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A General Mathematical Model for Analyzing the Performance

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A general mathematical model for analyzing the performance of fuel-cell membrane-electrode assemblies Huayang Zhu * , Robert J. Kee Engineering Division, Colorado School of Mines, Golden, CO 80401, USA Received 9 January 2003; accepted 27 January 2003 Abstract We have developed a general mathematical model to represent the membrane-electrode assembly (MEA) of fuel-cell systems. The model is used to analyze the effects of various polarization resistances on cell per
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  A general mathematical model for analyzing the performanceof fuel-cell membrane-electrode assemblies Huayang Zhu * , Robert J. Kee Engineering Division, Colorado School of Mines, Golden, CO 80401, USA Received 9 January 2003; accepted 27 January 2003 Abstract Wehavedevelopedageneralmathematicalmodeltorepresentthemembrane-electrode assembly (MEA)offuel-cellsystems.Themodelisused to analyze the effects of various polarization resistances on cell performance. The model accommodates arbitrary gas mixtures on theanode and cathode sides of the MEA. Moreover, it accommodates a variety of porous electrode and electrolyte structures. Concentrationoverpotentials are based on a dusty-gas representation of transport through porous electrodes. The activation overpotentials are representedusingtheButler–Volmerequation.Althoughthemodelisgeneral,theemphasisinthispaperisonsolid-oxidefuel-cell(SOFC)systemsforthedirect electrochemical oxidation (DECO) of hydrocarbons. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Model; MEA; Fuel cell; SOFC 1. Introduction Quantitative models can be valuable in the interpretationof experimental observations and in the development andoptimization of systems. The objective of the membrane-electrode assembly (MEA) model developed herein is toprovide a physically based model that can be used toevaluate the effects of design and operating alternatives.While the model can stand alone, the software is written in away that enables incorporation into larger system-levelmodels that also consider fluid flow and thermal manage-ment throughout a fuel-cell system. The model is general inthe sense that it can accommodate a variety of geometricalconfigurations. It can also represent proton-conducting oroxygen-ion-conducting electrolytes. The discussion andexamples in this paper, however, consider solid-oxidefuel-cell (SOFC) system.The great advantage of SOFC systems for highly efficientelectric powergeneration lies in its potential for direct use of hydrocarbon fuels, without the requirement for upstreamfuel preparation, such as reforming[1–4]. A typical layoutforaplanardesignoftheSOFCsystemisillustratedinFig.1.The membrane-electrode assembly, which is composed of an electrolyte sandwiched between the anode and the cath-ode, is itself sandwiched between metal interconnect struc-tures. The fuel and air channels are formed in theinterconnect structure. The electrolyte is a dense ceramic(e.g. yttria-stabilized zirconia (YSZ) or gadolinia-dopedceria (GDC)), which is impermeable to gas flow but is anoxygen-ion (O 2 À ) conductor. The composite electrodes areporous metal-loaded ceramics (cermets) (e.g. Ni–YSZ forthe anode and Sr-doped LaMnO 3 (LSM) for the cathode),which may be mixed electronic and ionic conductors thatcan extend the three-phase boundaries (TPB) and promotethe electrocatalytic reactions.The SOFC system involves complex transport, chemical,and electrochemical processes, with its operating perfor-mance strongly affected by the corresponding transportresistancesand the activation barriers.Fig.2illustrates someof these processes at the MEA level. During operation,oxygen from the air channel transports through the porouscathode, whereupon it is reduced to oxygen ion (O 2 À ) atgas–cathode–electrolyte three-phase boundaries. Theformed oxygen ion at the cathode–electrolyte interface isthen transported to the anode–electrolyte interface throughthe ion-conducting electrolyte. At the same time, fuel fromthe fuel channel transports through the porous anode to theanode–electrolyte interface, where it is oxidized electroche-mically atthegas–anode–electrolytethree-phase boundariesto products. The products (e.g. H 2 O and CO 2 ) are thentransported back to the fuel channel through the porousanode structure. Transport resistances of the gas-phasespecies in the porous electrodes and O 2 À in the electrolyte, Journal of Power Sources 117 (2003) 61–74 * Corresponding author. Tel.: þ 1-303-273-3890; fax: þ 