Modeling Cold Start in a Polymer-Electrolyte uelF Cell Ryan James … · 2018. 10. 10. · Modeling Cold Start in a Polymer-Electrolyte uelF Cell By Ryan James Balliet A dissertation

Modeling Cold Start in a Polymer-Electrolyte Fuel Cell

By

Ryan James Balliet

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Chemical Engineering

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor John Newman, ChairProfessor Clayton RadkeProfessor Thomas Devine

Fall 2010


c© 2010

By

Ryan James Balliet

Abstract


By

Ryan James Balliet

Doctor of Philosophy in Chemical Engineering

University of California, Berkeley

Professor John Newman, Chair

Polymer-electrolyte fuel cells (PEFCs) are electrochemical devices that create elec-tricity by consuming hydrogen and oxygen, forming water and heat as byproducts.PEFCs have been proposed for use in applications that may require start-up in envi-ronments with temperatures below 0 C. Doing so requires that the cell heat up, andwhen its own waste heat is used to do so, the process is referred to here as cold start.However, at low temperatures the cell's product water freezes, and if the tempera-ture does not rise fast enough, the accumulation of ice in the cathode catalyst layer(cCL) can reduce cell performance signicantly, extending the time required to heatup. In addition to reducing performance during cold start, under some conditions theaccumulation of ice can lead to irreversible structural degradation of the cCL.

The objective of this dissertation is to construct and verify a cold-start model for asingle PEFC, use it to improve understanding of cold-start behavior, and to demon-strate how this understanding can lead to better start protocols and material prop-erties. The macrohomogeneous model that has been developed to meet the objectiveis two-dimensional, transient, and nonisothermal. A key dierentiating feature is theinclusion of water in all four of the possible phases: ice, liquid, gas, and membrane. Inorder to predict water content in the ice, liquid, and gas phases that are present in theporous media, the thermodynamics of phase equilibrium are revisited, and a methodfor relating phase pressures to water content in each of these phases is developed.

Verication of the model is performed by comparing model predictions for cell behav-ior during parametric studies to measured values taken from various sources. In mostcases, good agreement is observed between the model and the experiments. Resultsfrom the simulations are used to explain the trends that are observed.

The veried cold-start model is deployed to determine a cold-start protocol and cCLproperties that enable better performance. Criteria include not only minimizing starttime but also exposure to high cCL ice pressures and cold-start energy while at thesame time maximizing power available from the cell during the cold-start process.

1

I dedicate this work to my parents and family, whose support helped see me throughthis endeavor, and to my wife, Judith, who is my sunshine.

i

Contents

1 An Introduction to the Problem of Cold Start 1

1.1 Polymer-Electrolyte Fuel Cells . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Advantages and disadvantages . . . . . . . . . . . . . . . . . . 1

1.1.2 Operability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Construction and operation . . . . . . . . . . . . . . . . . . . 4

1.1.4 Performance at normal operating temperature . . . . . . . . . 6

1.2 Cold Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Targets and performance gaps . . . . . . . . . . . . . . . . . . 8

1.2.2 The challenge of starting a cold PEFC . . . . . . . . . . . . . 10

1.2.3 The challenge of developing cold-start strategies . . . . . . . . 21

1.3 Objective and Approach . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Phase Equilibria and Frost Heave 34

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Phase Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 The chemical potential . . . . . . . . . . . . . . . . . . . . . . 39

2.3.2 Phase equilibria for pure water . . . . . . . . . . . . . . . . . 43

2.3.3 Corrections to the saturation vapor pressure . . . . . . . . . . 44

2.3.4 Corrections to the melting temperature . . . . . . . . . . . . . 47

2.4 Understanding PCI Flow and Frost Heave . . . . . . . . . . . . . . . 48

2.4.1 The driving force . . . . . . . . . . . . . . . . . . . . . . . . . 48

ii

2.4.2 Relevant Pore Properties . . . . . . . . . . . . . . . . . . . . . 50

2.4.3 Examples of PCI Flow and Frost Heave in Pores . . . . . . . . 53

2.5 Calculating Ice, Liquid, and Gas Saturations . . . . . . . . . . . . . . 61

2.5.1 Saturation-model development . . . . . . . . . . . . . . . . . . 61

2.5.2 Saturation-model results . . . . . . . . . . . . . . . . . . . . . 68


3 Two-Dimensional Cold-Start Model 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Cold-Start Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.2.1 Energy balance . . . . . . . . . . . . . . . . . . . . 76

3.2.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . 79

3.2.2.3 Diusion . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2.2.4 Rate expressions for phase change . . . . . . . . . . . 82

3.2.2.5 Transport of ions and water in the membrane . . . . 83

3.2.2.6 Membrane water content . . . . . . . . . . . . . . . . 84

3.2.2.7 Electron transport . . . . . . . . . . . . . . . . . . . 85

3.2.2.8 Electrode kinetics . . . . . . . . . . . . . . . . . . . . 85

3.2.2.9 Boundary conditions . . . . . . . . . . . . . . . . . . 86

3.3 Cold-Start Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 88


4 Model Verication Using Parametric Studies 91

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.1 Parameter types . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.2 Cold-start classication . . . . . . . . . . . . . . . . . . . . . . 93

4.2.3 Experimental cell congurations . . . . . . . . . . . . . . . . . 94

4.2.4 Simulating nonisothermal cold starts . . . . . . . . . . . . . . 96

iii

4.2.5 Experimental cold-start procedures . . . . . . . . . . . . . . . 99

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3.1 Aggregated results . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3.2 Isothermal parametric studies . . . . . . . . . . . . . . . . . . 103

4.3.2.1 The eect of start-up temperature . . . . . . . . . . 103

4.3.2.2 The eect of start-up current density . . . . . . . . . 104

4.3.2.3 The eect of initial membrane water content . . . . . 106

4.3.2.4 The eect of cathode-catalyst-layer thickness . . . . 107

4.3.3 Nonisothermal results . . . . . . . . . . . . . . . . . . . . . . . 108


5 Optimization of Procedural and Congurational Parameters 113

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Baseline Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.1 Rated power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.2 Galvanostatic and potentiostatic cold-start results . . . . . . . 114

5.3 The Eect of Operational Parameters . . . . . . . . . . . . . . . . . . 120

5.3.1 The eect of start potential . . . . . . . . . . . . . . . . . . . 120

5.3.2 The eect of initial relative humidity . . . . . . . . . . . . . . 122

5.3.3 The eect of start temperature . . . . . . . . . . . . . . . . . 124

5.4 The Eect of Cathode-Catalyst-Layer Congurational Parameters . . 124

5.4.1 The eect of catalyst-layer ionomer content . . . . . . . . . . 126

5.4.2 The eect of cathode catalyst layer porosity . . . . . . . . . . 127

5.4.3 The eect of specic interfacial area . . . . . . . . . . . . . . . 128

5.5 Improved Cold-Start Performance . . . . . . . . . . . . . . . . . . . . 129


6 Conclusions and Future Work 132

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2.1 Improvements to the model . . . . . . . . . . . . . . . . . . . 134

iv

6.2.1.1 Incorporate better kinetic expressions for freezing inporous media . . . . . . . . . . . . . . . . . . . . . . 135

6.2.1.2 Improve the membrane/ionomer water-uptake and trans-port model below 0 C . . . . . . . . . . . . . . . . . 135

6.2.1.3 Add a contact-angle distribution to the saturation cal-culations . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.2 Additional Studies . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2.2.1 The cold-start behavior of and strategies for cells us-ing thin catalyst layers . . . . . . . . . . . . . . . . 136

6.2.2.2 The eect of a cell's position in the cell stack on itscold-start performance . . . . . . . . . . . . . . . . 136

6.2.2.3 Mechanical modeling of the cell during cold-start . . 137

Bibliography 138

A Notation 146

v

List of Figures

1.1 Ragone plot comparing fuel-cells to an IC engine, various battery tech-nologies, and capacitors. Reprinted with permission from [1]. Copy-right 2008, American Institute of Physics. . . . . . . . . . . . . . . . 2

1.2 (a) Expanded view of a single PEFC. (b) Cells arranged in a stack, asthey are for many applications. . . . . . . . . . . . . . . . . . . . . . 5

1.3 Cell potential and power density as a function of current density for aPEFC at a temperature of 65 C, as predicted by the 0-D model. Fullyhumidied H2/air operation is assumed, with 1 bar of total pressurefor each gas. The kinetic and transport overpotentials for the hydrogenelectrode are neglected. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 The eect of temperature on (a) the ORR and (b) the ionic resistance ofNaon 117, as measured by Thompson et al. Reprinted with permissionfrom [17]. Copyright 2007, The Electrochemical Society. . . . . . . . 12

1.5 The eect of temperature on (a) oxygen partial pressure (in 1 bar ofsaturated air) and on (b) the oxygen limiting current as calculatedfrom equation 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 The eect of temperature and current density on cell potential usingthe 0-D model. The numbers next to each line correspond to theoperating temperatures, with units of C. . . . . . . . . . . . . . . . 14

1.7 Predictions from the 0-D model for (a) the maximum power densitywith temperature and (b) the fraction of the total loss in potential atmaximum power due to kinetic, ohmic, and transport limitations (forcase A as dened in (a)). . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8 Predictions from the 0-D model for the eect of temperature on (a) theenthalpy and standard potentials and (b) of temperature and currentdensity on kinetic losses. . . . . . . . . . . . . . . . . . . . . . . . . 17

1.9 Predictions from the 0-D model for the eect of temperature and cur-rent density on (a) ohmic and (b) mass-transport losses. . . . . . . . 18

vi

1.10 Fraction of product water that can be removed as vapor in the cathodeexhaust stream, at various stoichiometric ratios or oxygen, assumingsaturated gas at 1 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.11 (a) Cell potential as a function of time for a cell operating at a constanttemperature of -20 C and a constant current of 0.01 A/cm2. Theexperimental result is from [18]. (b) A cryo-SEM image taken of a cCLafter being operated until failure at 0.01 A/cm2 (as in (a)), and (c)an image of the same cCL after sublimation was used to remove theice. Images reprinted with permission from [18]. Copyright 2008, TheElectrochemical Society. . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.12 TEM images from of three dierent membrane-electrode assemblies(MEAs): (a) a fresh MEA before any cycling, (b) an aged MEA after150 galvanostatic cold-start cycles at -30 C and 0.3 A/cm2, and (c)an aged MEA after 110 galvanostatic cold-start cycles at -20 C and0.5 A/cm2. Reprinted with permission from [19]. Copyright 2008, TheElectrochemical Society. . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.13 Cumulative water produced prior to failure when a cell operates at aconstant current density without being allowed to heat up. Shown asa function of (a) start current density (at -20 C, from [18]), and (b)start temperature (at 0.1 A/cm2, from [20]). . . . . . . . . . . . . . . 23

1.14 Cumulative water produced prior to failure when a cell operates at aconstant current density without being allowed to heat up. Shown asa function of cCL thickness (at -20 C and 0.1 A/cm2, from [21]). . . 24

1.15 Predictions using the 0-D model for (a) heat generation and (b) powerdensity as a function of temperature and cell potential. . . . . . . . . 26

1.16 Predictions, using equations 1.10, 1.11, and the 0-D cell-performancemodel, for start time as a function of (a) cell potential (at a starttemperature of -20 C) and (b) temperature (at a start potential ofeither 0.4 or 0.6 V). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.17 Experimental results compared with predictions from the 0-D model forgalvanostatic cold starts from (a) -10 C and (b) -30 C. Experimentalresults from UTC Power.22 . . . . . . . . . . . . . . . . . . . . . . . . 29

1.18 Intended use for the 2-D cold-start model developed in subsequentchapters of the present work. . . . . . . . . . . . . . . . . . . . . . . . 31

vii

2.1 (a) Damage to a roadway caused by frost heave.27 (b) Ice lenses ina clay sample from under a heaved street in St. Peter, MN. (c) Icelenses grown in clay in laboratory frost-heave experiments by Taber.(d) Example of needle ice growing out of a log. (e) Delaminationof the cathode catalyst layer in a fuel cell from the membrane afterrepeated freeze-thaw cycles. Images in (b) and (c) reprinted with per-mission from [28]. Copyright 1930, The University of Chicago Press.Image in (d) reprinted with permission from [29]. Copyright 2006, An-nual Reviews. Image in (e) reprinted from [30], Copyright 2007, withpermission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Illustration of various terms when (a) PCI ow and pore lling areoccurring without frost heave, and (b), PCI ow and frost heave areoccurring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Phase diagram for pure water in the range of interest for cold-startsimulations of a polymer-electrolyte fuel cell. . . . . . . . . . . . . . . 40

2.4 System congurations considered in the phase equilibria discussion. In(a), (b), and (c) only at interfaces exist, while (d) and (e) containcurved interfaces. Gas-phase components are listed in parentheses. . 44

2.5 Corrections to the vapor pressure of pure liquid water, pV , at 65C,

relative to the vapor pressure with no gas-phase diluent and no in-terfacial curvature, psat,0

L . Positive values of 1/RLG correspond to aspherical drop of pure liquid immersed in gas that contains vapor andmay also contain diluent. Negative values correspond to a bubble of gasimmersed in liquid. Each line corresponds to a dierent gas pressure,pG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Corrections to the melting temperature, Tm, of a spherical drop of pureice, with radius RIL, immersed in liquid, relative to the triple point forpure water with no interfacial curvature, Tt. Each line corresponds toa dierent liquid pressure, pL. . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Ice and liquid pressures for three cases: (1) pI = pL (bold solid line),(2) pL (dot-dashed line) varies as pI is xed at 1 bar, and (3) pI (dashedline) varies as pL is xed at 1 bar. The markers (M and ) correspondto points specied for the cases in Figure 2.9. The inset shows thesame series as the main gure, assuming that the molar volumes forice and liquid are reversed. . . . . . . . . . . . . . . . . . . . . . . . 49

2.8 Cylindrical pores of radius rαβ containing a stationary spherical inter-face of radius Rαβ between uid phases α and β. Bold, straight arrowsindicate force vectors; thin, straight arrows indicate radii; and thin,curved arrows indicate contact-angle arcs. . . . . . . . . . . . . . . . 51

viii

2.9 (a) Schematic showing key characteristics and dimensions for the poretype used in the examples. (b) Two pores, one hot and one cold,containing liquid connected by a capillary. . . . . . . . . . . . . . . . 53

2.10 Equilibrium liquid, ice, and gas congurations for a hydrophilic pore(left) and a hydrophobic pore (right). In each successive subgure, (a)through (e), more heat is removed. . . . . . . . . . . . . . . . . . . . 55

2.11 Ice and liquid pressures for the HI and HO pores in Figure 2.10. Solidlines indicate the pressure of ice or liquid that exists in the pore, whiledashed lines indicate the pressures that would be required in order forthem to exist at a given temperature. The gas pressure is assumed tobe constant at 1 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.12 An unsaturated pore (a) with a liquid connection to pores with in-creasing levels of ice pressure, (b) through (d). . . . . . . . . . . . . 58

2.13 Examples of three methods for stopping PCI ow. . . . . . . . . . . . 60

2.14 Liquid saturation for fuel-cell media as a function of capillary pressureat temperatures where no ice is present. . . . . . . . . . . . . . . . . 63

2.15 Ice and liquid saturations in the CL as a function of capillary pressure. 64

2.16 Algorithm for determining liquid, ice, and gas saturations. . . . . . . 66

2.17 Liquid saturation for fuel-cell media at temperatures below the meltingpoint, assuming a constant ice pressure of 1 bar. . . . . . . . . . . . 68

2.18 Liquid saturation below the melting point for the MPL at dierenttotal saturations (liquid plus ice). . . . . . . . . . . . . . . . . . . . . 69

2.19 Liquid saturation predicted in the MPL when the saturation modelis used to account for the ice-gas and ice-liquid interfaces (series SL)compared with the prediction if the uncorrected liquid-saturation re-lationship to liquid capillary pressure from Figure 2.14 is used directly(series SLG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.1 The two-dimensional modeling domain used in this work. Solid linescorrespond to subdomain boundaries. . . . . . . . . . . . . . . . . . . 74

3.2 Simulated potential- and average-cell-temperature proles during astart-up from -10 C at 0.05 A/cm2. . . . . . . . . . . . . . . . . . . . 89

3.3 Predictions of water content in various phases and cell materials duringa start-up from -10 C at 0.05 A/cm2. . . . . . . . . . . . . . . . . . 90

4.1 Typical cell-temperature and cell-potential proles versus time for agalvanostatic start-up assuming (a) isothermal and (b) nonisothermalexperimental types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

ix

4.2 Illustrations of the cell used to generate nonisothermal cold-start data:(a) the cell assembly, (b) rigid insulation surrounding the cell package,and (c) the entire cell assembly, with endplates applying pressure tothe insulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Denition of the web region within the plate (P). . . . . . . . . . . . 98

4.4 Correlation plots for (a) cumulative water produced, mstartH2O, and (b)

initial cell potential, Vinit. . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Eect of start temperature. H2/air operation with a constant currentdensity of 0.1 A/cm2. For the simulation, λinit = 8.0. Experimentaldata from [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Eect of start current density. H2/O2 operation with a constant starttemperature of -20 C. For the simulation, λinit as shown in legend.Experimental data from [18]. . . . . . . . . . . . . . . . . . . . . . . . 105

4.7 Eect of start current density. H2/air operation with a constant starttemperature of -10 C. For the simulation, λinit = 3.7. Experimentaldata from [89]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.8 Eect of initial membrane water content as calculated from equation4.4. H2/air operation with a constant start temperature of -20 C.Current density of 0.04 A/cm2, constant after a linear ramp from 0A/cm2 during the rst 80 s of operation. Experimental data from [90]. 106

4.9 Eect of initial membrane water content as calculated from equation4.4. H2/O2 operation with a constant start temperature of -20 C.Constant current density of 0.05 A/cm2. Experimental data from [18]. 107

4.10 Eect of cathode-catalyst-layer thickness. H2/air operation with aconstant start temperature of -20 C. Constant current density of 0.1A/cm2. For the simulation, λinit = 7.4. Experimental data from [21]. . 108

4.11 Nonisothermal cold start from -10 C. H2/air operation with a con-stant current density of 0.6 A/cm2. For the simulation, λinit = 6.0.Experimental results from UTC Power.22 . . . . . . . . . . . . . . . . 109

4.12 Nonisothermal cold start from -20 C. H2/air operation with a constantcurrent density of 0.6 A/cm2. For the simulation, λinit is as listed inthe legend. Experimental results from UTC Power.22 . . . . . . . . . 110

4.13 Nonisothermal cold start from -30 C. H2/air operation with a con-stant current density of 0.6 A/cm2. For the simulation, λinit = 6.0.Experimental results from UTC Power.22 . . . . . . . . . . . . . . . . 111

5.1 Simulation results for the performance of the baseline cell at 75 Cusing H2/air, both at 1 bar of total pressure and with reactant relativehumidities of 84 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

x

5.2 Comparison of simulation results for a cold start from -20 C usingeither a galvanostatic (0.5 A/cm2) or potentiostatic (0.4 V) approach.Additional operating conditions include H2/air operation at 1 bar oftotal pressure and λinit = 6 mol H2O/mol SO−3 . . . . . . . . . . . . . 116

5.3 Additional results for the cold starts shown in Figure 5.2: (a) cellpotential and current density and (b) total waste heat generated bythe cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4 Ice (a) saturation and (b) pressure proles across the thickness of thecathode catalyst layer for the cold starts described in Figure 5.2 aswell as an additional galvanostatic cold start at 0.1 A/cm2. All prolescorrespond to the time when the average ice pressure in the cCL reachesa maximum during the cold start. They are all taken along the ribsymmetry line dened in Chapter 2. . . . . . . . . . . . . . . . . . . 119

5.5 The eect of start potential on (a) the time to 50 % of rated powerand the maximum fraction of the cCL that experiences an ice pressuregreater than 23 bar, as well as (b) the average power available from thecell prior to reaching 50 % power and the amount of energy requiredduring that same period. Results assume a start-up from -20 C usingpotentiostatic approach, H2/air operation at 1 bar of total pressure,and λinit = 6 mol H2O/mol SO−3 . . . . . . . . . . . . . . . . . . . . . 121

5.6 The eect of initial membrane and ionomer water content on the timeto 50 % of rated power and on the maximum fraction of the cCL thatexperiences an ice pressure greater than 23 bar. Results assume a start-up from -20 C using a potentiostatic approach, H2/air operation at 1bar of total pressure, and a cold-start potential of 0.65 V. . . . . . . . 123

5.7 The eect of initial start temperature on (a) the time to 50 % of ratedpower and the maximum fraction of the cCL that experiences an icepressure greater than 23 bar, as well as (b) the average power availablefrom the cell prior to reaching 50 % power and the amount of energyrequired during that same period. Results assume H2/air operation at1 bar of total pressure, λinit = 5 mol H2O/mol SO−3 , and a start-uppotential of 0.65 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.8 The eect of the volume fraction of ionomer present in the cCL on thetime to 50 % of rated power and on the maximum fraction of the cCLthat experiences an ice pressure greater than 23 bar. Results assume astart-up from -20 C using a potentiostatic approach, H2/air operationat 1 bar of total pressure, λinit = 5 mol H2O/mol SO−3 , and a cold-startpotential of 0.65 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xi

5.9 The eect of the porosity of the cCL on the time to 50 % of ratedpower and on the maximum fraction of the cCL that experiences anice pressure greater than 23 bar. Results assume a start-up from -20C using a potentiostatic approach, H2/air operation at 1 bar of totalpressure, λinit = 5 mol H2O/mol SO−3 , and a cold-start potential of0.65 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.10 The eect of the specic interfacial area of the cCL catalyst on thetime to 50 % of rated power and on the maximum fraction of the cCLthat experiences an ice pressure greater than 23 bar. Results assume astart-up from -20 C using a potentiostatic approach, H2/air operationat 1 bar of total pressure, λinit = 5 mol H2O/mol SO−3 , and a cold-startpotential of 0.65 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.11 For the improved cCL conguration, the eect of initial start tempera-ture on (a) the time to 50 % of rated power, and the maximum fractionof the cCL that experiences an ice pressure greater than 23 bar, and (b)the average power available from the cell prior to reaching 50 % power,and the amount of energy required during that same period. AssumesH2/air operation at 1 bar of total pressure, λinit = 5 mol H2O/molSO−3 , and a start-up potential of 0.65 V. . . . . . . . . . . . . . . . . 130

xii

List of Tables

1.1 Comparison of the overall eciency for PEFCs to that of Li-ion batteries. 3

1.2 Parameter values used in the 0-D model. . . . . . . . . . . . . . . . . 7

1.3 Automotive cold-start targets for an 80-kW cell stack, and status as of2005, as established by the Department of Energy.4 . . . . . . . . . . 10

1.4 Typical ice capacity and heat capacity for cell components based onthe properties listed in Chapter 3 of this work. . . . . . . . . . . . . . 25

2.1 Thermodynamic properties of pure water at the triple point.38,39 . . 41

2.2 Properties for the three possible uid interfaces.4044 . . . . . . . . . . 42

2.3 Contact angles for the dierent interfaces.45,50,51 . . . . . . . . . . . . 52

2.4 Physical properties used for determining saturations.50,60 . . . . . . . 62

3.1 For the baseline model conguration, subdomain dimensions and num-ber of elements as well as phases allowed. . . . . . . . . . . . . . . . 75

3.2 Thermodynamic and transport properties of the uid components. . . 77

3.3 Baseline thermodynamic and transport properties for the cell materi-als. Values for the M subdomain are taken to also apply to ionomer inthe CL subdomains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Energy-balance source terms. . . . . . . . . . . . . . . . . . . . . . . 79

3.5 Baseline physical properties for the porous media. . . . . . . . . . . . 79

3.6 Source terms for mass conservation (equation 3.10). . . . . . . . . . . 80

3.7 Stefan-Maxwell binary diusivities, where T is in degrees Kelvin.71 . . 82

3.8 Rate constants used in the porous media subdomains. . . . . . . . . . 83

3.9 Membrane property expressions and parameter values for the baselineconguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.10 Electrode and reaction properties for the baseline conguration. . . . 86

xiii

3.11 Boundary conditions used for fuel-cell simulations. . . . . . . . . . . 87

3.12 Values used to compute boundary conditions for the cold-start simu-lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1 Experimental congurations simulated by the model. All congura-tions contain microporous layers on both the anode and the cathode. 95

4.2 Congurational tting parameters used to simulate each type of cell.Values for all other parameters are as dened in Chapters 2 and 3,except for the membrane and cathode catalyst layer thicknesses, whichare shown in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . 101

xiv

Acknowledgments

I thank UTC Power for providing funding for this project. In particular, I thank toMichael Perry, Timothy Patterson, Badri Paravastu, and Tom Madden for their rolesin providing contract support, technical feedback, and experimental results. I thankNewman Lab members, past and present, for their support. I thank Adam Weberof LBNL, who has been, both in print and in person, an invaluable resource. Mostof all, I thank Professor Newman. He has provided me with patient instruction andmentoring while keeping my spirits up with thoughtful encouragement. It has beena pleasure and an honor to work with him.

xv

Chapter 1

An Introduction to the Problem of

Cold Start

1.1 Polymer-Electrolyte Fuel Cells

Polymer-electrolyte fuel cells (PEFCs) are electrochemical devices that create elec-tricity by consuming hydrogen and oxygen, forming water and heat as byproducts.Within the category of PEFCs, there are various types of cell designs. In this disser-tation, discussion is limited to acid-based PEFCs, operating on pure hydrogen fuel,with typical operating temperatures between 60 and 90 C.

1.1.1 Advantages and disadvantages

Systems that use PEFCs to generate electricity are of interest due to their low levels oflocal harmful emissions (zero if pure hydrogen is used and completely consumed), theirhigh eciency relative to internal-combustion (IC) engines, their ability to be refueledin a matter of minutes, their reasonable specic power, and their high specic energyrelative to current batteries. Figure 1.1 is a Ragone plot comparing the performance ofa typical fuel-cell system in the two latter areas to that of various battery technologies,capacitors, and the IC engine.1 Specic energy, which for transportation applicationsdetermines range, is plotted as a function of average specic power, which determinesacceleration. Currently available fuel-cell technology is assumed in this analysis, andthe mass of the hydrogen container is included in the calculation. On this basis, fuelcells fall short of IC engine performance. Nevertheless, they compare favorably tocurrent battery technology. Higher specic power may be possible with a battery orcapacitor, but that additional power comes at the expense of decreased energy.

For applications requiring energy storage with zero emissions and a renewable sourceof energy, a signicant drawback of PEFCs relative to batteries is their overall energy

1

Figure 1.1: Ragone plot comparing fuel-cells to an IC engine, various battery technolo-gies, and capacitors. Reprinted with permission from [1]. Copyright 2008, AmericanInstitute of Physics.

eciency, dened as the ratio of the electrical energy out of the fuel cell or batterywhen the hydrogen is consumed (or the battery is discharged) to the energy containedin the original resource that was used to create the hydrogen (or charge the battery).The overall eciency is the product of the eciency of electricity production (ifapplicable), the eciency of producing hydrogen or charging the battery, and theeciency consuming the hydrogen or discharging the battery.

Table 1.1 compares a PEFC system to a Li-ion battery system on the basis of e-ciency. Typical values for the three intermediate steps dened above are shown, asis the overall eciency. Additional losses due to the transmission of electricity orthe compression, distribution, and storage of the hydrogen are not considered in thepresent analysis. The second and third columns contain values based on using naturalgas as the original energy source. For the PEFC case, hydrogen is produced directlyfrom the natural gas using steam reforming. In the case of the Li-ion battery, thenatural gas is used to produce electricity via a combined-cycle power plant. In thisscenario, the PEFC system compares favorably to the battery system due to the higheciency of the steam reforming process. However, there are emissions of greenhousegases and other pollutants at the point of hydrogen (or electricity) production, andthese processes are not renewable because they depend on a fossil fuel.

When a renewable resource is used instead, as in the fourth and fth columns, theresults change considerably. In this case, the electricity produced from the renewable

2

Table 1.1: Comparison of the overall eciency for PEFCs to that of Li-ion batteries.

Typical eciency (%)

Energy source: Natural gas Wind

PEFC Li-ion battery PEFC Li-ion battery

Electricity production2,3 38 44 44

H2 production4 or charge5 70 94 60 94

H2 consumption4 or discharge5 55 90 55 90

Overall 39 32 15 37

is used to produce hydrogen via electrolysis or to charge the battery. Wind is usedas the renewable energy source in this analysis, but others could be used instead.Doing so would change the absolute values calculated for overall eciency, but notthe conclusion: if electricity generated from a renewable resource is available, storingthat energy in a battery and then discharging the battery is currently much moreecient than storing it in the form of hydrogen and then consuming the hydrogen ina fuel cell.

Another problem with PEFCs is their high capital cost; this issue has thus far lim-ited their use to demonstration programs and niche applications. The United StatesDepartment of Energy (DOE) cost target for fuel-cell systems for automotive appli-cations is $0.03/W. A study by Tsuchiya and Kobayashi in 2004 estimated the costof a PEFC to be $1.83/W.6 Multiplying this value by a specic power of 150 W/kgand dividing by a specic energy of 200 Wh/kg (based on Figure 1.1) produces anestimate for the capital cost of energy storage in a PEFC system: $1.37/Wh. Thisnumber includes only the cost of the PEFC system (including the hydrogen tank), notthe cost of any hydrogen-production equipment. For comparison, the cost of energystorage using current Li-ion battery technology is in the range of $0.30 to $0.60/Wh,7

meaning that the cost of power is in the range of $0.08 to 0.15/W, assuming a ratioof 4:1 in specic power to specic energy, a point consistent with the curve shownin Figure 1.1 and with the battery specications from several vehicles currently forsale.810

Durability is also an area in which PEFCs currently fall short of targets, at least forapplications in which both high cycle life and extended operational time are required.For the automotive application, the DOE lifetime target is 5000 operational hours,including all of the dierent types of cycles that are expected, such as power transientsand start-up and shutdown events. Such cycles have a signicant impact on theoperational life of a fuel cell.11,12 The DOE estimates that under these aggressiveconditions the technology as of 2005 was capable of a 1000-hour life.4

3

1.1.2 Operability

For the reasons described above, a signicant amount of current PEFC research isfocused on improving eciency, reducing cost, and improving durability. Anotherimportant issue for PEFCs, however, is operability. That is, their ability to performunder conditions (such as temperature, pressure, and relative humidity) outside of thenominal design envelope. In many applications, it is cost-prohibitive for the systemto control precisely the operating conditions for the cell at all times, and so the cellmust be able to tolerate these o-design modes, at least temporarily. In this sense,operability is a cross-cutting discipline within PEFC research: system eciency, cost,and durability are all at stake. Starting a cell from a low temperature, which is thetopic of the present work, is a prime example. Most of the time that the cell is in useits waste heat should be sucient to keep it operating in the nominal temperaturerange. However, for many applications it must rst be able to heat itself to that levelquickly, using a minimal amount of fuel, and without damage.

1.1.3 Construction and operation

Figure 1.2a shows an expanded view of the layers inside a conventional PEFC. Thetotal thickness of the layers shown is typically 2 to 3 mm. Each of the outermostlayers is an electronically conductive plate (P), which contacts the rest of the cellthrough ribs and partially encloses channels (CHs). The channels conduct gases (andoccasionally some liquid water) across the cell's active area.

Immediately adjacent to the ribs and channels are the gas-diusion layers (GDLs), andadjacent to them are the microporous layers (MPLs). Together, the GDLs and MPLscomprise the cell's diusion media (DM), whose primary functions are to permit gasand liquid to ow into and out of regions under the ribs and electrons to ow into andout of regions under the channels. Gas-diusion layers are typically highly porousstructures in cloth or paper form made of carbon bers. Usually, they are covered atleast partially with a coating of hydrophobic material, such as Teon R©. Microporouslayers, in contrast, are made of carbon particles that are either mixed or coated witha hydrophobic material. MPL porosities and pore sizes tend to be signicantly lowerthan those found in the GDL.

Adjacent to the MPLs are the catalyst layers (CLs), whose primary function is to fa-cilitate the fuel-cell half-reactions. At the anode (a), the hydrogen oxidation reaction(HOR) occurs,

2H2 → 4H+ + 4e−, (1.1)

while the oxygen reduction reaction (ORR) takes place at the cathode (c):

4H+ + 4e− + O2 → 2H2O. (1.2)

4

(b)single-

cellpackage

endstructure

cell stack

(-) (+)

(-)

(a)

Figure 1.2: (a) Expanded view of a single PEFC. (b) Cells arranged in a stack, asthey are for many applications.

