Model to Predict Temperature and Capillary Pressure Driven Water ...

Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲

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0013-4651/2009/156共6兲/B703/13/$25.00 © The Electrochemical Society

Model to Predict Temperature and Capillary Pressure Driven Water Transport in PEFCs After Shutdown Manish Khandelwal,a,* Sungho Lee,b and M. M. Mencha,**,z a

Fuel Cell Dynamics and Diagnostics Laboratory, Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Hyundai Motor Company, Research and Development Division, Yongin, 446-912 Korea

b

To enhance durability and cold-start performance of polymer electrolyte fuel cells 共PEFCs兲, residual water in the fuel cell components must be minimized during operation and after shutdown. A transient two-phase mathematical and computational model is developed to describe water redistribution in the PEFC components after shutdown, which for the first time includes thermo-osmotic flow in the membrane. The model accounts for capillary and phase-change induced flow in the porous media and thermo-osmotic and diffusive flow in the polymer membrane. In the porous media, liquid-water flow is dominated by capillary transport until irreducible saturation is achieved, after which water removal is dominated by phase-change induced flow. In the membrane, thermo-osmotic flow can significantly help or hinder water drainage from the catalyst layer, depending on the situation. During shutdown to the frozen state, residual water at the cathode can be controlled, and freeze damage can be avoided, through balancing the phase-change induced flux in the diffusion media with the net balance of thermo-osmosis and diffusion flux in the membrane. © 2009 The Electrochemical Society. 关DOI: 10.1149/1.3110742兴 All rights reserved. Manuscript submitted November 10, 2008; revised manuscript received February 9, 2009. Published April 14, 2009.

Fuel cell technology faces a number of technical challenges for automotive application that must be surmounted in order to compete against the internal combustion engine. Two key issues of fuel cell durability and cold-start performance have gained considerable attention.1-4 Apart from various design choices and operating conditions, one of the key parameters affecting durability and cold-start performance is the residual water redistribution in the fuel cell components. Recent studies performed by Mench and co-workers have shown that residual water plays a key role in freeze-damage4-7 and fuel cell cold-start ability.8-10 Khandelwal et al.8,9 have shown that if not purged, almost 15–20% of the total energy required to achieve successful cold start can be consumed in melting ice. Studies performed by Kim and Mench4 have identified various physical degradation modes and have shown that liquid water in contact with the catalyst layer 共CL兲 must be minimized to mitigate damage. Cho et al.11 showed that gas purge or solution purge can be used to reduce the residual water, preventing fuel cell degradation. Garzon et al.2 also investigated the impact of freeze–thaw cycling on fuel cell components and properties and recommended that a membrane water content of ⬃7–12 共corresponding to vapor-equilibrated membrane兲 represents an ideal situation for minimum freeze-damage. Neutron-imaging studies12-14 have shown images of liquid-water redistribution during normal operation and after fuel cell shutdown. Most of the imaging studies have focused on the design or material changes or operating condition variations. Currently, various purging methods are being utilized to minimize the residual water in the fuel cell. In practice, purge is traditionally restricted to a short duration due to the high parasitic energy requirement. Recently, various nonparasitic methods using temperature gradients15,16 have gained attention. Bradean et al.15 and Perry et al.16 have shown that a modest temperature gradient can induce significant liquid-water motion after shutdown. Perry et al.16 also observed that in some cases, the anode end cells can have substantially lower performance compared to cathode end cells due to the different direction of liquid-water transport during cool down to a frozen state. In spite of these experimental observations, fundamental understanding of this, and other nonparasitic modes of water drainage, is not yet completely understood. The current study is motivated by the need to understand the temperature-driven water-transport modes in polymer electrolyte

* Electrochemical Society Student Member. ** Electrochemical Society Active Member. z

E-mail: [email protected]

fuel cell 共PEFC兲 components and then identify the key parameters which control the liquid-water removal from the porous media 关including CL and diffusion media 共DM兲兴 into the gas channels. The objective of the present work is to develop a transient two-phase mathematical model to predict the water redistribution in the fuel cell components after PEFC shutdown. The novel feature of the model is inclusion of a directly measured thermo-osmosis relationship in the electrolyte membrane and the phase-change-related transport 共heat-pipe effect17兲 in the porous media, which is also recently termed as phase-change-induced 共PCI兲 transport.18 Ultimately, the developed model can be used as a tool to design guidelines for the material selection, to help understand water transport after PEFC shutdown, and to minimize the residual water in the fuel cell components. Temperature-Gradient-Driven Water Transport Modes in PEFC Components Membrane.— Thermo-osmosis in the polymer membrane is observed when a temperature difference exists across the membrane, even when the solutions on either side have the same species and concentration. The molar flux due to the thermal gradient nw,thermal can be represented as19 nw,thermal = − DT共T兲 ⵜ T

关1兴

where DT is the thermo-osmotic coefficient and depends on the hydraulic permeability, mean membrane temperature, and partial molar volume of the solvent.19,20 Recently, Kim and Mench21 presented directly measured rate laws for thermo-osmotic flux in several common PEFC membranes as a function of temperature and temperature gradient for fully liquid-saturated membranes. These relationships are used in the model presented here. A schematic of thermoosmosis phenomenon in a polymer membrane is shown in Fig. 1a. Positive thermo-osmotic diffusion coefficients imply water motion from the hot to cold side. Depending on the type of charged membrane, the thermo-osmotic coefficient may have positive or negative values. Tasaka et al.22 observed movement of water from the cold to hot side for a hydrophilic membrane and reversed flux for a hydrophobic membrane. They have also argued that the physical reason for this motion was based on the entropy change. Table I summarizes the results of direction of water transport for various polymeric membranes used in the fuel cell. To the best of the author’s knowledge, existing fuel cell modeling literature does not yet account for membrane thermo-osmosis. Porous media.— In porous media, water can be transported in both liquid and vapor phase. Under a temperature gradient, liquid

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water can evaporate at a hot location and move to a colder location in the vapor phase. The diffused vapor can condense as it moves along the temperature gradient due to the change in vapor pressure. This phenomenon has been called the heat-pipe effect,17,23 although the phenomenon is not strictly a heat pipe, because there is no capillary return of the condensed phase to the warm location. A more proper but cumbersome term is temperature-gradient-induced, phase-change-driven flow, as shown in Fig. 1b. Here, we refer to this phenomenon as PCI flow,18 with the understanding that it refers to phase-change-induced flow resulting from a temperature gradient, not just from humidity gradient. The phase-change-driven flow phenomenon has been studied by various researchers in the field of soil science over the past two decades23-26 and has recently gained attention in some fuel cell literature.15,17,18,27

TH TL liquid water

liquid water (a)

High temperature

Low temperature

Model Formulation

VAPOR

A schematic of the fuel cell and its components as well as and control volume for the model development is shown in Fig. 2. Conservation of mass and energy has been performed for each fuel cell component and combined with various interfacial boundary conditions31 as described.

