mm w ACL 00 Cathode catalyst layer wdth / mm w CCL 5 Electrolyte wdth / mm w E 25 Anode electrode thermal conductvty k A 3 /Wm K s Cathode electrode thermal conductvty /Wm K k C s 3.8 Electrolyte thermal conductvty / W m K k E.8 Anode convectve heat transfer coeffcent h A 57 /Wm 2 K Cathode convectve heat transfer coeffcent h C 49 /Wm 2 K Anode pre-exponental coeffcent / A m 2 g A [72] Cathode pre-exponental coeffcent / A m 2 g C [72] Anode actvaton energy / J gmol E A A 20,000 [72] Cathode actvaton energy / J gmol E C A 00,000 [72] FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 305

13 whch corresponds to a maxmum velocty at the fuel channel of 0.7 m s f a parabolc velocty profle s assumed. The reversble voltage has been adapted from expermental data (.7 V), snce the emprcal relaton used for our model s ncompatble wth the expermental data utlzed. The pre-exponental coeffcent and the actvaton energy for the exchange current densty used n the modfed Butler-Volmer equaton were obtaned from [72]. The fuel mxture used for the model valdaton s 30% pre-reformed and t conssts of 26.3% H 2, 49.3% H 2 O, 7.% CH 4, 2.9% CO and 4.4% CO 2 at the nlet to the fuel channel [7]. The polarzaton curve obtaned usng our model s compared to the expermental results and t s llustrated n Fgure 5. The model s shown to replcate the experment suffcently well. Small dscrepances are attrbuted to some geometrcal aspects beng unavalable and to some of the lterature-obtaned parameter referrng to dfferent condtons and materals than the ones used n the actual expermental setup. 4.2 Base Case Model Smulatons Our DIR-SOFC system s desgned to operate for both non pre-reformed and pre-reformed operatng condtons. The fuel mxture used n ths study s 0 % pre-reformed and t conssts of 8% H 2, 58% H 2 O, 28% CH 4, 2% CO and 4% CO 2 at the nlet of the fuel channel [0]. In Fgure 6, the H 2 and O 2 molar fractons at the anode fuel channel and at the cathode ar channel are llustrated. As H 2 and O 2 flow n the fuel and ar channels, respectvely, they dffuse through the porous electrodes where they react at the anode and cathode electrochemcally-actve catalyst layers respectvely, where the half-cell electrochemcal oxdaton (4) and reducton (6) reactons take place. In the porous electrodes, the formed gradents of the H 2 and O 2 molar fractons are larger than n the fuel and ar channel, because the dffuson s more dffcult n the porous electrodes than n the fuel and ar channels. Addtonally, at the fuel and ar channels the convectve effects are much larger compared to the porous Fg. 5 Polarzaton curve computed by the proposed model aganst expermental results [70]. electrodes, where dffuson prevals. Close to the nlet of the fuel channel, a steep ncrease of the H 2 partal pressure can be observed. Ths s due to the rapd MSR reacton (2) and s an ndcaton that ths occurs manly close to the fuel nlet. After all CH 4 s depleted, the produced H 2 s then gradually consumed along the length of the cell by the electrochemcal reacton (4), resultng n the decrease of ts molar fracton after reachng a maxmum value. The O 2 molar fracton at the cathode electrochemcally actve catalyst layer s decreased along the length of the ar channel due to the half-cell electrochemcal reducton reacton, as expected. Fgure 7 llustrates the anode fuel gas speces molar fractons along the length of the fuel channel. The mpact of the smultaneous occurrence of the MSR reacton, the WGS reacton and the electrochemcal oxdaton of H 2 and CO at the anode catalyst layer s presented. A rapd decrease n the CH 4 composton and a correspondng ncrease n the H 2 composton can be seen close to the nlet of the fuel channel. Ths s due to the very fast MSR reacton (2) and partally because of the WGS reacton. A decrease n the H 2 O composton can also be observed, whch can also be attrbuted to the MSR and WGS reactons as steam plays the role of the reactant n these reactons. However, H 2 and H 2 O are also partcpatng n the H 2 electrochemcal reacton (4), where they have an opposte role, although the effect of the MSR and WGS reactons cannot be balanced, due to the dfferent reacton rates. When CH 4 starts gettng depleted, H 2 and H 2 O compostons begn to decrease and ncrease respectvely, as now the effect of the electrochemcal reactons s domnatng. Consequently, the H 2 and H 2 O compostons exhbt a maxmum and a mnmum respectvely, close to the pont where CH 4 s depleted. Smlarly to H 2, at the nlet regon there s an ncrease n CO composton due to the MSR reacton, whch starts decreasng as soon as CH 4 s consumed and the WGS and the CO electrochemcal reactons domnate. These trends, presented n Fgure 7, come n qualtatve agreement wth results from [0] and [9], but vary quanttatvely due to the use of dfferent operatng parameters. In Fgure 8, the temperature dstrbuton n the DIR-SOFC, obtaned by the developed model, s llustrated. Inlet temperatures of T A IN ¼ TC IN ¼,73 K and base case condtons, as gven n Table 2, were used for ths smulaton. The steep temperature decrease, that results n a cold spot at the anode electrode near the fuel nlet, s a result of the excessve amount of heat consumed by the hghly endothermc MSR reacton (2). The DIR-SOFC temperature decreases n the frst 20% of ts length by 50 K n relaton to ts nlet value. CH 4 s depleted near the nlet due to the large MSR reacton (2) rate and as a result, the exothermc electrochemcal reactons (4) and (5) domnate the thermal energy generaton, leadng to a temperature ncrease all the way to the ext of the fuel and ar channels. Ths causes the formaton of thermal gradents that n turn gve rse to thermal stresses, as already dscussed. If these stres- 306 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,

