Materials for a Sustainable Energy Future IPAM Tutorials Sept 10-12, 2013 Continuum Models of PEM Fuel Cells Keith Promislow 1
PEM Fuel Cell: Macroview 2H 2 + O 2 2H 2 O + 1.2V Volts = Energy/electron This is the WRONG reaction 2
Opps 3
Half-Reactions in PEM Fuel Cells H 2 Pt 2H + + 2e Hydrogen Oxidation Reaction 4H + + 4e + O 2 Pt 2H 2 O + Heat Oxygen Reduction Reaction 4
Other Types of Fuel Cells 5
3D View of a PEM Fuel Cell y Fuel Oxidant x z Coolant The anode reaction The cathode reaction Anode Collector Plate Membrane Anode GDL anode GDL cathode Cathode + Catalyst layers H 2 2H + + 2e O 2 + 4H 3 O + + 4e 6H 2 O. 6 Consumption of O 2 and fuel along channels Gas diffusion in GDL Production of H 2 O and heat at cathode catalyst layers Build-up of double-layer charge at catalyst interface Ion and water motion in membrane Forms barrier for fuel, oxidant, e Conducts protons as H 3 O + Condensation/Evaporation of water Heat removal by coolant
Along-the-Channel Slice Dimensionless Quantities Cathode (Air) Stoich Oxidant Bipolar Plate + Water flux S c = Q O 2 I T /(4F ) Cathode GDL 250 µm Anode (Fuel) Stoich Membrane 25 50 µm S a = Q H 2 I T /(2F ) Anode GDL 250 µm Local Water transfer α = J w I/(2F ) Aspect ratio 2000 : 1 Bipolar Plate 0.5 1.0 meter Fuel 7
Coolant O 2 O 2 Bipolar + Plate O 2 + 4H O + 4e 3 6H2O Water Production O e 2 e O 2 H 2 O Diffusion Drag H O 3 + H2O Cathode GDL Catalyst PEM e H 2 H + 2 + 2 e H 2 O H e 2 H 2 H H 2O 2 2 e HO Catalyst Anode GDL H 2 H 2 H 2 Bipolar Plate 8
Ionomer Membranes: Network Formation 9
Slow Transients in Nafion Hydration In-situ MRI data of Nafion membrane exposed on one side to liquid water and on the other air at 20% RH. 10
WWW.POLYMERPHYSICS.ORG Membrane Macroscopic Properties FIGURE 5 Proton conductivities of 1100 EW Nafion and 555 EW SPEEK at 60 and 80 C as functions of hydrophilic volume fraction. The onset of the quadratic rise with volume fraction is identified as the critical fraction for percolation. The solid lines are fits to the data r ¼ r 0 (f þ (f _ ) c ) 2. Values of r 0 and (f þ ) c are listed in Table 2. that there is a threshold amount of sorbed water at which proton conductivity turns on and then increases with water activity. The water sorption data from Figure 2 was11com- bined with the conductivity measurements to identify the percolation threshold. Equation 9 is the volume fraction of the hydrophilic domains, (f ); this is given by the sum of FULL P 54 Adam Z. Weber and Figure 2.4 p H2 O p vap H2 O Activity showed two major differences in water sorption and pr conductivity between Nafion and SPEEK. 1. 1 Water sorption in Nafion had a positive excess volum mixing, while water sorption in SPEEK had a large n 0.8 tive excess volume of mixing. Saturated Liquid 2. The hydrophilic volume fraction percolation threshol 0.6 vapor water proton conductivity is much lower in Nafion tha SPEEK. 0.4 These differences in macroscopic properties results su 0.2 that there are differences in the hydrophilic domain m structures of Nafion and SPEEK. 0 The 0 spherical 2 4 cluster 6 8model 10 of 12hydrophilic 14 16 18 domains 20 22 prop by Gierke has been (mol the H common starting point to accoun 2 O/mol SO 3 ) the transport and mechanical properties of polymer ele lyte membranes. 16,17 This model is based on analog inverse micelles of hydrophilic groups in a hydrophobic vent. 52 54 To minimize the repulsive interaction en between the hydrophilic domains (sulfonic acid groups water) and the hydrophobic matrix, the hydrophilic gr are assumed to form spherical clusters. 