Condensation

Evaporation (b)

Figure 1. 共Color online兲 Schematic of thermal-driven water-transport mode in PEFC component 共a兲 thermo-osmosis in electrolyte membrane.21,31 For hydrophilic membrane, liquid water moves from cold to hot side, and the direction is reversed for hydrophobic membrane. 共b兲 Temperature-gradient PCI flow in the porous media. Water is evaporated at the CL and condensed near the gas channel, schematic adapted from Ref. 17.

Model assumptions 1. The DM and CLs are considered to be porous media, and the polymer electrolyte membrane 共Nafion兲 is considered to be watersolvent. All material properties are homogeneous but can be anisotropic. Swelling of the membrane due to water uptake is neglected in the model formulation.

Table I. Summary of temperature-gradient-related water-flow observations in literature. Observation

Reference

Water flux 共kg/m2s兲

Water flow direction

3.57 ⫻ 10−6 共for ⌬T ⬇ 10°C, Tavg = 35.2°C H+-form兲 7 ⫻ 10−5 共for ⌬T ⬇ 10°C, Tavg = 26.5°C兲 2.8 ⫻ 10−6 共for ⌬T ⬇ 1–2°C for each cell兲

Cold to hot

Material 19,22,28

Tasaka et al.

Nafion 417

Villaluenga et al.20

Nafion 117

Bradean et al.15

Fuel cell stack

Perry et al.16

Fuel cell

Not available

Hot to cold

Zaffou et al.29,30

Nonreinforced membrane 共Nafion 112, catalyzed Nafion 112兲 Teflon reinforced membrane and catalyzed Gore select membrane 共5510 series兲 Membrane only 共Nafion 117, Flemion SH50, Gore Select兲

—

No flow

—

Hot to cold

1.69 ⫻ 10−4 共N117兲, 2.55 ⫻ 10−4共SH 50兲, 3.29 ⫻ 10−4 共Gore兲共for ⌬T ⬇ 10°C, Tavg = 65°C兲 Flux depends on hydrophobicity of the DM

Cold to hot

Kim and Mench21

MEA and MEA with PTFE-reinforced DM and MPL 共both reinforced and nonreinforced兲

Cold to hot Hot to cold

Hot to cold

Comment Experiments performed with the membrane only. Experiments performed on a fuel cell stack. Details of MEA and DM are not available. Details of MEA are not available. Carbon cloth DM was used. Experiments performed in a fuel cell with SGL 10 BB DM.

Explicit experiments were performed to characterize the thermo-osmosis in membrane and heat-pipe effect in DM.


Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲 Bipolar plate Diffusion media Catalyst layer Membrane

Y Z

␧␳w

X

Domain A Domain B

⳵sw + ⵜ 共␳wuw兲 = Sm,w ⳵t

冉冊

uw = − krw

Figure 2. Schematic and control volume for the computational model. Computational domains with both land 共Domain A兲 and channel 共Domain B兲 boundary conditions are shown.

2. Gravity effects can be neglected, compared to the capillary effects in the porous media, because through scale analysis, ␳g共4␲r3 /3兲 Ⰶ 共2␴/r兲共␲r2兲 for the porous materials in PEFCs.7 The multiphase flow in porous media is treated as immiscible, so all continuous phases flow simultaneously through their own tortuous path in the available pore space. 3. The polymer electrolyte membrane is treated with a pure diffusion model. The diffusion model is chosen based on the availability of experimental water-diffusivity data. The water content of the membrane, ␭, is defined as the number of moles of water associated with a mole of sulfonic acid group in the membrane H2O/SO−3 . By using the gradient of ␭ as the driving force, the diffusion model can be used for the whole range of 0 ⬍ ␭ ⬍ ␭0 = 22. Actually, if the Nafion can be considered as a porous media with free water at a high water content 共i.e., 22 ⬎ ␭ ⬎ 14兲, through the capillary relation for a two-phase flow in porous media, we can show ⵜpw ⬇ 共 ⳵ pw /⳵sw兲 ⵜ sw ⬇ ⵜ ␭. Thus, the hydraulic-type model can be simplified to a diffusion-type model, providing unification of the two approaches.7 When the Nafion is fully saturated 共␭ = 14兲, the hydraulic-type model is actually the same as a single-phase flow in porous media. 4. In the porous media, water vapor and liquid water are considered to be in thermal equilibrium at the interface. The vapor pressure inside the pore is approximated as the same value as the interface pressure. 5. Vapor pressure in the porous media is small compared to the total pressure; therefore, total gas-phase pressure is approximated as the air pressure. 6. Total gas-phase pressure 共air pressure兲 in the porous media is considered to be constant. Influence of slight movement of air due to the diffusion of water vapor within the voids is also neglected. Diffusion of water vapor is described by Fick’s law. 7. The temperature-gradient-related phase-change transport is modeled using pure thermodynamics with no consideration of surface property effects. Considering the rough and partially hydrophobic nature of surface involved, there may be some influence of the surface properties that is yet unexplored.

Mass/flow equation.— The mass/flow equation is different between the porous media and Nafion membrane. For the porous media, the saturation relationship can be written as 关2兴

where sw and sg are the saturation of liquid water and gas phase, respectively. The saturation of the phase is defined as the ratio of volume occupied by the phase to the total available pore volume. From immiscible flow theory, the flow equation for liquid water can be written as32

关3兴

where ␧ is the porosity, ␳w is the liquid-water density, uw is the liquid-water velocity, and Sm,w is the mass-source term. The masssource term here is due to the vaporization/condensation to the water-vapor 共gas兲 phase. The water velocity in the porous media 共DM/CL兲 can be written from Darcy’s law as32

Gas channel

sw + sg = 1

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k ⵜ pw ␮w

关4兴

where krw is the relative permeability of the water phase, k is the permeability of the porous media, ␮w is the viscosity of water, and pw is the water-phase pressure. The relative permeability of the water phase is usually a function of water saturation and has a typical n , where n can vary from 1 to 9, as used by various form of krw = sw 5,33-36 authors, although a value of 3.0 is most commonly applied for fuel cells. In the present study, the value of 3.0 is used. Substituting the liquid-water velocity from Eq. 4 in Eq. 3, and simplifying the combined equation, we can show ␧␳w

⳵sw − ␳w ⵜ ⳵t

冋冉冊册 krw

k ⵜ pw = Scond ␮w

关5兴

where Scond represents the liquid-water generation due to the condensation. Similar to the liquid phase, the flow equation for the gas phase can be written as ␧␳g