14 Fg. 6 H 2 /O 2 molar fracton profles at the anode/cathode sde of the DIR-SOFC for base case nlet compostons. Fg. 7 Fuel speces molar fractons y dstrbuton along the length of the SOFC for base case nlet compostons. ses exceed the threshold values dctated by the ceramc and metal materals used for the constructon of the SOFC, t can result n reduced performance and even mechancal falure of the cell. Fgure 9 llustrates the average (along the SOFC length) electronc and onc voltage dstrbutons along both dffuson and catalyst layers and the electrolyte doman for base case condtons (as n Table 2) and for operatng potental V C = 0.6 V. As mentoned prevously, snce electrons cannot flow n the electrolyte and ons cannot flow through the dffuson layers of the electrodes, there s no electronc and onc potental at the respectve subdomans. The current denstes are defned by the gradent of the respectve onc and electronc voltage curves n Fgure 9. The actvaton and concentraton overpotentals are gven by Eq. (53) and Eq. (56) respectvely and as can be seen for the selected operatng condtons the anode overpotental s slghtly larger than the cathode one. The ohmc overpotental s almost zero at the electrodes as ther electronc conductvty s much hgher compared to the electrolyte onc conductvty. The onc potental dstrbuton n the electrolyte subdoman s lnear due to the constant onc current flux. The average current densty and the reversble voltage dstrbutons along the length of the cell are shown n Fgure 0. The current densty decreases to a mnmum of 6,200 A m 2 and then ncreases along the length of the cell. Its magntude s dependent on the temperature and on the H 2 and CO concentratons. The temperature dependence s more sgnfcant, as t s evdent from the trend observed n Fgure, whch agrees wth the temperature decrease close to the nlet of the DIR-SOFC. Accordngly, the reversble voltage E REV exhbts a peak value at the pont where the temperature s at ts lowest value as expected from the expresson: FUEL CELLS 6, 206, No. 3, ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem 307

15 Fg. 8 Temperature profle n the DIR-SOFC doman (MSR reacton endothermc effect). Fg. 9 Average (along the length) electronc and onc voltage dstrbutons over the wdth of the sold DIR-SOFC domans for base case operatng condtons. E REV ¼ E 0 þ RT n e F ln p! H 2 p 0:5 O 2 p H2 O (93) 4.3 Parametrc Investgaton A parametrc study of varous operatng and desgn parameters has been performed n order to examne ther effect on the performance of the DIR-SOFC. In Fgure, temperature dstrbutons along the length of the SOFC, for fuel and ar channel nlet temperatures of,073 K,,23 K, and,73 K and for a constant operatng voltage V C = 0.6 V, are depcted. As dscussed prevously, the temperature exhbts a mnmum close to the nlet of the fuel channel, where the rate of the endothermc MSR reacton (2) reaches ts hghest value. It can be seen that the temperature gradent s smoother for nlet temperature T IN =,073 K than for =,73 K. Ths can be attrbuted T IN manly to the fact that, for constant V C, hgher temperature operaton corresponds to hgher average current denstes, resultng n hgher exothermc electrochemcal oxdaton reacton rates and n turn n a more steep ncrease of the temperature along the cell. Furthermore, the MSR reacton rate ncreases wth ncreasng temperature, as expected from Eq. (), whch also affects the formaton of larger gradents. 308 ª 206 The Authors. WILEY-VCH Verlag GmbH & Co. KGaA, Wenhem FUEL CELLS 6, 206, No. 3,