These reactions are catalyzed, often by pure or alloyed Pt, within each layer, whichmust be accessible to both electrons and ions. For this reason the catalyst layerscontain both carbon particles, which provide an electronic connection to the MPL,and ionomer (ionically conductive polymer), which provides an ionic path to theproton-exchange membrane (M) located between the two CLs. The primary functionsof the M are to conduct cations from the anode to cathode, provide a barrier toelectronic current, and to separate the reactants from one another. Finally, pathwaysfor reactant gases and product water are also required, and thus the catalyst layerscontain ne pores to permit the transport of both liquid and gas.

Typically, cells are arranged in a stack, as illustrated in Figure 1.2b, in order to raisethe output voltage to a level convenient for the application of interest. The supercial

5

area necessary for the cell can then be determined based on the power requirementand the voltage that is expected from a cell at a given current density. The numberof cells in a stack varies widely based on the application, from 40 to 80 for 24 and 48V systems to 300 to 400 for 240 V systems. At both ends of the cell stack are theanode and cathode end structure. The functions of the end structure are to collectcurrent from the stack as well as to distribute the mechanical load that holds the stacktogether (often the anode and cathode end structures are connected via tie rods).

1.1.4 Performance at normal operating temperature

To model the cell potential, V, as a function of current density, i, the followingequation is often used:13,14

V = U +RT

Fln

(a0

1,2iORR0 δeff

CL

ipchan

O2

)−Reffi+

RT

Fln

(1− i

ilim

)(1.3)

The eect of the hydrogen electrode on V has been neglected, as it is generally quitesmall relative to the other terms when pure hydrogen is used as the fuel. Also,no potential loss due to electronic shorting or gas crossover is considered, as thesenormally aect potential at lower current densities than are of interest in the presentwork.

The rst term on the right side of equation 1.3 is referred to here as the open-circuitpotential. Technically, it is the open-circuit potential of a cell operating with 1 barof oxygen pressure at a given temperature, T, which is not necessarily the standardreference temperature.

The second term on the right represents the potential loss due to the kinetics ofthe ORR, assuming Tafel behavior and a rst-order dependence on oxygen partialpressure, pchan

O2. The superscript chan indicates that the value that is used is that

from the gas channel, which is dierent from that at the electrode due to oxygenmass-transport eects, which are captured in a separate term of the equation, and willbe discussed below. R and F are the universal gas constant and Faraday's constant,respectively, while a0

1,2, iORR0 , and δeff

CL are the specic interfacial area, exchange currentdensity, and the eective catalyst-layer thickness. The eective thickness is taken tobe less than the actual thickness of the catalyst layer because the reaction usuallytakes place primarily at the interface between the M and the CL due to the highconductivity of the electronically conductive phase relative to that of the ionicallyconductive phase.

In order to account for the change in pchanO2

from the inlet to the exit of the cell, theaverage of the inlet partial pressure and the exit partial pressure is used based onthe stoichiometric ratio for oxygen, γO2 . The stoichiometric ratio is dened as themolar ow rate of oxygen into the cell divided by the molar rate of consumption by

6

Table 1.2: Parameter values used in the 0-D model.

δeffCL γO2 δeff

M δeff

µm µm cm

4 2.0 70 0.237

ORR. At high stoichiometries, there is very little dierence between the inlet and exitpartial pressure, but PEFCs generally operate with stoichiometries between 1 and 2,where the dierences can be signicant.

The third term on the right side of equation 1.3 accounts for losses due to the internalresistance of the cell, referred here to as ohmic losses. Most of the eective resistanceof the cell, Reff , is due to the ionic resistances of the membrane and the ionomerin the catalyst layers, which are in series. Rather than estimating these resistancesseparately, for this simplied analysis, Reff is given by

Reff =δeff

M

κ, (1.4)

where κ is the ionic conductivity and δeffM is the eective thickness of the membrane.

δeffM is thicker than the actual cell membrane because it accounts for the ohmic dropthrough the catalyst layers as well.

The nal term in equation 1.3 accounts for the potential loss due to the transportof oxygen from the gas channel to the electrode. It is expressed as a function ofthe limiting current density, ilim, which is the current density of the cell when theoxygen partial pressure at the electrode goes to zero. The limiting current density iscalculated from14

ilim = 4FDO2

δeffcchan

O2(1.5)

where cchanO2

(calculated from pchanO2

) is the concentration of oxygen in the gas channel,DO2 is the eective diusion coecient for oxygen, and δeff is the eective pathlength for diusion, which accounts for the presence of the porous media, includingthe resulting tortuosity.

Of the parameters described above, the following are taken to be functions of tem-perature: U , iORR

0 , pchanO2

, κ, and DO2 . The oxygen partial pressure, pchanO2

, whichaects both the kinetic term and the transport term (through the ilim calculation)is a function of temperature because the cathode gas contains water vapor, and thepartial pressure of water vapor increases signicantly with temperature. Therefore,for a xed total gas pressure, pchan

O2decreases as the temperature increases.

Values for most of the parameters in the equations above, including their dependenceon temperature, are taken from Chapters 2 and 3 of this dissertation. Table 1.2summarizes the values used for those that are not used in those chapters.

7

Although equation 1.3 contains characteristic dimensions, it does not explicitly ac-count for spatial variations of any of the parameters. For this reason, it is referred toas a zero-dimensional (0-D) model for cell potential as a function of temperature andcurrent density. Figure 1.3 plots the prediction of the 0-D model for cell potential at65 C. Thinner lines illustrate the contribution of each type of potential loss (kinetic,ohmic, and transport) to the deviation in cell potential from the open-circuit value.At low current densities, kinetic eects dominate, while at intermediate current den-sities ohmic losses reduce the potential further. Finally, as the i approaches, ilim,mass-transport of oxygen to the electrode further limits the performance.

Cell power density, given byP = V i, (1.6)

and is shown in Figure 1.3 as well. A maximum in power density occurs at roughly1.7 A/cm2, close to ilim.

1.2 Cold Start

In the present work, cold start refers to starting an initially nonoperating PEFC froman initial temperature below 0 C. The signicance of this temperature is that at lowertemperatures ice can form in the cell, potentially limiting performance and causingdamage.

1.2.1 Targets and performance gaps

PEFCs are being considered for a number of applications that may require cold starts.These include backup power in remote locations, portable power for military appli-cations, and motive power for transportation. For the automotive sector, the DOEhas published specic cold-start targets for PEFCs,4 as shown in Table 1.3.

The unassisted start temperature is the minimum temperature from which the cellmust be able to start without using power from sources external to the vehicle, suchas plug-in electrical heaters. This requirement is not intended to preclude the use ofsuch devices, but it sets a standard that, even if they are not available, the cell shouldbe able to start up on its own and that it should not be damaged in the process. Thestart-time requirement denes how quickly 50 % power must be available from thetime that the start-up is initiated at -20 C.

The energy requirement denes the maximum amount of energy that can be expendedper shutdown and cold-start cycle. For reference, 5 MJ corresponds to 0.042 kgof hydrogen, assuming a change in Gibbs free energy for the fuel-cell reaction of-237.129 kJ/mol,15 while hydrogen vehicles typically store 3-5 kg of hydrogen onboard. Meeting this target, therefore, ensures that the vehicle will be able to complete

8

Figure 1.3: Cell potential and power density as a function of current density for aPEFC at a temperature of 65 C, as predicted by the 0-D model. Fully humidiedH2/air operation is assumed, with 1 bar of total pressure for each gas. The kineticand transport overpotentials for the hydrogen electrode are neglected.

9

Table 1.3: Automotive cold-start targets for an 80-kW cell stack, and status as of2005, as established by the Department of Energy.4

Minimum Maximum Maximum Minimumunassisted start time to 50 % start-up and durability

80-kW stack temperature power from -20 C shutdown energy with cycling

Target -40 C 30 s 5.0 MJ 5000 h

2005 Status -20 C 20 s 7.5 MJ 1000 h

a cold-start cycle even if the amount of fuel remaining inboard is low. Including theenergy for shutdown as well as start-up is important because a common strategy forachieving a successful cold start is drying the cell stack with gas upon shutdown, andthis purging process can require a substantial amount of energy.

No durability target has been dened by the DOE specically for cold start, but the5000 h lifetime goal includes all types of cycles.

As Table 1.3 indicates, not all of the cold-start targets are being met. Furthermore,it is important to recognize that the performance relative to a given metric is relatedto the others. For example, less energy could be expended by reducing the amountof time spent purging the cell of water. However, this may preclude the ability tostart unassisted from -40 C and may increase the start time from -20 C. On theother hand, doing so may be benecial from a durability perspective given the factthat repeated humidity cycles have been shown to cause membrane degradation.16

A further complication is cost. Even if reasonable cold-start performance is achievablewith current technology, today's cells are too expensive to be mass-produced for au-tomotive applications. As lower-cost materials are developed, such as catalyst layerswith reduced precious-metal loadings, their performance under cold-start conditionsmay be worse relative to the targets than the current technology. Therefore, in orderto meet the targets with low-cost materials, properties and procedures may have tobe reoptimized. Doing so will require a detailed understanding of cold-start behavior.Developing such understanding is the primary motivation for the present work.

1.2.2 The challenge of starting a cold PEFC

There are three primary reasons why meeting the cold-start targets in Table 1.3 isdicult. These will be discussed in turn.

Reduced performance at low temperatures. Cold starts require the cell to operate attemperatures that are 60 to 130 C below the nominal design point for most applica-tions. At lower temperatures, the kinetic rates of the HOR and ORR are adverselyaected. As an example, Figure 1.4a shows the result of an experimental study of

10

the ORR by Thompson showing the current density as a function of temperatureat a variety of overpotentials.17 In the gure, ix refers to the current density due tocrossover, η is the overpotential due to kinetics, equivalent to the absolute value of thesecond term on the right side of equation 1.3, and Ec is the activation energy. Theseresults illustrate the eect that reducing the temperature has on the performance ofthe catalyst. Achieving a current density of 1 mA/cm2 requires 80 mV of additionaloverpotential at -32 C relative to 40 C.

In addition to a decrease in catalyst activity, lower temperatures also result in lowerionic conductivity in the catalyst ionomer and membrane. Figure 1.4b shows themeasured resistance of Naon 117 as a function of temperature using two dierentmethods. The resistance at -30 C is nearly an order of magnitude higher than itis at 30 C. As discussed below, this increased resistance signicantly impacts cellpotential at low temperature.

The decreased catalytic activity and ionic conductivity with decreasing temperatureare oset to some degree by the increased partial pressure of oxygen in the gas channelsdue to the reduced saturation vapor pressure of water, as illustrated by Figure 1.5a.One bar of total pressure is assumed, as is ideal gas behavior. Equation 1.5 canbe used to calculate limiting current as a function of temperature as well, and theresults are shown in Figure 1.5b. A maximum limiting current exists at intermediatetemperatures. This is because DO2 increases with increasing temperature, while cchan

O2

decreases.

The 0-D model can be used to see more clearly how each of the eects describedabove aects the overall cell potential. For example, Figure 1.6 shows cell potentialas a function of current density at several temperatures ranging from -40 to 80 C.The shapes of the curves provide insight into the type of loss that limits performancethe most as the cell potential passes through 0 V. If the magnitude of the slope of agiven curve (dV/di) is high at an intermediate current density (for example, half thecurrent density at 0 V) relative to that at 60 C, the ohmic loss is high. If the slopeis nite as the curve passes through 0 V, the cell has not reached limiting currentdensity and is ohmically limited. If the slope appears innite, the cell is approachingilim and is oxygen-transport limited. For example, at -40 and -20 C the cell appearsto have a high ohmic drop, as expected from Figure 1.4b, and does not appear toreach limiting current, consistent with Figure 1.5b. The eect of the decrease in ilimat high temperatures is also apparent.

Figure 1.6 suggests that a maximum exists in cell performance as a function of tem-perature. For example, the cell is able to operate at a higher current density at 60C than at 80 C. To investigate this eect further, the maximum power density asa function of temperature was computed using the 0-D model; results are shown fortwo cases in Figure 1.7a. Case A shows the maximum power assuming that there isno limit to how low the cell potential may be during operation. In practice this isoften not true due to system-level considerations such as heat rejection and the volt-

11

(a)

(b)

Figure 1.4: The eect of temperature on (a) the ORR and (b) the ionic resistance ofNaon 117, as measured by Thompson et al. Reprinted with permission from [17].Copyright 2007, The Electrochemical Society.

12

(a)

(b)

Figure 1.5: The eect of temperature on (a) oxygen partial pressure (in 1 bar ofsaturated air) and on (b) the oxygen limiting current as calculated from equation 1.5.

13

Figure 1.6: The eect of temperature and current density on cell potential using the0-D model. The numbers next to each line correspond to the operating temperatures,with units of C.

age required by the power electronics. For this reason, Case B shows the maximumpower if the minimum usable voltage is 0.6 V.

In either case, a maximum in power is observed. At low temperatures the performanceis limited by the ohmic drop through the cell, while at high temperatures the limitingcurrent is reduced due to the decrease in oxygen partial pressure, as discussed above.As an illustration, Figure 1.7b shows, as a function of temperature for case A in (a),the fraction of the total potential loss caused by kinetics, ohmic resistance, and oxygenmass transport. The fraction of the total loss that is ohmic increases dramaticallywith decreasing temperature.

Because both the current density and temperature are changing in Figure 1.7b, it isdicult to judge how the dierent types of losses depend on either of these separately.To illustrate this dependence more clearly, Figures 1.8 and 1.9 plot all four of the termson the right side of equation 1.3 as a function of current density and temperature.The same change in potential has been used on the ordinate in each plot so that themagnitudes of the temperature dependence for each term can be readily visualized.

For example, it is readily apparent from Figure 1.8 that the increase in open-circuitpotential with decreasing temperature osets the increase in kinetic losses below 0C. In fact, while the kinetic loss is substantial at all temperatures and current den-sities, a strong temperature dependence is not observed. A minimum occurs at high

14

(a)I

unstableIII

stable

(b)

IImarginal

Figure 1.7: Predictions from the 0-D model for (a) the maximum power density withtemperature and (b) the fraction of the total loss in potential at maximum power dueto kinetic, ohmic, and transport limitations (for case A as dened in (a)).

15

temperatures because, while the iORR0 increases with temperature, pchan

O2decreases.

Likewise, 1.9b shows that transport losses below 0 C become signicant only at veryhigh current densities. On the other hand, 1.9a shows that the ohmic drop throughthe cell increases substantially with decreasing temperature, especially below 0 C.

As the above results show, the power output of the cell decreases signicantly attemperatures below 0 C, primarily due to the increased internal resistance of thecell. Therefore, reaching a reasonable power level (such as the DOE target of 50 %of maximum power) requires heating the cell up to near or above 0 C.

Flooding due to ice formation. If reduced ionic conductivity leading to high ohmiclosses were the only signicant eect of operating below 0 C, cold start would berelatively straightforward to analyze and design for. Unfortunately, this is not thecase. Below 0 C only a fraction of water produced by the ORR can be removedfrom the cell. Figure 1.10 illustrates this issue by plotting the fraction of the productwater that can be carried out of the cell in the cathode exhaust gas as a functionof temperature at several dierent current densities, assuming dry air supplied at astoichiometric ratio of 2.0 and an exhaust pressure of 1 bar.

In order for the cell to operate at steady state, the amount of water removed mustequal the amount produced. Figure 1.10 predicts that this will not be possible inthe vapor phase at reasonable stoichiometric ratios unless the temperature is greaterthan about 55 C. In practice this is a conservative estimate because above 0 Cliquid-phase removal of water can supplement vapor-phase removal to a signicantextent. However, below 0 C, the product water freezes, and the ice accumulates.

If the cell does not heat up fast enough, this accumulation can prevent the perfor-mance of the cell from increasing with temperature, as predicted in Figure 1.6, andcan even cause performance to decrease. This is because ice reduces the amount ofcatalyst area that is accessible to the oxygen (a0

1,2 in equation 1.3) and increases themass-transport resistance for oxygen in the cathode (corresponding to a higher valueof δeff in equation 1.5). Furthermore, as more and more ice forms near the interfacebetween the membrane and cathode catalyst layer, the reaction distribution acrossthe thickness of the catalyst layer shifts away from that interface. As a result, ioniccurrent must pass through a larger fraction of the total cathode catalyst layer (cCL)thickness, increasing the ohmic drop in the cathode catalyst layer. None of theseeects is captured in the simple model described above.

For these reasons, the performance predicted for temperatures below 0 C in Figure1.6 should be regarded as an upper bound that will likely apply only to short timesafter start-up, before product ice has had a chance to accumulate. As Figure 1.10shows, it is not possible to operate at steady state at useful current densities whenthe temperature is below 0 C. As an example, Figure 1.11 shows the result of anexperimental study by Thompson et al.18 In this experiment, a 50-cm2 cell was chilledto -20 C and started galvanostatically at 0.01 A/cm2. Instead of letting the cell heatup, the cell was maintained at -20 C.

16

(a)

(b)

Figure 1.8: Predictions from the 0-D model for the eect of temperature on (a)the enthalpy and standard potentials and (b) of temperature and current density onkinetic losses.

17

(a)

(b)

Figure 1.9: Predictions from the 0-D model for the eect of temperature and currentdensity on (a) ohmic and (b) mass-transport losses.

18

Figure 1.10: Fraction of product water that can be removed as vapor in the cathodeexhaust stream, at various stoichiometric ratios or oxygen, assuming saturated gas at1 bar.

In Figure 1.11a, the measured cell potential as a function of time is plotted using opensquares. The potential increases initially as the membrane hydrates, but eventuallyfalls to zero. For comparison, the cell potential predicted by the 0-D model is plot-ted as a solid line. Because the temperature and current density are constant withtime, the performance prediction is constant as well. Figures 1.11b and 1.11c showcryo-SEM images of a cathode catalyst layer that experienced the same experimentdescribed in (a). In (b), after the cell potential dropped to zero, it was maintained attemperatures well below 0 C as it was disassembled, mounted, and this image wastaken. Then, the ice was sublimated using dry gases, and the sample was imagedagain, with the result shown in (c). Comparison of the two images shows a signi-cant amount of ice in the pores of the catalyst layer after the galvanostatic cold-startexperiment.

To emphasize the limitations of the 0-D analysis, three regions are delineated inFigure 1.7. Region I, from -40 to 0 C, is labeled unstable because only a fractionof the water produced can be removed from the cell. Region III, above 55 C, islabeled stable because all of the product water can be removed in the vapor phase,as shown in Figure 1.10. The intermediate region is labeled marginal because in thisrange liquid water must be removed from the cell. In most solid-plate cell designs,this is accomplished by blowing the liquid water droplets out of the cell with thecathode gas-exhaust stream. Although this method can work, it can lead to slugs

19

(a)

(b) (c)

Figure 1.11: (a) Cell potential as a function of time for a cell operating at a constanttemperature of -20 C and a constant current of 0.01 A/cm2. The experimental resultis from [18]. (b) A cryo-SEM image taken of a cCL after being operated until failureat 0.01 A/cm2 (as in (a)), and (c) an image of the same cCL after sublimation wasused to remove the ice. Images reprinted with permission from [18]. Copyright 2008,The Electrochemical Society.

20

(a) (b) (c)

densification

densificationdelamination

Figure 1.12: TEM images from of three dierent membrane-electrode assemblies(MEAs): (a) a fresh MEA before any cycling, (b) an aged MEA after 150 galvano-static cold-start cycles at -30 C and 0.3 A/cm2, and (c) an aged MEA after 110galvanostatic cold-start cycles at -20 C and 0.5 A/cm2. Reprinted with permissionfrom [19]. Copyright 2008, The Electrochemical Society.

of water blocking gas channels, which in turn can cause reactant starvation anductuations in cell potential. In addition, although it may be possible to operatein this temperature range indenitely at low current densities by operating with arelatively high O2 stoichiometry, it may not be possible to reach the maximum powerpredicted by Figure 1.7 if the system cannot maintain high stoichiometries at highcurrent densities.

Structural damage due to ice formation. Meeting the performance target for coldstart is challenging for the reasons outlined above, but meeting the durability targetalso poses a problem.

As an example, Figure 1.12 shows TEM images of a new cathode catalyst layercompared with two others that have undergone over 100 cold starts each.19 In thiscase, the cells were allowed to heat up to normal operating temperature during eachcold start, unlike the experiment shown in Figure 1.11a. The aged samples in Figure1.12 show evidence of densication near the M/cCL interface, as indicated by darkregions. The sample in (c) shows delamination of the cCL from the M as well.In addition to this structural damage, the performance of this cell decreased withincreasing numbers of cycles.

1.2.3 The challenge of developing cold-start strategies

Developers of PEFCs face a number of problems when designing systems intended tostart from below 0 C, especially if aggressive targets, such as those in Table 1.3, are

21

used. These will be discussed in turn.

Cold-start behavior is sensitive to many parameters. As discussed above, in additionto temperature and current density, the cell potential is a function of the level of iceaccumulation in the cell during cold start. For this reason, a parameter of interestto developers is how much water can be produced in the cell at a given temperaturebefore the cell potential falls to zero. This is measured by starting a cell at a xedcurrent density and temperature below 0 C, as in Figure 1.11a. The temperatureis generally xed by owing coolant through the cell, removing the waste heat thatwould otherwise cause it to heat up.

Figure 1.13a plots the maximum amount of water that can be produced as a functionof the start current density for two dierent levels of initial membrane hydration.18

The hydration is expressed using λ, which is the number of water molecules persulfonic acid site in the membrane. In one case, the membrane was frozen with themaximum value of λ, while in the other λ was reduced by drying the cell prior tofreezing it. Reducing the initial level of hydration increases the amount of time thatthe cell is able to run before failure, indicating that the membrane is able to absorbproduct water during the start. In other words, in addition to accumulating as icein the pores, if the membrane is relatively dry initially, product water accumulatesin it as well. However, the results also show that increasing the start current densitysignicantly reduces the amount of water that can be produced prior to failure.

Figure 1.13b shows the eect of cold-start temperature on the maximum amount ofwater that can be produced, as measured by Ge and Wang.20 A strong correlation isobserved, with higher start temperatures corresponding to larger amounts of productwater before failure.

The examples above show the sensitivity of cold-start behavior to operational pa-rameters. That is, those parameters that dene the environment in which the cell isstarted. However, the congurational parameters that dene the cell's design have asignicant impact as well. One example, illustrated in Figure 1.14, shows the experi-mental results for charge passed prior to failure at -20 C and 0.1 A/cm2as a functionof catalyst-layer thickness. Thinner catalyst layers fail sooner.

Simple analyses often fall short of predicting actual cell behavior. Developers oftenbegin analyzing the problem of cold start by considering several gures of merit, suchas those shown in Table 1.4. In the table, the ice capacity refers to the maximumamount of water that can be stored as ice in each of the subdomains, Ω, calculatedfrom

mmaxI,Ω = δΩεΩρI , (1.7)

where δΩ and εΩ are the thickness and porosity of Ω, respectively, and ρI is the densityof ice. For the membrane, εM in equation 1.7 is replaced by εM , the volume fractionof water in the membrane, as dened in Chapter 3. Also, because the water in themembrane is not necessarily in the form of ice, ρI is replaced by ρL, the density of

22

(a)

(b)

Figure 1.13: Cumulative water produced prior to failure when a cell operates at aconstant current density without being allowed to heat up. Shown as a function of (a)start current density (at -20 C, from [18]), and (b) start temperature (at 0.1 A/cm2,from [20]).

23

Figure 1.14: Cumulative water produced prior to failure when a cell operates at aconstant current density without being allowed to heat up. Shown as a function ofcCL thickness (at -20 C and 0.1 A/cm2, from [21]).

liquid water. For reference, the amount of water produced at a current density of 0.1A/cm2 is 0.56 mg/cm2·min, calculated from Faraday's law, as applied to the ORR:

mprodH2O =

MH2Oi

2F, (1.8)

where MH2O is the molar mass of water.

The heat capacity in Table 1.4, denoted by Cp,Ω, is dened to be the energy requiredto raise the temperature of a unit area of the materials of construction in subdomainΩ by one degree:

Cp,Ω = εsρsCp,sδΩ + εMρM Cp,MδΩ, (1.9)

where Cp is the specic heat capacity. No gas, liquid, or ice is included in thiscalculation. In the table, a correction to the heat capacity of the plate has beenmade for the presence of gas channels by setting εs equal to 0.75, which constitutesan eective volume fraction of solid for a given thickness, δP. In order to calculatethe ice and heat capacities of the entire cell, the cell is taken to be composed of thenine layers shown in Figure 1.2a: two Ps, GDLs, MPLs, and CLs, and one membranelayer.

The values in Table 1.4 are rough estimates only, as they vary with dierent materialchoices. Nevertheless, several trends are correct. For example, most of the storage

24

Table 1.4: Typical ice capacity and heat capacity for cell components based on theproperties listed in Chapter 3 of this work.

parameter units P GDL MPL CL M cell

ice capacity mg/cm2 0.0 13.9 2.3 0.9 0.8 35.0

heat capacity mJ/cm2·K 958.5 48.6 31.6 7.4 52.5 2144.7

capacity for ice is in the GDLs. Also, the majority of the thermal mass is in thebipolar plates, because they are thick and nonporous (in this analysis). What is notclear from the table, but is true in practice, is that during cold start, at temperaturesmore than a couple of degrees below 0 C, the capacity of the MPL and GDL isgenerally not accessible to the product water. Only a relatively small amount of theproduct water is able to move out of the M and cCL.

Figure 1.11 illustrated how the 0-D model is able to predict initial performance fairlywell, but not cell potential during the start itself. Similarly, the simple analysis ofthe capacity of the cell for storing product water, given by equation 1.7, does notpredict the trends shown in Figure 1.13. In that gure, the solid horizontal linesin (a) and (b) correspond to the prediction for the capacity of the cathode catalystlayer. Neither the eect of current density nor the eect of temperature is predicted.The ice-capacity analysis, again shown as a solid line, does a better job predictingthe trend in 1.14, but there is still deviation for the thicker electrodes.

Predicting the experimental data in Figures 1.13 and 1.14 more accurately requires amore sophisticated model that at a minimum accounts properly for the uxes in andout of the cCL and M. At high current densities, the eect of the reaction distributionin the catalyst layer must also be accounted for, because the cell potential may fall tozero before all of the pores are lled due to the high ohmic drop through the cCL.18

The limits of simple analyses is further illustrated by trying to estimate the start timefor the cell, ∆tstart, using the 0-D model. To do so, the following energy balance mustbe solved:

Cp,celldT

dt= Qcell, (1.10)

where Qcell is the total amount of heat generated by the cell. The assumption from the0-D model of a single lumped temperature for the cell is preserved in this equation,and heat lost by the cell is neglected, because this analysis is intended to provide onlya lower bound for start time. Likewise, the heat required to melt any ice in the cellis not considered. Qcell is given by

Qcell = i(UH − V ) (1.11)

where UH is the enthalpy potential.15 Predictions for Qcell and power, P, as a functionof temperature and cell potential are shown in Figure 1.15.

25

(a)

(b)

Figure 1.15: Predictions using the 0-D model for (a) heat generation and (b) powerdensity as a function of temperature and cell potential.

26

(a)

(b)

Figure 1.16: Predictions, using equations 1.10, 1.11, and the 0-D cell-performancemodel, for start time as a function of (a) cell potential (at a start temperature of -20C) and (b) temperature (at a start potential of either 0.4 or 0.6 V).

27

Because V depends on T, equations 1.3, 1.10, and 1.11 must be solved simultaneously.By doing so, an estimate can be made for ∆tstart. Figure 1.16 shows the results ofsuch an analysis, assuming that the cell is started using a constant cell potential.In this analysis, ∆tstart is the time required to reach 50 % of rated power. Ratedpower is not necessarily the same as the the maximum power available from the cell,which is often signicantly higher. In automotive applications, rated power is basedon a trade-o between maximizing power density, which requires a lower voltage, andeciency, which requires a higher voltage. Eciency is important for two reasons.First, it improves fuel economy. Second, it reduces the amount of heat that mustbe rejected from the cell stack at a given power level, an important considerationwhen the operating temperature of the system is at least 40 C lower than that of aninternal-combustion engine.

In Figure 1.16, two cases are dened. In case A, it is assumed that there is nominimum voltage limit, and the rated power point is taken to correspond to themaximum power density for case A in Figure 1.7a. In case B, it is assumed that thereis a minimum voltage limit of 0.6 V and that rated power is that which is availableat 0.6 V at 75 C. This point can be obtained from the case B curve in Figure 1.7a.Based on these denitions and the results in Figure 1.15, the cell should be able toreach 50 % of rated power at -5 C in either case A or case B.

In (a), ∆tstart is plotted as a function of the start potential, assuming a start tempera-ture of -20 C. Increasing the start potential increases the amount of time required tostart due to the reduced amount of waste heat that is generated at a higher potential.

In (b), ∆tstart is predicted as a function of start temperature for the two cases. Aconstant cell potential of 0.4 V is used for case A based on the results in Figure 1.7showing that the maximum power available below 0 C occurs near this potential.For reference, the -20 C cold-start target from Table 1.3 is shown. In both cases, thecell is predicted to reach 50 % power within 30 s. However, as explained above, theseresults should be considered a lower-bound estimate for ∆tstart because they do notinclude the eects of heat loss or ice accumulation in the cell.

To illustrate further how important the latter eect can be, Figure 1.17 comparestransient 0-D model predictions to experimental results (open squares) for two dif-ferent single-cell cold starts performed at UTC Power.22 In (a), the temperature andcell-potential proles for a galvanostatic cold start from -10 C, using 0.6 A/cm2,are shown. In this case the 0-D model predicts the temperature prole below 0 Cwell, but does not capture the melting plateau observed in the experiment. The cell-potential prediction is quite good. In (b), the results for a -30 C start are shown. Inthis case, the initial cell potential is predicted by the model, but the potential at anysubsequent time is far o. In the experiment, the cell potential drops to below 0 V,whereas the model predicts a monotonic rise. In later chapters, it is shown that thisdrop is due to the increase in ionic resistance in the catalyst layer that occurs as thereaction distribution shifts farther and farther away from the interface between the

28

(a)

(b)

Figure 1.17: Experimental results compared with predictions from the 0-D model forgalvanostatic cold starts from (a) -10 C and (b) -30 C. Experimental results fromUTC Power.22

29

M and cCL as the pores near that interface ll with ice.

Cold-start experiments are expensive and time-consuming. Experiments are a valu-able means of gaining insight into cold-start behavior. However, they require specialequipment, such as environmental chambers, chillers, and, in some cases, custom cellhardware that can add signicant capital cost to a standard test stand. As a result,developers generally install cold-start capability in only a fraction of their test stands.Furthermore, even with the appropriate equipment, a thorough cold-start study cantake a signicant amount of time for two reasons.

First, the time required per cold start tends to be high. In a typical cycle, the cell isconditioned, and its performance is measured at normal operating temperatures. Itis then purged, frozen, held at the desired cold-start temperature, and then started.It is then returned to normal operating temperature (either via its own waste heator with external heating), and the cycle is repeated. It is not uncommon for sucha cycle to take several hours, although it can take much longer if the cell is frozenby allowing it to equilibrate with the environmental chamber rather than by activelycirculating chilled coolant.

Even if the per-cycle time is fairly low, the second complicating factor is the largenumber of parameters that aect cold start, as discussed above. To map the behaviorof a given conguration, parametric studies based on initial water content, start-upcurrent density, and start temperature are required. Of course, generally there aremany congurations that are of interest.

1.3 Objective and Approach

Because the experiments associated with cold start can be onerous and the resultscan be confusing, a means of both simulating and analyzing cold-start behavior isdesirable. However, as shown above, back-of-the-envelope analyses based on guresof merit and simple calculations can only take one so far toward this goal. Whilethey can be valuable, they generally provide either results with a very narrow rangeof applicability or an estimate of upper or lower bounds for parameters of interest.

The objective of the remainder of this dissertation is to construct and verify a physics-based cold-start model for a single cell and to apply it to improve understandingof cold-start behavior. Doing so will enable the specication of desirable materialproperties and start-up procedures. The intent is that the model can be used by fuel-cell developers alongside their experimental activities in order to help understandresults and perform trade studies in a manner that is substantially more economicalthan would be possible through experiments alone.