⳵sg + ⵜ 共␳gug兲 = Sm,g ⳵t

关6兴

where all the symbols are defined in the List of Symbols. The gasphase mass flux consists of pressure-driven Darcy flow and watervapor diffusion flux due to the phase-change-driven flow under a temperature gradient. As the gas-phase pressure is assumed to be constant, the Darcy flow is negligible. Hence, the gas-phase mass flux is due to the vapor-phase 共water vapor兲 diffusion only. The vapor-phase diffusion can be written from Fick’s law37 and can be simplified to obtain the mass-flux expression24,25 as ␳ gu g = D

冉冊

M w ⳵ Psat ⵜ T = DgT共T兲 ⵜ T R ⳵T T

关7兴

where D is binary diffusion coefficient of water vapor in the air, M w is the molecular weight of water, R is the universal gas constant, and Psat is the water saturation pressure. The water-vapor diffusion coefficient can also be modified for the DM/CL tortuosity using Bruggman’s relation,a D = D0共␧兲1.5, where ␧ is the porous media porosity. Substituting the gas phase velocity from Eq. 7 in Eq. 6 and simplifying the resulting gas-flow equation ␧␳g

⳵sg + ⵜ 共DgT ⵜ T兲 = Svap ⳵t

关8兴

where DgT is the equivalent diffusion coefficient of water vapor in the gas phase and Svap is the vaporization source term. The developed liquid and gas phase equation can be further simplified using the saturation relation. By substituting Eq. 8 into Eq. 2 and adding Eq. 5 共note that Svap = −Scond兲 a

The Bruggman relation assumes that a discrete and continuous liquid and gas phase exist despite the level of saturation. This can overestimate the saturation level required to achieve a given restriction of gas-phase transport in various situations especially if a connected liquid film exists in the DM.



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␧共␳w − ␳g兲

⳵sw − ␳w ⵜ ⳵t

冋冉冊册 krw

flux. The governing equation for the water transport in the membrane can be written as

k ⵜ pw − ⵜ 共DgT ⵜ T兲 = 0 ␮w

⳵cw = − ⵜ nw = ⵜ 共D␭ ⵜ cw兲 + ⵜ 共DT ⵜ T兲 ⳵t

关9兴 Using the approximation ␳w Ⰷ ␳g, the governing equation for the flow in the porous media can be written as ␧␳w

⳵sw − ␳w ⵜ ⳵t

冋冉冊册 krw

k ⵜ pw − ⵜ 共DgT ⵜ T兲 = 0 关10兴 ␮w

In Eq. 10, the second and third terms represent the capillary transport and diffusion flux due to the phase-change effect, respectively. Note that in this formulation, the PCI flow is driven purely by thermodynamic saturation pressure gradients. It does not include surface-energy effects that may also play an important role but have yet to be investigated in fuel cell components. Future study is aimed at elucidating the impact of surface properties on the expected phase-change-related flux. An additional characteristic equation is required to estimate the liquid-water pressure and saturation 共to solve Eq. 10兲. The capillary pressure relation between the gas-phase pressure and water-phase pressure is32 pc = pg − pw = ␴ cos ␪冑␧/kJ共se兲

关11兴

where ␴ is the surface tension of the air–water interface, ␪ is the contact angle, se is the effective saturation 关共sw–sirr兲/共1–sirr兲兴, and sirr is the irreducible saturation. The most widely used J-Leverett function, J共se兲,23 in the fuel cell literature7,33-35 is

J共se兲 =

再

pc = 共293/T兲6␥共T兲20.4C

冑

The molar water concentration satisfies the relation43,45 cw = ␭wcacid, where ␭w is the membrane water content. The ratio of membrane density to its electronic weight, ␳m /EW, is cacid. Equation 15 can therefore be simplified as

冉冊

⳵␭w EW ⵜ 共DT ⵜ T兲 = ⵜ 共D␭ ⵜ ␭w兲 + ⳵t ␳m

␧C K共snw兲 k

Energy equation.— The energy equation may have different forms in various fuel cell components. For the porous media, the generalized energy equation can be written as D共␳CpT兲 ⵮ = ⵜ 共k ⵜ T兲 + Qph Dt

2 K共snw兲 = wt % PTFE共0.0469 − 0.00152 共wt %兲 − 0.0406snw 3 + 0.143snw 兲 + 0.0561 ln snw

␪ ⬎ 90 共hydrophobic兲

nw = − D␭ ⵜ cw − DT ⵜ T

关14兴

where D␭ is the membrane water-diffusion coefficient, cw is the molar water concentration, and DT is the membrane thermo-osmotic diffusion coefficient. In Eq. 14, the first term on the right side represents the diffusion flux and the second term the thermo-osmotic

冎

关12兴

change in the porous media. Neglecting the advection transport due to the vapor diffusion, Eq. 17 can be simplified as

⳵共␳CPT兲 + ␳wCp,w ⵜ 共uwT兲 = ⵜ 共k ⵜ T兲 + SvapHgl ⳵t

关18兴

where Hgl is the latent heat for the vaporization of water. Thermal transport properties for the DM can be obtained by volume averaging ␳CP = 共␳wCP,wsw + ␳gCP,gsg兲␧ + ␳pmCP,pm共1 − ␧兲

关13b兴

where wt % is the percentage of polytetrafluoroethylene 共PTFE兲 content, snw is the non-wetting phase saturation, and C is the compression/stress relationship. Some authors35,41 have combined both equations 共Eq. 10 and 11兲, i.e., substituting the liquid-pressure derivative from Eq. 11 in Eq. 10, to obtain a single governing equation in terms of sw and its gradient. However, there can be some numerical issues associated with this methodology, as according to a purely capillary-flow-based formulation, a saturation discontinuity 共saturation jump兲 exists at the CL兩DM, CL兩microporous layer 共MPL兲, or MPL兩DM interface.36,41,42 In the current model, Eq. 10 and 11 are solved simultaneously for pw and sw. For the Nafion domain, water transport is modeled with a pure diffusion model,43-45 including thermo-osmosis.19,20 The molar water flux in the membrane can be written as

关17兴

where ␳, Cp, and k are porous media density, specific heat, and thermal conductivity. In Eq. 17, the left term represents the rate of change of energy in the control volume 共i.e., thermal energy storage兲, the first term on the right is the conduction heat transfer, and the last term represents the heat released/absorbed due to the phase

1.417se − 2.120s2e + 1.263s3e

where snw ⬍ 0.5 关13a兴

关16兴

With the appropriate boundary and initial condition, Eq. 16 can be solved for ␭w in the membrane.