55 63 As wat sorbed the spheres expand and begin to touch each oth the percolation threshold, the hydrophilic clusters fo continuous path through the matrix, and the proton con tivity increases exponentially with the volume fractio hydrophilic domains. Water-uptake isotherm at 25 C showing the effect of Schröd (left) Conductivity of Nafion is sensitive to pre-treatment, water uptake, and temperature, but generally exceeds saturated-vapor that of other reservoir ionic conducting is significantly membranes. lower than that in a liq (right) Water uptake of Nafion reservoir. increases This is with an important ambient RH, issue and since isfuel discontinuous cells are often ope from saturated vapor to liquid humidified water. gases, resulting in situations where there is liquid w cathodic side of the membrane and only water vapor on the a With this introduction, one can now dissect the isotherm and rel membrane microstructure. The general structure of Nafion, and ionomers in general, as of water content has The been threshold the source for percolation of many depends studies, on the as recentl shape o objects. 46,48,64 It has been proven that for randomly by Mauritz and Moore [22] and Kreuer et al. [4]. The experimenta
Catalyst Layer Carbon support material forms agglomerates (left) which are interpenetrated by ionomer material (right). 12
Platinum Catalyst At the anode H 2 is catalyzed into H +, which is conducted by the ionomer and e which travels through the Pt and carbon support. At the cathode H + from the ionomer, e from the external circuit, and O 2 from the GDL meet on the Pt surface, completing the reaction. 13
Flow field Quad serpentine Bipolar plate Solid Hybrid (one WTP) MEA and 3M Anode Catalyst Cross-sectional view of Membrane-Electrode Assembly (MEA) showing the ionomer flanked by the anode and cathode catalyst layers. The cathode catalyst is a traditional carbon-support/pt agglomerate while the anode catalyst is the 3M Nanostructured thin film (NSTF) from Pt-Ni alloy. 14
Butler-Volmer Equations Open Circuit Potential Potential Jumps with Current c Ea U= E + E = E c a o E c E a a m U E c c c a a Anode Membrane Cathode Anode Membrane Cathode Over-potentials η a and η c describe non-equilbrium loss of voltage ( ) ec [ ( ) ( co αc F I c = i o,c exp c O,ref RT η c exp (1 α )] c)f η c RT ( ) ea [ ( ) ( ch αa F I a = i o,a exp c H,ref RT η a exp (1 α )] a)f η a RT 15
Cell Voltage Polarization Curve The membrane resistance is a function of membrane water content R m = R m (λ), The catalyst oxygen conc. is a function of current and water content (flooding) C O (cat) = C O (ch) Iδ. The cell voltage is given by U = E o R m (λ)i η(i, C O (ch)). Fuel Cell Effeciency 0.8 Fuel Cell Power 1.2 0.7 Voltage 1 0.8 0.6 0.4 0.2 Power Density Watts/cm 2 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 Current Density Amps/cm 2 0 0 0.2 0.4 0.6 0.8 1 1.2 Current Density Amps/cm 2 Below the red line is the Cheetah regime Fuel Cell Power: 0.5 Watts/cm 2 300 cm 2 = 150 Watts/cellt 16
Cathode Overpotential Polarization Curve 1.2 Activation Polarization Theoretical Potential 1 Cell potential (V) 0.8 0.6 Ohmic Losses Overpotential Knee 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Current density (A/cm 2 ) At non-zero currents the cathode over potential takes the form η c Tafel Slope {}}{ RT α c F ( I ln i o,c ( c O,ref )) c O ( ch ) δi 17
1+1D Unit Cell Models Quasi-Steady Along the Channel Oxidant Bipolar Plate + Water flux Cathode GDL Membrane Anode GDL Bipolar Plate 0.5 1.0 meter Fuel 250 µm 25 50 µm 250 µm Cathode Oxygen flux Q c,o (x) Cathode Vapor flux Q c,v (x) Cathode Nitrogen flux Q c,n (x) Water cross-over/proton α(x) (An. Cath.) Anode Hydrogen flux Q a,h Anode Vapor flux Q a,v Coolant Temp. T col Reservoir Temp. T r Ignore build up of liquid water in channels. Exploit aspect ratio: couple 1D, steady-state transport for gas phases in channel with time dependent, 1D transport through MEA and accumulation of heat in coolant phase. 18