1-303-273-3602. E-mail address: hzhu@mines.edu (H. Zhu).0378-7753/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-7753(03)00358-6  as well as the activation energy barriers for electrochemicalreactions can result in various polarizations. They are repre-sented as concentration overpotentials, activation overpo-tentials, and ohmic overpotentials. The main contribution tothe ohmic polarization is from the transport resistance of O 2 À in the electrolyte. The concentration polarization is dueto the transport resistance of the gaseous species throughporous composite electrodes, and the activation polarizationis related to the charge-transfer processes at the electrode – electrolyte interfaces. These various polarizations are func-tions of both operating conditions and the physical proper-ties of the cell. The operating conditions includetemperature, pressure, and fuel and oxidizer concentrations.The cell properties involve materials, macro- and micro-structures of the electrolyte and the composite electrodes,which include porosity, tortuosity, permeability, and thick-ness of the anode and the cathode, the ionic conductivityandthicknessoftheelectrolyte,theactiveareaandactivityoftheelectrode – electrolyte interface.To understand these various resistances and enhance thecell performance, parametric studies for these various polar-izations in the electrolyte and composite electrodes havebeen widely made through experiments[5,6], theoreticalanalysis[7],and numerical models[8 – 10]. The theoreticalanalyses and numerical models for the activation polariza-tions require an understanding the elementary thermal – electrochemical reaction mechanisms and the microstruc-ture of the electrode. Due to the complex and uncertaindetails of the reaction processes (particularly in case of hydrocarbons), most modeling in the literature has focusedon using H 2 as the fuel. Chan et al.[8]used the intrinsiccharge-transfer resistance to represent the activation polar-ization. Tanner et al.[11]have derived an effective charge-transfer resistance for the composite electrode, which is afunction of the microstructural parameters of the electrode,intrinsic charge-transfer resistance, ionic conductivity of theelectrolyte, and the electrode thickness. This effectivecharge-transfer resistance model was later used to analyzethe activation overpotentials[9,10].Adler et al.[12]derived a charge-transfer resistance model for the composite cath-ode, which is a function of the tortuosity, porosity, interfacearea, oxygen self-diffusion coef  fi cient and oxygen exchangecoef  fi cient. The activation polarizations for a particularelectrode may also be measured by the electrochemicalimpedance spectroscopy (EIS)[5,6,13]. Previous porous-media gas-phase transport models have been developed topredict concentration overpotentials[8 – 10]. These includemolecular diffusion, Knudsen diffusion, and pressure-drivenDarcy fl ow. However, these have only been applied in fuel-cell systems with binary gas-phase mixtures.The model developed in this paper represents a unit-cellstructure (Fig. 2), considering the effects of various over-potentials on the cell performance. As illustrated inFig. 2the cell is anode-supported, with a very thin electrolyte andcathode, and both electrodes are porous ceramic – metalcomposite structures. Channels formed in an interconnectstructure carry the fl ow of fuel and air. Depending on theposition of the unit cell within a fuel-cell system, thecomposition in both the fuel and air channels varies depend-ing on initial composition and depletion. Fig. 1. Typical layout of a planar SOFC. The left-hand panel illustrates the system at the scale of flow channels (in mm), while the right-hand panel illustratesthe small-scale structure (in m m) of the porous cermet electrode.Fig. 2. Illustration of the components of the MEA unit cell.62 H. Zhu, R.J. Kee/Journal of Power Sources 117 (2003) 61–74  2. Model formulation The overall performance of SOFC systems is greatlyaffected by ohmic, concentration, and activation polariza-tions. Thus, the operating cell potential ( E  cell ) is lower thanthe Nernst potential ( E  ). The operating cell potential ( E  cell )can be formally expressed as: E  cell ¼ E  À Z conc ; a À Z act ; a À Z ohm À Z conc ; c À Z act ; c À Z interface À Z leakage ; (1)where Z conc ; a and Z conc ; c are the concentration overpotentialsat the anode and the cathode due to the gas-phase speciesdiffusion resistance, Z act ; a and Z act ; c the corresponding acti-vation overpotentials, Z ohm the ohmic overpotential in theelectrolyte, Z interface the interface overpotential due to thecontact resistance atmaterialboundaries, and Z leakage the cellpotential loss due to the electric current through the electro-lyte. Subsequent sections in the paper develop models foreach oftheseoverpotentials, all of which are functions ofthecurrent density i e . 