Figure 1.18 illustrates how the model is intended to be used. In the input phase,parameters related to the cell conguration are entered, along with the operational

30

Geo

met

ric c

onfig

urat

ion

Phys

ical

and

che

mic

al p

rope

rties

Tran

spor

t and

kin

etic

pro

perti

es

Con

figur

atio

nalp

aram

eter

s

2-D

cel

lco

ld-s

tart

mod

elIn

itial

tem

pera

ture

Cur

rent

den

sity

Rel

ativ

e hu

mid

ity

Ope

ratio

nal p

aram

eter

s

V Tt

Cel

l per

form

ance

Key

fact

ors

affe

ctin

g st

art

λ

S I, c

CL

t

Inpu

tS

imul

atio

nO

utpu

t

Figure

1.18:Intended

use

forthe2-Dcold-start

modeldeveloped

insubsequentchapters

ofthepresentwork.

31

parameters that dene the experiment of interest. In the simulation phase, the modelsolves the governing equations assuming the properties and boundary conditions de-ned by the input parameters. In the output phase, the model produces both aprediction of macroscopic observables, such as cell potential and temperature, as wellas insight into the parameters that determine them, such as the amount of ice in theporous media and the amount of water in the membrane. From these results, thephysics that determine cell behavior can be more thoroughly understood.

The model that has been developed to meet the objectives is two-dimensional, tran-sient, and nonisothermal. While the dimension of primary importance to consider isthe x -direction in Figure 1.2, using two dimensions provides insight into dierences inthe the y-direction due to the dierent transport boundary conditions imposed underthe rib relative to those under the channel. As discussed above, a fuel cell cannotoperate at steady state at the temperatures of interest. Therefore, making the modelcapable of simulating transients is necessary. Temperature variations within the cellare important because these can result in transport of water in both the liquid andvapor phases.

A macrohomogeneous framework has been chosen for the model. Hallmarks of thisapproach are averaging and superposition. Averaging refers to the fact that thedetailed physical structure of each phase is averaged when determining both howmuch of a given phase is present and what its transport properties are. For example,the amount of gas present per unit volume in a dry GDL is equal to its porosity (amacroscopic quantity), and the permeability of gas through the GDL is taken to bea function of that porosity. Superposition refers to the assumption that all phasesthat have been dened in a given domain are present simultaneously in a given unitvolume. For example, in the catalyst layer two continuous phases are dened to carrycharge: a solid, electronically conductive phase, and an ionomer phase that conductsprotons.

A key dierentiating factor of the macrohomogeneous model developed in the presentwork is the inclusion of water in all four of the possible phases: ice, liquid, gas, andmembrane. In order to predict water content in the ice, liquid, and gas phases thatare present in the porous media, the thermodynamics of phase equilibrium have beenrevisited, and a method for relating phase pressures to water content in each of thesephases has been developed.

This approach to predicting water content in the dierent phases, presented in thecontext of the related issue of the phenomenon of frost heave, is the subject of Chapter2. In Chapter 3, the remainder of the cold-start model is developed, and an exampleof a simulation result is reviewed. Verication of the model using experimental datafrom a variety of sources is presented in Chapter 4. In Chapter 5, the veried modelis applied to the question of optimization of materials and procedures for cold start.Chapter 6 summarizes key results and provides a perspective on various ways thatthe model could be improved and studies of interest that could be completed.

32

1.4 Summary and Conclusions

A 0-D performance model has been used to explore the eect of temperature on cellperformance in the absence of ice accumulation. In addition, this performance modelhas been combined with a simple energy balance to estimate the time required to startup from a cold temperature. Such an analysis is appropriate for estimating the initialpotential of a cell at a given cold-start temperature, the maximum potential of a cell ata given temperature, the magnitude of the kinetic, ohmic, and oxygen transport lossesat a given temperature, and the minimum start time that is probably achievable.

Using this approach, it has been shown that the type of potential loss that increasesthe most at temperatures below 0 C is ohmic due to the poor ionic conductivityof the membrane and ionomer. In addition, it has been shown that, in the absenceof any heat loss and neglecting the eects of ice accumulation, a cell with a typicalconguration should be able to start to 50 % power from -20 C in less than 10 s,even if the cell potential is not allowed to drop below 0.6 V.

To predict more accurately the performance of the cell during cold start requiresa more sophisticated approach. The objective of the remainder of this dissertationis to construct and verify a physics-based cold-start model for a single cell and toapply it to improve understanding of cold-start behavior. Doing so will enable thespecication of desirable material properties and start-up procedures.

33

Chapter 2

Phase Equilibria and Frost Heave

2.1 Introduction

Freezing in porous media has been studied for decades in soil science, motivated inpart by the phenomenon called frost heave that can have a destructive impact oninfrastructure. Frost heave occurs when a porous medium is frozen nonuniformly, asin winter when soil is frozen from the earth's surface downward. During frost heavein soils, a layer of pure ice, known as an ice lens, forms and grows, displacing the soilabove the lens. Liquid water from below the ice lens moves under a pressure gradientto the base of the lens, where it freezes.

An example of frost-heave damage to a road is shown in Figure 2.1a, and a sample ofclay containing ice lenses, taken from under a street damaged by signicant heaving,is shown in Figure 2.1b. Taber was among the rst to demonstrate the growthof ice lenses in the laboratory, and showed in 1929 that this phenomenon is not aconsequence of the expansion of water upon freezing; it occurs with materials thatcontract as well.23 Figure 2.1c shows a clay sample containing ice lenses grown duringone of Taber's experiments. Ice lenses may form on the surface of a porous mediumas well, as is the case with the needle ice shown growing out of a log in Figure 2.1d.

Everett24 used equilibrium thermodynamics to develop a proposed mechanism forfrost heave. According to the hypothesis, at temperatures near but below the meltingpoint, liquid water in larger pores is frozen while that in smaller pores is not. Theequilibrium liquid pressure will likely be lower at the cold side of the medium thanon the hot side due to the lower temperature. Due to this pressure dierence, liquidows through the ne, unfrozen pores from the hot side toward the cold side, whereit freezes.

Dirksen and Miller25 as well as Hoekstra26 showed that a signicant amount of waterredistribution can occur in unsaturated soil prior to the onset of frost heave. Anunsaturated porous medium is one that contains an appreciable amount of gas in a

34

(a) (b) (c)

(d) (e)

PEMaCL

cCL

clay

icelenses

clay

icelenses

log

needle ice

frost-heavedamage

void

Figure 2.1: (a) Damage to a roadway caused by frost heave.27 (b) Ice lenses in a claysample from under a heaved street in St. Peter, MN. (c) Ice lenses grown in clay inlaboratory frost-heave experiments by Taber. (d) Example of needle ice growingout of a log. (e) Delamination of the cathode catalyst layer in a fuel cell from themembrane after repeated freeze-thaw cycles. Images in (b) and (c) reprinted withpermission from [28]. Copyright 1930, The University of Chicago Press. Image in (d)reprinted with permission from [29]. Copyright 2006, Annual Reviews. Image in (e)reprinted from [30], Copyright 2007, with permission from Elsevier.

given unit volume, while a saturated medium does not. In the experiments, cylindricalsamples of soil were lled to an initially uniform level of liquid saturation (a levelsubstantially less than 100 %) and subsequently frozen from one end. As soon asfreezing began at the cold side of the sample, water migrated from the hot sidetoward the freezing zone. The rate of water ow to the cold side was far greater thancould be explained due to vapor-phase diusion of water alone. Furthermore, if thetotal water saturation of the soil in the freezing zone reached a critical level (around90 %), an ice lens formed.

As explained in detail in a later section in this chapter, higher ice saturations are theresult of higher ice pressures. Frost heave, therefore, is the eventual result of watermovement in a freezing porous medium if the ice pressure reaches a high enoughlevel and favorable conditions persist. In the present work, such water movement isreferred to as phase-change-induced (PCI) ow.

Recently, the term PCI ow has also been used in the fuel-cell literature to describediusion of water vapor caused by dierences in saturation pressure in the presence

35

of a thermal gradient.31 Unless vapor PCI ow is specied, however, all references toPCI ow here are to liquid owing toward colder regions of a porous medium whenice and liquid coexist there.

PCI ow and frost heave are of interest to polymer-electrolyte fuel-cell (PEFC) de-velopers who are designing systems with cold-weather applications because the sameconditions that lead to these phenomena in soils are present. A fuel cell typicallyconsists of layers of porous media that contain at least some water. As water in thecell freezes (e.g., after being shut down), or as the cell starts up from a frozen con-dition, ice and liquid can coexist, and therefore temperature gradients could inducePCI ow, and possibly frost heave.

If frost heave were to occur, it could irreversibly alter the pore structure of the mate-rials or cause separation to occur between layers, especially if the process resulting infrost heave were repeated many times. An example of layer separation (also known asdelamination) that has been observed after repeated cycles of freezing and thawing isshown in Figure 2.1e. Even if PCI ow were to occur without leading to frost heave,water could redistribute in ways that aect cell operation. For example, if ice wereto accumulate in the cathode catalyst layer during freezing, this could reduce theeective catalyst area, thereby lowering performance during a subsequent cold start.On the other hand, PCI ow from the cathode catalyst layer during cold start couldimprove the chances of a successful start-up. For these reasons, it would be desirableto have a means of predicting whether or not PCI ow and frost heave are likely tooccur in fuel-cell systems.

Understanding these phenomena in the context of fuel cells, however, is challenging.To begin with, because they are designed to facilitate transport of gas-phase species,the materials are generally unsaturated, which adds complication due to the presenceof interfaces between liquid or ice and the gas phase. Second, the media are generallyof mixed wettability, meaning that in a given unit volume both hydrophilic and hy-drophobic pores may exist. Third, a cell is a composite structure made up of layersof porous materials that, although they may be adjacent to one another, can havevery dierent physical and chemical properties depending on the function that theyare serving. Finally, these phenomena are relevant not only as a nonoperating cellfreezes and thaws while seeking thermal equilibrium with its environment, but alsoas the cell starts up from a frozen condition, during which sources of both heat andwater exist within the cell.

A 1-D model for predicting both PCI ow and frost heave in a PEFC has been de-veloped by Mench and coworkers.32 Various parametric studies have been performedusing the model.33,34 In addition, Balliet et al. have presented a 2-D PCI ow modelfor a PEFC.35 While these models do provide insight into how water might redis-tribute in a freezing cell, the papers cited do not thoroughly explain the fundamentaldriving force for PCI ow and frost heave. Furthermore, although they each discussmethods for estimating the equilibrium liquid-water content once freezing has begun,

36

they do not address how to predict the equilibrium ice and water-vapor contents aswell. These are both of importance for developers seeking to simulate not only freez-ing and thawing of a nonoperating cell, but also start-up of an operating cell from afrozen condition.

The objectives of the present work are two-fold. First, to dene clearly the thermo-dynamic mechanism that drives PCI ow and frost heave, examining in particular theimplications for PEFC porous media. Second, to extend this reasoning to develop acomplementary, quantitative model for predicting ice, liquid, and gas saturations inthe various types of PEFC porous media at all temperatures and pressures of interestfor typical applications.

2.2 Terminology

In the present work, PCI ow refers to liquid ow toward a cold region of a porousmedium that contains both liquid and ice from a warm region that must contain liquidbut may or may not contain ice. As is explained later in greater detail, under certainconditions the liquid pressure in the presence of ice in the cold region is lower thanthat of liquid in the warmer region, causing liquid to ow toward the cold region,where it freezes. As long as the additional ice formed in the cold region does not raisethe liquid pressure to the same level as in the hot pore, a driving force will persist forow.

As an example, when PCI ow occurs, the ice formed in the cold region may moveinto pores previously occupied by gas, provided such pores exist and their structureis strong enough to withstand the ice pressure inherent in lling them. Displacementof gas by either ice or liquid in this way is referred to in this work as pore lling todistinguish it from frost heave. Figure 2.2a shows an example of PCI ow from ahot pore to a cold pore where pore lling is occurring. The two pores are at xedtemperatures, meaning that ice is melting in the hot pore and freezing in the coldpore. Friction between the ice and the pore walls is neglected. The total amount ofwater in the hot pore is decreasing, while it is increasing in the cold pore.

Note that, in a fuel cell, pore lling can occur without PCI ow. If there is a sourcefor water production, as in the cathode catalyst layer, pore lling can occur regardlessof whether or not a temperature gradient exists.

If the ice pressure in the medium is high enough, pores may fracture. When thishappens and a layer of pure ice grows, displacing the surrounding medium, the icelayer is referred to as an ice lens. Alternatively, an ice lens may grow at the interfacebetween adjacent media or at the interface between a porous medium and an openvolume that contains only gas (such as that between soil and open air). Neitherof these scenarios requires pore fracture, and may not necessarily cause permanentdeformation of the medium.

37

liquid ice gas

capillary connection

poredraining

hot pore, TH

cold pore, TC(TC < TH)

(a)

PCIFlow

porefilling

liquid ice gas


poredraining

hot pore, TH

cold pore, TC(TC < TH)

(b)

PCIFlow

frostheave

icelens

pore-wallfracture

Figure 2.2: Illustration of various terms when (a) PCI ow and pore lling are occur-ring without frost heave, and (b), PCI ow and frost heave are occurring.

For the purposes of the present work, the criterion in the soil for predicting ice-lens initiation and continued growth is that the ice pressure equals the overburdenpressure. The overburden is the combined pressure exerted by the medium itself(the maximum is related to the tensile strength of the medium) and any externalforces constraining the medium. For example, for the needle ice growing on the log inFigure 2.1d, the medium exerts no pressure to constrain the ice lens. Therefore theoverburden is the weight of the ice lens divided by the area of the lens/log interfaceplus the atmospheric pressure.

A more complicated criterion for ice-lens initiation is sometimes used in the frost-heave modeling literature for soils. The ice and liquid pressures are combined intoa parameter called the pore pressure, and that value (instead of the ice pressurealone) is compared to the overburden, as proposed by Miller36 and discussed furtherby O'Neill and Miller.37 Each phase's contribution to the pore pressure is propor-tional to the amount of the phase that is present. Applying this criterion requires anunderstanding of the proper partitioning of the stress imparted to the pores by eachphase, as a function of the amount of each that is present, which generally leads toempirical relationships that depend on the type of medium. Furthermore, the poreand ice pressures tend to be fairly close since so little liquid exists in the region of themedium where an ice lens might grow. Finally, using the pore pressure as opposed tothe ice pressure as a criterion may be considered a renement that could improve theaccuracy of a model for a particular system but is not essential to understanding thethermodynamic mechanism that results in frost heave. For these reasons, the simplercriterion of comparing the ice pressure to the overburden is applied below.

If an ice lens is formed due to PCI ow, then the growth of the lens and any associated

38

displacement of the surrounding medium or media is referred to here as frost heave.Figure 2.2b shows an example of PCI ow feeding the growth of an ice lens in atwo-pore system.

It is worth reiterating the distinctions between the key terms. PCI ow in the presentwork refers to liquid moving toward a cold region in a porous medium that containsliquid and ice. PCI ow can occur without frost heave, provided that the ice formedin the cold region can occupy previously empty pores, a process known as pore lling.Pore lling may also occur by other means, such as the production of water in thecathode catalyst layer. If the ice pressure exceeds the overburden pressure, an icelens may form. If the formation and growth of the ice lens is due to PCI ow, itsgrowth and displacement of the surrounding medium or media is referred to as frostheave. Frost heave cannot occur without PCI ow.

2.3 Phase Equilibria

Within the temperature and pressure range of interest, H2O may be present in ice,liquid, and/or gas phases. In the ice and liquid phases, it is assumed that only onecomponent, H2O, is present. Therefore, the subscripts (or superscripts) I and L referto H2O in the ice and liquid phases, respectively, as well as to the phases themselves.In the gas phase, components other than H2O, such as reactants and inerts, hereafterreferred to as diluents, may also be present. The subscripts (or superscripts) V andD refer to H2O and diluents in the gas phase, respectively. The gas phase is referredto with the subscript (or superscript) G. yi is used to refer to the mole fraction ofcomponent i in the gas phase.

The phase behavior of pure water is reviewed rst, and expressions for the lines ofcoexistence on the p − T phase diagram shown in Figure 2.3 are derived. Later,corrections to these expressions for the presence of gas-phase diluents and curvedinterfaces are considered.

2.3.1 The chemical potential

Consider any of the phase boundaries in Figure 2.3. When two phases, α and β,containing component i, are equilibrated, the chemical potentials, µi, of i in eachphase, are equal:

µαi = µβi , (2.1)

where the chemical potential is given by

µi = Hi − T Si, (2.2)

39

Figure 2.3: Phase diagram for pure water in the range of interest for cold-start sim-ulations of a polymer-electrolyte fuel cell.

40

Table 2.1: Thermodynamic properties of pure water at the triple point.38,39

Vi,t Hi,t Cp,i

i cm3/g J/g J/g·K

ice (I ) 1.0909 -333.6 2.040

liquid (L) 1.0002 0 4.217

vapor (V ) 206146 2500.8 1.854

and the (partial) molar enthalpy and entropy of i are Hi and Si, respectively. Com-bining these equations yields

Hαi − H

βi = T

(Sαi − S

βi

). (2.3)

The total dierential of the chemical potential is

dµi = −SidT + Vidp, (2.4)

where Vi is the (partial) molar volume of i. Applying this to equations 2.1 and 2.3yields the Clapeyron equation,

dp

dT=Sαi − S

βi

V αi − V

βi

=1

T

Hαi − H

βi

V αi − V

βi

, (2.5)

which is an exact equation for the slope of any one of the three phase-boundary linesin Figure 2.3, and can be expressed in the equivalent forms

dlnp

d (1/T )= −T

p

Hαi − H

βi

V αi − V

βi

, (2.6)

anddlnp

dT=

1

pT

Hαi − H

βi

V αi − V

βi

. (2.7)

Table 2.1 gives the physical properties for water needed to calculate these three prin-cipal slopes at the triple point, Tt = 273.16 K and pt = 0.006112 bar. As an example,dlnp/dT is given in Table 2.2, along with values for the surface tensions, which willbe needed later when considering curved interfaces and phase equilibria in porousmedia.

Further approximations can be applied to these equations in some cases. For thevapor-liquid and vapor-ice lines in Figure 2.3 the Clausius-Clapeyron equation resultsfrom applying these approximations:

41

Table 2.2: Properties for the three possible uid interfaces.4044

Interface dlnpdT

(K−1) at Tt, pt γαβ (N/m)

liquid-gas (LG) 0.07266 0.076

ice-liquid (IL) -22030 0.033

ice-gas (IG) 0.08235 0.109

1. VV is much larger than either VL or VI .

2. The vapor is an ideal gas, VV = RT/p.

3. The enthalpy of vaporization, ∆HV i = HV − Hi, where i is I or L, is a constantalong the phase boundary.

The Clausius-Clapeyron equation reads:

dlnp

dT=

∆HV i

RT 2or

dlnp

d (1/T )= −∆HV i

R. (2.8)

To derive expressions for the chemical potential for water in each of the three phases,the chemical potential of pure H2O at the triple point, µt, is taken to be the referencechemical potential for L, I, and V. The value of µt is taken to be zero. Next, thedenition of the heat capacity is used,

Cp,i =

(∂Hi

∂T

)p,nj

. (2.9)

Constant composition, nj, is included for the treatment of diluents in the gas phase,which will be discussed later. The heat capacities are taken to be constant, although atemperature dependence could be included for temperatures far from the triple point.A pressure dependence can safely be neglected for the range of conditions of interest.Integration of equation 2.9 thus gives

Hi = Hi,t + Cp,i (T − Tt) . (2.10)

The pressure dependence of the enthalpy is(∂Hi

∂p

)T,nj

= Vi − T(∂Vi∂T

)p,nj

. (2.11)

For I and L, Vi is taken to be constant, and the second term vanishes. For V ,assumption 2 above is used, the two terms cancel, and ∂Hi/∂p = 0.

42

From equation 2.4, the temperature dependence of µi can be written as(∂µi∂T

)p,nj

= −Si (2.12)

or (∂ (µi/T )

∂T

)p,nj

= −Hi

T 2. (2.13)

The latter form is generally more convenient because Hi is more frequently foundtabulated and more nearly constant than Si. Equation 2.13 can now be integrated toget an expression for µi in any of the three phases:

µi =T

Ttµi,t + Hi,t

(1− T

Tt

)+ Cp,i

(T − Tt − T ln

(T

Tt

))+ C(p), (2.14)

where equation 2.10 has been used to eliminate Hi. To nd the integration constant,C(p), for L and I, (

∂µi∂p

)T,nj

= Vi (2.15)

is integrated to give C(p) = Vi (p− pt). For V , integration gives C(p) = RT ln(pyipt

).

In summary, the resulting equations for the chemical potentials of H2O in the threephases are:

µL = HL,t

(1− T

Tt

)+ Cp,L (T − Tt − T ln (T/Tt)) + VL(pL − pt), (2.16)

µI = HI,t

(1− T

Tt

)+ Cp,I (T − Tt − T ln (T/Tt)) + VI(pI − pt), (2.17)

and

µV = HV,t

(1− T

Tt

)+ Cp,V (T − Tt − T ln (T/Tt)) +RT ln

(pyVpt

). (2.18)

2.3.2 Phase equilibria for pure water

Expressions for the phase boundaries in Figure 2.3 now follow from equations 2.1 and2.16-2.18. Consider the system shown in Figure 2.4a, which contains pure H2O vaporin the gas phase in equilibrium across a planar interface with pure L or I. In this case,p = pi = psat,0

i , where psat,0i is the saturation vapor pressure of liquid or ice. For this

system,

ln(psat,0i

pt

)− Vi

RT

(psat,0i − pt

)= −∆HV i

RT

(1− T

Tt

)− ∆Cp,V i

RT

(T − Tt − T ln

(TTt

)) (2.19)

43

I

(b)

L or I

G GG

RLG

L

I

RIL

(e)

(V) (V, D)(V, D)

(d)

L or I

or RIG

(a) (c)

L L or I

Figure 2.4: System congurations considered in the phase equilibria discussion. In(a), (b), and (c) only at interfaces exist, while (d) and (e) contain curved interfaces.Gas-phase components are listed in parentheses.

Likewise, for coexistence of pure ice and liquid across a planar interface, as in Figure2.4b,

∆VLI (p− pt) = −∆HLI

(1− T

Tt

)−∆Cp,LI

(T − Tt − T ln

(T

Tt

)). (2.20)

In these equations, ∆Kji = Kj − Ki where Ki is a (partial) molar property for i.

2.3.3 Corrections to the saturation vapor pressure

The saturation vapor pressure of a condensed phase (in this case, liquid or ice) willchange when the pressure of the condensed phase deviates from psat,0

i . In the contextof a fuel cell, one reason why this occurs is that diluents are present in the gas phase,as shown in Figure 2.4c. In this case, pi = pG = psat

i +pD, where psati is the saturation

vapor pressure with diluent (D) present.

To determine how much psati diers from psat,0

i , the equilibrium criterion from equation2.1 is applied to i and V in Figure 2.4c:

Hi,t

(1− T

Tt

)+ Cp,i (T − Tt − T ln (T/Tt)) + Vi(pG − pt)

= HV,t

(1− T

Tt


(psati

pt

) (2.21)

The terms on the left side of the equation comprise the condensed-phase chemicalpotential, µi, while those on the right comprise µV . The same procedure is used forFigure 2.4a, which yields:

Hi,t

(1− T

Tt

)+ Cp,i (T − Tt − T ln (T/Tt)) + Vi(p

sat,0i − pt)

= HV,t

(1− T

Tt


(psat,0i

pt

) (2.22)

Assuming that the two systems are at the same temperature, if equation 2.22 issubtracted from equation 2.21, rearranging gives

ln

(psati

psat,0i

)=

ViRT

(pG − psat,0

i

), (2.23)

44

which is the Poynting correction to the saturation vapor pressure for i, where i iseither L or I.

The second case of interest occurs when the interface between the phases is curved, asshown in Figure 2.4d, which shows a spherical drop of i with radius RiG, equilibratedwith V in the gas phase. In this case, a pressure dierence exists between i and Gwhich is supported by the interfacial surface tension, γiG. The pressure dierenceacross a curved interface is given by the Laplace equation, which for a sphere takesthe form

pi − pG =2γiGRiG

. (2.24)

In order to assess how much a change in liquid pressure resulting from a curvedinterface aects the saturation vapor pressure, apply the same method that was usedto derive the Poynting correction. Applying equation 2.1 to Figure 2.4d,

Hi,t

(1− T

Tt

)+ Cp,i (T − Tt − T ln (T/Tt)) + Vi(pi − pt)

= HV,t

(1− T

Tt


(pGyVpt

) , (2.25)

where the terms on the left side of the equation comprise the condensed-phase chem-ical potential, µi, while those on the right comprise µV . Subtracting equation 2.21(which arises for the system in Figure 2.4c) from equation 2.25 and rearranging yields

pi − pG =RT

Viln

(pVpsati

)−(pV − psat

i

), (2.26)

where pV is the vapor partial pressure, pGyV . To arrive at this equation, assumptionsare made that the systems in Figure 2.4c and 2.4d are at the same temperature andthat p

(c)D = p

(d)D , but p

(c)G 6= p

(d)G because pV 6= psat

i .

The second term on the right of equation 2.26 is much smaller than the rst term,and is therefore safely neglected. Using this assumption, combining equations 2.24and 2.26 results in the Kelvin equation,

ln

(pVpsati

)=

2ViγiGRTRiG

, (2.27)

which relates the vapor pressure to the radius of the drop of i. A smaller radiusresults in a higher vapor pressure, while at large RLG, pV → psat

i and the interfaceapproaches a planar shape. Finally, a negative value for RiG corresponds to a sphereof G inside of i instead of the other way around and a vapor pressure that is smallerthan psat

i .

To understand the relative magnitudes of the corrections for dilution and curvedinterfaces, equations 2.23 and 2.27 can be combined to give the vapor pressure for

45

Figure 2.5: Corrections to the vapor pressure of pure liquid water, pV , at 65 C,relative to the vapor pressure with no gas-phase diluent and no interfacial curvature,psat,0L . Positive values of 1/RLG correspond to a spherical drop of pure liquid immersedin gas that contains vapor and may also contain diluent. Negative values correspondto a bubble of gas immersed in liquid. Each line corresponds to a dierent gaspressure, pG.

the drop of i shown in Figure 2.4d relative to the saturation vapor pressure for pureV as in Figure 2.4a:

ln

(pV

psat,0i

)=

ViRT

(pG − psat,0

i

)+

2ViγiGRTRiG

. (2.28)

The rst term on the right side of this equation corrects for the presence of diluentsin the gas phase, while the second term corrects for the eect of curvature.

For a liquid-gas system at 65 C, Figure 2.5 plots the combined correction from theright side of equation 2.28 as a function of 1/RLG at a number of gas pressures. Forreference, fuel cells operate with gas pressures as high as 3 bar, and the porous fuel-cell media contain distributions of pore sizes, with a signicant fraction of the porevolume occurring within a radius range of 0.05 to 5 µm.45 The black line in Figure2.5 corresponds to a condition where no diluent is present (pG = psat,0

L ), while thegray lines correspond to higher gas pressures, indicating that diluent is present. Theinset gure shows the results near 1/RLG = 0, which corresponds to an innite radius

46

Figure 2.6: Corrections to the melting temperature, Tm, of a spherical drop of pureice, with radius RIL, immersed in liquid, relative to the triple point for pure waterwith no interfacial curvature, Tt. Each line corresponds to a dierent liquid pressure,pL.

of curvature (a planar interface). At this value, the vapor pressure correction is dueentirely to the presence of diluents.

2.3.4 Corrections to the melting temperature

Just as the pores in a fuel cell can support a pressure dierence between a condensedphase and a gas across a curved interface, they can also support a pressure dierencebetween the two condensed phases, ice and liquid, an eect that changes the meltingtemperature, Tm, for ice.

For the moment, interaction with a porous medium is neglected. Consider rst thesimplied system in Figure 2.4e, where a spherical drop of ice is in equilibrium withliquid at Tm. From the Laplace equation,

pI − pL =2γILRIL

, (2.29)

where RIL is the radius of the drop.

47

From equations 2.1, 2.16, and 2.17,

VI(pI−pt)− VL(pL−pt) = ∆HLI

(1− T

Tt

)+∆Cp,LI

(T − Tt − T ln

(T

Tt

)). (2.30)

The second term on the right side is much smaller than the rst, and is thereforesafely neglected. Applying this assumption while combining equations 2.29 and 2.30and solving for the freezing-point depression, Tm − Tt, yields

Tm − Tt =∆VLITt∆HLI

(pL − pt)−2γILVITtRIL∆HLI

. (2.31)

The rst term on the right side of this equation gives the correction to the melting-point temperature due to deviations of the liquid pressure from the triple-point pres-sure, and the second term gives the correction due to curvature. If the rst term isneglected, the expression is known as the Gibbs-Thomson equation. If the secondterm is neglected, the expression approximates equation 2.20, which gives the melt-ing temperature for water with no curved interfaces away from the triple point, as inFigure 2.4b.

Figure 2.6 shows the amount of freezing-point depression, Tt − Tm, as a function of1/RIL at various values of pL. The results indicate that the eect of curvature onthe melting temperature is signicant within the range of pore sizes expected in thefuel-cell materials.

2.4 Understanding PCI Flow and Frost Heave

2.4.1 The driving force

It is now possible to understand the driving force for PCI ow and frost heave inporous media. When the liquid and ice pressures are equal, equation 2.30 givesthe phase boundary between bulk ice and liquid. Figure 2.7 shows this boundary,based on the thermodynamic properties for water in Table 2.1, as a dark solid linewith a negative slope. When the liquid and ice pressures are not equal, equation 2.30dictates that the pressure dierence between them increases as the magnitude of T−Ttincreases. The dot-dashed line in Figure 2.7 illustrates this: it is the liquid pressurein a system equilibrated with ice, where the ice is maintained at a constant pressureof 1 bar. The reverse scenario is described by the dashed line, which corresponds tothe pressure of ice equilibrated with liquid at a constant 1 bar. The markers shownin the gure are discussed in a later section.

Ice existing to the right side of the phase boundary is superheated, a condition whichis very unstable and requires special experimental methods to induce.46,47 For these

48

Supercooled liquid(metastable)

Superheated ice(unstable)

Figure 2.7: Ice and liquid pressures for three cases: (1) pI = pL (bold solid line), (2)pL (dot-dashed line) varies as pI is xed at 1 bar, and (3) pI (dashed line) varies as pLis xed at 1 bar. The markers (M and ) correspond to points specied for the casesin Figure 2.9. The inset shows the same series as the main gure, assuming that themolar volumes for ice and liquid are reversed.

reasons, superheated ice is not considered further in the present work. On the leftside of the boundary, the liquid is supercooled, a phenomenon that occurs readily. Inthis region, the ice pressure is always higher than the liquid pressure.

The dot-dashed line in Figure 2.7 illustrates that, when liquid coexists with constant-pressure ice, as they may in a porous medium, the liquid at the lower temperature isat a lower pressure. Liquid can ow from hot to cold as long as a ow path and adriving force exist. One way for the driving force to disappear is for the ice pressurein the cold region to increase, as shown by the dashed line. However, in this case,if the overburden is not high enough to withstand the increased ice pressure, frostheave will occur.

The inset in Figure 2.7 shows that the identical phenomenon would arise even ifthe molar volume of ice were lower than that of liquid. To generate this result,Equation 2.30 is solved assuming the properties of water in Table 2.1, except withthe molar volumes for ice and liquid reversed. As expected, the phase-boundary linehas a positive slope in this case. However, at temperatures lower than this boundary,the liquid pressure is always lower than the ice pressure, and the magnitude of the

49

dierence increases with decreasing temperature. This result is consistent with theempirical fact that frost heave occurs in uids which do not expand when frozen, suchas benzene and helium.23,48

In summary, the driving force for PCI ow and frost heave is the same and can bederived from equilibrium thermodynamics. When ice and liquid coexist at tempera-tures lower than the phase-boundary line on a p-T diagram, the ice pressure is alwayshigher than the liquid pressure, and the dierence in pressure increases with decreas-ing temperature. Consequently, if the ice pressure is not high enough and the liquidin the hot and cold regions are connected by a ow path, PCI ow will occur: liquidwill move from the hot region to the cold region due to the liquid-pressure dierence.When the liquid reaches the cold region, it freezes, increasing the amount of ice there,which may manifest itself as frost heave if the ice pressure is high enough.