Channel Equations at Equilibrium The through-plane current I = I(x) drives the gas composition in channel dq c,o dx dq c,v dx dq c,n dq a,h dx = I(x)L w 4F, C c,o = P c = (1 + α(x)) I(x)L w 2F, C c,v = Min = 0, C dx c,n = P c = ± I(x)L w 2F, C a,h = P a dq a,v dx = ±α I(x)L w 2F, C a,v = Min { Pc Q c,o RT c (x) Q c,o +Q c,v +Q c,n, Q c,v RT (x) Q c,o +Q c,v +Q c,n, P sat(t c ) RT c (x) Q c,n RT c (x) Q c,o +Q c,v +Q c,n, Q c,h RT a (x) Q a,h +Q a,v, { P a Q a,v RT a (x) RT a (x) The coolant temperature is transient and coupled to resevoir t (ρct col) + x (ρct col v g ) = N T L w, dt r dt = q(t out col T r ) r(t r T amb ) }, } Q a,h + Q, P sat(t a ), a,v T col (0, t) = T r (t), 19
GDL: Transient Multiphase flow Degenerate transport of liquid water Disparate time scales: 10 6 s for pressure 10 3 s for liquid flow C o Oxygen Molar Conc. T Temperature C v Vapor Molar Conc. β Liquid Vol. Frac. C n Nitrogen Molar Conc. C Total Gas Conc. Conservation of Energy and Mass: (Γ denotes liquid-vapor phase change rate) t ((1 β)c) + {}}{ y( CU g ) = Γ, N {}} T { t (ρct ) + y(( ρcu)t κ y T ) = h lg Γ, { N }} o { t ((1 β)c o) + y ( C o U g + J o ) = 0, { N }} v { t ((1 β)c v) + y ( C v U g + J v ) = Γ, N {}} l { t (c lβ) + y ( βc l U l ) = Γ. N g 20
Constitutive Relations: P g = CRT, Ideal Gas Law U g = Kk rg(β) µ g y P g, Darcy s Law-Gas U l = Kk rl(β) µ y P l, l Darcy s Law-Liquid [J i ] = M 1 [ y C i ] Maxwell Stefan Flux P c = P g P l = L(β), Leveret-like Capillary Pressure. Γ = H(β)(C v C sat (T )), Condensation-Saturation Capillary pressure and relative permeability form nonlinear diffusivity f(β) = βk rl (β)l (β). f( ) Hydrophobic (Teflonated) x 100 t (β) D y(f(β) y β) = Γ/c l * Hydrophillic (sand) 21
Non-dimensional Form of GDL equations Collect the fluxes and unknowns N = (N g, N T, N o, N v, N l ) t and V = (C, T, C o, C v, β) t The full problem can be written as ( ( M( V ) V )τ + D( V ) y V ) y = S Γ, where S = ( 1, 1, 0, 1, 1) t is the scaled stoichiometry vector for phase change and the matrix D is given by R g k rg T C R g k rg C 2 0 0 0 δ l C lg T 2 δ l C lg CT + R T 0 0 R c T f(β) ( D = Rg k rg T 1 ) ( Co R C g k rg CC o 1 0 0 Rg k rg T 1 ) Cv R C g k rg CC v 0 1 0 R l βk rl (β)t R l βk rl (β)c 0 0 R c f(β)/δ l The local current production I is the driving force: fluxes at membrane proportional to I. A large charge flow is a small molar flux: I = O(10 2 ) 22
Membrane-Catalyst Layer Coupling (Boundary conditions) Membrane equilibrium hydration levels function of GDL RH level r, c (r) = 0.043+17.81r 39.85r 2 +36.0r 3. Flux out of membrane proportional to disequilibrium disequilibrium Flux {}}{{}}{ γ(c w ) (c w,a c w,a) = J GDL w,a = J m w,a + I F, γ(c w )(c w,c c w,c ) = J GDL w,c = J m w,c + 3I 2F. Heat Production in Catalyst ( T hrc Q heat = 4F γ 1 controls membrane water loss. ) + η c I c h v γ(c T c T (r)). 23
Bipolar Plate Flow Field Channel Fuel Anode GDL Dry Dry Dry Membrane Dry Cathode GDL Mixed Two Phase Dry z y Air Dry Two Phase Bipolar Plate + Coolant Break PDE into Dry, Two-phase, and a Boundary Layer regime O ( I H ). Exact solutions in Dry and Two-phase yield flux imbalance at wet-dry interface. Leads to explicit ODE for slow front evolution. K.P., J. Stockie, B. Wetton, Proc. Roy. Soc. London: Series A 462 (2006). 24
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1 Two phase verses Dry Polarization Curve 0.8 Voltage 0.6 0.4 0.2 Two phase Single phase 0 0 0.5 31 1 1.5 Current Amp/cm 2
Fuel Cell Stacks A single 60cm 30cm cell, operating at 1 Amp/cm2 and 0.6 Volts produces about a kilowatt in power. Automotive applications, which require almost a MegaWatt of power require Stacks. 32