1.417共1 − se兲 − 2.120共1 − se兲2 + 1.263共1 − se兲3 ␪ ⬍ 90 共hydrophilic兲

A recent study performed by Kumbur et al.36 has shown that the above form of the Leverett function is not entirely suitable for the fuel cell DM. An alternative experimentally measured capillary pressure relationship is given as38-40

关15兴

关19兴

Using scale analysis, it can be shown that the transient term in Eq. 8 is negligible compared to the diffusion term for a porous media.24,25 Therefore, Eq. 18 can be rewritten as

⳵共␳CPT兲 + ␳wCp,w ⵜ 共uwT兲 = ⵜ 共k ⵜ T兲 + 关ⵜ共DgT ⵜ T兲兴Hgl ⳵t 关20兴 ⇒

⳵共␳CPT兲 + ␳wCp,w ⵜ 共uwT兲 = ⵜ 关共k + DgTHgl兲 ⵜ T兴 ⳵t = ⵜ 共knet ⵜ T兲

关21兴

Here, knet is the net thermal conductivity, which includes conduction heat transfer and heat transfer due to the PCI flow. From the formulation, it should be noted that the phase-change effect contributes in enhancing the heat removal from the fuel cell components toward the gas channel 共cold side兲 and will play a role as long as saturation is above zero and there is a temperature gradient in the DM. For the electrolyte domain, conduction is the dominant mode of thermal transport. In the absence of any heat generation from phase change, the energy equation can be written as



⳵共␳CpT兲 + ␳wCp,w ⵜ 共uwT兲 = ⵜ 共k ⵜ T兲 ⳵t

关22兴

where all symbols are defined in the List of Symbols. The membrane thermal properties can also be evaluated as a mixture of dry membrane and water by volume averaging.7 Boundary and interface conditions.— The net mass flux in the porous media can be written as

冉冊

˙ ⬙ = ␳wuw + ␳gug = − ␳w krw m

k ⵜ pw − DgT ⵜ T ␮w

关23兴

In the DM, CL, and membrane, both water and thermal transport equations are solved. The boundary/interface conditions need to be specified in the thru-plane direction 共x direction兲 only. Boundaries in other directions are symmetric; hence, all gradients on these boundaries are zero. For the water transport, the boundary conditions for the DM are as follows ˙⬙ = 0 m

关24a兴

pw = f关Pch共t兲,T兴

关24b兴

at the DM兩BP at the DM兩GC

For an ideal scenario, the capillary pressure at the DM/gas-channel 共DM兩GC兲 interface should be zero,7 which facilitates maximum capillary drainage. During shutdown, water droplets may emerge and reside at the DM兩GC interface, which can create some water pressure in the stationary droplet, suppressing additional droplet growth. However, the functional form of the water pressure at the DM兩GC boundary is difficult to know. The boundary condition used here therefore simulates the maximum water removal during shutdown, ignoring the impact of DM surface droplet formation. At the other porous material interfaces 共DM兩CL, DM兩MPL, MPL兩CL兲, mass flux 共Eq. 23兲 and water-pressure continuity is imposed. At the membrane兩CL interface, mass continuity and the following condition for water content is imposed:7 water content

␭w = 14 + 8sw

关25兴

The interface condition for water content can also be set to include the effect of Schroeder’s paradox, which is explained by Zawodzinski et al. and others.46-48 For this case, ␭w = 22 when the CL is equilibrated with liquid water 共sw ⬎ 0兲, and ␭w = 14 when sw = 0. For thermal transport, a constant temperature or convective boundary condition can be specified at the DM boundary 共at both land/channel case兲. For all other interfaces in the computational domain, temperature and flux continuity is imposed. Properties and parameters.— Details of all modeling and material parameters are listed in Table II. Reported values for DM watertransport properties are limited or vary to a large degree. For example, reported values for DM permeability range from 10−11 to 10−15 m2,5,33,35,49-51 and no direct experimental measurement has yet been shown to demonstrate irreducible saturation in fuel cell DM. Of course, these properties vary from one DM type to another and can change the quantitative values in the results shown in this article. However, qualitative trends in the results shown are similar, and the range of parametric values chosen are in a reasonable range. Numerical implementation and solution technique.— Based on the analytical formulation described, a numerical code was developed and one-dimensional simulations were performed. In the numerical code, either a channel or a land boundary condition can be specified to investigate the water transport. The control-volume method was used to discretize the governing equation in the computational domain. A single-domain approach was used to solve the energy equation and water transport in porous media. Numerical equations were derived in terms of interface temperature/pressure and were obtained by integrating the governing equation over the control volume.52 For the time marching, an implicit method along

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Table II. Properties and parameters for the model simulation. Symbol

Parameter

tDM tCL tm kp,DM

DM thickness CL thickness Membrane thickness DM permeability

kp,CL kDM kCL km

CL permeability DM thermal conductivity CL thermal conductivity Membrane thermal conductivity DM density Membrane density DM specific heat Membrane specific heat DM porosity CL porosity Irreducible saturation Membrane water diffusion coefficient Membrane thermo-osmotic diffusion coefficient

␳DM ␳m Cp,DM Cp,m ␧DM ␧CL sirr D␭ DT Mw EW Hgl

Value

Unit

400 10 20 3 ⫻ 10−12 共Ref. 49 and 50兲 3 ⫻ 10−14 共Ref. 5兲 0.4 共Ref. 59兲 0.2 共Ref. 59兲 0.2 共Ref. 59兲

␮m ␮m ␮m m2

450 共Ref. 60兲 1980 共Ref. 44兲 710 共Ref. 61兲 1170 0.8 0.3 0.35, assumed 2 ⫻ 10−10 共Ref. 62兲 −1.04 ⫻ 10−3 e共−2297.3/T兲共Ref. 21兲 0.018

m2 W/mK W/mK W/mK kg/m3 kg/m3 J/kgK J/kgK — — — m2 /s mol/msK

Molecular weight of kg/mol water vapor Membrane equivalent weight 1.1 kg/mol Latent heat of vaporization 2260 ⫻ 103 共Ref. 61兲 J/kg

with a Newton–Raphson method was used to solve the numerical equations. Numerous test cases were also developed to verify the each term 共transient, diffusion, and advection term兲 in the final governing equations. Time and grid-sensitivity analysis were also performed to ensure that the solution is independent of chosen grid size and time step. Details of numerical implementation, verification study, and grid-sensitivity analysis are presented elsewhere.53 Result and Discussion The direction of water flow after shutdown strongly depends on the temperature gradient inside the fuel cell.15,16 After shutdown, water transport is governed by capillary and phase-change mechanisms in DM and CL and diffusion and thermo-osmosis in the membrane. Each water-transport mechanism was investigated with a different computational domain in the following section. The first subsection presents investigation of water transport in the DM only, and additional simulations described later in this article are presented on a full membrane electrode assembly 共MEA兲. PCI Flow in DM.— To investigate the characteristics of phase change and temperature gradient on water and thermal transport in DM, the computational domain was changed to a four-layer sandwiched DM as shown in Fig. 3. All simulations are presented with constant temperature and land boundary condition with TL = Tavg + ⌬T/2, TR = Tavg − ⌬T/2, where ⌬T is the temperature across the DM. Initial temperature and saturation is specified as value of TL and 0.2 unless otherwise specified. Comparison of water transport due to capillary flow and PCI flow.— To numerically investigate the relative contributions of capillary and PCI flux of water in the DM, simulations were performed with initial saturation of 0.5 共⬎sirr = 0.35兲, Tavg = 75°C, and ⌬T = 10°C

Constant Temperature (TL)

DM1

DM2

DM3

DM4

Constant Temperature (TR)

Figure 3. Computational domain to investigate the PCI effect in DM 共fourlayer sandwiched structured with land boundary condition兲.