Examples are constructed in a later section which further illustrate this process andclarify the relationship between PCI ow and frost heave. First it is necessary to denethe material properties used to apply the phase-equilibrium relationships derivedabove to pores.

2.4.2 Relevant Pore Properties

The role of the porous medium in the phenomenon of PCI ow and frost heave has sofar been intentionally deemphasized in order to bring into sharper focus the fact thatthe driving force that gives rise to them is thermodynamic in nature and does notdepend on the material containing the uid. However, to determine how much PCIow and frost heave occurs with a given driving force, as well as to predict how muchliquid and ice are present in a medium at a given temperature and pressure, materialproperties must be accounted for. Specically, the chemical properties of interestare a set of contact angles, while the physical properties of interest are those thatdene the shape(s) of the pores and the distribution of pore sizes within the medium.Contact angles and pore shape are discussed here in order to provide the necessarybackground for a set of simple examples. Pore-size distribution will be discussed in alater section.

Contact angles. For the present work, a hydrophilic (HI) pore is one where the contactangle formed between the solid phase, s, that comprises the pore wall, and the gasphase, in a pore lled with liquid and gas, is less than 90. An example is shownin Figure 2.8a, which depicts a stationary liquid-gas interface with a contact angle,θsG,L, where the L in the subscript indicates that liquid is the phase included in theangle. The pore is assumed to be cylindrical with a radius of rLG while the interfaceis spherical with a radius of RLG. The relationship between the two radii is

RLG = − rLGcosθsG,L

. (2.32)

50

rLG rIL rIGRILRLG

GL L I I

(a) (b) (c)

s s

θsG,L

γsL γsG

γLG γIL

γsI

θsL,I

γsL γsI

θsG,I

RIG

G

γsG

γIG

s

Figure 2.8: Cylindrical pores of radius rαβ containing a stationary spherical interfaceof radius Rαβ between uid phases α and β. Bold, straight arrows indicate forcevectors; thin, straight arrows indicate radii; and thin, curved arrows indicate contact-angle arcs.

If the pore were hydrophobic (HO), θsG,L would be greater than 90.

Figure 2.8 also denes θsL,I and θsG,I , the contact angles applicable to pores lledwith ice and liquid or ice and gas, respectively. In these cases,

RIL = − rILcosθsL,I

and RIG = − rIGcosθsG,I

. (2.33)

The three contact angles may be related to one another as follows. Assume that I,L, and G are fully deformable uids (they cannot support internal stresses), while sis a smooth solid phase that can support internal stresses. To nd γIGcosθsG,I , startwith force balances from Figure 2.8a, b, and c:

γsG = γsL + γLGcosθsG,L, (2.34)

γsL = γsI + γILcosθsL,I , (2.35)

andγsG = γsI + γIGcosθsG,I . (2.36)

Eliminate γsG by combining 2.36 and 2.34:

γsI + γIGcosθsG,I = γsL + γLGcosθsG,L. (2.37)

Substitute 2.35 for γsL:

γsI + γIGcosθsG,I = γsI + γILcosθsL,I + γLGcosθsG,L. (2.38)

Subtract γsI from both sides:

γIGcosθsG,I = γILcosθsL,I + γLGcosθsG,L. (2.39)

The result is the Bartell-Osterhof equation.49

51

Table 2.3 summarizes the contact angles used in the present work. The rst rowcontains values for θsG,I consistent with estimates for the GDL and MPL by Weberand Newman.45,50 In the second row, θsL,I , is taken to be 180. For the HI case,this is consistent with the measurements of Liu et al.,51 which were made for thecase of an ice-liquid interface in a glass capillary. Skapski also reported that the ice-liquid interface formed a hemisphere on glass.52 Values for θsG,I are calculated basedon equation 2.39. Note that, from this equation, in the case of a hydrophilic pore(θsG,L < 90), a special case may arise in which θsG,I > 90. In other words, althoughthe pore is hydrophilic, it is also icephobic. This case is discussed further in a latersection.

Note that the values for θsL,I are consistent with the prediction from equation 2.31(examples shown in Figure 2.7) that, in order for ice and liquid to coexist at tem-peratures below the phase boundary, the ice pressure must be greater than the liquidpressure. As a result, the radius of curvature of an ice-liquid interface must always bepositive, as in the illustration in Figure 2.4e. In other words, there is no stable case ofa liquid drop immersed in ice. As pointed out by Scherer,53 if θsL,I were less than orequal to 90, ice would be able to propagate through the material without resistanceat the temperatures on the phase-boundary line (there would be no curvature-inducedfreezing-point depression).

Table 2.3: Contact angles for the dierent interfaces.45,50,51

Angle θHI () θHO ()

solid-gas, liquid (sG,L) 53 104

solid-liquid, ice (sL,I ) 180 180

solid-gas, ice (sG,I ) 83 118

Pore size and shape. The size of a pore, along with its contact angle, dictate theamount of pressure dierence required to displace one uid with another. For example,for the cylindrical pore shown in Figure 2.8b containing ice and liquid, the pressuredierence between ice and liquid required for ice to penetrate the pore is given bycombining equations 2.29 and 2.33:

pI − pL = −2γILcosθsL,IrIL

. (2.40)

Smaller pores, corresponding to smaller rIL, require larger pressure dierences inorder for ice to displace the liquid contained inside.

The shape of a pore must also be considered. For example, Figure 2.9a shows a cross-section of a pore made up of a cylinder or radius r0 attached to the wide end of a

52

L IRIL

θsL,I

βrIL r0

CasepL

(bar)pI

(bar)rIL

(μm)

1.00 3.00

3.271.20

|vC|

A 0.78 >0

B 1.00 0

CasepL

(bar)pI

(bar)rIL

(μm)

1.00 4.50

4.911.13

|vH|

A 0.86 >0

B 1.00 0


vC

(b)(a)

pist

on

hot pore: TH - Tt = -0.0192 K

cylinderblunt cone

pore detail

Sectionβ(°)

r0

(μm)

cone 10

0

5.00

cylinder 5.00

L I vHL I

cold pore: TC -Tt = -0.0251 K

Figure 2.9: (a) Schematic showing key characteristics and dimensions for the poretype used in the examples. (b) Two pores, one hot and one cold, containing liquidconnected by a capillary.

cone. Ice and liquid occupy the pore, with the interface between them located withinthe cone. The interface is assumed to be a section of a sphere of radius RIL, just asin Figure 2.8b. The cone radius where the liquid meniscus meets the pore wall, rIL,is related to the radius of interfacial curvature by

RIL = − rILcos (θsL,I − β)

. (2.41)

where β is the half-angle of the cone.54 If the conical portion of the pore containsliquid and gas or ice and gas instead of ice and liquid, the corresponding relationshipsare

RLG = − rLGcos (θsG,L + β)

and RIG = − rIGcos (θsG,I + β)

, (2.42)

respectively. By combining equations 2.29 and 2.41, the cone radius can be relatedto the ice-liquid pressure dierence:

pI − pL = −2γILcos (θsL,I − β)

rIL. (2.43)

Note that, when β = 0, the cone becomes a cylinder, and equation 2.43 matchesequation 2.40. On the other hand, as β increases, rIL decreases.

2.4.3 Examples of PCI Flow and Frost Heave in Pores

The following examples are based on variations and combinations of the pore geometryshown in Figure 2.9a. The ice pressure is set by a piston on the right side of thecylinder. Friction between the walls and the liquid, ice, and piston, is neglected. The

53

parameters β and r0 are constant at 10 and 5 µm, respectively. The uid properties

are taken from Table 2.1 while the contact angles are taken from Table 2.3.

Flow between pores. In Figure 2.9b, two pores are shown with tips connected by a necapillary. For the purposes of this example, the capillary serves as a liquid connectionbetween the pores. The radius of the capillary is taken to be much less than rIL, sothat ice cannot enter the capillary under the conditions considered. The pore on theleft is taken to be 0.0251 K below Tt while the pore on the right is taken to be 0.0192K below Tt. Two cases, A and B, are listed in the tables below each pore. These casescorrespond to those with the same labels in Figure 2.7, although for clarity only thepoints for the phase with a varying pressure are shown in Figure 2.7; the pressure ofthe opposing phase is always 1 bar.

The radius of the cone where the ice-liquid meniscus meets the pore wall, rIL, iscomputed from equation 2.43 and listed in the tables in Figure 2.9. Note that rILis larger for the warmer pore. Since rIL is proportional to RIL (see equation 2.41),this is consistent with Figure 2.6. Also, it demonstrates the role of pores in PCI owand frost heave, as presented by Everett:24 pores prevent ice from forming until thetemperature is low enough, enabling ow of liquid water toward a growing ice crystal.

In case A, the ice in each pore is maintained at a constant pressure, and a liquidpressure dierence exists due to the temperature dierence between the pores. As aresult, ice melts in the warmer pore and ows to the colder pore where it freezes. Thetotal amount of ice increases in the colder pore and decreases in the warmer pore.Therefore, the piston in the cold pore has a nonzero velocity, vC , in the directionshown by the associated arrow, and the piston in the warmer pore also has a nonzerovelocity, vH .

Case A can be considered analogous to one of two scenarios in a porous medium.In an unsaturated medium, meaning that gas exists in addition to liquid and ice,case A would correspond to lling of pores previously occupied by gas with ice, thatis, increasing the level of ice saturation in the cold part of the medium. In sucha scenario, frost heave does not occur because additional ice can simply move intounoccupied pores. Alternatively, in a saturated medium (no gas present), case Awould correspond to frost heave. A layer of pure ice separates the cold part of themedium and grows because all of the medium's pores are lled (there is nowhere elsefor the ice to accumulate) and the ice pressure is equal to the overburden. Note thatthese two scenarios can also occur sequentially, as has been observed experimentallyin soils. The cold region may start unsaturated, become saturated due to PCI ow,and then frost heave may occur because the overburden pressure is reached.

In case B, the ice pressure in the cold pore has been increased, which has the eect ofincreasing the liquid pressure as well. As a result, there is no liquid-pressure dierencebetween the pores, no liquid ow between them, and vC= vH = 0. Note that rILchanges only slightly from case A to case B, and is smaller in the cold pore in both

54

L

(a)

(b)

L

CaseT-Tt

(K)pL

(bar)pI

(bar)rLG

(μm)rIL

(μm)

0.80

1.03

4.500.95

1.17 4.50

---

3.50

rIG

(μm)

HI -0.0188 5.00

HO -0.0205 ---

CaseT-Tt

(K)pL

(bar)pI

(bar)rLG

(μm)rIL

(μm)

0.73

1.02

3.000.95

1.21 ---

---

---

rIG

(μm)

HI -0.0247 5.00

HO -0.0247 3.13

hydrophobic (HO) case

L I

G

hydrophilic (HI) case

G

L

(c)

G

L I

G

G

(d)

(e)

I

I I

I

L

L

L

I

LI

I

CaseT-Tt

(K)pL

(bar)pI

(bar)rLG

(μm)rIL

(μm)

0.82

1.03

------

--- ---

5.00

3.50

rIG

(μm)

HI -0.0060 ---

HO -0.0060 ---

CaseT-Tt

(K)pL

(bar)pI

(bar)rLG

(μm)rIL

(μm)

0.82

1.03

5.000.95

1.16 5.00

---

3.50

rIG

(μm)

HI -0.0178 5.00

HO -0.0194 ---

CaseT-Tt

(K)pL

(bar)pI

(bar)rLG

(μm)rIL

(μm)

0.82

1.03

5.000.95

1.16 5.00

---

3.50

rIG

(μm)

HI -0.0178 5.00

HO -0.0194 ---

G

G

G

G

G

Figure 2.10: Equilibrium liquid, ice, and gas congurations for a hydrophilic pore(left) and a hydrophobic pore (right). In each successive subgure, (a) through (e),more heat is removed.

cases. As mentioned above, such a result requires that pI can be maintained by thecontainer for the ice. Otherwise the container will fracture and frost heave willoccur.

The eect of wettability. The example in Figure 2.9b provides insight into the mecha-nism behind PCI ow and frost heave, which applies regardless of whether the mediumis saturated or unsaturated. However, for simplicity the gas phase is not shown. Next,the eect of the presence of gas is explicitly considered. In Figure 2.10a, two pores,one hydrophilic and one hydrophobic, are shown. Between them is a table listing theirtemperatures, the ice and liquid pressures, and the radius of the various menisci. Bothcontain liquid and gas and are at a temperature and pressure that are on or to theright of the ice-liquid phase boundary. The gas pressure is taken to be 1 bar through-out this example. Figures 2.10b through 2.10e show the growth of an ice crystal asheat is withdrawn from each of the pores and will be discussed in turn.

In Figure 2.10b, the temperature has been lowered to the point where rIL is equal tor0, meaning that ice can penetrate the cylindrical section of the pore, and nucleation

55

has arbitrarily been taken to have begun on the right side of each pore. As shownin the corresponding table, the temperature at which rIL = r0 is not the same in thetwo pores due to the dierent liquid pressure in each one. In Figure 2.10c, more heathas been withdrawn and the crystals have grown, but the temperature is the same asin (b).

In Figure 2.10d, more heat has been withdrawn, and the ice crystals have grown intothe conical sections of each pore such that rIL is the same in both cases. In order forthis to happen, the temperature must be lower than in cases (b) and (c) since rIL islower. In the HI pore, the ice pressure is the same as in (b) and (c) because rIG hasnot changed. However, the ice-liquid pressure dierence has increased because of thelower temperature. Consequently, the liquid pressure is lower than in (a)-(c). In theHO pore, the situation is reversed. The liquid pressure is unchanged because rLG hasnot changed, but the ice pressure has increased because of the lower temperature.

In Figure 2.10e, the temperature is lower than in (d), meaning that rIL and theliquid pressure have decreased further in the HI pore. In the HO pore at the sametemperature, all of the liquid has frozen in the HO pore. Note that the ice pressurein the HO pore has been taken to be constant at the pressure at which rIG = rLG.As a result, rIG is dierent from rLG.

The p − T paths taken by each of the pores is illustrated in Figure 2.11. The ice-liquid phase boundary is shown as a thick, solid black line with a negative slope. Thetwo thinner black lines correspond to the liquid and ice pressures in the HI pore atvarious temperatures, while the two thin gray lines correspond to the liquid and icepressures in the HO pore. When the lines have a solid pattern, the indicated phaseexists. If a line is dashed, it does not. Instead, it represents the pressure at whichthe phase would have to be in order to exist. Whether it exists or not is a functionof the properties of the pore.

As discussed above, in the HI case, once ice and liquid coexist, the ice pressureis constant because rIG is constant and the liquid pressure decreases. In the HOcase, the liquid pressure is constant while it coexists with the ice because rLG isconstant. Although it will not necessarily be the case in a real system that rIGor rLG is constant, these examples serve to illustrate that in unsaturated systems(that is, when gas is present in the medium as a contiguous phase with a xedpressure), whether hydrophilic or hydrophobic, the pressure of the condensed phasein contact with the gas is set by the contact angle θsG,i, where i is the condensedphase. Furthermore, determining whether ice penetrates a pore containing gas andliquid depends on θsG,L, θsG,I , and θsL,I .

In contrast, if no gas is present, such as in Figure 2.9b, only θsL,I is necessary to deter-mine the size of pore that ice can penetrate at a given temperature (given the ice andliquid pressures). In fact, in a saturated system, the hydrophilicity or hydrophobicityof a pore is essentially irrelevant because these denitions are based on θsG,L. This

56

Figure 2.11: Ice and liquid pressures for the HI and HO pores in Figure 2.10. Solidlines indicate the pressure of ice or liquid that exists in the pore, while dashed linesindicate the pressures that would be required in order for them to exist at a giventemperature. The gas pressure is assumed to be constant at 1 bar.

57

(a)

(b)

L I

(c)

L I

G

G

(d)

L I

L I


vH

vC

cold pores: TC -Tt = -0.0247 K

vC

vC

hot pore: TH - Tt = -0.0188 K

pL

(bar)pI

(bar)rIL

(μm)rIG

(μm)

0.95 4.50 5.00

|vH|

0.80 >0

pL

(bar)pI

(bar)rIL

(μm)rIG

(μm)

0.95 3.00 5.00

|vC|

0.73 >0

pL

(bar)pI

(bar)rIL

(μm)rIG

(μm)

0.98 3.04 ---

|vC|

0.77 >0

pL

(bar)pI

(bar)rIL

(μm)rIG

(μm)

1.01 3.08 ---

|vC|

0.80 0

Figure 2.12: An unsaturated pore (a) with a liquid connection to pores with increasinglevels of ice pressure, (b) through (d).

hypothesis is supported by the experiments of Sage and Porebska,55 who measuredfrost heave in saturated samples of hydrophilic and hydrophobic soils. When theparticle-size distribution of the HI and HO samples matched, the amount of heaveobserved was very similar. Fuel-cell media are generally not saturated, however, andestimates for all three contact angles are required.

Finally, this example demonstrates that, just as in a saturated system, in unsaturatedsystems the liquid in larger pores freezes at higher temperatures than smaller ones,regardless of whether the pores are hydrophilic or hydrophobic.

Flow from an unsaturated pore. As mentioned above, experimental results have shownthat frost heave can be induced in a porous medium that is not initially saturatedbut contains a xed amount of water.25,26 In this case, the water content in the coldregion of the medium increases while that in the hot region decreases. If the coldregion becomes saturated, frost heave can occur, despite the fact that the hot regionis not saturated.

Figure 2.12 illustrates how this can happen. In (a), an unsaturated pore containsliquid, ice, and gas at a xed temperature, TH . If this pore is connected to anotherunsaturated pore at a lower temperature, as in (b), PCI ow occurs from the hotpore to the cold one. The ice pressure in this case is the same in both pores becauserIG is the same.

If the cold pore becomes saturated, as represented in (c) by the presence of the piston

58

next to the ice crystal instead of gas, then the ice pressure can increase, which alsoraises the liquid pressure. However, a driving force for liquid ow to the cold pore canstill exist, even with this increase. If the ice pressure equals the overburden pressure,as is assumed in (c), the ice crystal can continue to grow, meaning that frost heave isoccurring. Only if the ice pressure reaches a high enough level, depicted in (d), willthe ice crystal in the cold pore stop growing. In this case, the liquid pressure in thehot and cold pores is the same, and thus the driving force has been eliminated.

In this example, the ice pressures are relatively modest, and one might reasonablyquestion whether frost heave would actually occur in a real porous medium underthese conditions. However, this result is due to the temperatures that have been cho-sen in order to allow for illustrations consistent with other examples. The maximumice pressure that could theoretically be achieved is calculated by rearranging equation2.30 to give

pmaxI = pt+

VLVI

(pL−pt) +∆HLI

VI

(1− T

Tt

)+

∆Cp,LIVI

(T − Tt − T ln

(T

Tt

)). (2.44)

Therefore, if the cold pore's temperature in Figure 2.12d decreases to 0.115 K belowTt while maintaining TH (which sets pL) at the hot pore, the maximum ice pressure is2.00 bar. It should be noted that good agreement has been found between equation2.44 (usually without the specic-heat correction) and experimental measurements ofpmaxI for both H2O

56,57 and 4He,48 although the magnitude of deviations from theorydo increase with lower temperature.

Stopping PCI ow. There are three ways in which PCI ow (and thus, frost heave,if it is occurring), can be stopped. These are illustrated in Figure 2.13 and will bediscussed in turn.

The rst method is for the the ice pressure in the cold pore to be raised high enoughsuch that its liquid pressure is equal to that of the hot pore. As discussed above,Figure 2.12d shows one example of how this might occur: the pore is saturated andthe overburden equals pmax

I , meaning that the material can withstand the maximumice pressure. An alternative scenario is shown in Figure 2.13a, where the cold poreis comprised of a hydrophobic section on the left and a hydrophilic section on theright. In this case, although the medium is unsaturated, the ice pressure is not highenough to penetrate the hydrophobic pore, even at pmax

I . Such a scenario is relevantto fuel-cell systems due to the presence of both HI and HO pores in a single medium.Note that in order for frost heave not to occur this still requires that the pore be ableto withstand pmax

I .

As discussed above, this method of stopping PCI ow can require very high overbur-den pressures as the temperature in a cold region of a medium is reduced relative tothe hot region. In practice, high overburden pressures generally do not develop dueto a combination the two remaining methods.

59

(a)

(b)

(c)

Method of stopping PCI flow

PoreT-Tt

(K)pL

(bar)pI

(bar)

H -0.0188 0.80 0.95

-0.0247 0.80C 1.01

Decrease the hotpore’s liquid pressure.

Disconnect the hotand cold pores.

Cold (C) pore Hot (H) pore

PoreT-Tt

(K)pL

(bar)pI

(bar)

H -0.0188 0.73 ---

-0.0247 0.73C 0.95

PoreT-Tt

(K)pL

(bar)pI

(bar)

H -0.0188 0.80 0.95

-0.0247 0.73C 0.95

L I GLGIncrease the cold pore’s ice pressure.

L GLIG

LIG L I Gx x

I

Figure 2.13: Examples of three methods for stopping PCI ow.

The second method of stopping PCI ow is to lower the liquid pressure in the hotpore. From equation 2.44, this has the eect of lowering pmax

I . Figure 2.13b showsone way in which this could occur: the hot pore is drained of water until the liquidpressure is low enough to match that of the cold pore. In this case, there is liquidbut no ice in the hot pore, because ice cannot penetrate deep enough into the coneat this temperature. The source of water at a higher pressure has been depleted, andtherefore PCI ow stops, despite the fact that a temperature dierence still exists.This scenario is common in unsaturated systems,25 and is, therefore, of interest tofuel-cell porous media.

The third method of stopping PCI ow is to disconnect the hot and cold pores. InFigure 2.13c this is represented in a literal way by a broken ow path between hot andcold pores that contain a fair amount of liquid. Two means by which disconnectioncan occur in real media are now discussed. They are not mutually exclusive.

The rst means is as follows. As the temperature is lowered when ice and liquid arepresent, the magnitude of the pressure dierence between them increases. This is thedriving force for PCI ow and frost heave. However, a higher pressure dierence alsomeans that the ice can penetrate smaller pores (see equation 2.43). Therefore, at lowertemperatures the amount of liquid present in a medium is much smaller. Because theeective permeability of the medium is generally related to the amount of liquid, theresistance to ow from hot to cold increases with decreasing temperature. Once alow enough temperature is reached, what liquid remains no longer forms contiguouspathways, and PCI ow cannot occur, even if a large temperature dierence exists.This point is revisited in a more quantitative way below, once a method for estimatingwater content in a real medium has been developed.

60

The second means of disconnection applies to composite systems. That is, systemsthat contain layers of media with dierent properties, such as fuel cells. If the hotand cold regions of the system are separated by a medium that is less susceptible toPCI ow than the regions themselves, the amount of water redistribution from hot tocold can be signicantly reduced. Gieselman et al.58 demonstrated this by measuringwater content and frost heave in an initially unsaturated soil sample with and withouta hydrophobic layer separating the hot region from the cold region. If present, thehydrophobic layer was 10 % of the overall sample thickness and was placed near thehot region of the sample. Although water still redistributed within the cold regionwith or without the HO layer, the layer did reduce substantially the amount of newwater that owed from the hot region to the cold region during the freezing process.

2.5 Calculating Ice, Liquid, and Gas Saturations

The examples above are constructed to provide insight into the nature of PCI ow andfrost heave and the role of pores in these phenomena. In particular, the quantitativerelationship among the uid-phase pressures and the meniscus radii for each uidpair is emphasized. However, real porous media have a distribution of pore sizes andcontact angles, and the one- and two-pore models above do not yield quantitativeestimates for the amount of each phase that one should expect in a porous mediumat a given temperature and pressure. This section addresses this gap by developinga detailed model for saturation.

2.5.1 Saturation-model development

The porosity is the fraction of a unit volume of medium that is available to be lledwith uid. The saturation of phase α in a porous medium is dened as the fractionof the porosity occupied by α. We seek a model capable of predicting SI , SL, andSG based on the temperature and the pressures of these phases. To do so, we startwith the model for SLG, the liquid-phase saturation in a porous medium that alsocontains gas, developed by Weber, Darling, and Newman.45 This method assumesthat the pores in a medium can be represented by bundles of cylindrical capillaries.Two log-normal pore-radii distributions are used, and pores may be hydrophilic orhydrophobic, with the relative amount of each set by fHI, the fraction of hydrophilicpores. HI and HO pores are assumed to have the same pore-size distributions.

To compute SLG, hydrostatic equilibrium between the liquid and gas phases is as-sumed. For the purposes of the present work, hydrostatic equilibrium means that theLaplace equation is satised and that the interface between the two phases is sphericalin shape, as in the simplied examples in the sections above. In other words, equa-

61

Table 2.4: Physical properties used for determining saturations.50,60

Description Symbol GDL MPL CL

Fraction hydrophilic pores fHI 0.20 0.20 0.20

PSD properties

Characteristic radii (µm) r0,1 6.00 2.00 0.20

r0,2 0.70 0.05 0.05

Characteristic spreads s1 0.60 0.75 1.20

s2 1.00 1.00 0.50

Fraction in distribution 1 fr,1 0.95 0.50 0.50

tion 2.24 applies. Everett refers to this condition as Laplace equilibrium, althoughno particular interface shape is assumed in his treatment.59

Combining equations 2.24 and 2.32 yields an expression for rhLG, the critical radius ofa pore of type h, where h can be either HI or HO:

rhLG = −2γLGcosθhsG,LpL − pG

. (2.45)

The critical radius represents the largest liquid-lled pore in a hydrophilic mediumor the smallest liquid-lled pore in a hydrophobic medium.

The saturation level of the pores of type h is related to the critical radius of type hby

ShLG =∑k

fr,k2

[1 + ϑherf

(lnrhLG − lnr0,k

sk√

2

)], (2.46)

where k is the pore-size-distribution (PSD) number (either 1 or 2), fr,k is the fractionof the overall porosity that is in PSD k, ϑh is 1 for HI or -1 for HO pores, r0,k is thecharacteristic radius of distribution k, and sk is the characteristic spread (a measureof the breadth of the PSD where a low number indicates that most of the pores areclose in radius to r0,k). The hydrophilic and hydrophobic saturations are combinedto give the overall saturation of the medium:

SLG = fHISHILG + (1− fHI)S

HOLG . (2.47)

The relationships between liquid saturation and liquid capillary pressure (pL − pG)that result from this method for the gas-diusion layers (GDLs), microporous layers(MPLs), and the catalyst layers (CLs), based on the PSD properties in Table 2.4, areshown in Figure 2.14. The saturation corresponding to a capillary pressure of zero is

62

Figure 2.14: Liquid saturation for fuel-cell media as a function of capillary pressureat temperatures where no ice is present.

the fraction of hydrophilic pores, fHI. The order of pore lling with increasing liquidcapillary pressure is (1) small, HI (2) large, HI (3) large, HO and nally (4) small,HO.

If a given pore is lled with ice and gas or ice and liquid instead of liquid and gas, thesame assumptions used for equation 2.45 are applied (hydrostatic equilibrium and aspherical interface), and the critical pore radii are related to the pressure dierencesbetween the phases by

rhIG = −2γIGcosθhsG,IpI − pG

and rhIL = −2γILcosθhsL,IpI − pL

. (2.48)

Equations 2.45 and 2.48 are combined to give the equivalent liquid-gas pressure dif-ference for a given pore radius based on the ice-gas and liquid-ice pressure dierencesthat would be present if those phases occupied the pore instead:

pL − pG = (pI − pG)γLGcosθhsG,LγIGcosθhsG,I

and pL − pG = (pI − pL)γLGcosθhsG,LγILcosθhsL,I

. (2.49)

Equation 2.49 is used to translate ShLG, the liquid saturation in the pores of type h ina liquid-gas system, into ShIG and ShIL, which are the saturations in an ice-gas system

63

Figure 2.15: Ice and liquid saturations in the CL as a function of capillary pressure.

and an ice-liquid system, respectively. This is done by calculating ShLG based on theliquid capillary pressures from equation 2.49, as shown in the brackets below:

ShIG = ShLG

[(pI − pG)

γLGcosθhsG,LγIGcosθhsG,I

]and

ShIL = ShLG

[(pI − pL)

γLGcosθhsG,LγILcosθhsL,I

].

(2.50)

To reiterate, the parameters inside the brackets are not multiplied by ShLG; they arearguments of the function ShLG [pL − pG], which is dened by equation 2.45 and 2.46.

The method above for predicting ShIL from ShLG was rst introduced by Koopmansand Miller61 for the case of saturated, hydrophilic soil. Verication was completedon several dierent types of soil, including colloidal and noncolloidal. In colloidalsoil, adsorption forces dominate over capillary forces, while the reverse is true fornoncolloidal soil. The porous media of interest in the present work are taken to benoncolloidal. The results conrm that, in noncolloidal systems that are hydrophilicand saturated, the ratio γLGcosθHI

sG,L/γILcosθHIsL,I can be used to translate the liquid-

gas saturation curve (SLG) into the ice-liquid freezing curve (SIL). Black and Ticeprovided further verication.62 This approach is extended in the present work toestimated liquid SHO

IL and ShIG as well.

As an example, SLG and SIG as a function of liquid-gas or ice-gas capillary pressure,respectively, are compared in Figure 2.15, assuming CL properties. The dotted line

64

represents the fraction of hydrophilic pores, fHI. For the solid lines, saturations belowa capillary pressure of zero are equal to fHIS

HIαG while those above zero are equal to

fHI + (1− fHI)SHOαG . The order of lling for ice is the same as outlined above for the

liquid-gas system.

The thin solid line corresponds to SIG for the case where the HI pores in the CL areicephilic, that is θHI

sG,I < 90, as in Table 2.3. However, if θHIsG,I > 90, the hydrophilic

pores will be icephobic. This means that although liquid capillary pressures belowzero are sucient to ll these pores with liquid, positive ice capillary pressures mustbe present to ll them with ice. The dashed line in Figure 2.15 shows the expectedSIG assuming icephobic HI pores, where θHI

sG,I = 100 . In this case, below a capillarypressure of zero, no ice is present, while above zero, ice lls the HI and HO pores,which are both icephobic. Larger pores are lled rst, followed by smaller pores.

The results in Figure 2.15 have implications for start-up of a fuel cell from belowfreezing. Because water is produced in the cathode catalyst layer and ice can ac-cumulate there, the ice pressures in that layer can be very high. As the capillarypressure increases from zero, the ice must penetrate smaller and smaller hydrophobicpores. As shown, to reach a saturation of 0.5 with the CL properties listed in Table3.5, the ice capillary pressure must be 9.3 bar. Such high ice pressures could cause anumber of problems, including irreversible damage to the pores in the CL, delamina-tion between the cCL and the adjacent layers, and ice lens growth within the cCL. Itis important to reiterate, however, that the root cause is not frost heave as dened inthe present work. Frost heave requires a source of water in a hot region of a porousmedium feeding ice crystal growth in a cold region. Instead, what occurs is simplyforce-lling of the cCL, which is the hot region, with product water from the fuel-cellreaction.

Regardless of whether the CL is icephilic or icephobic, these results predict that theamount of pressure required to ll the CL with ice to a given saturation is higherthan that required to ll it with liquid (except at fHI for the icephilic case, where allthe HI pores are lled and all the HO pores are empty, corresponding to an innitepore radius).

The relationships derived above provide the basis for determining saturations in sys-tems containing two phases: liquid and gas, ice and gas, or ice and liquid. However,in fuel-cell media, all three phases may be present at once. For this reason, a method-ology for determining ice, liquid, and gas saturations under all conditions of interestmust be dened. The approach proposed here is to compute rst the pairwise satura-tions, as dened above, compare them using a set of logical conditions to determinewhich actually prevails, and then combine them to predict all saturations.