2.1. Global electrochemical reaction A global electrochemical reaction can be written as: n f  0 X K k  ¼ 1 n f  ; k  w k   ! þ n o 0 X K k  ¼ 1 n o ; k  w k   ! ? X K k  ¼ 1 n 00 f  ; k  w k  þ X K k  ¼ 1 n 00 o ; k  w k  ; (2)where w k  is the chemical symbol for the k  th species (whichmayparticipateasafuel,anoxidizer,orboth), n f  0 and n o 0 thestoichiometric coef  fi cients for the fuel and oxidizer mix-tures, and n 00 f  ; k  and n 00 o ; k  the stoichiometric coef  fi cients of the k  th product species in the fuel and air channels, res-pectively. Each species w k  has a primary identity as a fuel,an oxidizer, a product, or an inert, and this identity playsan essential role in balancing the reaction to determinestoichiometric coef  fi cients and in assigning charge trans-fer. The global reaction can be expressed in compact formas: X K k  ¼ 1 n k  0 w k  ? X K k  ¼ 1 n 00 k  w k  ; (3)where n k  0 ¼ n f  0 n f  ; k  þ n o 0 n o ; k  and n 00 k  ¼ n 00 f  ; k  þ n 00 o ; k  .Fuel and oxidizer mixtures are separated by the anode – electrolyte – cathode MEA structure. Thus, in writing a glo-bal electrochemical reaction and calculating the Nernstpotential, it is convenient to represent the fuel mixtureand the oxidizer mixture separately as reactants. Thisseparation is accomplished by writing the LHS of thereaction as two summations. The fuel channel mixturecomposition is represented in terms of the species molenumbers as n f  ; k  . Similarly, n o ; k  is used to represent themixture of the composition of the oxidizer mixture in theair channel (e.g. for air, n o ; O 2 ¼ 0 : 21 and n o ; N 2 ¼ 0 : 79).The RHS of the global reaction is also separated into twoterms. Depending on whether the cell is an oxygen-ionconductor or a proton conductor, the product species appearon the anode or cathode side of the MEA. Thus, followingreaction balancing, the location of the product speciesaffects the product stoichiometric coef  fi cients n 00 f  ; k  and n 00 o ; k  . There is more discussion on this point following thediscussion on reaction balancing.Since the electrochemical reactions involve charge trans-fer it is necessary to specify the charge transfer associatedwith each reactant  species. For this purpose, we introducetwo sets of parameters (  z f  ; k  and z o ; k  ), which specify thecharge transferred per mole of the k  th fuel and oxidizerspecies. For a species that is inert relative to the electro-chemical charge-transfer process, z ¼ 0. In our convention,the product species (e.g. H 2 O and CO 2 ) also are assigned  z ¼ 0. Consider two examples for assigning z . Since O 2 inthe air channel is reduced to O 2 À by obtaining four electrons(i.e. O 2 þ 4e À ! 2O 2 À ), the parameter z o ; O 2 ¼ 4. Becausethe direct electrochemical reaction of CH 4 with O 2 À at theanode to produce H 2 O and CO 2 (i.e. CH 4 þ 4O 2 À ! CO 2 þ 2H 2 O þ 8e À ) releases eight electrons, z f  ; CH 4 ¼ 8.The total number of electrons transferred by the globalelectrochemical reaction is represented as: n e ¼ X K k  ¼ 1 n f  0 n f  ; k   z f  ; k  ¼ X K k  ¼ 1 n o 0 n o ; k   z o ; k  : (4)There may be reasons to incorporate oxidizer species intothefuelchannelorfuelspeciesintotheoxidizerchannel.Forexample, oxygen in the fuel channel may promote bene fi cialpartial oxidation of the fuel or it may work to inhibit depositformation. Such species, while in fl uencing thermal chem-istry in the gas or on catalytic surfaces, do not participatedirectly in the electrochemical reaction. Thus, they areconsidered inert in the global electrochemical reaction.For example, if O 2 (i.e. an oxidizer species) appears as partof the fuel composition (i.e. in the fi rst term of the globalreaction(2)), then z f  ; O 2 ¼ 0. The added oxygen in the fuelchannel cannot work directly to produce electric energy — onlyfuel oxidization with oxidizer from the cathode channelcan produce power. Thus, any oxidizer in the fuel channelmust reduce the cell potential because of its diluting effect.Beforeproceedingtoevaluatethecellpotential,theglobalreaction must be balanced to determine the stoichiometriccoef  fi cients. Although balancing reactions is a well knownprocess, a few words of explanation are warranted. Abalance equation is written for each element (e.g. H, C,O, N), requiring that the reaction conserves the elementbalance.Theprocessleadstoasystemoflinearequationsforthe stoichiometric coef  fi cients ( n f  0 , n o 0 , and n 