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Connected liquid water

Pore cross-section Figure 4. 共Color online兲 Steady-state liquid-saturation distribution in fourlayer sandwiched DM for different DM permeability 共showing balance of capillary and PCI flux兲, Tavg = 75°C, ⌬T = 10°C, initial liquid saturation swi = 0.5.

with varying DM permeability. Specifying initial liquid saturation higher than the irreducible saturation with a land boundary condition provides an opportunity to investigate both modes of water transport. Capillary pressure 共Eq. 11兲 reduces to zero for liquid saturation below the irreducible saturation,32 implying, as expected, that capillary transport ceases as soon as saturation falls below the irreducible value. Figure 4 shows the steady-state saturation profile and temperature distribution in the domain for DM permeability values matching the range found in various literature. The liquid water evaporates at higher temperatures 共on left兲 and diffuses and condenses at lower temperatures 共right side兲 due to a concentration gradient 共resulting from a saturation pressure gradient兲 in the vapor phase. Due to the liquid-saturation difference, a backward capillary flow is also developed. At steady state 共as simulated and shown in Fig. 4兲, a balance is achieved between forward PCI flux and opposing capillary flux. The maximum variation in saturation is less than 0.001 for all permeability values, implying that the change in saturation due to phase change in the presence of capillary transport is negligible. Thus, in the presence of capillary transport, the water flux due to PCI flux can be neglected. This observation is consistent with the scale analysis performed by Wang and Wang54 and Cho.55 In this simulation with land boundary, capillary transport was initiated by the saturation gradient formed due to PCI flux. For the same initial condition with a channel boundary, water drains from the DM into the channel due to capillary transport to the irreducible saturation value, followed by subsequent drainage by phase-change flux into the colder channel. This drainage rate strongly depends on the DM property, relative permeability relation, and initial saturation, as discussed by He et al.5 Water drainage in porous DM can be divided into two regimes,26,56 The first is dominated by capillary transport 共funicular regime56兲 and the second by vapor diffusion 共pendular regime26兲, as shown in Fig. 5. Extensive work has been done to observe both regimes in soil science literature.23,26,56-58 However, to the authors’ knowledge, no specific study has been performed in the fuel cell literature in the context of water transport in porous DM. After PEFC shutdown, water transport is initially dominated by capillary transport 共Fig. 5a兲, and saturation in DM decreases to the irreducible saturation, except where blocked by lands or channel-level droplets on the DM surface. The remaining residual water can only be removed by purge of residual channel droplets to allow further drainage to the irreducible value or evaporation26,55 共Fig. 5b兲. This evaporation can be achieved either by phase-change effect driven by temperature gradient or by passing purge gases in the gas channel

(a) Isolated water droplets

Pore cross-section

(b) Figure 5. Schematic showing the two different regimes of water transport in fuel cell DM and a pore cross section. 共a兲 Funicular regime: flow dominated by capillary transport and 共b兲 pendular regime: water can be removed only by evaporation or convective flow.

without a temperature gradient.57,58 The time required to change regimes from capillary-dominated to phase-change-dominated 共i.e., funicular to pendular兲, tcr, is important for fuel cell engineers and designers. Numerous studies to estimate tcr have been done in the soil science or drying of porous media literature.56 Computationally, the key controlling parameter is the relative permeability, krw, which is modeled as a power-law function of the liquid saturation. Depending on the power chosen, the capillary drainage time can vary by a large degree 共a few seconds to 10–15 min5兲. The tcr is also a strong function of initial saturation and DM properties. To estimate the critical time, two-dimensional simulation with both land and channel boundaries is required but is beyond the scope of the current work. Thermal transport due to PCI flux.— It has already been shown in the model formulation that the heat absorbed and released due to water evaporation/condensation can be combined with the conduction heat transfer term 共Eq. 21兲, significantly enhancing the heat transfer from the hot CL to the generally colder bipolar plate surface. The equivalent thermal conductivity due to the PCI effect can be written as



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Figure 6. 共Color online兲 Variation of equivalent thermal conductivity due to the PCI flux with the temperature in the porous DM. Thermal conductivity of various commercially available DM is also shown.

kequ = DgTHgl = D

冉冊

M w ⳵ Psat Hgl R ⳵T T

关26兴

where all the symbols are defined in the List of Symbols. The variation of equivalent thermal conductivity with temperature is shown in Fig. 6. It is evident from the plot that the equivalent conductivity changes from 0.1 to 0.9 W/mK with a change in the cell temperature from room temperature 共300 K兲 to a typical operating temperature of 353 K. The typical thermal conductivity of SGL series DM and CL is 0.2–0.5 W/mK.59 Therefore, the effective conduction heat transfer across the DM and CL can be increased up to 100–200% when phase-change flow is considered. Neglecting this mode of heat transfer in fuel cell shut down or normal operation may result in poor prediction of the cell-component temperatures. Effect of land/channel boundary conditions.— Figure 7 shows a transient profile of liquid-saturation distribution in the four-layered sandwiched DM with land boundary condition for two different temperature differences 共⌬T = 10 and 2°C兲. These two temperature differences provide two extreme cases 共⌬T = 0.5 and 2.5°C across each DM兲 to investigate the PCI effect in the DM. Initial liquid saturation of 0.2 was specified and average temperature was maintained at 75°C in both scenarios. With a land boundary condition, liquid water evaporates on the left side and diffuses toward the right 共due to lower temperature/concentration兲, and it can condense anywhere inside the DM or on the right wall. A schematic to explain the vapor diffusion and evaporation/condensation with land boundary is shown in Fig. 9a. As the configuration marches forward in time, liquid saturation on left side decreases to zero and a dry front moves toward the right. Condensed water on the right side increases the liquid saturation and eventually increases to a value greater than the irreducible saturation. As the liquid-saturation value reaches the irreducible saturation value, a dominating backward capillary flow is developed, and eventually, a steady-state condition can be achieved when evaporation and vapor diffusion toward the right are balanced by the backward capillary flux. It has already been shown that a small liquid-saturation difference above irreducible is sufficient to create sufficient capillary pressure to balance the vapor-diffusion flux due to PCI flux 共Fig. 4兲. In Fig. 7, a small liquid-saturation difference exists in the plateau region on the right side in the DM. Comparing Fig. 7a and b, it can also be observed that with a change in temperature gradient across DM, only the time scale of the diffusion process 共phase-change flux兲 changes. For ⌬T = 10°C, steadystate condition is achieved in almost 300 s as shown in Fig. 7a. However, for ⌬T = 2°C, steady-state condition was not achieved will 900 s 共Fig. 7b兲. At steady state, the liquid-saturation profile is