Specically, the saturations of type h from equations 2.46 and 2.50 are used to deter-mine the combined liquid and ice saturations, SL and SI , using the algorithm shownin Figure 2.16. In this gure, the total condensed-phase saturation, ST , is the sum of

65

Hydrophilic (HI) Hydrophobic (HO) Combined (HI + HO)

Tota

l (T

= L

+ I)

Liqu

id (L

)Ic

e (I)

Pore type

Uns

atur

ated

Med

ium

HOHOHOLTI SSS −=HIHIHI

LTI SSS −=

( ) HOHI

HIHI 1 TTT SfSfS −+=

( ) HOHI

HIHI 1 LLL SfSfS −+=( )0,max HOHOHO

ILTL SSS −=( )HIHIHI ,min ILLGL SSS =

LTI SSS −=

( )HOHOHO ,max IGLGT SSS =

( )1,1min HIHIHI,LGIG

BT SSS +−=

( )( )HIHIHIHI, ,,max,0if LGIGLGLIA

T SSSppS >−=

( )BT

ATIsGT SSS HI,HI,HI

,HI ,,90if o<= θ

1

2

4

5

6

7

3

Gas

(G)

HOHO 1 TG SS −=HIHI 1 TG SS −= TG SS −=1

Primary output

Sat

urat

ed M

ediu

m

Liqu

id (L

)Ic

e (I) HOHO 1 LI SS −=HIHI 1 LI SS −=

( ) HOHI

HIHI 1 LLL SfSfS −+=HOHO 1 ILL SS −=HIHI

ILL SS =

LI SS −=1

Primary output

Figure 2.16: Algorithm for determining liquid, ice, and gas saturations.

the liquid and ice saturations. The gas-phase saturation is determined from the factthat the saturations must sum to unity:

SG = 1− ST = 1− SI − SL. (2.51)

The algorithm shown in Figure 2.16 is divided into two subalgorithms separated by ahorizontal gray line. The equations above the gray line apply to a saturated medium:that is, one that contains no appreciable amount of gas (SG is taken to be zero).Below the line are the equations for an unsaturated medium. Both algorithms areconsistent with one another: the unsaturated algorithm collapses to the saturatedone when SHI

LG = SHIT = SHO

T = 1, as is the case when SG = 0.

Several operations that have not been discussed above are numbered in the unsatu-rated algorithm. The basis for each of these operations is as follows.

1. This operation chooses the appropriate subroutine based on whether the HIpores are icephilic or icephobic.

2. In this case, the HI pores are icephilic. If the ice pressure is greater thanthe liquid pressure, then it is possible that ice penetrates some pores, and so

66

operation 3 applies. If not, ice cannot penetrate any pores, and so the totalsaturation is SHI

LG.

3. If the ice pressure is high enough relative to the liquid pressure, ice can occupypores that are larger than those that the liquid can occupy (when gas is theopposing phase in either case). Therefore, the phase in contact with the gas isice, and liquid occupies smaller pores. Otherwise, liquid is in contact with thegas, and there is no ice. The saturation of the phase in contact with gas equalsthe total saturation (assuming the HI pores are icephilic).

4. In this case, the HI pores are icephobic. If pI−pL is small but positive, then themedium could contain liquid in the ne pores, gas in the intermediate pores,and ice in the large pores. In such a case, the ice saturation is given by 1-SHIIG, not S

HIIG, because gas is wetting relative to ice. This must be added to

the liquid saturation to give the total saturation. However, at higher positivevalues of pI − pL, ice penetrates into the intermediate pores, displacing gas,and eventually the HI pores are saturated, meaning that the total saturation isunity and only liquid and ice exist.

5. If ice can penetrate the pores lled with liquid in the HI pores, then the icecontacts the gas in the larger pores and the liquid will exist in ner pores, witha saturation given by SHI

IL. Otherwise, no ice will be present and the liquidsaturation will be SHI

LG.

6. In the HO pores, if pI − pL is negative, no ice exists, and the saturation is givenby SHO

LG . If pI−pL is positive but small relative to pL−pG, than ice may occupythe larger pores while liquid occupies the intermediate ones, contacting the gaswhich is in the smallest ones. If pI − pL is large relative to pL − pG, than noliquid exists because ice can penetrate all of the pores that liquid is able to ll.In this case ice occupies the larger pores, contacting gas in the smaller pores.The total saturation of HO pores is equal to the saturation of whichever phaseis contacting the gas.

7. The liquid saturation in the HO pores is the dierence between the total satu-ration (assuming that the liquid phase contacts the gas) and the ice saturation.However, the minimum value is zero, which occurs if pL−pG < 0 or ice contactsthe gas directly (see operation 6).

When applying the algorithm shown in Figure 2.16, the liquid, ice, and gas pressuresare required to compute the pairwise saturations ShLG, S

hIG, and S

hIL using equations

2.46 and 2.50. These pressures are determined as a function of position in a mediumby solving dierential transport equations. Alternatively, phase equilibrium could beassumed, meaning that equation 2.1 is satised, which in turn means that equations

67

Figure 2.17: Liquid saturation for fuel-cell media at temperatures below the meltingpoint, assuming a constant ice pressure of 1 bar.

2.26 and 2.30 apply. This is consistent with what Everett refers to as Kelvin equilib-rium.59 Finally, if all three phase are present, a combination of these two approachescould be used (phase equilibrium might be assumed between only two of the threephases).

2.5.2 Saturation-model results

Figure 2.17 shows the equilibrium liquid saturations as a function of temperature forthe GDL, MPL, and CL properties listed in Table 3.5, assuming constant ice and gaspressures of 1 bar. The water in the GDL, which contains a higher proportion of largepores, freezes at a high temperature relative to the CL. In the MPL, which contains amix of large and ne pores, the liquid in the large pores freezes at high temperaturesand that in the small pores freezes at much lower temperatures.

As shown in Figure 2.6, whether or not water in a given pore freezes depends notonly on the radius of curvature, but also on the liquid pressure: for a given radius,freezing occurs at a lower temperature for higher liquid pressures. In addition, higherice pressures (which correspond to higher liquid pressures) are required at higher total(liquid plus ice) saturations in a given medium. Therefore, at a given temperature,the amount of liquid present is higher for higher total saturations. This is illustrated

68

Figure 2.18: Liquid saturation below the melting point for the MPL at dierent totalsaturations (liquid plus ice).

in Figure 2.18, which shows the predicted liquid saturation for ve dierent levels of(constant) total saturation.

Horizontal lines near T − Tt = 0 in Figure 2.18 indicate that no ice exists (SL = ST ).In other words, the freezing point has been depressed. Both types of freezing-pointdepression predicted by equation 2.31 and illustrated by Figure 2.6, pressure-inducedand curvature-induced, are accounted for in all curves, but are most evident at thetwo extremes of ST shown. At a total saturation of 0.8, the temperature oset infreezing is due to the fact that the liquid pressure is high relative to pt. At a totalsaturation of 0.05, the oset is due to the fact that ice cannot penetrate the ne poreoccupied by the liquid until the temperature is reduced far enough below Tt.

Some recent fuel-cell models35,63 assume that the liquid-saturation curve, SLG (pL − pG),as in Figure 2.14, applies as the liquid water in the medium freezes. However, theSLG (pL − pG) relationship is predicated on the existence of a liquid-gas interface,which sets the pressure of the liquid. As discussed above, Koopmans and Millershowed that this is not necessarily be the case for a noncolloidal porous medium, asit is tantamount to assuming that the ratio γLGcosθHI

sG,L/γILcosθHIsL,I is unity.

61 Themodel in the present work accounts for the fact that, once ice exists in the medium,a liquid-gas interface may no longer exist, and may be replaced by ice-liquid and/orice-gas interfaces.

69

Figure 2.19: Liquid saturation predicted in the MPL when the saturation model isused to account for the ice-gas and ice-liquid interfaces (series SL) compared with theprediction if the uncorrected liquid-saturation relationship to liquid capillary pressurefrom Figure 2.14 is used directly (series SLG).

As an example, in Figure 2.19 the prediction of liquid saturation in the MPL predictedby the full saturation model (the series labeled SL) is compared to what is expectedif the SLG (pL − pG) relationship is always valid (the series labeled SLG). The latterseries is constructed by assuming gas and ice pressures of 1 bar (consistent with theother cases), calculating the liquid pressure from equation 2.30, and computing thesaturation based on equations 2.45 and 2.46.

There is a clear dierence in the amount of liquid water that is expected at a giventemperature for the two series. Because the liquid capillary pressure is the same at agiven temperature in both cases, this dierence stems from the assumption of whatphase is contacting the liquid. The SLG case assumes the existence of a liquid-gasinterface while the MPL case accounts for the fact that there is an ice-liquid interface.Although the dierence may appear small, when modeling PCI ow and frost heavethe results can be quite sensitive to SL. This is due primarily to the fact that theeective permeability of the material is a strong function of the liquid saturation. Forexample, often it is considered proportional to the cube of SL. Therefore, the amountof ow possible for a given liquid-pressure dierence can vary signicantly with SL.

In summary, the method described above enables estimation of liquid, ice, and gassaturations (and their distributions between HI and HO pores) for a given mediumbased on the chemical properties shown in Tables 2.2 and 2.3, the assumption of

70

hydrostatic equilibrium, and the physical properties shown in Table 3.5. The approachextends the work of Weber, Darling, and Newman50 to the case where ice exists ina porous medium in addition to liquid and gas. Similarly, it extends the work ofKoopmans and Miller61 to a medium where gas exists in addition to ice and liquid,and also to a medium of mixed wettability. In addition, the uid properties shownin Table 2.1 may be used in combination with equations 2.26 and 2.30 to estimatethe saturations at a given temperature and pressure when the system is also in phaseequilibrium.


In the present work, a thermodynamic mechanism for PCI ow, whereby liquid owsfrom hot to cold in a freezing porous medium, has been derived. The mechanism doesnot depend on the volume expansion of ice relative to liquid water, and applies equallyto uids that contract upon freezing. When ice and liquid are in phase equilibrium attemperatures below the phase-boundary line, the liquid pressure is always lower thanthe ice pressure, and the dierence in pressure increases with decreasing temperature.As a result, liquid in a cold part of a medium is likely to be at a lower pressure thanthat in the hot region. This liquid-pressure dierence can drive liquid to ow fromthe hot region to the cold region, where it freezes.

Frost heave, where a layer of pure ice grows from the edge of or within a porousmedium (or between adjacent porous media), has been shown to be a possible, butnot inevitable, result of PCI ow. In order for frost heave to occur, the pressure ofthe ice must be equal to the overburden, which is the maximum constraining pressureavailable. Frost heave can be stopped by increasing the overburden, lowering theliquid pressure in the hot region, or eliminating the ow path from the hot region tothe cold region.

Pores play a critical role in PCI ow and frost heave because they support the pressuredierence between the ice and liquid phases, enabling liquid to remain stable in asupercooled state. Because a larger pressure dierence, which corresponds to a lowertemperature, is required for ice to penetrate ner pores, liquid in larger pores freezesat higher temperatures than in smaller pores. The smaller pores also provide the owpath for liquid water moving toward the cold region.

The thermodynamics developed to explain PCI ow and frost heave has been utilizedto develop a method for predicting ice, liquid, and gas saturations in a porous mediumat all temperatures and pressures relevant to PEFC operation. The method accountsfor both pore-size distribution and hydrophilicity and hydrophobicity and can beused to model either saturated or unsaturated media. With the saturation model,predictions are given for the amount of liquid water present in each type of fuel-cellporous medium at low temperatures. The results show that liquid in a structure with

71

larger pores, such as a GDL, is expected to freeze at higher temperatures than liquidin a medium, such as an MPL, that contains a signicant amount of ne pores. Inaddition, the amount of liquid water present at a given temperature below freezingis shown to increase with increasing total (liquid plus ice) saturation, due to theincreased pressure of the liquid.

Irreversible damage to the cCL not due to frost heave is likely during cold start iftoo much ice, formed by the fuel-cell reaction, is allowed to accumulate. Increasingthe ice saturation requires that ice penetrate small hydrophobic pores, which in turnrequires higher ice pressure and can put large amounts of stress on the pores in thecCL. Although the damage from any one start-up may be relatively small, over thecourse of many freeze/start cycles signicant structural damage can occur. The modeldeveloped here provides a framework for predicting the saturation as a function ofice capillary pressure, and can therefore be used to impose design and operationalconstraints that minimize the accumulation of ice during start-up.

72

Chapter 3

Two-Dimensional Cold-Start Model

3.1 Introduction

In the PEFC literature, several models have been presented that simulate the cold-start behavior of single cells. Mao and Wang64 and Wang65 both present models topredict cathode-catalyst-layer ice content and cell performance as a function of timeduring start-ups from below 0 C. In both cases, the entire cell is taken to be at alumped temperature, ice is assumed to accumulate only in the membrane and thecathode catalyst layer, and the rate of ice production is assumed to be independentof position in the catalyst layer.

Models of greater complexity have also been developed. Ahluwalia and Wang66 andMeng67 present two-dimensional multiphase models for cold start, while Wang andcoworkers have constructed a three-dimensional multiphase model.68,69 These multi-dimensional models all account for accumulation of ice outside the cathode catalystlayer and (with the exception of Ahluwalia and Wang) are spatially nonisothermaland allow for nonuniform ice production within the cathode catalyst layer.

The models discussed above neglect the presence of liquid water within the cell below0 C. However, at temperatures near but below the melting point, liquid water canexist due to curvature-induced melting, sometimes called the Gibbs-Thomson eect.Once enough pores have thawed, transport may occur in the liquid phase.

Jiao and Li63 have presented a three-dimensional multiphase model that attemptsto account for such melting by assigning each of the porous media in the cell atemperature below which all the water is expected to freeze and above which all thewater is expected to melt. The characteristic freezing temperature for a given mediumis calculated using the Gibbs-Thomson equation (discussed below) assuming a singlecharacteristic pore size for that medium. This approach neglects the distribution ofpore sizes in the media, which results in a continuous, rather than binary, equilibrium

73

Channel

CH

Plate

P

GDL GDL

MPL | CL | M | CL | MPL

Rib Symmetry Plane

Channel Symmetry Plane

2.593

x

1.000

0.500

All dimensions in mm

2.0931.593

1.0000.500

y

Rib Rib

Channel

CH

Plate

P

anode (a) side cathode (c) side

Figure 3.1: The two-dimensional modeling domain used in this work. Solid linescorrespond to subdomain boundaries.

liquid water content as a function of temperature below 0 C, as discussed in Chapter2.

In the present work, a two-dimensional, nonisothermal, multiphase model is presentedthat predicts ice, liquid, and vapor content and transport across the operating tem-perature range of an automotive fuel cell. The distribution of pore sizes in each ofthe layers of the cell is utilized to estimate the amount of liquid and ice present at agiven temperature.

3.2 Cold-Start Model

3.2.1 Overview

Figure 3.1 depicts the modeling domains used in the present work. Solid lines repre-sent boundaries between subdomains, whose acronyms are in bold. In the x -directionthe domain spans from the outside of an anode plate to the outside of the cathodeplate of the same cell. In the y-direction, the domain spans from the midplane of thechannel to the midplane of an adjacent rib. Symmetry is assumed at each of thesemidplanes. Dimensions for each layer are listed in Table 3.1.

74

Table 3.1: For the baseline model conguration, subdomain dimensions and numberof elements as well as phases allowed.

Subdomains, Ω

P CH GDL MPL CL M

Thickness, δx (µm) 1000 500 250 20 14 25

Height, δy (µm) 1000 500 1000 1000 1000 1000

Number of elements in x 20 10 10 10 10 10

Number of elements in y 10 5 10 10 10 10

Allowable phases, α s G s,G,L,I s,G,L,I s,M,G,L,I M

Within each subdomain, denoted by Ω, various phases, denoted by α, are consideredwithin the model. These may include a rigid, electronically conductive solid phase(s), an ionically conductive phase (M ), a gas phase (G), a liquid phase (L), and anice phase (I ). The volume fraction of a phase, εα, is dened as the fraction of a unitvolume occupied by α. The volume fractions of the phases must sum to unity,

εs + εM + εG + εL + εI = 1. (3.1)

Table 3.1 summarizes the phases that the model accounts for in each subdomain.Note that, although liquid water or ice may be present in the channels, and thisfact may be considered in determining certain boundary conditions, the only phaseexplicitly considered in the governing equations for the channels is gas. Also, themathematical treatment of the ionomer in the CL is identical to that of the ionicallyconductive polymer in the membrane, although they may have dierent properties,such as equivalent weight. For this reason they are both denoted by M .

The gas phase may contain several components, denoted by i, including reactant(R), inert (D), and H2O vapor (V ). The reactants are H2 and O2 on the anode andcathode, respectively, while the only inert considered in this work is N2. No crossoverof gas-phase components from one side of the cell to the other is allowed.

The liquid and ice phases are assumed to be pure, and the subscripts L and I areused to represent the component H2O in the these phases. The ionically conductivephase contains water as a component, denoted by 0, in addition to polymer.

The model has been implemented using commercial nite-element software, COMSOLMultiphysics 3.5a, using a mesh containing a total of 1100 rectangular elements. Abreakdown of the number of elements present in each subdomain is provided in Table3.1.

75

3.2.2 Governing equations

3.2.2.1 Energy balance

The model assumes local thermal equilibrium among all phases. The resulting generalenergy-balance equation used in the model to nd the temperature, T , is∑

α

εαραCp,α

(∂T

∂t+ vα · ∇T

)+∇ · q = Qv +Qs +Qf +Qjle +Qrxn, (3.2)

where εα, ρα, Cp,α, and vα are the volume fraction, density, specic heat capacity,and velocity of phase α. The conductive heat ux, q, is dened by

q = −keffT ∇T. (3.3)

To account for the various phases that may be present in a given subdomain, theeective thermal conductivity, keff

T , is assumed to be a combination of the bulk thermalconductivities for the various phases, kT,α:

keffT ≡

∑α

εαταkT,α. (3.4)

where τα is a tortuosity factor, assumed to be equal to ε−0.5α .70

Other relationships for the eective thermal conductivity have been derived,71 suchas

keffT

kT,α= 1 +

3εβ(kT,β+2kT,αkT,β−kT,α

)− εβ

, (3.5)

which applies to a system containing spheres of phase β uniformly distributed in acontinuous phase of α. In the context of a fuel-cell porous medium with two phases(solid and gas), if the continuous phase is taken to be the solid and the dispersedphase is taken to be gas, then kT,α >> kT,β. In this case, the dependence of keff

T onεα is quite close for equations 3.4 and 3.5.

For the gas phase, which may contain several components, the overall density, specicheat capacity and thermal conductivity are necessary. Assuming ideal-gas properties,

ρG =pGRT

∑i

yiMi, (3.6)

andCp,G =

∑ωiCp,i, (3.7)

where yi, Mi, and ωi are the mole fraction, molar mass, and mass fraction of speciesi. For the thermal conductivity, the semiempirical formula

kT,G =∑i

yikT,i∑j yjΦi,j

(3.8)

76

Table 3.2: Thermodynamic and transport properties of the uid components.

Values at or near Tt as reported by

[39], which are assumed constant

Values at Tt, pt [A] with T and p except as noted.

Mi Vi,t Hi,t Cp,i kT,i µi

i g/mol cm3/g J/g J/g·K W/m·K µPa·s

H2O (I ) 18.015 1.0909 -333.6 2.040 1.880 ∞

H2O (L) 1.0002 0 4.217 0.569 1750 [B]

H2O (V ) 206146 2500.8 1.854 0.018 8.02

H2 2.016 14.176 0.169 8.39

N2 28.013 1.042 0.024 16.57

O2 31.999 0.917 0.025 19.18

A H2O triple point: Tt = 273.16 K, pt = 0.006112 bar from [38].

Values for ice from [40]. Values for liquid and vapor from [38].

B Model uses t to data from [39]: µL = 1750exp[−16000

R

(1Tt− 1

T

)].

is used,71 where

Φi,j =1√8

(1 +

Mi

Mj

)−1/2[

1 +

(kT,ikT,j

)1/2(Mj

Mi

)1/4]2

. (3.9)

Equations 3.8 and 3.9 provide an approximation rooted in the kinetic theory of gases.72

The same equations can be used to determine the viscosity of multicomponent gasmixtures.73

Table 3.2 gives the thermodynamic and transport properties for the uid componentsthat may be present in the cell. For the solid phase contained in each subdomain,Table 3.3 summarizes the corresponding properties.

The right side of equation 3.2 contains source terms. From left to right, they accountfor heating (or cooling) due to evaporation/condensation, sublimation/deposition,freezing/melting, joule heating, and the heat of reaction. Not all of the terms applyto all of the subdomains. Table 3.4 gives the denition of each term for each applicablesubdomain. ∆Hv, ∆Hs, and ∆Hf , refer to the heats of vaporization, sublimation,and fusion, respectively. The rates of evaporation, sublimation, and freezing in theporous media, Rv, Rs, andRf , and the rate of evaporation from the ionomer, Rv,M , aredened in a later section. The electronic and ionic current densities are represented

77

Table 3.3: Baseline thermodynamic and transport properties for the cell materials.Values for the M subdomain are taken to also apply to ionomer in the CL subdomains.

Symbol Units P GDL MPL CL M

ρs g/cm3 1.8 1.8 2.2 2.2 2.0

[74], [A] [75] [76], [A] [76], [A] [77]

Cp,s J/g·K 0.71 0.71 0.71 0.71 1.05

[A] [A] [A] [A] [B]

kT,s W/m·K 5 3 3 3 0.2

[74], [78], [A] [78] [78] [78] [78]

ksatΩ cm2 3.4× 10−8 6.7× 10−12 1.6× 10−11

[C] [C] [C]

σs S/cm 200 120 120 120

[79], [80] [81] [81] [81]

A Values for pyrolytic graphite from [39].

B Value for Teon R© from [82].

C Calculated from equation 3.13.

78

Table 3.4: Energy-balance source terms.

Ω Qv Qs Qf Qjle Qrxn

BP i1·i1σeff

GDL, MPL −∆HvRv −∆HsRs ∆HfRfi1·i1σeff

CL −∆Hv(Rv +Rv,M) −∆HsRs ∆HfRfi1·i1σeff + i2·i2

κeff irxnh (ηs,h + Πh)

M i2·i2κeff

by i1 and i2, respectively, while the electronic and ionic eective conductivities areσeff and κeff , respectively. Finally, irxn

h is the overall rate reaction h, while ηs,h andΠh are the surface overpotential and Peltier coecient, respectively.

3.2.2.2 Convection

Within subdomains containing porous media, the ice, liquid, and gas phases aresubject to the following mass-conservation equation:

∂(ραεα)

∂t+∇ · nconv

α = sv,α + ss,α + sf,α + srxn,α. (3.10)

The convective mass ux, nconvα , is determined from Darcy's law,

nconvα = −ραk

effα

µα∇pα, (3.11)

where µα and pα are the phase dynamic viscosity and pressure, respectively. For allsimulations presented here, the dynamic viscosity of ice is assumed to be very high,resulting in an ice ux of zero. The dynamic viscosity of the gas phase is computedusing equations 3.8 and 3.9, replacing kT,i by µi, the dynamic viscosity of pure i.71

Values for the dynamic viscosities used in the model are found in Table 3.2.

Table 3.5: Baseline physical properties for the porous media.

Description Symbol Units GDL MPL CL Source(s)

Porosity ε 0.70 0.24 0.70

C-K Diameter dCK µm 7.600 2.05 0.25 [83], [A]

C-K Constant kCK 4.060 9.375 9.375 [83], [B]

A For MPL and CL: dCK = 2 [fr,1r0,1 + (1− fr,1)r0,2].

B For MPL and CL: from [84].

79

Table 3.6: Source terms for mass conservation (equation 3.10).

sv,α ss,α sf,α srxn,α

Ω L G G I I L L G

GDL, MPL −Rv Rv Rs −Rs Rf −Rf 0 0

aCL −Rv Rv +Rv,M Rs −Rs Rf −Rf 0 − irxnHORMH2

2F

cCL −Rv Rv +Rv,M Rs −Rs Rf −RfirxnORRMH2O

2F− irxn

ORRMO2

4F

The eective permeability of the porous medium is assumed to be given by

keffα = kr,αk

satΩ , (3.12)

where ksatΩ is the permeability of the subdomain when it is lled with uid. Gostick

et al.83 show that the Carman-Kozeny equation,

ksatΩ =

d2CKε

3

16kCK(1− ε)2, (3.13)

does a reasonable job of predicting the saturated permeability of GDLs as a functionof ber diameter, dCK , and the porosity of the medium, ε. In this equation, kCK isan empirical constant. In the present work, the same equation is used for the MPLand CL as well, except that dCK is assumed to related to the characteristic pore sizesin these subdomains. Table 3.5 gives the values used for ε, dCK , and kCK in eachsubdomain, while Table 3.3 gives the resulting values for saturated permeabilities.

The relative permeability for phase α, kr,α, is assumed to be

kr,α = S3α, (3.14)

where Sα is the level of saturation of α in the subdomain:

Sα =εαε. (3.15)

The saturations for liquid, ice, and gas in the porous media are calculated using themethod developed in Chapter 2.

The right side of equation 3.10 contains source terms that account for mass gained(or lost) by phase α due to, from right to left, evaporation/condensation, sublima-tion/deposition, freezing/melting, and the cell reactions. Not all of the terms applyto all of the subdomains or phases. Table 3.6 gives the denition of each term foreach applicable subdomain and phase. Mi is the molar mass of component i, whileF is Faraday's constant.

80

3.2.2.3 Diusion

In the gas phase, the components can move by diusion in addition to convection.For the reactant and inert components, the mass balance reads

∂(ρGωi)

∂t+∇ · ndiff

i = srxn,i, (3.16)

where ωi is the mass fraction of i and the diusive mass ux is given by an invertedform of the Stefan-Maxwell multicomponent-diusion equation,

ndiffi = −ρGωi

n∑j

Deffij (∇yj + (yj − ωj)∇pG/pG) + ρGωivG. (3.17)

In this equation, yj is the mole fraction of component j, and vG is the velocity ofthe gas phase, obtained by dividing the convective mass ux given by Darcy's law(equation 3.11) by ρG.

The eective diusion coecients Deffij take into account the fact that the components

are diusing not only through the gas phase, but also through the pores of the mediumcontaining the gas. Specically,

Deffij =

εGτG

pDij

pG. (3.18)

The quantity pDij is the product of a pressure of 1 bar and the binary diusioncoecient at that pressure in a bulk gas phase (i.e., not inside a porous medium).When the Stefan-Maxwell equations are inverted, as in equation 3.17, the resultingbinary diusion coecients are not the same as the Stefan-Maxwell binary diusioncoecients, Dij. The relationship between the two for a three-component system istaken to be71

D12 =

ω1(ω2+ω3)y1D23

+ ω2(ω1+ω3)y2D13

− ω23

y3D12

y1

D12D13+ y2

D12D23+ y3

D13D23

(3.19)

which emphasizes the fact that the diusion coecients used in equation 3.17 are afunction of composition. Table 3.7 lists the values for pDij that are used in the model.

A source term for production of gas-phase components appears on the right side ofequation 3.16. This term applies only in the anode and cathode catalyst layers andis equal to srxn,G as given in Table 3.6.

Finally, the fact that the mass fractions of the components must sum to unity,

∑i

ωi = 1, (3.20)

is used to nd ωV .

81

Table 3.7: Stefan-Maxwell binary diusivities, where T is in degrees Kelvin.71

i, j pDij, bar·cm2/s value at Tt

H2, N2 0.049(

T64.83

)1.8230.681

H2, H2O 0.247(

T146.82

)2.3341.050

N2, H2O 0.241(

T285.81

)2.3340.217

O2, N2 0.052(

T139.59

)1.8230.178

O2, H2O 0.291(

T316.14

)2.3340.207

3.2.2.4 Rate expressions for phase change

Water is assumed to be able to evaporate from or condense into pores and, in thecase of the catalyst layer, from or into the ionomer. The two rate expressions used toaccount for these types of phase change are

Rv = kv(µL − µV ) (3.21)

andRv,M = kref

v,Mexp(4.48a0)(µ0 − µV ), (3.22)

respectively. The exponential dependence of the latter rate on the activity of waterin the membrane, a0, is consistent with the results of Kientiz et al.85 Similarly, therates of freezing/melting and sublimation/deposition in the porous media are takento be given by

Rf = kf (µL − µI) (3.23)

andRs = ks(µI − µV ) (3.24)

respectively. Expressions for the chemical potentials of liquid, vapor, and ice in theporous media, µL, µV , and µI , respectively, are given in Chapter 2. The equation fordetermining the chemical potential of water in the membrane, µ0, is discussed in thenext section.

The values assumed for the phase-change rate constants kv, krefv,M , kf , and ks are sum-

marized in Table 3.8. Phase equilibrium is taken to exist for water in the porous

82

Table 3.8: Rate constants used in the porous media subdomains.

Symbol Units GDL MPL CL

kvgJ· mol

cm3·s 1.0× 10−3 1.0× 10−3 1.0× 10−3

krefv,M

gJ· mol

cm3·s 5.7× 10−7

ksgJ· mol

cm3·s 0.0 0.0 0.0

kfgJ· mol

cm3·s 1.0 1.0 1.0

media. Therefore, high values are used for kv, and kf , such that the chemical poten-tials of the ice, liquid, and vapor are very close to one another under all conditions.The chemical potential of water in the membrane, however, is not taken to alwaysbe equilibrated with the vapor phase. Instead, kref

v,M is used as a tting parameter, asdiscussed in Chapter 4.

3.2.2.5 Transport of ions and water in the membrane

The membrane model developed by Weber and Newman has been adapted to simulatetransport within the ionomer of the catalyst layer and the bulk membrane.77 Therevised version used in the present work can be summarized by four equations. Therst two are

∇ · i2 = irxnh , (3.25)

i2 = −κV∇Φ2 −κV ξVF∇µ0, (3.26)

where κV is the ionic conductivity, ξV is the electroosmotic coecient. The thirdequation is

∂(ρLε0εM)

∂t+∇ ·N0 = −Rv,M , (3.27)

where ε0 and εM are the volume fraction of water in M and the volume fraction of Min the subdomain, respectively. The ux of water, N0, is given by the fourth equation,

N0 = −κV ξVMH2O

F∇Φ2 −MH2O

(αV +

κV ξ2V

F 2

)∇µ0, (3.28)

where αV is a transport coecient. The source term in equation 3.27 applies only inthe catalyst-layer subdomains.

In all of these expressions, the subscript V indicates a vapor-equilibrated value. Noliquid-phase transport within the membrane is considered, unlike in the full modeldeveloped in [77]. A second change is the addition of the accumulation term in

83

Table 3.9: Membrane property expressions and parameter values for the baselineconguration.

Symbol Units Expression or value Source(s)

κV S/cm 0.26(ε0 − 0.06)1.5ε1.5M exp

[15000R

(1Tt− 1

T

)]f1(pI) [77]

ξV λεM [77]

αV mol2/J·cm·s(

λVLλ+VM

)Dµ0

RT(1− λλ+1)

ε1.5M [77]

Dµ0 cm2/s 9.37× 10−6ε0exp[

20000R

(1Tt− 1

T

)][77]

ρM,0 g/cm3 2.0 [77]

εM CLs: 0.112, M: 1.0 [18]

EW g/equiv CLs: 900, M: 1100 [18]

Ci n0/nSO−3C1: 35.12, C2: -43.49, C3: 21.96 [86]

equation 3.27 to allow for transient simulations. No accumulation term appears inequation 3.25 because double-layer charging is neglected. Expressions for determiningthe membrane transport properties (κV , ξV , αV ) are found in Table 3.9.

In the table, the expression for κV contains a function of the ice pressure that isnot included in the original equation in the reference that is cited. The ice-pressurecorrection applies to the CLs only, and is taken to be

f1(pI) =

1 pI − p0

I ≤ 0

exp [−ACR(pI − p0I)] pI − p0

I > 0, (3.29)

where ACR is a constant that determines the strength of the pressure dependence,and p0

I is a threshold value for the ice pressure. Once the ice pressure exceeds thisthreshold, the ionic conductivity decreases. Values used for these parameters in thischapter are 0.35 and 28 bar, respectively.

3.2.2.6 Membrane water content

The dependence of the membrane transport properties (κV , ξV , and αV ) on watercontent and temperature is assumed to be as described by Weber and Newman.77

The volume fraction of water in the membrane is given by

ε0 =λV0

VM + λV0

, (3.30)

84

where λ is the number of moles of H2O per mole of SO−3 in the membrane and εM isthe volume fraction of membrane in a given subdomain. The partial molar volume ofthe membrane is

VM =EW

ρm,0, (3.31)

where EW is the equivalent weight of the membrane and ρM,0 is its dry density.77

To relate λ to the activity of water (the eect of temperature is neglected), theapproach dened by Gallagher et al. is used.86 First, the activity of water in themembrane is dened to be

a0 ≡pVpsatL

, (3.32)

and λ is then calculated from

λ = C1a30 + C2a

20 + C3a0, (3.33)

an empirical expression in which the constants Ci are tting parameters. Values for Cias well as for the other parameters required for determining membrane water contentare found in Table 3.9.

