Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 BostonFigure1. 2D computational domain and grid(a) anode gas channel, (b) anode gas diffuser, (c) anode catalyst layer, (d) membrane,(e) cathode catalyst layer, (f) cathode gas diffuser, (g) cathode gas channelProtons diffuse in a solid state across the membrane,solely under the influence of an electrolyte potentialgradient i.e. via migration only. In fact for PBI, thedrag co-efficient of water is virtually zero ,indicating that liquid water is not carried across themembrane together with the protons. As a result, theproblems associated with the presence of liquid waterare avoided with PBI.Most of the work done in fuel cell modeling hasbeen for Nafion ® membranes . This paper presentsa mathematical model of a PEMFC using a PBImembrane. We investigate 2 dimensional effects dueto variations in transport and polarization properties.+−( g) ⎯→2H( s) + eH221+−2g + 2Hs + 2e2( ) ( ) ⎯→H O( g)O22 Model DevelopmentFigure 1 shows the 2-D computationaldomain consisting of the gas channels, gas diffusers,catalyst layers and membrane regions. Figure 2shows a schematic of the cathode catalyst region.Protons, flowing through the solid matrix of the PBImembrane, react with gaseous oxygen and electronsto produce water vapor. The following are the halfcell equations for the anode and cathode reactions.Figure 2. Schematic of the cathode catalyst layer
Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 BostonBecause the protons are in a solid state,there is a solid-fluid “phase change” which occursduring the electrochemical reactions. Thisphenomenon, absent in low temperature PEMFCs,must be accounted for. This is accomplished by usinga non-conservative form of the continuity equation.∇ ⋅( ρ u) = = ∑S mSThe term on the RHS accounts for creationof fluid mass at the cathode catalyst layer, and theconsumption of fluid mass at the anode catalyst layer.The RHS is zero outside of the catalyst regions whereno electro-chemical reactions take place. Note thattotal mass is always preserved.The gas mixtures are assumed to behaveideally, so the mixture density (not constant) is givenby the ideal law relationship.−1P ⎛ ωi⎞ρ = ⎜∑⎟RT ⎝ Mi ⎠The porous media and catalyst regions areassumed to be isotropic and macro-homogeneous,and the membrane is assumed to be impermeable togas flow. So there is no fluid flow across the PBImembrane, and flow in the porous catalyst anddiffuser regions are governed by Darcy’s Law.μ∇ P = − ukpFluid flow in the gas channels are describedby the compressible Navier-Stokes equation.ρ u ⋅∇u= −∇P+ ∇ ⋅ ( μ ∇u)The mixture velocity is specified as aboundary condition at the gas channel inlets since thesupply flow rates are known, while atmosphericpressure is specified at the outlets. At the interfacebetween the gas channels and the gas diffusers, thepressure and velocity variables are coupled to allowcontinuity between the Non-Isothermal Flow and theDarcy’s Law modules. At all other boundaries, noslip,impervious or symmetry conditions apply.Flow of the multi-component gas species isgoverned by the Maxwell-Stefan equations.However, because a non-conservative form of thecontinuity equation is used, the species conservationequations must be re-derived. In the derivation, anextra term appears on the RHS.ρ u ⋅ ∇ωi=⎡N⎪⎧M ⎛ ∇M⎞⎪⎫⎤∇ ⋅ ⎢ρε ωi∑ Di, j ⎨ ⎜∇ωi+ ωi⎟⎬⎥+ Si− ωiSm⎢j =1⎣⎪⎩ Mj ⎝ M ⎠⎪⎭⎥⎦The concentration of each species isspecified at the gas channel inlets, while convectiveflux is specified at the outlets. At every otheriboundary, symmetry conditions apply in theMaxwell-Stefan module.Thermal generation and heat transfers withinthe cell are governed by the energy equation. Theprincipal means of heat transfer expected areconduction and convection. Heat generations are dueto ohmic heating and heat of reaction.ρ cpu ⋅∇T= ρ cp∇ ⋅ ( DT∇T) + ST= Sohm+ SrxnThe temperature of the mixture is specifiedat the gas channel inlet, while a convective fluxboundary condition is specified at the outlets. At allother boundaries, insulation conditions are specifiedin the Conduction and Convection Heat Transfermodule.Due to the flow of electrical currents in boththe solid and electrolyte regions, there are potentialvariations across each of these phases. Chargeconservation is given by the following equations.∇ ⋅∇ ⋅iis=∇ ⋅eff( −σs∇φs) = − jeff( −σ∇φ) = je= ∇ ⋅e e+These equations are modeled using twoConductive Media DC modules. The potential isspecified at the gas channel/gas diffuser interfaces,while insulation conditions are specified at thecatalyst layer/membrane interfaces and all otherboundaries.The quantity j is the spatial rate of reaction.This term is defined by the Butler-Volmer equation.j =γ i⎛ ⎞ ⎧⎫ref⎡⎤ ⎡⎤⎜xiP α Fα Fa i ⎟0 ⎨exp⎢( φs− φe) ⎥ − exp⎢−( φs− φe)⎥⎬⎝ pref⎠ ⎩ ⎣ RT ⎦ ⎣ RT ⎦⎭This quantity provides the coupling betweenall the variables, since the source terms are given interms of the reaction rate.MH2SH= j22 FMO2SO= j24 FMH 2OSH O= − j22 F⎛ T ΔS⎞Srxn= − j⎜φe−φs−⎟⎝ n F ⎠The only source term not given in terms of jis the ohmic heating term, nevertheless it does dependon the local current densities.Sohm=iσ2seffsi+σ2eeffe
Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Boston3 ResultsCell Potential (V)0.90.80.70.18.104.22.168.20.100 1000 2000 3000 4000 5000 6000 7000 8000Current Density (A/m 2 )Figure 3. Polarization curveThe cell is modeled at 150 0 C (423 K) andatmospheric pressure. The inlet gases consist ofhumidified hydrogen and air (oxygen and nitrogen).The average velocity of inlet gases supplied to eachgas channel is 0.1 m/s.The refined grid consisted of 5088 elements(see Figure 1). With 7 modules used for 10independent variables, the problem entailed 48565degrees of freedom. The stationary non-linear solverwas used with a general solution form. Weakconstraints were added to the solid phase potentialequations in order to accurately determine the cellcurrent density for a specified cell potential.The polarization curve, shown in Figure 3, isgenerated using the parametric non-linear solver. Thisfigure shows clearly the activation and ohmicoverpotential regions. Noticeably absent, however, isa distinct concentration overpotential region. This isdue to the absence of liquid water, which blocks thegas pores in low temperature PEMFCs, resulting inconcentration limitations. With water present only inthe vapor phase, the cell performance is limited solelyby ohmic and activation phenomena.Figure 4 shows the concentration variationsof hydrogen and oxygen in the anode and cathoderegions respectively at optimum operating conditions(cell potential = 0.4 V, current density = 3120 A/m 2 ).It shows a decrease in reactant concentration in thedirection of gas channel flow, which is due to speciesconsumption in the electrochemical reactions. It alsoshows a variation in the direction perpendicular to themembrane electrode assembly (MEA), which isnecessary for species diffusion to take place. There isa sharp decrease in oxygen concentration across thecathode catalyst layer, whereas the correspondingdecrease in hydrogen concentration is not significant.This is because of the slower kinetics of the cathodereaction. Notice that the membrane region isuncolored since no fluid flows there. The arrowsdepict the velocity vector field. Note that the velocityof flow is much greater in the gas channels than in theMEA indicating that flow in the channels areconvection dominated whereas flow in the porousMEA is diffusion dominated.Figure 4. Concentration profiles at optimum conditions
Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 BostonFigure 5. Temperature distribution at optimum conditionsFigure 5 shows the temperature variation inthe cell at optimum conditions. Once again there arevariations in temperature parallel to andperpendicular to the gas channel flow. The reasonsare the same as above. There is a temperature risealong the gas channels as heat is carried away by thegas streams. The variation perpendicular to the MEAoccurs because of uneven heat generation within thecell. The area of most significant heat generation isthe cathode catalyst region. The hottest point islocated at the cathode catalyst region under the pointof outlet. A temperature rise of 22 K is observed atthese operating conditions. The lengths of the arrowsshow that the dominant heat transfer processes areconduction in the membrane, and convection in thegas channels.4 ConclusionsThe modeling of the high temperaturePEMFC entailed a highly coupled system ofequations, consisting of 10 independent variables and48565 degrees of freedom. Numerical convergencewas obtained by solving the problem in stages. Firstthe two potential equations (solid and electrolyte)were solved using initial value estimates. Secondly,the two flow equations (Darcy’s Law and Non-Isothermal Flow) were added and solved using the“current solution”. Finally the thermal and speciesequations were added and solved using the “currentsolution”. Solution times on a Pentium 4, 1 GB, 3.4GHz processor were as long as 8 minutes.Acknowledgements The authors are grateful for the FIUGraduate School Dissertation Fellowship, and to GasTechnology Institute (Contract Number 8390) for theirsupport of this work.References1. Bouchet R, Siebert E. Proton conduction in aciddoped polybenzimidazole. Solid State Ionics1999; 118:287-99.2. Bouchet R, Miller S, Duclot M, Souquet JL. Athermodynamic approach to proton conductivityin acid-doped polybenzimidazole. Solid StateIonics 2001; 145:69-78.3. Li Q. Hjuler HA, Bjerrum NJ. Phosphoric aciddoped polybenzimidazole membranes:physiochemical characterization and fuel cellapplications. Journal of Applied Electrochemistry2001; 31:773-9.4. Cheddie D, Munroe N. Review and Comparisonof Approaches to PEMFC modeling. Journal ofPower Sources 2005; 147:72-84.1.