00 k  ). Because of an essential linear dependence stemming from the fact thatspeciesarecomposedofelementsinaspeci fi edway,onecanarbitrarily set n f  0 ¼ 1. Then depending on the number of product species, only some of the n 00 k  are non-trivial. Spe-ci fi cally, the number of product species (and hence the  H. Zhu, R.J. Kee/Journal of Power Sources 117 (2003) 61  –  74 63  number n 00 k  coef  fi cients) is one fewer than the number of elements. Once the overall product stoichiometric coef  fi -cients n 00 k  are known, then either: n 00 f  ; k  ¼ n 00 k  or n 00 o ; k  ¼ n 00 k  ; (5)depending on the type of cell (i.e. oxygen ion or protonconducting) and hence the location of the product species.Inert species (i.e. reactant  species for which z f  ; k  ¼ 0 and  z o ; k  ¼ 0) are initially excluded from the balancing proce-dure. Once the balance is completed, the inert species arereintroduced with the same stoichiometric coef  fi cients onthe reactant and product sides of the global reaction. Theentire balancing process is completely automated in thesoftware. 2.2. Nernst potential and the Nernst equation Based on the chemical potential balance at open circuit,the Nernst equation provides that: E  ¼ E   þ RT n e F  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ ln p k   p 0   ; (6)where E  is the Nernst electric potential, E   the ideal Nernstpotential at standard conditions (  p 0 ¼ 1 atm), p k  the partialpressure of the k  th species, T  the temperature, R ¼ 8 : 314 J/ (mol K) isthe universalgas constant,and F  ¼ 96485 : 309 C/ mol is Faraday ’ s constant. The ideal Nernst potential at thestandard conditions is given as: E   ¼À D G  n e F  ; (7)where D G  is the change in standard-state Gibbs free energybetweenproductsandreactantsoftheglobalreactionEq.(3).Speci fi cally: D G  ¼À X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ m  k  ; (8)where m  k  is the standard-state chemical potential of the k  thspecies. The standard-state thermodynamic properties of ideal gases depend on temperature. The needed thermody-namic properties are readily available in databases such as inC HEMKIN [14].As an alternative to using the species partialpressures, the Nernst equation can also be represented interms of the species molar concentrations ½  X  k   as: E  ¼ E   þ RT n e F  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ ln ½  X  k  þ RT n e F  ln RT  p 0  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ : (9)Fig. 3illustrates the Nernst potentials for three fuels as afunction of fuel utilization (at T  ¼ 750 8 C and p ¼ 1 atm).Considerthesituationforbutanewheretheglobalreactionis:C 4 H 10 þ 132 O 2 ! 5H 2 O þ 4CO 2 : (10)If the fuel is 50% utilized then the fuel mixture is composedof 1/2 moles of C 4 H 10 , 5/2 moles of H 2 O, and 4/2 moles of CO 2 .The Nernst potentials are highest forthe hydrocarbons,assuming direct electrochemical oxidation. The fi gure alsoshows that the Nernst potential for hydrogen is reduced mostas a function of fuel depletion. The potentials illustrated inFig. 3are calculated presuming that the oxidizer is pure air(i.e. no depletion of the oxygen in the air). 2.3. Concentration polarization For open-circuit conditions (i.e. zero current fl ow), thespecies concentrations at the electrolyte interface (three-phase boundary) are the same as those in the bulk channel fl ow. Denoting the molar species concentrations in thechannels as ½  X  k   à , the Nernst potential is written as: E  à ¼ E   þ RT n e F  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ ln ½  X  k   à þ RT n e F  ln RT  p 0  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ : (11)However, when the current is fl owing there must be con-centration gradients across the electrode structures. This isbecause the diffusion processes that supply the species fl uxto the electrochemical reactions are driven by concentrationgradients. Consequently, species concentrations at the three-phase boundaries ½  X  k   s are different from the bulk concen-trations. Evaluated at the three-phase boundaries, the corre-sponding Nernst equation is written as: E  s ¼ E   þ RT n e F  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ ln ½  X  k   s þ RT n e F  ln RT P 0  X K k  ¼ 1 ð n f  0 n f  ; k  þ n o 0 n o ; k  À n 00 k  Þ : (12) Fig. 3. Nernst potential for three fuels-air systems as a function of percentage of the fuel utilization. As the fuel is ‘‘ utilized ’’ it is converted tostoichiometric products that dilute the fuel on the anode side. The air is notdepleted in this system.64 H. Zhu, R.J. Kee/Journal of Power Sources 117 (2003) 61  –  74
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