3.2.2.7 Electron transport

Transport of electrons, denoted by subscript 1, is governed by the current balance

∇ · i1 = −irxnh , (3.34)

where i1, the electronic current density, is

i1 = −σeff∇Φ1, (3.35)

where Φ1 is the electronic potential. The eective electronic conductivity, σeff , istaken to be given by

σeff =εsτ 1.5s

σs. (3.36)

Values for the solid-phase electronic conductivity in each subdomain are summarizedin Table 3.3.

3.2.2.8 Electrode kinetics

The overall reaction current density for reaction h is given by

irxnh = a0

1,2SGih, (3.37)

85

Table 3.10: Electrode and reaction properties for the baseline conguration.

Anode and HOR Cathode and ORR

Symbol Units Value Value Source(s)

a01,2 1/cm 1.53× 105 1.53× 105 [18]

αha 1 0.6 [60]

αhc 1 1 [60]

ih0 A/cm2 iref0 exp

[17000R

(1Tt− 1

T

)]iref0 exp

[55000R

(1Tt− 1

T

)][17,87]

iref0 A/cm2 2.15× 10−2 2.00× 10−10 [17,87]

U θh V 0 1.250 + 0.231

(1− T

Tt

)[15]

Πh V -0.012×T/Tt -0.220×T/Tt [88]

where a01,2 is the specic interfacial area between the ionically and electronically con-

ductive phases and ih is the transfer current density. The transfer current density isrelated to the electrode overpotential, ηh, through expressions of Butler-Volmer form:

iHOR = iHOR0

[(pH2

pref

)exp

(αHORa F

RTηHOR

)− exp

(−α

HORc F

RTηHOR

)], (3.38)

and

iORR = iORR0

[exp

(αORRa F

RTηORR

)−(pO2

pref

)exp

(−α

ORRc F

RTηORR

)]. (3.39)

In these equations, i0,h is the exchange current density, αha and α

hc are the anodic and

cathodic transfer coecients, and the overpotential is determined from

ηs,h = Φ1 − Φ2 − U θh , (3.40)

where U θh is the standard potential for reaction h. Table 3.10 summarizes the param-

eters related to fuel-cell reaction kinetics used in the model.

3.2.2.9 Boundary conditions

Table 3.11 summarizes the boundary conditions used to solve the partial dierentialequations presented above. The thermal boundary conditions are specied by h andT cool, which represent the heat-transfer coecient for the coolant and the coolant tem-perature, respectively. Simulations can be run by setting either the current density,icell, or the cell potential, Φcell.

86

Reactant and diluent mass concentrations at the CH/GDL interface, ωCHR and ωCH

D ,are calculated based on the dry mole fraction of reactant present in the gas, yCH

R,dry,and its relative humidity, RHCH. Specically,

ωCHi =

yCHi Mi∑j y

CHj Mj

, (3.41)

yCHR = (1− yCH

V )yCHR,dry, (3.42)

yCHV =

min(psatL , psat

I )

pCHG

RHCH

100, (3.43)

andyCHD = 1− yCH

R − yCHV . (3.44)

The liquid pressure at the CH/GDL interface is calculated assuming equilibrium withthe vapor phase:

pCHL = pt +

RT

VLlnpVpsatL

− ∆HV i

VL

(1− T

Tt

)− ∆Cp,V i

VL

(T − Tt − T ln

(T

Tt

)). (3.45)

Table 3.11: Boundary conditions used for fuel-cell simulations.

Boundary condition

Interface Variable (a and c unless otherwise noted)

outside edge of P

T keffT ∇T = −h(T − T cool)

Φ1 a: 0 V, c: icell or Φcell

CH/GDL

pL pCHL

pG pCHG

ωR ωCHR

ωD ωCHD

MPL/CL

Φ2 ∇Φ2 = 0

µ0 ∇µ0 = 0

In summary, the primary inputs for a given simulation are: h, T cool, yCHR,dry, RH

CH,

pCHG , and icell or Φcell. All of these parameters are taken to be constant in the y-direction; anode and cathode values need not be the same.

87

Table 3.12: Values used to compute boundary conditions for the cold-start simulation.

Symbol Units Boundary value

T cool C 65

h W/cm2 ·K if(TcCL < 65C,0, 1.0)

icell A/cm2 0.05

pCHG bar 1

RHCH % 100

yCHR,dry a: 1, c: 0.2

3.3 Cold-Start Simulation

Using the model described above, it is possible not only to predict cell performanceduring start-up from a wide variety of operating conditions, but also to understandclearly the factors that determine the performance. For example, Figures 3.2 and 3.3show results from a simulated start-up from -10 C, at a constant current densityof 0.05 A/cm2, assuming adiabatic boundary conditions. The initial ice saturationin all porous-media layers is 0.2, and the initial membrane water content, λ, is 10.4mol H2O per mol SO−3 , corresponding to an assumption of minimal dry-out of thecell before freezing. The anode and cathode gas channels are assumed to contain H2

and air, respectively, each at a total pressure of 1 bar and a relative humidity of 100% throughout the simulation. Table 3.12 summarizes the values used to computethe necessary boundary conditions. Note that TcCL is the average temperature ofthe cathode catalyst layer. The value for h is within the range expected for forcedconvection using liquid.39

Figure 3.2 shows the predicted cell potential and average temperature prole for thecase described above. Although the simulation was run until the cell reached steadystate at 65 C (a condition achieved 10 minutes after start-up is initiated), only theinitial portion of the start-up, which includes the cell's transition through the freezingpoint, is shown. Figure 3.3 shows the water content in critical subdomains during thestart. Select times appear for reference as vertical dotted lines in both gures.

Prior to tA, the average cell temperature increases by about 7 C, but the cell potentialstays within 2 mV of its initial value. On the one hand, the increase in average λ duringthis period, which corresponds to increasing average membrane conductivity, tendsto increase cell potential by reducing internal resistance. However, ice simultaneouslyaccumulates in the cathode catalyst layer, which in turn increases the overpotentialrequired to maintain a given current density. This eect, combined with the decreasein the standard cell potential with increased temperature, counteracts the decreasedohmic drop, resulting in a small net drop in potential.

88

Figure 3.2: Simulated potential- and average-cell-temperature proles during a start-up from -10 C at 0.05 A/cm2.

After tA, ice begins to melt in the cathode catalyst layer, and the resulting liquiddrains into the microporous layer. The cell potential rises quickly until tB due tothe decreasing total saturation (ice plus liquid) in the cathode catalyst layer. Liquidow from the CL to the MPL begins at about -2.9 C. This is possible because theCL and MPL contain ne pores in which ice melts at temperatures below 0 C, asshown in Figure 2.17. Once enough of these pores contain liquid, the eective liquidpermeability becomes high enough to permit substantial ow.

During the period between tB and tC , the remaining ice in the cell melts, and thetemperature and cell potential remain relatively constant. Most of the ice that meltsduring this period is in the GDLs, which have pore-size distributions that containmore large pores than the MPLs and CLs. Ice in these larger pores melts much closerto 0 C than ice in smaller pores. Although it appears that the GDL ice has allmelted about 15 s prior to tC , this is because only the average GDL ice saturation isshown. Small amounts of ice persist in the coldest parts of the GDL until tC . Onceall of the ice in the cell has melted, the temperature and potential continue to rise.The increasing potential results from the increasing exchange current density withincreasing temperature.

89

Figure 3.3: Predictions of water content in various phases and cell materials duringa start-up from -10 C at 0.05 A/cm2.


A two-dimensional, nonisothermal, transient model is developed which is capable ofpredicting polymer-electrolyte fuel-cell performance during cold start. Transport ofwater in the vapor, liquid, and membrane phases at temperatures both above andbelow 0 C is included. In the porous media, the saturation levels of ice and liquid,as well as their pressures, are predicted rigorously based on the chemical potential ofeach phase and the physical and chemical properties of the cell materials.

Results indicate that the cell potential during start-up is inuenced strongly by thelevel of membrane hydration, with an increased level corresponding to higher per-formance, as well as by the total saturation (liquid plus ice) in the cathode catalystlayer, with higher saturation leading to lower performance. Liquid water transportnear but below 0 C plays an important role in determining the cell potential as afunction of time during start-up because it can allow for rapid drainage of the cathodecatalyst layer once enough ne-pore ice has melted.

90

Chapter 4

Model Verication Using Parametric

Studies

4.1 Introduction

In Chapters 2 and 3 of this dissertation, a cold-start model for polymer-electrolyte fuelcells is developed. The purpose of the model is to provide insight into the physics thatdrives critical parameters during start-up, including cell temperature and potential.In turn, this insight can lead to improved specications for materials and procedures.

Before deploying the model for these purposes, it is prudent to compare its predic-tions to experimental observations. Doing so is primarily a means of verifying thatsimulation results are reasonably accurate and that the physical phenomena relevantto the problem have been included. The assumption underlying this activity is that ifthe model is capable of predicting cell behavior assuming a variety of congurationsand operating procedures, then it can be used with an acceptable level of condenceto extrapolate behavior under conditions or congurations for which experimentaldata are not available.

A practical challenge that confronts users of this model, however, is the large numberof parameters that have been incorporated in order to deliver the desired level ofdelity and exibility. Therefore, a second important objective of the present chapteris to dene which model parameters are important to adjust when simulating a givenconguration and what parameters are appropriate to use when tting experimentaldata. Clearly, minimizing the number of adjustable parameters is important in orderfor the model to be useful. As discussed below, only a small fraction of the totalnumber of parameters is used for tting, and, once these parameters are set, a widerange of operational modes can be simulated without further adjustment.

91

4.2 Approach

In order to verify the model, parametric experimental studies of cold-start behaviorhave been simulated, and the two sets of results have been compared. Based on thesecomparisons, the model's ability to simulate experiments is evaluated below based onquantitative accuracy for key parameters such as start time as well as its qualitativeaccuracy regarding, for example, where ice forms during the start-up.

4.2.1 Parameter types

In all, a total of 40 experimental cold starts, including 11 cell congurations from 6dierent sources, operating over a wide range of conditions, has been t using themodel. A conguration is dened to be the set of parameters that dene a given celldesign that is constant in all simulations of that cell. Examples of congurationalparameters include geometric properties such as the thickness of the anode GDL,physical properties such as the porosity of the cathode catalyst layer, chemical prop-erties such as the contact angle for the hydrophobic pores in the anode microporouslayer, and electrochemical performance characteristics such as the ORR exchangecurrent density of the cathode catalyst layer. In contrast, operational parametersare those that dene the conditions of the particular test being simulated. For ex-ample, the initial cell temperature and the applied current density are operationalparameters.

Fitting parameters are dened in this work as those that are adjusted in order tomatch better an experimental data set. Generally, parameters chosen for this purposeare those whose eect on the results is signicant and for which a relatively low levelof condence exists a priori for a specic value (although a probable range may beknown). A small number of both congurational and operational parameters is usedas tting parameters, but, as described above, congurational parameters dene thecell and are taken to be constant (for a given cell type) while operational parametersdene the test and may vary from one simulation to the next. In other words, whilea given congurational parameter may be used to t the data, all simulation resultsfor a given cell reported below assume the same value found using the t. On theother hand, operational parameters used for tting are not necessarily the same fromcell to cell or from one simulation to the next.

All but nine of the congurational parameters are set equal to the baseline valuesdened in Chapters 2 and 3. Those that are not include the thickness of the cathodecatalyst layer and the thickness of the membrane, which are set equal to the thick-nesses in the experimental cell (for the membrane, the dry thickness is used). Theremaining seven are used as tting parameters.

The operational parameters used as tting parameters are the dry mole fraction ofoxygen at the cCH/GDL interface, yCH

O2,dry, and the relative humidity of the gas at the

92

V

Tt

i

V

Tt

iFailure

(a) (b)

Figure 4.1: Typical cell-temperature and cell-potential proles versus time for agalvanostatic start-up assuming (a) isothermal and (b) nonisothermal experimentaltypes.

CH/GDL interfaces, RHCH, which is always assumed to be equal on the anode andcathode sides, and the initial value of which sets the initial water content of the cellbecause the chemical potential of water is assumed to be equal in all phases prior tothe start-up. All other operational parameters, such as start-up current density, areset to the values reported for the experiment.

4.2.2 Cold-start classication

The nomenclature used in existing fuel-cell literature is employed to describe the typeof experiment being simulated. Specically, cold-starts in which the cell is not allowedto heat up are referred to as isothermal. This is usually accomplished by owinglarge amounts of coolant through the cell hardware. It is important to recognize thatthis type of experiment is not truly isothermal, despite the name, as there can besignicant temperature variations within the cell, even if the edges of the cell arepinned at the coolant temperature. When the cell is allowed to heat up, which isthe type of start of interest for practical applications, the experiment is referred toas nonisothermal.

Figure 4.1 illustrates typical cell temperature, T , and potential, V , proles for isother-mal and nonisothermal experiments, assuming tests in which the current density, istart,is constant (i.e., a galvanostatic experiment). In the isothermal case, as shown in (a),the temperature is nearly constant with time, and the cell is typically able to operateat a useful potential before failing. For the purposes of this chapter, cell failure isdened to occur when the cell potential reaches an arbitrary lower limit. Dierent

93

research groups dene dierent lower limits: in the experimental studies discussedbelow, the limit used ranges from 0.0 to 0.3 V. The time dierence between the startof the test and the failure is referred to as ∆tstart. The mass of water produced perunit area is given (using Faraday's law) by

mstartH2O =

MH2Oistart

2F∆tstart, (4.1)

whereMH2O is the molar mass of water and F is Faraday's constant. These equationsassume 100 % Faradaic eciency for the oxygen-reduction reaction (ORR). The per-formance of the cell when current is initially applied is referred to as Vinit. Note thatthe maximum cell potential during the test is not necessarily equal to Vinit, althoughit is shown that way in this particular illustration.

During a typical nonisothermal experiment, the cell temperature rises once current isapplied, as shown in Figure 4.1b. If enough ice is present in the cell as it approaches0 C, a plateau in temperature relative to time may appear as melting occurs. Oncemelting is complete, the cell temperature continues to rise. Note that a meltingplateau may not necessarily be present, depending on the amount of residual waterleft in the cell prior to start-up and the amount of product water produced below 0 Cduring the start itself. Once the cell reaches a specied temperature, owing coolantis introduced into the cell at a temperature of Tcool, and the cell temperature changesto match that temperature and stabilizes shortly after. The illustrated prole of Vcell

is an example of desirable cell performance during the experiment. The potentialstarts at a fairly high level and rises as the start-up proceeds. However, dependingon congurational parameters and test conditions, the potential can decrease, evenas the temperature increases, and may even fail. For a nonisothermal start, ∆tstart

is dened in the present chapter to be the time required for the cell temperature toreach 5 C.

4.2.3 Experimental cell congurations

Table 4.1 summarizes the available information for all of the dierent cell congura-tions that were simulated. All experiments were performed on single cells. Except asnoted, cell materials are symmetricthat is, the materials used on the anode were thesame as those used on the cathode. Isothermal and nonisothermal data were obtainedas described below.

Isothermal cold starts. All experimental data for isothermal start are taken from theliterature. In Table 4.1, the rst author's name is used to denote the congurationused in their experiments.

Nonisothermal cold starts. Nonisothermal data have been provided by UTC Powerfor purposes of verication. These types of experiments are more complicated to

94

Table4.1:

Experim

entalcongurationssimulatedbythemodel.Allcongurationscontain

microporouslayers

onboth

theanodeandthecathode.

Cellactive

Mdry

a/c

Groupor

area

thickness

Ptloading

1stAuthor

(cm

2)

GDLtype

MEAtype

Mtype

(µm)

(mg/cm

2)

Chacko

89

5SGL10BB

Gore57

series

Gore

180.4/0.4

Ge2

05

SGL20BB

Lab

fabricated

Goreselect

180.4/0.4

Nandy21

5Toray

TGP-H-060

Lab

fabricated

Naon

212

500.4/0.1to

0.8

Tajiri90

25Toray

Japan

GoreTex

Gore

300.4/0.4

Thom

pson18

50SGL21BC

Lab

fabricated

Naon

1100

EW

250.42/0.42

UTCPow

er320

SGL21DC

Gore57

series

Gore

180.4/0.4

95

perform than isothermal starts, primarily because special cell hardware is required.This is because conventional single-cell hardware, where the current collection andload follow-up functions are combined into one massive steel component, contains toomuch thermal mass adjacent to the cell package. A single cell is not able to heat itselfunder most conditions of interest because too much heat is lost to the surroundinghardware. This is not the case in the cell stacks used in most applications, wherethe amount of waste heat available relative to the thermal mass is far greater andthe cells that are not near the end of the stack are relatively thermally isolated fromend-structure heat losses.

To address the problem of how to perform single-cell nonisothermal starts, UTCPower developed a cell that separates the thermal mass associated with the axial-loadfollow-up system (which holds the cell sandwich together) from that of the currentcollector, which is much smaller. Figure 4.2a illustrates the dierent layers present inthe UTC Power cell hardware. The core of the cell, made of gas-ow plates, diusionmedia (DM), and a membrane-electrode assembly (MEA) is in the center, contactedon either side by electronically conductive coolant plates which in turn contact a thincurrent collector. Adjacent to both current collectors is a resistive heater. The anodeand cathode heaters can be controlled independently of one another and may be usedduring both freezing and cold start to apply thermal gradients to the cell, if desired.Rigid insulation, pictured in Figure 4.2b, separates the cell assembly from the largestainless-steel pressure plates that are used to maintain the axial load on the cell. Asolid model of the fully assembled cell is shown in Figure 4.2c. The low thermal massof the cell assembly, combined with controlled application of heat to the anode andcathode, enables a much more accurate simulation of the thermal environment thatthe cell experiences in a cell stack.

4.2.4 Simulating nonisothermal cold starts

To simulate a cell operating within the UTC cell hardware, the model developed inChapters 2 and 3 has been modied slightly for the nonisothermal simulations dis-cussed below. This is necessary for three reasons. First, despite the fact that painshave been taken to minimize the thermal mass of the hardware, it is still higher thanthat of the cell package dened in the model. Second, water is used as coolant dur-ing normal (above 0 C) operation, and some water remains in the channels uponshutdown, despite a short purge of the coolant channels. This water freezes as thecell is cooled and causes a melting plateau (see Figure 4.1b) to appear in the tem-perature prole during start-up. Finally, anode and cathode heaters are used in theexperiments, a source of heat not accounted for previously.

In order to account for the additional thermal mass of the cell hardware, the valueof the specic heat capacity of the regions labeled web in Figure 4.3, referred tohere as Cweb

p , is increased beyond the baseline value dened in Chapter 3. Similarly,

96

Cur

rent

col

lect

orR

esis

tive

heat

er

Coo

lant

cha

nnel

Cel

l ass

embl

y

Rig

id in

sula

tion

Cel

l Ass

embl

y

Gas

cha

nnel

DM

, ME

A

Gra

phite

pla

te

(a)

(b)

(c)

Ext

erna

l air

man

ifold

Figure 4.2: Illustrations of the cell used to generate nonisothermal cold-start data:(a) the cell assembly, (b) rigid insulation surrounding the cell package, and (c) theentire cell assembly, with endplates applying pressure to the insulation.

97

Gas channels

DM, MEAPlate (P)

RibWeb

RibWeb

Figure 4.3: Denition of the web region within the plate (P).

in order to account for residual ice in the cell hardware, the variable εwebH2O, which is

the volume fraction of ice in the web prior to start-up, is dened in both the anodeand cathode web subdomains. It is understood that in reality the web's specic heatcapacity is the same as the rib's and that the volume fraction of ice is zero, but thisapproach is used as a means of accounting for the eects of increased thermal massand residual ice without dening additional subdomains in the model. Values forCwebp and εweb

H2O used to simulate all nonisothermal starts in the UTC Power hardwareare 9.94 J/g·K and 0.30, respectively, determined by tting the temperature prole ofthe start-up from -10 C discussed below.

Heat from the anode and cathode heaters is simulated by applying heat-ux boundaryconditions to the outer edges of both the anode and cathode plate subdomains. Inthe model, the heat ux for each boundary is taken to be identical to that applied inthe experiments:

Qbdy = istart(1.48− V )Ahtr, (4.2)

where Ahtr is referred to as the heater factor, which varies during the start-up accord-ing to the following schedule:

Ahtr =

3 TcCH < −0.5C

1 −0.5C ≤ TcCH ≤ 0.5C

1.5 TcCH > 0.5C

. (4.3)

In the experiments, the temperature measurement for control of the heaters, TcCH, istaken via a single thermocouple inserted into a cathode gas channel, situated in the

98

center of the cell's active area. In the model, the temperature used is the averagetemperature of the cCH subdomain.

4.2.5 Experimental cold-start procedures

There are dierences in the ways that the dierent research groups cited in Table 4.1performed their cold-start experiments. However, most of the cold-start experimentswere performed according to the following procedure. Key exceptions are discussedlater.

First, the cell is operated at normal operating temperature, pressure, and relativehumidity. The intent of this step is twofold. The rst is to return the cell to a knownlevel of hydration. The second is to measure the performance of the cell in order totrack whether the performance changes as a result of the cold-start experiments.

Next, the cell is shut down (the load is removed and reactant gas ow is stopped)and a purge is performed by owing gas, usually nitrogen, on the anode and cathode.The purpose of the purge is to remove excess water from the gas channels (whichotherwise forms ice that could block reactant transport upon start-up) and to set alevel of hydration within the cell's porous media and membrane. Purge proceduresvary widely. Sometimes a xed relative humidity is used, and the purge continues foran extended period in order to equilibrate the membrane's level of hydration with thegases. In other cases, dry gas ows for a xed period of time before every test or untila certain level is reached for the high-frequency resistance (HFR) of the cell. TheHFR increases during the purge because the membrane becomes drier and thereforeless ionically conductive. Sometimes the purge is completed in stages.

The variability in purge procedures as well, as the substantial uncertainty associatedwith the question of how a certain purge procedure relates to the level of hydrationin the cell once it is frozen, poses a problem for simulating the experiments. This isso because, as discussed further below, both start time and the cell potential proleduring start are strong functions of how much water is present in the cell before itstarts. This is why RHCH is used as a tting parameter.

After the cell is purged, it is cooled to the intended start-up temperature, typically inan environmental chamber. After cooling is complete, reactant gases are turned on,and the start-up current density is applied. If the start-up is isothermal, coolant owsat high rates at the start-up temperature at the same time, and the test proceedsuntil the cell potential falls to the lower limit. If the start-up is nonisothermal, nocoolant ows until the cell temperature reaches a predetermined sepoint. Once thislevel is reached, coolant is introduced, and the cell is allowed to stabilize at the coolanttemperature.

Further detail on the procedures followed during the isothermal experiments can befound in the works cited in Table 4.1. For the nonisothermal starts performed by

99

UTC Power, the purge takes place in two stages. First, 5 standard liters per minute(SLPM) of dry nitrogen are used to purge the anode, and 10 SLPM are used to purgethe cathode for three minutes while coolant maintains the cell temperature at 50 C.Next, the cell is cooled to room temperature, and the anode and coolant channels arepurged for three minutes with 5 SLPM and 10 SLPM, respectively, of dry nitrogen.

Two notable exceptions to the procedure as described above are as follows. Chackoet al.89 performed successive isothermal cold starts at -10 C without heating thecell between experiments. This was accomplished by subliming the ice from the cellat -10 C after failure by owing dry gas through it. Once the HFR climbed backto a preset value (indicating that the cell had been dried out), another cold startwas completed. The second exception is that both Tajiri et al.90 and Chacko et al.relied on cell-hardware thermal mass and an environmental chamber to maintain thedesired cold-start temperature. In other words, liquid coolant did not ow throughthe hardware during cold start.

4.3 Results and Discussion

Comparisons between experimental data and simulation results are presented below attwo levels. First, key characteristics of the cell potential during start-up are comparedon an aggregated basis, which allows the quality of the ts to be quantied. Second,trends observed in various parametric studies are discussed in more detail in orderto establish that the simulation results are physically reasonable and are in harmonywith additional experimental observations.

4.3.1 Aggregated results

Table 4.2 summarizes the seven congurational parameters varied in each of theexperimental studies.

A snapshot of key results for all simulations are shown in Figure 4.4. A comparisonbetween the value for mstart

H2O predicted by the model as a function of the measuredvalue is shown in (a). For reference, dashed lines representing 20 % error are included.A similar comparison is shown in (b) for Vinit. Both ts appear to be fairly good.However, the agreement is better for Vinit than for mstart

H2O, for two reasons. First,predicting Vinit is a relatively easy task, since it is determined primarily by the cell'sohmic drop, as described in Chapter 1. In contrast, predicting mstart

H2O over a range ofconditions is challenging because one must correctly predict the rates of membranehydration as well as the rate of accumulation of ice in the cathode catalyst layer overan extended time span. The second reason why the Vinit t is better is that, becauseit is such a strong function of membrane hydration, it has been used for many of the

100

Table4.2:

Congurational

ttingparam

etersusedto

simulate

each

typeof

cell.Values

forallother

param

etersareas

dened

inChapters

2and3,exceptforthemem

braneandcathodecatalyst

layerthicknesses,whichareshow

nin

Table

4.1.

Symbol

Units

Chacko

Ge

Nandy

Tajiri

Thom

pson

UTCPow

er

HORexch.curr.dens.,ir

ef 0A/cm

26.

00×

10−

41.

00×

10−

55.

00×

10−

41.

00×

10−

45.

00×

10−

46.

00×

10−

4

ORRspec.intfc.

area,a

0 1,2

1/cm

2.30×

106

1.53×

104

3.06×

105

1.53×

104

3.06×

105

2.30×

106

Mdi.coe.,D

ref

µ0

cm2/s

9.37×

10−

69.

37×

10−

63.

20×

10−

69.

37×

10−

63.

20×

10−

69.

37×

10−

6

Mevap.rate

const.,k

ref

v,M

g J·

mol

cm3·s

1.00×

10−

61.

00×

10−

68.

00×

10−

71.

00×

10−

68.

00×

10−

71.

00×

10−

6

CLporosity,ε C

L0.5

0.5

0.7

0.5

0.7

0.5

thresholdicepress.,p0 I

bar

1414

2014

2228

contact

res.

const.,A

CR

0.35

0.35

0.22

0.35

0.22

0.22

101

(a)

(b)

Figure 4.4: Correlation plots for (a) cumulative water produced, mstartH2O, and (b) initial

cell potential, Vinit.

102

Figure 4.5: Eect of start temperature. H2/air operation with a constant currentdensity of 0.1 A/cm2. For the simulation, λinit = 8.0. Experimental data from [20].

simulations as a means of setting the tting parameter RHCH. In other words, RHCH

was varied until the predicted value of Vinit matched the experimental value.

4.3.2 Isothermal parametric studies

The sections below review in greater detail the results of the isothermal simulationsthat have been completed for comparison to the parametric experimental studiesfound in the papers listed in Table 4.1.

4.3.2.1 The eect of start-up temperature

As shown in Figure 4.5, the amount of water that can be produced before failure de-creases with start temperature. Below about -4 C, this eect is due to the decreasingdiusion coecient for water in the membrane and gas phases (and the lower concen-tration in the gas phase), which limits the amount of water can be removed from thecathode catalyst layer, where it is produced. As a consequence, at lower temperaturesmore of the water produced in the cCL tends to stay there, which results in a shortertime to failure.

103

Above -4 C, the amount of water produced before failure rises dramatically. Modelresults indicate that some of the ice in the cCL, which is about 0.5 C higher thanthe coolant temperature, melts. As a result, water can move into the cMPL in theliquid phase, eectively increasing the amount of ice capacity in the cell. At stillhigher temperatures, a signicant fraction of the cMPL pores, and some of the cGDLpores, melt as well, meaning that liquid water can be stored in the GDL, which has avery high capacity. In the experimental study, it was shown that the cell was able tooperate at -1 C indenitely. Simulation results predict that the cell should be ableto operate for at least an hour at this temperature, and indenitely at temperaturesat or above -0.5 C (at 0.1 A/cm2).

The experimental study used for comparison conrms that the increase in waterproduced before failure is due to liquid water ow. The authors used a transparentwindow and small holes punched through the cathode diusion media to observe thesurface of the cCL in several locations. Below -3 C no water of any kind was observedon the cCL surface. At and above this temperature, however, liquid water emergedfrom the cCL and formed droplets.

4.3.2.2 The eect of start-up current density

Figure 4.6 shows that, as the start current density increases, the amount of water thatcan be produced before failure decreases signicantly. At higher current densities, theORR reaction distribution is highly skewed toward the M/cCL interface due to thelow ionic conductivity of the ionomer. As a result, as the pores in this area ll up, thecurrent shifts farther from this interface, which increases the ohmic drop through theionomer. Eventually, the resulting ohmic loss is high enough to cause the voltage tofall to the lower limit, even if not all of the pores are lled with ice. This predictionis supported by cryo-SEM images of the cCL taken during this experimental studyshowing that, after the cell failure at 0.01 A/cm2, a signicant fraction of the poresappears to be lled with ice, whereas, after failure at 0.2 A/cm2, there is still a largeamount of open pore volume. Furthermore, additional cryo-SEM images taken priorto failure show that at 0.01 A/cm2 the cCL appears to ll evenly (across the thickness).At 0.1 A/cm2, a distinct ice front is observed, indicating that the cCL lls near theM/cCL interface rst, and then proceeds toward the cCL/cMPL interface.87

The above observation is true whether starting with a well-hydrated membrane orone that is relatively dry. As discussed in Chapter 3, membrane hydration is oftendescribed by λ, the moles of H2O per mole of SO−3 in the membrane. At -20 C, andat a relative humidity of 100 %, the approach used in the present work (based on thework of Gallagher et al.86) predicts a λ of 8.2, whereas a relative humidity of 34 %corresponds to 3.5. As shown here, starting with a lower λ increases the amount oftime that the cell can operate prior to failure, for reasons that are discussed below,but also lowers performance.

104

Figure 4.6: Eect of start current density. H2/O2 operation with a constant starttemperature of -20 C. For the simulation, λinit as shown in legend. Experimentaldata from [18].

Figure 4.7: Eect of start current density. H2/air operation with a constant starttemperature of -10 C. For the simulation, λinit = 3.7. Experimental data from [89].

105

Figure 4.8: Eect of initial membrane water content as calculated from equation 4.4.H2/air operation with a constant start temperature of -20 C. Current density of 0.04A/cm2, constant after a linear ramp from 0 A/cm2 during the rst 80 s of operation.Experimental data from [90].

Although the results in Figure 4.6 are for -20 C with H2/O2, Figure 4.7 shows thata similar eect of current density is observed at -10 C with H2/air as well. Modelresults indicate that the steep increase in the amount of water that can be producedat very low current densities is due to the increase in the rate that water that canbe removed in the gas and membrane phases relative to the rate at which it is beingproduced.

4.3.2.3 The eect of initial membrane water content

Figures 4.8 and 4.9 illustrate that decreasing the amount of water present in themembrane initially increases the amount of water that can be produced prior tofailure, although Vinit decreases considerably at low values of λ. At lower λ there ismore capacity for water uptake in the membrane but also a lower ionic conductivity,which results in higher ohmic losses. The sharp drop in mstart

H2O that appears in themodel result near λ = 14 is due to the presence of ice in the hydrophilic pores of thecathode catalyst layer initially. At lower values of λ these pores are empty initially,meaning that there is more room for the accumulation of product ice during thestart-up. Hydrophilic pores are taken to comprise 20 % of the total pore volume, asdiscussed in Chapter 2.

106

Figure 4.9: Eect of initial membrane water content as calculated from equation4.4. H2/O2 operation with a constant start temperature of -20 C. Constant currentdensity of 0.05 A/cm2. Experimental data from [18].

In these gures, the initial water content has been computed from the model ofSpringer et al.,91 which is consistent with the method that the authors used to reporttheir results:

λ = 0.043 + 17.81a− 39.85a2 + 36.0a3, (4.4)

where a is the activity of water relative to ice. As discussed in Chapter 3, this isdierent from the value of λ used in the model, which is computed based on the ac-tivity of water relative to liquid water, consistent with the experimental observationsof Gallagher et al.86 As a result, the value of λ used in the model is always less thanthat reported by the authors.