Figure 7. 共Color online兲 Transient liquid-saturation distribution in four-layer sandwiched DM at different times Tavg = 75°C, land boundary condition 共a兲 ⌬T = 10 and 共b兲 2°C.

the same for both cases. The time to reach steady-state condition can be dramatically different, depending on the temperature gradient. Figure 8 shows the same simulations as above 共Tavg = 75°C, ⌬T = 10 and 2°C兲, only with a channel boundary condition that allows drainage of the DM. Zero capillary pressure is specified at the channel boundary, simulating a channel DM surface free of droplets. Similar to the observation of the previous simulation of land boundary, water evaporates on left side of DM and diffuses toward the right side. Diffused vapor can condense in the DM or can escape the domain from the right side 共channel兲. Vapor does not diffuse from the left channel due to the elevated temperature specified at the left boundary. On the right boundary, vapor can diffuse due to saturation gradient and evaporation due to the heat flux 共because of constant temperature gradient兲. For ⌬T = 10°C, all the liquid water in the DM has been evaporated and diffused in the gas channel in less than 240 s, as shown in Fig. 8a. However, for ⌬T = 2°C, liquid water has been removed in a portion of the sandwiched DM as shown in Figure 8b. In Fig. 8a, a small kink between the dry 共sw = 0兲 and wet 共sw ⬎ 0兲 region may be due to computational effects. A schematic to explain the vapor diffusion and evaporation/condensation with channel boundary is shown in Fig. 9b. For the channel boundary, a dry front can move from either direction, and the rate of movement depends on the specified chan-


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Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲 Land: TL

Land: TH TH > TL

Left Boundary

Diffusion Flux

Center Node

Right Boundary

(a)

Channel: TH TH > TL

Channel: TL Diffusion Flux Evaporation due to temperature gradient

Left Boundary

Center Node

Right Boundary (b)

Figure 9. 共Color online兲 Schematic showing the evaporation/condensation and vapor diffusion in various nodes in the porous DM: 共a兲 land boundary and 共b兲 channel boundary.

Figure 8. 共Color online兲 Transient liquid-saturation distribution in four-layer sandwiched DM at different times Tavg = 75°C, channel boundary condition 共a兲 ⌬T = 10 and 共b兲 2°C.

nel boundary. In the current simulation, no convective evaporation 共gas purge, etc.兲 on the channel boundary condition is considered, i.e., liquid water can evaporate only because of the temperature gradient imposed on the computational domain. So, the drying front from the warmer side moves faster compared to the colder right side. If convective drying is used in the same scenario, the dry front from the right can also move faster, depending on the temperature, relative humidity, and flow of gas.63 For further details about convective drying of porous media, the reader can refer to Ref. 56-58. Comparing Fig. 8a and b, it can be observed that the rate of drying directly depends on the applied temperature gradient. The same observation was also made for simulation with land boundary condition. Based on results shown in Fig. 7 and 8, it can also be inferred that when specifying land or channel boundary for identical temperature conditions in one-dimensional simulation, the results have a similar qualitative trend. For the channel, water vapor can escape the control volume. So, there is a continual decrease in the average liquid-saturation value. For a land boundary, water vapor condenses and a backward capillary flow is developed. In a two-dimensional simulation or actual fuel cell, lateral capillary flow in the plane of the DM can be present as well. Another key aspect of the PCI flux phenomenon is the increase in the thermal transport due to the phase-change term. In the above simulations, a temperature profile at t = 10 s only has been shown

共Fig. 7 and 8兲. The temperature field is fully developed in the MEA and DM in less than 2–3 s due to the low thermal mass of fuel cell components. As time passes, the slope of the temperature profile changes in the dry domain 共sw = 0兲 compared to rest of the DM domain 共where sw ⬎ 0兲 due to the absence of a phase-change term in the drier region. The change in the slope affects the rate of vapor diffusion. Results shown so far for land and channel cases have identical temperature boundary conditions. In an actual fuel cell, the DM underneath the land is typically colder compared to the DM near the channel boundary. So, there should be a preferential location for condensation of diffused water vapors underneath the land as compared to the vapor diffusion in the channel. To validate these observations and qualitatively validate the model results, neutronradiography 共NR兲 imaging was used. In this study, the NR images are used as a qualitative comparison to validate the model trend. The NR imaging experiments were performed at the National Institute of Standards and Technology 共NIST兲 test facility, where a small fieldof-view 共2.5 cm2兲 high resolution 共⬃10 ␮m兲 neutron imaging system exists. Details of the test fixture and experimental methodology are presented elsewhere.27 To observe the PCI effect as simulated, four layers of commercial DM 共SGL series 10BB/10BA/10BA/ 10BB兲 were used for the experimentation. In a nonoperating condition, anode-side gas channels were filled with liquid water, and a temperature gradient was applied by two coolant recirculation baths.27 The anode bipolar plate 共left side in Fig. 10兲 was maintained at 70°C, and the cathode bipolar plate 共right side兲 was maintained at 60°C. MPLs were turned to be facing both channel sides to prevent liquid-water penetration into the DM 共breakthrough pressure is measured to be ⬃6 kPa兲 but allow vapor diffusion. Figure 10 shows the NR images at 1, 5, and 16 min after the start of the experiment. After 1 min of experimentation 共Fig. 10a兲, it can be clearly seen that water has condensed on the cathode side from the anode channel. The condensed water moved underneath the cathode land or began to drain inside the channel 共top/bottom wall of cathode gas channel兲. Without an imposed temperature gradient, no water transport across the cell was observed. This observation is quite similar to that predicted from the one-dimensional model. At t = 5 min, more water has condensed underneath the land and drained inside the cathode gas channel 共Fig. 10b兲. From Fig. 10c, it can be seen that the cathode gas channel is almost filled with liquid water after 16 min of the experiment. In the NR imaging experiments, the anode gas channel was always filled with liquid water, i.e., it has an infinite source of water to evaporate. However, in the numerical simulation, the amount of water is limited by the specified initial condition. Thus, no dry front movement can be observed from the NR imaging.


Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲 Anode 4 layered Cathode BP BP DM

70 ºC ∆T = 10 ºC

(a)

60 ºC

Liquid filled channel

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Condensed/ drained water

MPL on outside of 4 layer sandwiched DM

(b)

(c)

Figure 10. 共Color online兲 NR images showing the PCI effect across a fourlayer sandwiched DM: SGL series 10BB/10BA/10BA/10BB, Tavg = 65°C, ⌬T = 10°C, 共a兲 t = 1, 共b兲 5, and 共c兲 16 min.

Effect of average temperature.— Figure 11 shows the simulated liquid-saturation profile in four-layered sandwiched DM for different average temperatures 共with ⌬T = 10°C兲 at t = 60 and 180 s. In these simulations, constant temperature 共TL = Tavg + ⌬T/2, TR = Tavg − ⌬T/2兲 and land boundary conditions were specified. As the average temperature decreases, vapor diffusion decreases drastically. For example, at an average temperature of 15°C, a dry front has not yet started after 180 s. At 75°C, it almost reaches a steady-state condition 共balance of phase-change and capillary flux兲. In the PCI phenomenon, vapor diffusion is governed by the saturated concentration 共saturation pressure兲 gradient. Due to the high saturation pressure gradient at around 70–80°C as compared to room temperature,64 vapor diffusion 共and phase-change flux兲 is greater, for the same temperature gradient, at high average temperatures. Effect of DM thermal mass.— Figure 12 shows the liquid-saturation profile in the four layered sandwiched DM for different thermal mass after 10 s for Tavg = 75°C and ⌬T = 10°C. It can be observed from the plot that there is negligible change in the saturation profile with an increase in the DM thermal mass. This is because the temperature profile in the domain is fully developed in all three simulations within 2–3 s. Water transport here is governed by the temperature distribution, so no significant effect can be observed on the saturation profiles. Therefore, in practical applications, water motion by temperature-driven flux is controlled by coolant and bipolar-plate temperature during shutdown. Impact of thermo-osmotic flux.— To investigate the effect of thermo-osmosis on water transport, one-dimensional simulations were performed on full cell geometry. Simulation dimensions and other parameters are shown in Table II. The average temperature was maintained at 75°C, and two different ⌬T, 10 and 2°C, across the fuel cell were used. All simulations are presented with constant temperature and land boundary condition with TL = Tavg + ⌬T/2, TR = Tavg − ⌬T/2. It has already been shown that the trends for water transport with land and channel boundary conditions are similar. Moreover, with land boundary condition, it is relatively easy to monitor the water transport, as no vapor or liquid can escape the system. Initial temperature, liquid saturation in DM兩CL, and membrane water content is specified as the value of TL, 0.15, and 15.2, unless otherwise specified. Three different scenarios, positive thermo-osmotic diffusion coefficient 共water moves from hot to cold side兲, zero thermo-osmotic diffusion coefficient, and negative thermo-osmotic diffusion coefficient 共water moves from cold to hot

Figure 11. 共Color online兲 Liquid-saturation distribution in four-layer sandwiched DM with varying average temperature ⌬T = 10°C and land boundary condition 共a兲 after 60 s and 共b兲 after 180 s.

side, as is the normal case for fuel cell membranes27兲 are used to investigate the water transport in fuel cell subjected to constant temperature gradient. A typical range of thermo-osmotic diffusion coefficient is 10−6 to 10−7mol/msK,21 depending on the membrane average temperature. The expression shown in Table II is for a W. L. Gore membrane.21 Transient liquid-saturation/water-content distribution and temperature profile for a positive thermo-osmotic diffusion coefficient 共flow from hot to cold兲 is shown in Fig. 13a and b, respectively. For this simulation, the value of diffusion coefficient shown in Table II with positive sign is used. On the anode side, due to the PCI induced flux, water evaporates and condenses near the anode CL兩membrane interface, increasing the liquid saturation at the anode CL. However, on the cathode side, water evaporates from the cathode CL and condenses near the right land walls. So, the liquid saturation in the cathode CL decreases with time. Now, a concentration difference across the membrane is formed due to the PCI water flux on the anode/cathode side. A positive diffusion flux transports water from


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(a)

Figure 12. 共Color online兲 Liquid-saturation distribution in the four-layer sandwiched DM after 10 s with different thermal mass of DM.

the anode to the cathode side. At the same time, due to the positive thermo-osmotic diffusion coefficient, the membrane also transports water from anode CL layer to cathode CL. In summary, the direction of PCI water flux in DM兩CL and net water flux in membrane due to diffusion and thermo-osmosis are both in the positive direction. So, net water transported in PEFC would also be in the positive direction 共or from hot to cold兲, and eventually all the water would be drained from anode to cathode side 共or from hot to cold兲. Numerical simulation indicates that water from the anode side was drained in almost 50 s under the simulated conditions. Also, the PCI flux on the cathode side is less than the net membrane water flux 共both diffusion and thermo-osmotic flux兲. So, in this scenario, i.e., with thermo-osmotic flux running counter to that experimentally observed, the cathode CL is dried later than the other fuel cell components. From Fig. 13b, it can also be seen that the decrease in the net thermal conductivity in the dry region 共due to the absence of phasechange term兲 increases the temperature gradient in the dry zone. This may slightly increase the vapor diffusion in other fuel cell components for sw ⬎ 0. Figure 14a and b shows the transient liquid-saturation/watercontent profile and temperature distribution in a fuel cell without any thermo-osmotic diffusion coefficient. Similar to the previously discussed simulation, saturation on the anode CL increases, and saturation decreases on the cathode CL, both due to PCI flux. However, the water transport from anode to cathode is now limited only by diffusion across the membrane. In this scenario, diffusion flux is lower than the PCI water flux on either side. So, the membrane limits the rate of water transport from anode to cathode. This may lead to drying of the cathode CL, and transported water by membrane diffusion may evaporate instantaneously, keeping the cathode CL dry. As the net flux is still positive, all the water from the anode is eventually drained to the cathode under these conditions. The trend of transient temperature profile without thermo-osmosis is shown in Fig. 14b and is similar to the case with positive thermoosmotic coefficient 共shown in Fig. 13b兲. The temperature gradient is increased in the dry zone due to the decrease in thermal conductivity. Figure 15a and b shows the transient liquid-saturation/watercontent profile in a fuel cell with a negative thermo-osmotic diffusion coefficient for ⌬T = 10 and 2°C, respectively. Negative thermo-osmotic diffusion coefficients exist in PEFC membranes, as detailed by Kim and Mench.21 In both cases, Tavg is maintained at 75°C. With negative thermo-osmotic diffusion coefficient, the direction of thermo-osmotic flux in the membrane is from the cold to hot

(b)

Figure 13. 共Color online兲共a兲 Transient liquid-saturation and water-content distribution. 共b兲 Transient temperature distribution in the one-dimensional fuel cell simulation with positive thermo-osmotic diffusion coefficient, Tavg = 75°C, ⌬T = 10°C, land boundary condition.