Carbon Corrosion Oxygen Reduction/Reverse Oxygen Reduction (o) O 2 +4p + +4e 2H 2 O, Hydrogen Oxidation/Hydrogen Evolution (h) H 2 2p + +2e, Carbon Oxidation (c) C+2H 2 O 4p + + 4e + CO 2 Common Electrode potential, for reaction r=o, h, c, at anode/cathode E a/c = E r,ref + N r (C o, C h ) + η r (i r ), V cell = E c E a R Ω I, ( ) ( αr Fη i r = i r,ref {exp r RT exp (1 α )} r)fη r RT, Small current i r η r /R r, Large current η r α RT r F ln i r i r,ref Each reaction competes to provide the local current I(x) I(x) = i c (x) + i o (x) + i h (x) Oxidation reactions (producing electrons) i > 0 and η > 0, Reduction reactions (consuming electrons) i < 0 and η < 0 33
Kinetic Parameters Reaction Nerst E ref i ref α r Carbon Ox. (c) - 0.207V 1.1 10 5 A 0.324 m 2 Hydrogen Ev. (h) C ref H 2 = 100 mol 0 η m 3 h = R h i h R h = 0.10Ω-cm 2 Oxygen Red. (o) C ref O 2 = 40.9 mol 1.28V 9.3 10 4 A 1.0 m 3 m 2 Rev. Ox. Red. i o > 0 - η o = R o i o R o = 0.01Ω-cm 2 Oxygen/Hydrogen cross over, A = 3 10 3 m/s, J r = A(C r,a C r,c ) Unknowns Anode and Cathode catalyst layer conc. C o,a, C h,a C o,c, C h,c Partial currents i o,a, i h,a, i c,a i o,c, i h,c, i c,c Local current I(x) I(x) Anode and Cathode potentials E an E cat 34
Base case Coflow 0.2 Cathode Currents 0.8 Electrochemical Potentials 0 0.7 E c oxidation currents A/cm 2 0.2 0.4 0.6 i hc, approx 2.3 ma/cm 2 voltage 0.6 0.5 0.4 0.3 0.8 i 0.2 1 0.1 E a 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length An/Cat 2.2/2.0 barg, stoich=1.2/1.8, I T = 1 A/cm 2, V cell = 0.635V. Small hydrogen oxidation current (2.3 ma/cm 2 ) on cathode due to hydrogen crossover. This current is present at open circuit and causes the drop in open circuit voltage from 1.28 to 0.95V. 35
Partial Anode Understoich (idle) 0.01 Cathode Currents 0.05 Anode Currents 0 i hc 0.04 i oxidation currents A/cm 2 0.01 0.02 0.03 oxidation currents A/cm 2 0.03 0.02 0.01 i i oa 0.04 0 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length An/Cat 2.2/2.0 barg, stoich=1.1/1.8, I T = 30 ma/cm 2, V cell = 0.845V. Hydrogen cross-over drives cell to anode understoich. Oxygen reduction at anode (from crossover) and Reverse Oxygen reduction at cathode. Elevated cathode pot leads to sig. carbon corrosion. (1 mg of carbon/hour = 6 10 4 A). 36
Partial Anode Understoich (idle) 1.4 Electrochemical Potentials 1.2 x 10 4 Cathode Carbon Oxidation Current 1.2 1 1 voltage 0.8 0.6 0.4 E c oxidation current A/cm 2 0.8 0.6 0.4 0.2 E a 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length An/Cat 2.2/2.0 barg, stoich=1.1/1.8, I T = 30 ma/cm 2, V cell = 0.845V. Hydrogen cross-over drives cell to anode understoich. Oxygen reduction at anode (from crossover) and Reverse Oxygen reduction at cathode. Elevated cathode pot leads to sig. carbon corrosion. (1 mg of carbon/hour = 6 10 4 A). 37
Anode Understoich Coflow 1.4 Electrochemical Potentials 1.8 x 10 4 Anode Carbon Oxidation 1.2 E a 1.6 1 1.4 voltage 0.8 0.6 0.4 E c oxidation current A/cm 2 1.2 1 0.8 0.6 0.2 0.4 0 0.2 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 channel length An/Cat 2.2/2.0 barg, stoich=0.8/1.8, I T = 1 A/cm 2, V cell = 0.575V. Current near inlet is limited only by hydrogen mass transfer on anode. On cathode oxygen reduction (at mass transfer limit) plus hydrogen evolution. After depletion of hydrogen, anode and cathode pot. rise, leading to reverse oxygen reduction on anode, hydrogen oxidation and oxygen reduction on cathode. 38
Fuel Cell Issues Would like higher temperature operation to avoid CO poisoning, reduce the Cheetah effect, and improve the performance of non-precious metal (cheaper) catalysts. Requires the development of new (cheaper, higher temperature) polymer electrolyte membranes Lower Pt loading (mg/cm 2 ). Catalyst layers can degrade when run at high voltage, mostly due to understoich (Carbon Corrosion). Pt dissolution is also a long-term degradation worry. Liquid water builds up in catalyst layers and clogs oxygen transport (Flooding). Also freezethaw issues. 39