4.3.2.4 The eect of cathode-catalyst-layer thickness

Because the amount of water that can be produced before failure is a function of theamount of pore volume in the cathode catalyst layer that is available for ice to occupy,one might expect that increasing the cathode-catalyst-layer thickness increases mstart

H2O.Figure 4.10 shows that this is true as the cCL gets thicker, but that the additionalcapacity gained per unit thickness decreases. Model results show that this is because,beyond about 10 µm, not all of the pores near the cCL/cMPL interface can be lled

107

Figure 4.10: Eect of cathode-catalyst-layer thickness. H2/air operation with a con-stant start temperature of -20 C. Constant current density of 0.1 A/cm2. For thesimulation, λinit = 7.4. Experimental data from [21].

prior to failure due to ohmic eects similar to those described above when discussingthe eect of start current density.

4.3.3 Nonisothermal results

Figures 4.11 through 4.13 show experimental and simulated cell potential and tem-perature as a function of time for three nonisothermal cold-start experiments, frominitial temperatures of -10, -20, and -30 C, respectively.

In these experiments, the current density is quite high: 0.6 A/cm2, higher than inany of the isothermal cases discussed above. As a result, the amount of cell wasteheat (and as a result, external heat input, see equation 4.2) is quite high, and thecell heats up rapidly. For Tinit = -10 C, the cell reaches 0 C in about 5 s, and toreach 10 C takes about 20 s. The cell potential increases steeply during this timeperiod, due primarily to membrane and ionomer hydration, which leads to higherionic conductivity and a lower ohmic loss.

In the -20 C case, the time to reach 0 C increases by roughly a factor of two relativeto the -10 C case, as expected given that the amount of heat available is similar, thespecic heat capacity is the same, and the necessary change in temperature is twiceas large. The simulation predicts initial performance that is worse than that observed

108

Figure 4.11: Nonisothermal cold start from -10 C. H2/air operation with a constantcurrent density of 0.6 A/cm2. For the simulation, λinit = 6.0. Experimental resultsfrom UTC Power.22

in the experiment. It is likely that this is due to a higher level of initial membranehydration than that assumed for the simulation, which is taken to be the same in allthree cases (λinit= 6.0).

However, even the experiment exhibits a (shallow) maximum in performance at about3 s, and the performance does not start to climb monotonically until after 5.0 s. Sim-ulation results indicate that this behavior near time zero, where the cell potentialincreases slightly, plateaus or decreases, and then rises again is a result of the com-peting eects of membrane/ionomer hydration and reaction-distribution shift withinthe cCL. At this high current density, almost all of the reaction takes place near theM/cCL interface due to the low ionic conductivity of the ionomer. During the rstsecond or two of operation, hydration of the membrane and ionomer occurs, reduc-ing the ohmic drop. However, pores within the cCL simultaneously ll up near theM/cCL interface, and, once they are full, the reaction shifts further away, increasingthe ohmic drop through the thickness of the cCL, and cell potential plateaus or drops.It is only once the rate of ohmic gain due to membrane/ionomer hydration exceedsthe rate of ohmic loss due to the reaction-distribution shift that the potential startsto rise again.

109

Figure 4.12: Nonisothermal cold start from -20 C. H2/air operation with a constantcurrent density of 0.6 A/cm2. For the simulation, λinit is as listed in the legend.Experimental results from UTC Power.22

110

Figure 4.13: Nonisothermal cold start from -30 C. H2/air operation with a constantcurrent density of 0.6 A/cm2. For the simulation, λinit = 6.0. Experimental resultsfrom UTC Power.22

111

As shown in Figure 4.13, the magnitude of this cell-potential oscillation increaseswith decreasing temperature (for a xed istart and λinit). In the -30 C case, the cellpotential decreases to about -0.2 V before increasing again. This is due to the lowerionic conductivity at this temperature, which increases the likelihood that the cellpotential will fall rather than rise or plateau, as observed experimentally at -10 and-20 C, respectively.

The potential prole predicted by the model as the cell temperature passes through0 C is signicantly better than that observed in the experiment. In fact, in theexperiment the cell potential remains near 0 V until the cell reaches about 20 C(note that, as a result, it also heats up faster than the model predicts, in contrastto the good agreement between thermal proles shown in Figures 4.11 and 4.12).Several factors could contribute to this poor performance. Because the temperatureis measured at only one point in a relatively large active area of 320 cm2, it could bethat a signicant amount of the cell's area is still at a lower temperature than thatmeasured. In addition, there could be reactant blockage in some of the gas channelsthat limits access to part of the active area. Another possibility is some form ofhysteresis. For example, perhaps the pathways in the cathode's porous media thatare required for liquid-phase water removal are disrupted during the purge and coldstart procedures and are unable to reform until a certain liquid pressure is achieved.


The cold-start model developed in Chapters 2 and 3 is used to simulate both isother-mal and nonisothermal cold-start experiments. Good ts to cell performance areachieved using a relatively small number of tting parameters: seven congurationaland two operational. In addition, results of parametric experimental studies arematched using the model and the physical bases for trends observed in these studiesare explained.

For example, consistent with experimental observations, it is shown that near, butbelow 0 C, liquid water ow out of the cathode catalyst layer enables the cell tooperate for a signicantly longer time than it could if no liquid ow occurred. It isalso shown that increasing the current density during start-up reduces the amountof pore space that can be utilized during cold start. Increasing the thickness of thecathode catalyst layer beyond a certain point has little eect on the time that the cellcan operate prior to failure due to the high ionic resistance associated with drawingcurrent far away from the membrane/cathode catalyst layer interface.

These results verify that the model can be a valuable tool for rapidly evaluatingprocedural and congurational improvements for cold-start applications.

112

Chapter 5

Optimization of Procedural and

Congurational Parameters

5.1 Introduction

In Chapters 2 and 3 of this dissertation, a 2-D cold-start performance model is de-veloped. In Chapter 4, the model is veried using isothermal and nonisothermalcold-start data. In the present chapter, the veried model is deployed to determinea cold-start protocol and a set of cathode-catalyst-layer parameters that enables op-timal performance relative to the DOE's automotive cold-start objectives, which aresummarized in Table 1.3.

5.2 Baseline Performance

The baseline cell type for the simulations in this chapter is the UTC Power con-guration dened in Chapter 4. All of the simulations use these parameters unlessotherwise specied.

5.2.1 Rated power

In order to determine the time required to reach 50 % power, 100 %, or rated powermust rst be dened. For the purposes of this chapter, it is the power available fromthe cell at 1.0 A/cm2 and an operating temperature of 75 C using pure hydrogenand air and anode and cathode total pressures of 1.0 bar. Figure 5.1 shows that, forthe baseline cell, this power level is 0.64 W/cm2. As discussed in Chapter 1, ratedpower is not necessarily the same as the the maximum power available from the cell,

113

Figure 5.1: Simulation results for the performance of the baseline cell at 75 C usingH2/air, both at 1 bar of total pressure and with reactant relative humidities of 84 %.

which is often signicantly higher. In automotive applications, rated power is basedon a trade-o between maximizing power density, which requires a lower voltage, andeciency, which requires a higher voltage. Eciency is important for two reasons.First, it improves fuel economy. Second, it reduces the amount of heat that mustbe rejected from the cell stack at a given power level, an important considerationwhen the operating temperature of the system is at least 40 C lower than that of aninternal-combustion engine.

5.2.2 Galvanostatic and potentiostatic cold-start results

As discussed in Chapter 1, in order for the cell to reach 50 % of rated power whenstarting from -20 C, it must heat up. When cell self-heating is the cold-start strategy,current must be drawn from the cell. A question of interest is what an appropriateprotocol is for doing so, given the constraints on both the cell and the system, whichwill be described in greater detail below. Two options are considered here: galvanos-tatic and potentiostatic.

114

During a galvanostatic cold start, a constant current density is applied to the cell. Ingeneral, during this type of cold start the cell potential starts low, but eventually risesas the waste heat raises the cell's temperature. Consequently, the heat-generation ratestarts at a high value but then decreases with time. During a potentiostatic start,in contrast, the cell potential is held constant. The current density starts low, buteventually rises. As a result, the heat-generation rate starts low but increases withtime.

There are many other methods for starting the cell, including ramped-voltage starts,constant- or ramped-power starts, and constant- or ramped-resistance starts. Exam-ining all of these is beyond the scope of the present work. However, in some ways thegalvanostatic and potentiostatic examples discussed above represent bounding cases:either the potential starts low and increases while the heat generation starts high anddecreases, or the current density and heat generation start low and gradually increasewith increasing temperature.

The specic protocols used to perform simulations of these two types of starts are asfollows. For the galvanostatic start,

i =

istart TcCH < −3C

0.75 A/cm2 −3C ≤ TcCH ≤ 15C

1.00 A/cm2 TcCH > 15C

. (5.1)

where i is the current density, istart is the start current density that will be denedfor each experiment, and TcCH is the average temperature of the cathode gas channel.For the potentiostatic start,

V =

Vstart TcCH < −3C

0.6 V −3C ≤ TcCH ≤ 25C

varies, i = 1.00 A/cm2 TcCH > 25C

. (5.2)

where V is the cell potential, Vstart is the start potential that will be dened for eachexperiment. In both cases, there is a step from the initial potential or current-densitysetpoint once TcCH reaches -3 C to a xed value. At this temperature and thesexed values listed, a fully hydrated cell should be able to reach 50 % of rated power.Therefore, stepping to these setpoints is an attempt to reach 50 % output poweras quickly as possible regardless of istart or Vstart. For example, if Vstart = 0 V, thepower output of the cell is 0 W/cm2. Therefore, operating at this potential is notdesirable for a time period that is any longer than necessary to heat the cell. Inthe potentiostatic case, once the temperature reaches 25 C, the simulation switchesto galvanostatic mode, setting 1.0 A/cm2. This is done so that the cell approachesnormal operating temperature (75 C) in roughly the same way, regardless of whichprocedure is used. Potentiostatic refers therefore to the boundary condition applied

115

Figure 5.2: Comparison of simulation results for a cold start from -20 C using eithera galvanostatic (0.5 A/cm2) or potentiostatic (0.4 V) approach. Additional operatingconditions include H2/air operation at 1 bar of total pressure and λinit = 6 molH2O/mol SO−3 .

during the portion of the cold-start which is of most interest here: that below 0 C,where the presence of ice aects start performance and cell durability.

Figure 5.2 compares the cell power output and temperature as a function of time fora cell started using each of the two methods, where istart = 0.5 A/cm2 or Vstart = 0.4V . Similar performance is observed. In both cases, 50 % power is reached in 10 sor less, while rated power is achieved in about 30 s, and 75 C is reached within 50s. Once the cell reaches this temperature, the heat-transfer coecient and ambienttemperature at the outside edge of each plate are set to 1 W/cm2·K and 75 C inorder to simulate introduction of coolant, and the cell is allowed to reach steady statebefore the simulation is terminated.

Figure 5.3 shows the cell potential, current density, and heat output for each case as afunction of time, illustrating the trends described above. For the potentiostatic case,the current density starts low and then increases quite quickly as the cell approaches-3 C. This increase corresponds to melting of the ice that has accumulated in thecathode catalyst layer (cCL). As a result, the heat-generation rate also starts low

116

(a)

(b)

Figure 5.3: Additional results for the cold starts shown in Figure 5.2: (a) cell potentialand current density and (b) total waste heat generated by the cell.

117

and then increases. The process repeats itself during the second step of the process,when 0.6 V is set. For the galvanostatic start, the cell potential starts low and thenincreases near -3 C, and the heat generation decreases accordingly. The processrepeats itself when the current density is raised to 0.75 A/cm2.

Although the galvanostatic start-up reaches each power level a couple of seconds fasterthan the potentiostatic case, this is a dierence could easily be overcome by loweringVstart slightly. In other words, the advantage is not inherent. The question of interestis, therefore, not which of these types of start-up is faster than the other, but whetherone has an advantage with respect to the ice pressure that builds up in the cell duringthe cold-start process, which might be expected to relate to durability.

Figure 5.4 addresses this question. Results from two galvanostatic simulations areshown, the one discussed above where the start temperature, Tstart, is -20 C andistart = 0.5 A/cm2, as well as one where Tstart is -30 C and istart = 0.1 A/cm2.These correspond to the conditions used by Yang et al.,19 who performed repeatednonisothermal cold starts on multiple cells, measuring performance loss and analyzingTEM images of the cells before operation and after teardown. For the cell subjectedto istart = 0.5 A/cm2, after 110 cold starts a severe reduction in performance at 1.0A/cm2 and 70 C was observed, and upon teardown severe delamination of the cCLfrom the M as well as densication of the cCL was observed near the M, as shownin Chapter 1 of this work. In contrast, a cell subjected to 100 cycles at istart = 0.1A/cm2 showed no performance loss and no morphological changes.

In Figure 5.4a, the ice saturation prole through the thickness of the cCL (in theregion under the rib, along the rib-symmetry line dened in Chapter 3) is shown.The x-position in the cCL is normalized by the thickness of the layer (14 µm), andthe interface with the M is at a normalized x value of 0, while the interface with theMPL is at a value of 1. The saturation prole for istart = 0.1 A/cm2 is more uniformthan that for istart = 0.5 A/cm2, where ice accumulates near the M/cCL interface.This occurs because the ORR reaction distribution shifts toward this interface athigh current densities due to the low ionic conductivity of the ionomer in the cCL, asdiscussed in Chapter 4. Figure 5.4b illustrates that the high saturation in this regioncorresponds to a higher ice pressure (about 10 bar higher at the M/cCL interface), asexpected from Chapter 2. Therefore, an explanation for the results of Yang et al. isthat operating at a high current density is more likely to damage the cCL (and thatthe damage should be concentrated at the M/cCL interface) due to the nonuniformreaction distribution.

Results of the potentiostatic start are also shown for comparison in Figure 5.4, butthese are very similar to those of the high-current galvanostatic start. Therefore itdoes not appear that either procedure has an advantage over the other in terms ofdurability, at least when they are performed in such a way that the time to a givenpower is roughly the same.

118

(a)

(b)

Figure 5.4: Ice (a) saturation and (b) pressure proles across the thickness of thecathode catalyst layer for the cold starts described in Figure 5.2 as well as an addi-tional galvanostatic cold start at 0.1 A/cm2. All proles correspond to the time whenthe average ice pressure in the cCL reaches a maximum during the cold start. Theyare all taken along the rib symmetry line dened in Chapter 2.

119

5.3 The Eect of Operational Parameters

In the following sections, the potentiostatic start-up procedure dened by equation 5.2is used to investigate the eect of key operation parameters on cold-start performance.Based on these results, a cold-start procedure is dened that balances both start timeand potential for structural damage to the cCL.

5.3.1 The eect of start potential

In this chapter, the time to 50 % of rated power, ∆tstart, is reported as a means ofjudging performance relative to the targets in the rst two columns in Table 1.3. Ifthe cell oscillates about 50 % power before climbing to higher levels, the latest timeat which the power passes through the 50 % threshold is reported, in order to ensurethat the cell's performance is stable at the reported time. If the cell is not able tooperate indenitely at or above 50 % power, the start is considered to have failed,and no time is reported, regardless of whether the 50 % power threshold was evercrossed.

Figure 5.5a shows ∆tstart, as a function of Vstart. As discussed in Chapter 1, operatingat a lower potential results in a higher level of heat generation, and thus one wouldexpect that the shortest start-up time should occur at the lowest potential. This iswhat the simulations predict, with the values for start time increasing monotonicallyfrom 6.75 s at 0 V to 432 s at 0.9 V. On the same plot, the maximum fraction of thecCL that experiences an ice pressure higher than 23 bar, fmax, cCL

high pI, is also plotted. The

threshold of 23 bar is based on the results discussed above, where a model predictionof pI <23 bar corresponds to an experimental result of no damage to the cCL. Theresults below show a maximum in fmax, cCL

high pI, with the 0.4 V case from Figure 5.4 at the

maximum. Examination of the simulation results indicates that the maximum occursbecause at lower potentials the amount of time that the the ice is able to accumulateis lower than at higher potentials, while at high potentials the current density is lowand the ice accumulates uniformly within the cCL rather than being shifted towardthe M/cCL interface, causing a high-pressure region there.

Two other parameters of interest are plotted in Figure 5.5b. First, the average poweroutput of the cell during the time period prior to reaching 50 % of rated power, Pstart,which is important because it is often advantageous to have some power available fromthe fuel-cell stack during the cold start. Depending on the start-up strategy beingimplemented, this power may be used to run auxiliary equipment, heat components,melt ice in other parts of the system, or provide some motive power to the vehicle.The second parameter is the amount of energy required during the time period priorto reaching 50 % of rated power, Estart, which relates directly to the DOE energyrequirement. By dividing the 80-kW stack power output target by the rated powerof 0.64 W/cm2, the total required stack active area can be determined: 12.5 m2. The

120

(a)

(b)

Figure 5.5: The eect of start potential on (a) the time to 50 % of rated power andthe maximum fraction of the cCL that experiences an ice pressure greater than 23bar, as well as (b) the average power available from the cell prior to reaching 50 %power and the amount of energy required during that same period. Results assumea start-up from -20 C using potentiostatic approach, H2/air operation at 1 bar oftotal pressure, and λinit = 6 mol H2O/mol SO−3 .

121

energy requirement in Table 1.3 can then be converted to an areal basis. Doing soresults in a maximum allowable combined start-up and shutdown energy of 40 J/cm2.

To calculate the values for cold-start energy shown, the number of moles of hydrogenconsumed until 50 % power is reached is calculated (assuming a hydrogen stoichiome-try of 1.0), and this value is converted into energy expended using ∆G for the fuel-cellreaction. This approach does not account for the energy used during shutdown, whichis included in the requirement. Shutdown energy can be signicant, especially if agas purge is used to dry the cell stack before allowing it to freeze. Therefore, cal-culating Estart is primarily a means of comparing the various start-up scenarios toone another, rather than judging whether the requirement will be met. On the otherhand, the calculation can be used to eliminate cold-start scenarios that exceed therequirement even without any shutdown energy, or, for cases where Estart is less thanthe requirement, to estimate how much energy is available for shutdown.

The results show that a maximum in average cell output power occurs at 0.4 V (32% of rated power), while the amount of energy required increases signicantly aboveabout 0.65 V, but at lower potentials is about 6 J/cm2. The increase at high potentialsreects the fact that heat loss from the cell is included in the calculations. That is,the parameter h dened in Chapter 3, which is the heat-transfer coecient that isapplied to the outside edges of the plates, is 1.7×10−4 W/cm2 · K. Although thisresults in very little heat lost relative to the amount generated at low-to-moderatepotentials, at high potential the fraction lost is signicant.

5.3.2 The eect of initial relative humidity

Typically, a gas purge is used during the shutdown procedure to remove water from thecell, increasing its capacity to absorb product water during cold start. One measureof the extent to which the stack has been purged is the initial water content in theM and the ionomer in the cCLs, as measured by λinit, the moles of H2O per mole ofSO−3 present prior to the start-up. Figure 5.6 shows the eect of λinit on ∆tstart andfmax, cCL

high pI. At lower initial water contents, the start time increases. This is because

the initial ionic conductivity of the cell is much lower, resulting in a lower currentdensity for a given cell potential. On the other hand, the fraction of the cCL exposedto high ice pressures decreases. This is due to a combination of the lower currentdensities providing increased uniformity in the reaction rate and the greater fractionof the product water that is able to move into the membrane. When λinit=λmax = 8,very little water moves into the membrane, and consequently it builds up in the cCLpores, which results in higher ice pressures.

122

Figure 5.6: The eect of initial membrane and ionomer water content on the timeto 50 % of rated power and on the maximum fraction of the cCL that experiencesan ice pressure greater than 23 bar. Results assume a start-up from -20 C using apotentiostatic approach, H2/air operation at 1 bar of total pressure, and a cold-startpotential of 0.65 V.

123

5.3.3 The eect of start temperature

Based on the results in Figures 5.5 and 5.6, a Vstart and λinit of 0.65 V and 5.0,respectively, are chosen for the remainder of the simulations presented below. Using0.65 V, the heat generated is high enough that the start time is well within 30 s andthe heat loss does not aect signicantly the energy required for start-up. Decreasingthe initial water content from 6.0 (in Figure 5.5) to 5.0 should decrease the fraction ofthe cCL exposed to high ice pressures. Finally, a signicant amount of power shouldbe available from the cell for use in the system.

Figure 5.7 shows the results when this procedure is applied to the baseline cell con-guration at dierent values of Tstart. In (a), the time to start increases from 9 s at-3 C to 89 s at -40 C. As expected, fmax, cCL

high pIis quite low (0.02) at -20 C, although

it does increase to 0.89 at -40 C. This increase at lower temperatures is due to theincreased time that the cell must spend at temperatures low enough that ice can form.At temperatures greater that -20 C, none of the cCL is expected to experience highice pressures.

At -20 C, (b) shows that an average of 23 % of rated power is available as thecell is heating up. However, as expected due to the strong dependence of the ionicconductivity on temperature, the average power out of the cell decreases with de-creasing Tstart. On the other hand, the energy required increases due to the greatertemperature dierence that must be traversed before the cell can reach 50 % of ratedpower.

5.4 The Eect of Cathode-Catalyst-Layer Congu-

rational Parameters

The 2-D model can be used to examine the eect of congurational parameters in anyof the layers of the fuel cell. However, for the current study the focus is limited to thelayer that is impacted most signicantly by the accumulation of ice during cold start:the cathode catalyst layer. In Chapter 4 it is shown that much of the accumulation ofproduct water during cold start occurs in the cCL, especially at lower temperatures.This accumulation can aect cell performance both by blocking oxygen's access tothe catalyst and by shifting the reaction in the cCL in such a way as to increase itseective ionic resistance. For these reasons, the cCL properties play a signicant rolein determining overall cell performance.

To demonstrate this, three cCL properties have been chosen for parametric studies:the ionomer loading, εM, the porosity ε, and the specic interfacial area of the catalyst,a0

1,2. In practice these properties may not necessarily be completely independent. Forexample, changing the ionomer loading may also change the specic interfacial area.

124

(a)

(b)

Figure 5.7: The eect of initial start temperature on (a) the time to 50 % of ratedpower and the maximum fraction of the cCL that experiences an ice pressure greaterthan 23 bar, as well as (b) the average power available from the cell prior to reaching50 % power and the amount of energy required during that same period. Resultsassume H2/air operation at 1 bar of total pressure, λinit = 5 mol H2O/mol SO−3 , anda start-up potential of 0.65 V.

125

Figure 5.8: The eect of the volume fraction of ionomer present in the cCL on thetime to 50 % of rated power and on the maximum fraction of the cCL that experiencesan ice pressure greater than 23 bar. Results assume a start-up from -20 C using apotentiostatic approach, H2/air operation at 1 bar of total pressure, λinit = 5 molH2O/mol SO−3 , and a cold-start potential of 0.65 V.

However, a feature of the model is that it allows the user to decouple the eects ofeach of these changes. All other parameters, including those of the aCL retain theirbaseline values during these studies, except as noted below.

In these studies, when a congurational change results in a cell potential at 1.0 A/cm2

and 75 C that is dierent from the baseline, which changes the rated power, thisdierence is accounted for when calculating ∆tstart. For example, decreasing a0

1,2 bytwo orders of magnitude decreases 50 % of rated power from 0.32 W/cm2 to 0.25W/cm2. Therefore, ∆tstart in the revised case represents the time required for the cellto reach the latter value, not the former.

5.4.1 The eect of catalyst-layer ionomer content

In the model, the ionomer content of the catalyst layer is specied by the volumefraction of ionomer in that layer, εM. Because εM, ε, and εs (the volume fraction ofthe solid, electronically conductive phase) must sum to unity, changing εM implies achange to at least one of the other two. For the present study, because the electronicconductivity is over three orders of magnitude higher than the ionic conductivity, andchanging the porosity has signicant eects (as discussed below), εs is the parameter

126

Figure 5.9: The eect of the porosity of the cCL on the time to 50 % of rated powerand on the maximum fraction of the cCL that experiences an ice pressure greater than23 bar. Results assume a start-up from -20 C using a potentiostatic approach, H2/airoperation at 1 bar of total pressure, λinit = 5 mol H2O/mol SO−3 , and a cold-startpotential of 0.65 V.

that is changed, while ε is held constant at the baseline value of 0.50. Similarly,during the parametric study based on ε, εM is held constant at its baseline value of0.11 while εs varies.

Figure 5.8 shows the eect of changing the ionomer content on ∆tstart and fmax, cCLhigh pI

.Increasing the ionomer content increases the eective conductivity of the cathodecatalyst layer, resulting in a higher current density at the start potential of 0.65V. As a a result, the start time decreases as ionomer is added. In addition, thefraction of the cCL exposed to high ice pressures also decreases, despite the highercurrent densities. Examination of the model results indicates that this happens fortwo reasons. First, increasing εM adds capacity for absorbing product water. Second,the higher eective conductivity in the cCL results in a more uniform reaction-ratedistribution. At very low values of εM , the distribution is shifted toward the M/cCLinterface, creating a region of high ice pressure there.

5.4.2 The eect of cathode catalyst layer porosity

Figure 5.9 shows the eect of changing the porosity on ∆tstart and fmax, cCLhigh pI

. Increasingthe porosity has very little eect on the time to power, although a higher porosity

127

Figure 5.10: The eect of the specic interfacial area of the cCL catalyst on the timeto 50 % of rated power and on the maximum fraction of the cCL that experiencesan ice pressure greater than 23 bar. Results assume a start-up from -20 C using apotentiostatic approach, H2/air operation at 1 bar of total pressure, λinit = 5 molH2O/mol SO−3 , and a cold-start potential of 0.65 V.

is slightly better than a lower porosity. This weak dependence is a symptom of thefact that the performance of the cell is not limited by the through-plane transport ofoxygen, but rather by the transport of ions. On the other hand, the porosity doeshave a strong eect on the fraction of the cCL exposed to high ice pressure. This isbecause increasing ε adds capacity for absorbing product water. Therefore, even ifthe reaction distribution is not particularly uniform, the amount of ice that can beheld near the M/cCL interface before high ice pressures are reached is extended.

5.4.3 The eect of specic interfacial area

The specic interfacial area is a measure of the amount of catalyst area availablefor the oxygen-reduction reaction (ORR) in the cathode catalyst layer. The unitsare cm−1, derived from cm2 of Pt surface area per cm3 of catalyst layer volume.While a0

1,2 is generally a function of catalyst loading (in mg/cm2), for a given loadingit can also change with dierent processing techniques, the oxidation state of thecatalyst, and with time as the cell ages. Figure 5.10 shows the eect of a0

1,2 on ∆tstart

and fmax, cCLhigh pI

, where the baseline value is 2.3×106 cm−1. Decreasing the area bytwo orders of magnitude (the range shown) results in an increase in start time from

128

23 to 51 s due to the reduced current density possible at 0.65 V with less catalystarea. Although this result may appear to imply a large amount of margin againstperformance loss over the life of the fuel cell, it should be noted that the amount ofPt area loss with operating hours, especially with large amounts of cycling, can bequite high.12 Furthermore, the baseline value corresponds to a catalyst loading thatis at least an order of magnitude too high for practical use, at least in the automotiveapplication.16 Reducing a0

1,2 results in a more uniform reaction rate distribution due

to the lower current density, and as a result fmax, cCLhigh pI

is zero for most of the rangeshown.

5.5 Improved Cold-Start Performance

Based on the results shown in Figures 5.8, 5.9, and 5.10. An improved cCL congu-ration is proposed, where εM = 0.20, ε = 0.60, and εs = 0.20 (for the baseline case,the values are 0.11, 0.5, 0.39, respectively). The intent of these changes is primarilyto decrease the exposure of the cCL to high ice pressure at very low temperatures,the key issue identied with the baseline conguration, as shown in Figure 5.7. Thespecic interfacial area of the catalyst is unchanged, and all other parameters retaintheir baseline values.

Figure 5.11 shows the cold-start performance that is expected for this improved con-guration as a function of start temperature. Axis scales that match those in Figure5.7 have been used in order to ease comparison with the baseline case. In (a), theresults show improvements in both ∆tstart and f

max, cCLhigh pI

. For example, at -40 C, thestart time decreases from 89 to 70 s, and the fraction of the cCL exposed to high icepressure decreases from 0.86 to 0.53. These benets are derived from the combinedeects of increased capacity for absorbing product water (in both the ionomer andthe pores) as well as the more uniform reaction rate distribution in the cCL due tothe higher eective ionic conductivity. For the same reasons, the results in (b) areequal to or better than those of the baseline.


The cold-start model developed in Chapters 2 and 3 and veried in Chapter 4 isdeployed to determine a reasonable cold-start protocol and improved cathode catalyst-layer properties that enable better performance relative to the DOE automotive cold-start objectives. Criteria include not only minimizing start time but also exposureto high cCL ice pressures and cold-start energy while at the same time maximizingpower available from the cell during the cold-start process.

129

(a)

(b)

Figure 5.11: For the improved cCL conguration, the eect of initial start temperatureon (a) the time to 50 % of rated power, and the maximum fraction of the cCL thatexperiences an ice pressure greater than 23 bar, and (b) the average power availablefrom the cell prior to reaching 50 % power, and the amount of energy required duringthat same period. Assumes H2/air operation at 1 bar of total pressure, λinit = 5 molH2O/mol SO−3 , and a start-up potential of 0.65 V.

130

Starting a cell with the UTC Power conguration described in Chapter 4 and aninitial membrane and ionomer content of 5 mol H2O/mol SO−3 potentiostatically at0.65 V and then stepping to 0.6 V once the cell reaches -3 C results in a time to 50 %power of 23 s from -20 C, with minimal exposure of the cCL to high ice pressure. Inaddition, on average 23 % of rated power is available from the cell prior to reaching 50% power, and only 6.5 J/cm2 are used during this period, leaving 33.5 J/cm2 availablefor the shutdown procedure.

Increasing the porosity and the volume fraction of ionomer from 0.5 to 0.6 and from0.11 to 0.2, respectively, reduces the exposure of the cCL to high ice pressure whenstarting from very low temperatures: 52 % from -30 C and 38 % from -40 C. Starttime from these temperatures is also reduced by 16 and 22 %, respectively. Thesechanges increase the ionic conductivity of the cCL and increase its capacity to absorbproduct water. As a result, the average current density during start-up is higher, butthe reaction rate is distributed more evenly across the thickness of the layer, reducingthe amount of ice that accumulates near the M/cCL interface.

These results show how the 2-D model can be used both to predict and to understandbetter the cell behavior observed during cold start and how this insight can be usedto improve performance.

131

Chapter 6

Conclusions and Future Work

6.1 Conclusions

When starting a polymer-electrolyte fuel cell (PEFC) from temperatures below about-3 C, it is necessary for the cell to heat up in order for it to produce a substantialfraction (0.5 or higher) of the power that it can at normal operating temperatures. Ifthe cell must heat itself up using its own waste heat (i.e., no external source of heatis available), then current must be drawn from the cell. Doing so generates waterin the cathode catalyst layer (cCL) through the oxygen reduction reaction (ORR), asubstantial portion of which freezes in the pores of the cCL, although some water maymove out of the cCL into the adjacent membrane (M) and microporous layer (MPL).If the cell does not heat up fast enough, this ice accumulation can eventually inhibitthe performance of the cell enough such that the amount of heat it can produce isless than the amount of heat being lost to the environment, meaning that the cell'stemperature will not rise without an external source of heat and the cold start hasfailed.

A zero-dimensional (0-D) model is developed to predict the performance of the cellwithout accounting for any ice accumulation for the purpose of determining both thelower bound for start time of a cell and examining the factors that limit the cell'sperformance even when no ice is present. It is shown that, at -20 C, the poweravailable at an operating potential of 0.6 V is only 22 % of the amount produced at75 C (referred to as rated power). In addition, the primary reason why the cell'smaximum power output is so low is the decrease in ionic conductivity associated withlow-temperature operation. It is also shown that, in the absence of ice accumulation,the cell should be able to reach 50 % of rated power in roughly 10 s when startingfrom -20 C, assuming a thermal mass of 2.1 J/cm2·K for the cell, consistent withcurrent automotive technology.

In order to understand better the role that ice formation plays during cold start, a

132

more sophisticated model is required. For this reason, a two-dimensional (2-D) cold-start model for polymer-electrolyte fuel cells is developed based on incorporatingice-content calculations and the temperature dependence of key properties into atraditional macrohomogeneous framework. A new approach for predicting equilibriumlevels of ice and liquid saturations at a given temperature is presented that is basedon combining the theory of freezing in soils developed by Everett24 and the approachto calculating saturation in a liquid-gas system developed by Weber, Darling, andNewman.45 Using this new method, curvature-induced melting and phase-change-induced ow are accounted for, and the ice pressure, which is related to the durabilityof the cell materials, is estimated at all positions and times within the porous media.

To illustrate the importance of including these phenomena, a simulation is presentedfor a typical nonisothermal cold start in which residual ice remains in the cell prior toa start-up from -10 C. The results indicate that the cell is able to operate stably foran extended period near but below 0 C as the ice in the gas-diusion layers (GDLs)melts. This is possible because the ice in the ne pores of the cCL and cMPL meltsat a lower temperature, which prevents the catalyst layer from lling up.

The model is veried by comparing its predictions to experimental results across arange of operating conditions and congurations. In the process, cell behavior duringcold start during various parametric studies is analyzed. It is shown that, at xed celltemperatures above roughly -3 C, the amount of time that a cell can operate beforefailure increases substantially relative to lower temperatures due to curvature-inducedmelting in the ne-pores of the cCL and cMPL, as described above. Furthermore, ata given xed temperature, operating at higher current densities reduces signicantlythe amount of water that can be produced by the cell prior to failure. In fact, notall of the pores in the cCL are necessarily lled at the time of failure at high currentdensities. Instead, failure occurs because relative to the low-current-density case theORR distribution shifts toward the M/cCL interface due to the low eective ionicconductivity of the cCL. As a result, ice lls the pores at the interface between theM and the cCL rst, and, once this region is lled with ice, the reaction must moveaway from the interface, which in turn results in a large ohmic loss which eventuallycauses the cell potential to fall to 0 V (assuming that the current density is constant).It is also shown that decreasing the amount of water present in the membrane andionomer prior to cold start increases the amount of time that the cell can operateprior to failure because doing so increases the capacity for product water. Similarly,increasing the thickness of the cCL increases the amount of water that can be storedalthough, at higher current densities, the additional capacity is not usable beyond acertain thickness due to the ohmic eects described above.

The veried cold-start model is deployed to determine a reasonable cold-start pro-tocol and cathode-catalyst-layer properties that enable better performance. Criteriainclude not only minimizing start time but also exposure to high cCL ice pressures(which can cause structural damage) and cold-start energy while at the same time

133

maximizing power available from the cell during the cold-start process.

Starting a cell with an initial membrane and ionomer content of 5 mol H2O/mol SO−3potentiostatically at 0.65 V and then stepping to 0.6 V once the cell reaches -3 Cresults in a time to 50 % power of 23 s from -20 C, with minimal exposure of thecCL to high ice pressure. In addition, on average 23 % of rated power is availablefrom the cell prior to reaching 50 % power and only 6.5 J/cm2 are used during thisperiod, leaving 33.5 J/cm2 available for the shutdown procedure.

Increasing the porosity and the volume fraction of ionomer from 0.5 to 0.6 and from0.11 to 0.2, respectively, reduces the exposure of the cCL to high ice pressure whenstarting from very low temperatures: 52 % from -30 C and 38 % from -40 C. Starttime from these temperatures is also reduced by 16 and 22 %, respectively. Thesechanges increase the ionic conductivity of the cCL and increase its capacity to absorbproduct water. As a result, the average current density during start-up is higher, butthe reaction rate is distributed more evenly across the thickness of the layer, reducingthe amount of ice that accumulates near the M/cCL interface.

When the 2-D model and the 0-D model are used to simulate a cold start underidentical conditions (an initial temperature of -20 C and a start potential of 0.60 V),a start time of 16.5 s is predicted based on the 2-D model, which is a 65 % increaseover the 0-D result. The dierence is due to the fact that, in the 2-D case, theaccumulation of ice in the cCL limits the cell's current density to a nearly constantvalue, despite the fact that the temperature is increasing, until a temperature ofroughly -3 C is reached in the cCL and the ice it contains begins to melt, at whichpoint the current density increases very rapidly. In contrast, in the 0-D case thecurrent density increases monotonically with temperature. This result emphasizesthe importance of accounting for the accumulation of ice in the cell when trying topredict start time.

6.2 Future Work

There is a variety of ways in which the framework developed in the present workcould be used for further fruitful research. Some of these involve improvements toaspects of the existing model while others are studies of interest that have not yetbeen completed. Examples of each of these are discussed in turn. This list is notexhaustive, but instead includes items judged to be of a high priority.

6.2.1 Improvements to the model

The following is a list of ways in which the physics included in the model could beupdated for greater delity.

134

6.2.1.1 Incorporate better kinetic expressions for freezing in porous me-dia

In the present work, the rate expression and rate constant used for freezing andmelting in the porous media are identical, and the value of the rate constant is veryhigh, eectively corresponding to a condition of phase equilibrium between ice andliquid water. However, while melting is generally a thermodynamic phenomenonthat is likely well-described using phase-equilibrium calculations, freezing is oftenkinetically controlled, involving both nucleation and crystallization steps. In fact,supercooled liquid water at -10 C has been observed in-situ in an operating fuelcell,92 and supercooling of water in GDLs has been reported.93 Yet, no quantitativestudy of the kinetics of freezing in the CLs and MPLs currently exists in the fuel-cellliterature. In the absence of such studies, phase equilibrium is assumed in the presentwork as a means of establishing a point of reference. As future investigation shedsfurther light on this subject, the model should be revised as necessary.

6.2.1.2 Improve the membrane/ionomer water-uptake and transport modelbelow 0 C

The present model assumes that the amount of nonfrozen water in the membranebelow 0 C can be estimated by using a water-uptake curve from 30 C but com-puting the activity relative to liquid water rather than ice. Transport properties aredetermined from this calculation of water content. This approach is in agreementwith the results of the experimental study of Gallagher et al.86 However, the work ofGallagher et al. as well as the work of Thompson et al.94 suggests strongly that, ifthe membrane is frozen quickly enough, ice can form inside of it. This phenomenon isnot incorporated into the model in the present work because the relationship betweenthe chemical potential of water in the membrane and ice formation inside of it is notknown. Furthermore, how the presence of ice aects the transport properties in themembrane is also not known. As future investigation sheds further light on theseareas, the model should be revised as necessary.

6.2.1.3 Add a contact-angle distribution to the saturation calculations

The present model for saturation assumes that the porous media contain hydrophilicand hydrophilic pores, each with its own associated liquid-gas contact angle. Recently,a revised approach to this type of model has been published that incorporates acontinuous, rather than binary, contact-angle distribution.95 This change improvesthe ability of the saturation model to predict the liquid saturation in a porous mediumlled with liquid and gas as a function of capillary pressure. Because the cold-startmodel's prediction of equilibrium ice and liquid content depends on the liquid-gas

135

saturation curve, as explained in Chapter 2, incorporating this change should lead toimproved accuracy.

6.2.2 Additional Studies

The following is a list of studies that could be completed using the present model asa starting point.

6.2.2.1 The cold-start behavior of and strategies for cells using thin cat-alyst layers

One approach to reducing cost in a PEFC is to reduce the amount of precious-metalcatalyst used for the half-reactions. To do this, electrodes that are much thinnerthan the 10 to 15 µm catalyst layers that are presently available may ultimately bedesirable. For example, the thickness of the nanostructured thin-lm (NSTF) catalystlayer developed by 3M is less than 1 µm, and can even be less than 0.5 µm at a lowPt loading of 0.1 mg/cm2.96 A potential drawback for these designs in the contextof cold start is a signicant reduction in ice capacity in the electrode, which mayreduce the time that the cell is able to operate prior to failure. However, as discussedin Chapter 4, nonisothermal cold-start performance is often signicantly better thanthat predicted by isothermal experiments because of the ability of water to move outof the cCL as the cell heats up. Furthermore, there may be modications to thelayers adjacent to the electrode that could mitigate any problems. The cold-startmodel could be used in its present form to investigate both of these possibilities.In addition, the model for the catalyst layers could be changed while leaving theremainder of the model intact. This may be desirable if, for example, the behavior ofsuch thin catalyst layers is not well described by porous-electrode theory.

6.2.2.2 The eect of a cell's position in the cell stack on its cold-startperformance

The thermal environment that a cell experiences during freezing and during coldstart is inuenced strongly by its position in the cell stack, as shown by Sundaresanand Moore as well as Khandelwal, Lee, and Mench.97,98 Not only may one cell'saverage temperature be signicantly colder than another, but a temperature gradientmay exist across a given cell as well. As a result of these combined eects, theperformance of the cells at the ends of the stack is often substantially dierent fromthose in the center, as has been shown experimentally by Patterson.99 Because themodel is capable of simulating water movement in both liquid and vapor phases overa range of temperatures, it could be used to understand better how the thermalenvironment aects cold-start performance.

136

6.2.2.3 Mechanical modeling of the cell during cold-start

A key feature of the present model is that it provides an estimate of the ice pressureat all positions and times. This provides information regarding the conditions underwhich the ice pressure may exceed the structural properties of the cell materials.However, neither deformation of cell materials nor separation of layers due to ice-lens growth is included. Such mechanical modeling could be used to help understanddegradation mechanisms and produce estimates of the cold-start cycle life of a fuelcell. The eects of dierent congurational and operational parameters on structuraldegradation could be considered, just as their eects on performance is analyzed inthe present work.

137

Bibliography

[1] V. Srinivasan, Batteries for Vehicular Applications. 2008.

[2] Environmental Assessment of Plug-In Hybrid Electric Vehicles Volume 1: Na-tionwide Greenhouse Gas Emissions, tech. rep., Electric Power Research Insti-tute, 2007.

[3] Siemens, Product Specications for Wind Turbine, model SWT-2.3-82 VS, http://www.energy.siemens.com/br/pool/hq/power-generation/

wind-power/E50001-W310-A123-X-4A00_WS_SWT-2.3-82%20VS_US.pdf, 2010.

[4] Hydrogen, Fuel Cells & Infrastructure Technologies Program; Multi-Year Re-search, Development and Demonstration Plan. U.S. Department of Energy, 2007.

[5] C. E. Thomas, Fuel Cell and Battery Electric Vehicles Compared, InternationalJournal of Hydrogen Energy, 34, no. 15, 60056020, 2009.

[6] H. Tsuchiya and O. Kobayashi, Mass Production Cost of PEM Fuel Cell ByLearning Curve, International Journal of Hydrogen Energy, 29, no. 10, 985990, 2004.

[7] Vehicle Electrication, tech. rep., Deutsche Bank, 2010.

[8] G. Berdichevsky, K. Kelty, J. B. Straubel, and E. Toomre, The Tesla RoadsterBattery System, tech. rep., Tesla Motors, 2006.

[9] Nissan, Product Specications for the 2011 Nissan Leaf, http:

//www.nissanusa.com/leaf-electric-car/index#/leaf-electric-car/

specs-features/index, 2010.

[10] Chevrolet, Product specications for the 2011 Chevy Volt, http://www.

chevrolet.com/volt/?evar3=volt_hom_mhp, 2010.

[11] C. A. Reiser, L. Bregoli, T. W. Patterson, J. S. Yi, J. D. L. Yang, M. L. Perry,and T. D. Jarvi, A Reverse-Current Decay Mechanism for Fuel Cells, Electro-chemical and Solid State Letters, 8, no. 6, A273A276, 2005.

138

[12] R. M. Darling and J. P. Meyers, Mathematical Model of Platinum Movement InPEM Fuel Cells, Journal of the Electrochemical Society, 152, no. 1, A242A247,2005.

[13] J. Newman, Optimization of Potential and Hydrogen Utilization In an Acid FuelCell, Electrochimica Acta, 24, no. 2, 223229, 1979.

[14] Allen J. Bard and Larry R. Faulkner, Electrochemical Methods: Fundamentalsand Applications. John Wiley & Sons, 2001.

[15] J. Newman and K. E. Thomas-Alyea, Electrochemical Systems. John Wiley &Sons, New York, 2004.

[16] M.F. Mathias, R. Makharia, H.A. Gasteiger, J.J. Conley, T.J. Fuller, C.J. Gittle-man, S.S. Kocha, D.P. Miller, C.K. Mittelsteadt, T. Xie, S.G. Yan, and P.T. Yu,Two Fuel Cell Cars In Every Garage?, The Electrochemical Society Interface,Fall, 2435, 2005.

[17] E.L. Thompson, J. Jorne, and H.A. Gasteiger, Oxygen Reduction ReactionKinetics in Subfreezing PEM Fuel Cells, Journal of the Electrochemical Society,154, no. 8, B783B792, 2007.

[18] E. L. Thompson, J. Jorne, W. B. Gu, and H. A. Gasteiger, PEM Fuel Cell Op-eration at -20 Degrees C. I. Electrode and Membrane Water (Charge) Storage,Journal of the Electrochemical Society, 155, no. 6, B625B634, 2008.

[19] X. G. Yang, Y. Tabuchi, F. Kagami, and C. Y. Wang, Durability of MembraneElectrode Assemblies Under Polymer Electrolyte Fuel Cell Cold-Start Cycling,Journal of the Electrochemical Society, 155, no. 7, B752B761, 2008.

[20] S. H. Ge and C. Y. Wang, Characteristics of Subzero Startup and Water/Ice For-mation on the Catalyst Layer in a Polymer Electrolyte Fuel Cell, ElectrochimicaActa, 52, no. 14, 48254835, 2007.

[21] A. Nandy, F. M. Jiang, S. H. Ge, C. Y. Wang, and K. S. Chen, Eect of Cath-ode Pore Volume on PEM Fuel Cell Cold Start, Journal of the ElectrochemicalSociety, 157, no. 5, B726B736, 2010.

[22] B. Paravastu, Solid Plate Freeze Testing Data From FCM 78374 (unpublishedwork), tech. rep., UTC Power, 2010.

[23] S. Taber, Frost Heaving, Journal of Geology, 37, no. 5, 428461, 1929.

[24] D. H. Everett, Thermodynamics of Frost Damage To Porous Solids, Transac-tions of the Faraday Society, 57, no. 9, 1541, 1961.

139

[25] C. Dirksen and R. D. Miller, Closed-System Freezing of Unsaturated Soil, Proc.Soil Sci. Soc. Am., 30, 168173, 1966.

[26] P. Hoekstra, Moisture Movement in Soils Under Temperature Gradients withCold-Side Temperature Below Freezing, Water Resour. Res., 2, no. 2, 241, 1966.

[27] A. DiMillio, A Quarter Century of Geotechnical Research, FHWA-RD-98-139,1999.

[28] S. Taber, The Mechanics of Frost Heaving, J. Geol., 38, 303317, 1930.

[29] J. S. Wettlaufer and M. G. Worster, Premelting Dynamics, Annual Review ofFluid Mechanics, 38, 427452, 2006.

[30] S. Kim and M. M. Mench, Physical Degradation of Membrane Electrode As-semblies Undergoing Freeze/Thaw Cycling: Micro-Structure Eects, Journal ofPower Sources, 174, no. 1, 206220, 2007.

[31] S. Kim and M. M. Mench, Investigation of Temperature-Driven Water Transportin Polymer Electrolyte Fuel Cell: Phase-Change-Induced Flow, Journal of theElectrochemical Society, 156, no. 3, B353B362, 2009.

[32] S. H. He and M. M. Mench, One-Dimensional Transient Model for Frost Heave InPolymer Electrolyte Fuel Cells, J. Electrochem. Soc., 153, no. 9, A1724A1731,2006.

[33] S. He, J. H. Lee, and M. M. Mench, 1D Transient Model for Frost Heave InPEFCs - III. Heat Transfer, Microporous Layer, and Cycling Eects, J. Elec-trochem. Soc., 154, no. 12, B1227B1236, 2007.

[34] S. H. He, S. H. Kim, and M. M. Mench, 1D Transient Model for Frost Heavein Polymer Electrolyte Fuel Cells - II. Parametric Study, J. Electrochem. Soc.,154, no. 10, B1024B1033, 2007.

[35] R. J. Balliet, K. E. Thomas-Alyea, and J. S. Newman, Water Movement DuringFreezing in a Polymer-Electrolyte-Membrane Fuel Cell, ECS Transactions, 16,no. 285, 2008.

[36] R.D. Miller, Frost Heaving in Non-Colloidal Soils, Proceedings of the ThirdInternational Conference on Permafrost, 707713, 1978.

[37] K. O'Neill, The Physics of Mathematical Frost Heave Models - A Review, ColdRegions Science and Technology, 6, no. 3, 275291, 1983.

[38] R. W. Bain, NEL Steam Tables 1964, 1964.

140

[39] F. P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer.John Wiley & Sons, New York, 1996.

[40] D. R. Lide, CRC Handbook of Chemistry and Physics, 90th Edition (InternetVersion 2010). CRC Press/Taylor and Francis, Boca Raton, FL, 90th ed.

[41] S. C. Hardy, Grain-Boundary Groove Measurement of Surface Tension BetweenIce and Water, Philosophical Magazine, 35, no. 2, 471484, 1977.

[42] W. B. Hillig, Measurement of Interfacial Free Energy for Ice/Water System,Journal of Crystal Growth, 183, no. 3, 1998.

[43] L. Granasy, T. Pusztai, and P. F. James, Interfacial Properties DeducedFrom Nucleation Experiments: A Cahn-Hilliard Analysis, Journal of Chemi-cal Physics, 117, no. 13, 61576168, 2002.

[44] W. M. Ketcham and P. V. Hobbs, An Experimental Determination of SurfaceEnergies of Ice, Philosophical Magazine, 19, no. 162, 1161, 1969.

[45] A. Z. Weber, R. M. Darling, and J. Newman, Modeling Two-Phase Behavior inPEFCs, J. Electrochem. Soc., 151, no. 10, A1715A1727, 2004.

[46] K. Baumann, J. H. Bilgram, and W. Kanzig, Superheated Ice, Zeitschrift FurPhysik B-Condensed Matter, 56, no. 4, 315325, 1984.

[47] M. Schmeisser, H. Iglev, and A. Laubereau, Maximum Superheating of BulkIce, Chemical Physics Letters, 442, no. 4-6, 171175, 2007.

[48] M. Hiroi, T. Mizusaki, T. Tsuneto, A. Hirai, and K. Eguchi, Frost-Heave Phe-nomena of He-4 On Porous Glasses, Physical Review B, 40, no. 10, 65816590,1989.

[49] F. E. Bartell and H. J. Osterhof, Determination of the Wettability of a Solidby a Liquid - Relation of Adhesion Tension to Stability of Color Varnish andLacquer Systems, Industrial and Engineering Chemistry, 19, no. 1, 12771280,1927.

[50] A. Z. Weber and J. Newman, Eects of Microporous Layers in Polymer Elec-trolyte Fuel Cells, J. Electrochem. Soc., 152, no. 4, A677A688, 2005.

[51] Z. H. Liu, K. Muldrew, R. G. Wan, and J. A. W. Elliott, Measurement offreezing point depression of water in glass capillaries and the associated ice frontshape, Physical Review E, 67, no. 6, 2003.

[52] A. Skapski, R. Billups, and A. Rooney, Capillary Cone Method For Determi-nation of Surface Tension of Solids, Journal of Chemical Physics, 26, no. 5,13501351, 1957.

141

[53] G. W. Scherer, Crystallization in Pores, Cement and Concrete Research, 29,no. 8, 13471358, 1999.

[54] F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure. AcademicPress, Inc., New York, 2nd ed., 1992.

[55] J.D. Sage and M. Porebska, Frost Action in a Hydrophobic Material, Journalof Cold Regions Engineering, 7, no. 4, 99113, 1993.

[56] F.J. Radd and D.H. Oertle, Experimental Pressure Studies of Frost Heave Mech-anisms and the Growth-Fusion Behavior of Ice, 2nd International Symposiumon Permafrost, 377383, 1973.

[57] T. Takashi, T. Ohrai, H. Yamamoto, and J. Okamoto, Upper Limit of Heav-ing Pressure Derived By Pore-Water Pressure Measurements of Partially FrozenSoil, Engineering Geology, 18, no. 1-4, 245257, 1981.

[58] H. Gieselman, J. L. Heitman, and R. Horton, Eect of a Hydrophobic Layeron the Upward Movement of Water Under Surface-Freezing Conditions, SoilScience, 173, no. 5, 297305, 2008.

[59] D. H. Everett, Thermodynamics of Multiphase Fluids in Porous Media, Journalof Colloid and Interface Science, 52, no. 1, 189198, 1975.

[60] A.Z. Weber, Gas-Crossover and Membrane-Pinhole Eects in Polymer-Electrolyte Fuel Cells, Journal of the Electrochemical Society, 155, no. 6, B521B531, 2008.

[61] R. W. R. Koopmans and R. D. Miller, Soil Freezing and Soil Water Character-istic Curves, Proceedings. Soil Science Society of America, 30, 680685, 1966.

[62] P. B. Black and A. R. Tice, Comparison of Soil Freezing Curve and Soil-WaterCurve Data for Windsor Sandy Loam, Water Resources Research, 25, no. 10,22052210, 1989. Times Cited: 20.

[63] K. Jiao and X. G. Li, Three-Dimensional Multiphase Modeling of Cold StartProcesses in Polymer Electrolyte Membrane Fuel Cells, Electrochimica Acta,54, no. 27, 68766891, 2009.

[64] L. Mao and C. Y. Wang, Analysis of Cold Start in Polymer Electrolyte FuelCells, J. Electrochem. Soc., 154, no. 2, B139B146, 2007.

[65] Y. Wang, Analysis of the Key Parameters in the Cold Start of Polymer Elec-trolyte Fuel Cells, Journal of the Electrochemical Society, 154, no. 10, B1041B1048, 2007.

142

[66] R. K. Ahluwalia and X. Wang, Rapid Self-Start of Polymer Electrolyte Fuel CellStacks From Subfreezing Temperatures, Journal of Power Sources, 162, no. 1,502512, 2006.

[67] H. Meng, A PEM Fuel Cell Model For Cold-Start Simulations, Journal ofPower Sources, 178, no. 1, 141150, 2008.

[68] L. Mao and C. Y. Wang, A Multiphase Model for Cold Start of Polymer Elec-trolyte Fuel Cells, J. Electrochem. Soc., 154, no. 3, B341B351, 2007.

[69] F. M. Jiang, W. F. Fang, and C. Y. Wang, Non-Isothermal Cold Start of PolymerElectrolyte Fuel Cells, Electrochimica Acta, 53, 610621, 2007.

[70] A. Z. Weber and J. Newman, Modeling transport in polymer-electrolyte fuelcells, Chemical Reviews, 104, no. 10, 46794726, 2004.

[71] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena. JohnWiley & Sons, Inc., New York, 2002.

[72] E. A. Mason, Thermal Conductivity of Multicomponent Gas Mixtures, Journalof Chemical Physics, 28, no. 5, 10001001, 1958.

[73] C. R. Wilke, A Viscosity Equation for Gas Mixtures, Journal of ChemicalPhysics, 18, no. 4, 517519, 1950.

[74] SGL Carbon, Product Specications for Separator Plate, Model L10818Z1F,

[75] SGL Carbon, Product Specications for GDL, Model GDL 34 BA,

[76] SGL Carbon, Product Specications for GDL, Model GDL 34 BC,

[77] A. Z. Weber and J. Newman, Transport in Polymer-Electrolyte Membranes -II. Mathematical Model, J. Electrochem. Soc., 151, no. 2, A311A325, 2004.

[78] M. Khandelwal and M. M. Mench, Direct Measurement of Through-Plane Ther-mal Conductivity and Contact Resistance In Fuel Cell Materials, Journal ofPower Sources, 161, no. 2, 11061115.

[79] R. B. Mathur, S. R. Dhakate, D. K. Gupta, T. L. Dhami, and R. K. Aggarwal,Eect of Dierent Carbon Fillers on the Properties of Graphite Composite Bipo-lar Plate, Journal of Materials Processing Technology, 203, no. 1-3, 184192,2008.

[80] Y. Fu, M. Hou, D. Liang, X. Q. Yan, Y. F. Fu, Z. G. Shao, Z. J. Hou, P. W.Ming, and B. L. Yi, The Electrical Resistance of Flexible Graphite as FloweldPlate in Proton Exchange Membrane Fuel Cells, Carbon, 46, no. 1, 1923, 2008.

143

[81] Toray, Product Specications for GDL, Model TGP-H-090,

[82] Toray, Product Specications for TEFLON PTFE ber,

[83] J. T. Gostick, M. W. Fowler, M. D. Pritzker, M. A. Ioannidis, and L. M. Behra,In-Plane and Through-Plane Gas Permeability of Carbon Fiber Electrode Back-ing Layers, Journal of Power Sources, 162, no. 1, 228238, 2006.

[84] J. D. Seader and E. J. Henley, Separation Process Principles. John Wiley &Sons, Inc., 2nd ed., 2006.

[85] B. Kientiz, H. Yamada, N. Nonoyama, and A. Z. Weber, Interfacial WaterTransport Eects in Proton-Exchange Membranes, Journal of Fuel Cell Scienceand Technology, 8, no. 1, 2011.

[86] K. G. Gallagher, B. S. Pivovar, and T. F. Fuller, Electro-osmosis and WaterUptake in Polymer Electrolytes in Equilibrium with Water Vapor at Low Tem-peratures, Journal of the Electrochemical Society, 156, no. 3, B330B338, 2009.

[87] E. L. Thompson, J. Jorne, W. B. Gu, and H. A. Gasteiger, PEM Fuel CellOperation at -20 Degrees C. II. Ice Formation Dynamics, Current Distribution,and Voltage Losses Within Electrodes, Journal of the Electrochemical Society,155, no. 9, B887B896, 2008.

[88] A. Z. Weber and J. Newman, Coupled Thermal and Water Management inPolymer Electrolyte Fuel Cells, J. Electrochem. Soc., 153, no. 12, A2205A2214,2006.

[89] C. Chacko, R. Ramasamy, S. Kim, M. Khandelwal, and M. Mench, Character-istic Behavior of Polymer Electrolyte Fuel Cell Resistance During Cold Start,Journal of the Electrochemical Society, 155, no. 11, B1145B1154, 2008.

[90] K. Tajiri, Y. Tabuchi, F. Kagami, S. Takahashi, K. Yoshizawa, and C. Y. Wang,Eects of Operating and Design Parameters on PEFC Cold Start, Journal ofPower Sources, 165, no. 1, 279286, 2007.

[91] T.E. Springer, T.A. Zawodzinski, and S. Gottesfeld, Polymer Electrolyte FuelCell Model, Journal of the Electrochemical Society, 138, no. 8, 23342341, 1991.

[92] Y. Ishikawa, H. Harnada, M. Uehara, and M. Shiozawa, Super-Cooled Wa-ter Behavior Inside Polymer Electrolyte Fuel Cell Cross-Section Below FreezingTemperature, Journal of Power Sources, 179, no. 2, 547552, 2008.

[93] T. Dursch, C. J. Radke, and A. Z. Weber, Ice Formation in Gas-Diusion Lay-ers, ECS Transactions, 33, no. 1, 11431150, 2010.

144

[94] E.L. Thompson, T.W. Capehart, T.J. Fuller, and J. Jorne, Investigation of Low-Temperature Proton Transport in Naon Using Direct Current Conductivity andDierential Scanning Calorimetry, Journal of the Electrochemical Society, 153,no. 12, A2351A2362, 2006.

[95] A. Z. Weber, Improved Modeling and Understanding of Diusion-Media Wetta-bility on Polymer-Electrolyte-Fuel-Cell Performance, Journal of Power Sources,195, no. 16, 52925304, 2010.

[96] M. K. Debe, A. K. Schmoeckel, G. D. Vernstrorn, and R. Atanasoski, HighVoltage Stability of Nanostructured Thin Film Catalysts for PEM Fuel Cells,Journal of Power Sources, 161, no. 2, 10021011, 2006.

[97] M. Sundaresan and R. M. Moore, PEM Fuel Cell Stack Cold Start ThermalModel, Fuel Cells, 5, no. 4, 476485, 2005.

[98] M. Khandelwal, S. H. Lee, and M. M. Mench, One-Dimensional Thermal Modelof Cold-Start in a Polymer Electrolyte Fuel Cell Stack, Journal of PowerSources, 172, 816830, 2007.

[99] T. W. Patterson, J. O'Neill, and M. L. Perry, Final Technical Report, 2006.PEM Fuel Cell Freeze Durability and Cold Start Project. Report DE-FG36-06GO86042., 2007.

145

Appendix A

Notation

Abbreviations

CH channel

CL catalyst layer

DM diusion media or medium

DOE United States Department of Energy

GC gas channel

GDL gas-diusion layer

HFR high-frequency resistance

HOR hydrogen oxidation reaction

IC internal combustion

M membrane

MEA membrane-electrode assembly

MPL microporous layer

ORR oxygen reduction reaction

P plate

PCI phase-change-induced

PEFC polymer-electrolyte fuel cell

PM porous media or medium

PSD pore-size distribution

RH relative humidity

Roman

a01,2 specic interfacial area, 1/cm

A heater factor

ci concentration of species i, mol/cm3

C constant

146

Cp,Ω heat capacity of subdomain Ω, J/cm2·KCp molar heat capacity, J/mol·KCp,α specic heat capacity of phase α, J/kg·KdCK Carman-Kozeny diameter, cm

Di Fickian diusion coecient for species i, cm2/s

Dij binary diusion coecient for species i in j, cm2/s

Dij Stefan-Maxwell binary diusion coecient, cm2/s

E energy, J/cm2

EW equivalent weight of membrane, g/equiv

fk fraction of pores in distribution k

F Faraday's constant, C/mole e−

h heat-transfer coecient, W/cm2·KHi (partial) molar enthalpy of species i, cm3/mol

Hi specic enthalpy of species i, J/g

∆Hij dierence in enthalpy between species i and j, J/g

i supercial current density, A/cm2

ih0 exchange current density for reaction h, A/cm2

i current density, A/cm2

k permeability, cm2

kCK Carman-Kozeny constant

kf rate constant for freezing, g/cm3·skr relative permeability

kT thermal conductivity of phase, W/cm·KKi generic (partial) molar property of species i

mi mass ow or production rate, g/cm2·sMi molar mass of species i, g/mol

ni mass ux of species i, g/cm2·spi partial pressure of species i, bar

pα total pressure of phase α, bar

P supercial power density, W/cm2

q heat ux, W/cm2

Q supercial heat ux, W/cm2

r pore radius, cm

r0,k characteristic radius for distribution k, cm

R ideal-gas constant, J/mol·K or radius, cm

RΩ ohmic resistance, Ω· cm2

147

Rk rate of freezing/melting, vaporization/condensation, orsublimation/deposition, g/cm3·s

RH relative humidity, %

sk characteristic spread for distribution k

Sα saturation of phase α

Sαβ saturation of phase α in a binary system of α and β

Si (partial) molar entropy of species i, J/mol·Kt time, s

T temperature, K

U open-circuit potential, V

UH enthalpy potential, V

vα velocity of phase α, cm/s

V cell potential, V

Vi (partial) molar volume of species i, cm3/mol

Vi specic volume of species i, cm3/g

x distance, cm

y distance, cm

yi mole fraction of i

z distance, cm

Greek

α phase, transfer coecient, or transport coecient, mol2/J·cm·sβ phase

γαβ surface energy between phases α and β, N/cm

δ thickness, cm

εα volume fraction of phase α

ε bulk porosity

η overpotential, V

θ contact angle,

κ ionic conductivity, S/cm

λ membrane water content, mol H2O/mol SO−3µi chemical potential of species i, J/mol, or dynamic viscosity, µPa·sξ electro-osmotic coecient

Πh Peltier coecient for reaction h, V

ρ density, g/cm3

σ electronic conductivity, S/cm

τα tortuosity of phase α

Φ potential, V

148

ωi mass fraction of species i

Ω subdomain

Subscripts and

Superscripts

0 reference value or membrane water

1 electronically conductive phase

2 ionically conductive phase

a anode or anodic

bdy boundary

c cathode or cathodic

C cold

chan channel

conv convective

cool coolant

di diusive

e eective

f freezing/melting

G gas phase

h reaction type or pore type (HI or HO)

htr heater

H hot

HI hydrophilic

HO hydrophobic

i species

I ice phase

init initial

j species

k pore-size-distribution number or pore type (HI or HO)

L liquid phase

lim limiting

M membrane phase

max maximum

min minimum

s sublimation/deposition or solid

sat saturated

t triple-point value

v vaporization/condensation

149

V water vapor or vapor-equilibrated value

θ standard value (1 bar and 298.15 K)

Ω subdomain

150

Documents

Modeling Cold Start in a Polymer-Electrolyte uelF Cell Ryan James … · 2018. 10. 10. · Modeling Cold Start in a Polymer-Electrolyte uelF Cell By Ryan James Balliet A dissertation