side, i.e., from cathode CL to anode CL, as simulated here. However, diffusion flux is from the anode to the cathode side. So, the direction of net water flux in the membrane depends on the relative magnitude of diffusion flux 共positive direction兲 and thermo-osmotic flux 共negative direction兲. For a given membrane, the magnitude of these fluxes depends solely on a coupled balance between the concentration and temperature gradient effects, respectively. As the temperature field is developed in the domain, thermo-osmotic flux removes water at a constant rate from the cathode to the anode side of the membrane. However, due to PCI flux in DM兩CL, liquid saturation on the anode CL increases. On the cathode CL, the liquid saturation decreases with time. The magnitude of the diffusion flux increases as the concentration gradient increases across the membrane. Figure 15a shows a scenario 共⌬T = 10°C兲 when the diffusion flux cannot overcome the opposing thermo-osmotic flux due to high temperature gradient. In this case, as soon as the cathode CL is dried, no water can be transported from anode to cathode or vice versa. Thus, in this scenario the membrane becomes a barrier for water drainage, and anode and cathode DM/CL behave as two disjoint porous media domains. The thermo-osmotic fluxes deducted by Kim and Mench21 were for fully liquid-saturated membranes, and it is likely that for partially dry membranes, thermo-osmotic flux is reduced somewhat. Figure 15b shows another scenario 共⌬T = 2°C兲 when diffusion flux is able to overcome the opposing thermo-osmotic flux. It has already been shown the in the porous media 共DM兩CL兲 that decreasing the



(a)

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(a)

(b)

Figure 14. 共Color online兲共a兲 Transient liquid-saturation and water-content distribution. 共b兲 Transient temperature distribution in the one-dimensional fuel cell simulation with zero thermo-osmotic diffusion coefficient, Tavg = 75°C, ⌬T = 10°C, land boundary condition.

average temperature only slows the rate of water-vapor diffusion, implying that diffusion flux in the membrane can reach the same order as the previous scenario of ⌬T = 10°C. However, thermoosmotic flux in the membrane is drastically reduced due to the decrease in the temperature gradient across the membrane. As we can see from the plot, initially liquid saturation near the anode increases and then begins to decrease as water diffuses from the anode to the cathode side. At the same time, the liquid saturation at the cathode CL reduces to zero, as liquid water is transported to the anode CL by thermo-osmotic flux along with the water movement toward the land by PCI flux. Also, the net rate of water transport across the membrane 共from anode CL to cathode CL兲 is lower than the amount of water which can be removed from PCI effect at the cathode CL. Thus, the cathode CL remains dry. Another important aspect is that all the water from the anode cannot be transported to the cathode side. At steady state, a balance of diffusion flux and thermo-osmotic flux in the membrane is achieved and no more water can be transported. So, in this scenario, the cathode CL can achieve the dry condition, and the anode CL always has some residual water. The amount of residual water can be estimated by the flux balance. For example, in the ⌬T = 2°C scenario, thermo-osmotic flux is −2.65 ⫻ 10−4 kg/m2 s 共⌬Tmembrane ⬇ 0.2°C兲. From the diffusion flux relation D␭M w共␳m /EW兲 ⵜ ␭w and with sw = 0 共␭w = 14兲 on the cathode CL side, it can be found that residual liquid saturation on the anode side should be around 0.102. ⌬Tcell in a PEFC stack can vary

(b) Figure 15. 共Color online兲 Transient liquid-saturation and water-content distribution in the one-dimensional fuel cell simulation with negative thermoosmotic diffusion coefficient, Tavg = 75°C, land boundary condition 共a兲 ⌬T = 10 and 共b兲 2°C.

from 0 to 2°C depending on the cell location in the stack. But, this may only be true for the conventional stack design. However, with the change in the stack design 共e.g., Bradean et al.15兲 and shutdown protocol 共e.g., forced cooling on one end during shutdown65兲, temperature gradient across the cells can be substantially increased during PEFC stack shutdown. Thus, temperature-driven water drainage can be potentially used to minimize the residual water in the fuel cell components. A detailed study to investigate the impact of stack design on water drainage 共including thermal-driven water-transport mode兲 has been presented else where.65 A note on freeze/thaw damage mitigation.— Depending on the location of a particular fuel cell in a stack, the anode or cathode side may be the colder location during shutdown, and this scenario can exist. The goal is to fully drain the electrode at shutdown before a frozen condition is reached. Based on the competing mechanisms of water removal from the electrode, either a thermal or a masstransport solution can be achieved. If a temperature gradient is kept so that the electrode fully drains before cold conditions 共a high enough gradient to move water but not enough for thermo-osmotic flux to overcome diffusive drainage兲, or so that the electrode does not fill in the first place 共zero temperature gradient兲, proper shutdown can be achieved. As a mass-transport solution, the flow to the


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Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲 tative comparison with the results of neutron-imaging was also presented. Based on one-dimensional simulations performed, the following conclusions can be made.

Figure 16. 共Color online兲 Mass flux across fuel cell component for different temperature differences and water-content differences 共membrane thickness = 20 ␮m, DM thickness = 400 ␮m兲.

cold electrode can be restricted so that this fillage and drainage process does not occur in the first place. These solutions are believed to be those achieved by Ballard and UTC patents and publication,15,16 which described ways to eliminate freeze/thaw damage at anode end cells in stacks but did not explain the fundamental processes involved. With the development of new stack design and/or shutdown protocol, a modest temperature gradient can be maintained across fuel cells, particularly in a stack. A key to minimize the residual water at the cathode CL is to maintain the relative balance of PCI flux in DM and net balance of thermo-osmotic flux and diffusion flux in the membrane. Figure 16 shows the mass flux for all three modes of water transport for varying water content and temperature difference at average temperature of 75°C. Other properties used are shown in Table II. Such design plots can be helpful for the stack designer to compare the relative magnitude of thermal-driven water flux in fuel cell components to optimize the stack design or shutdown protocol. For example, if ⌬T = 0.5°C exists across the membrane, and thermo-osmotic and diffusion flux are in the opposite direction, then the membrane acts as a barrier to water transport for ⌬␭membrane ⬍ 2 共thermo-osmotic flux ⬎ diffusion flux兲. However, water can be transported if membrane water-content difference is higher than 2, i.e., ⌬␭membrane ⬎ 2 共diffusion flux ⬎ thermo-osmotic flux兲. Thus, by maintaining the temperature gradient across the fuel cell component, one can drastically enhance the water removal and minimize the residual water at the cathode CL. In this work, a two-phase model to analyze the thermal-driven flow in a fuel cell component was presented. In porous media, the model was based on capillary flow and thermodynamic phasechange effects, i.e., vapor evaporates and diffuses solely due to the saturation pressure gradient. However, in a real scenario, the kinetics of evaporation and condensation can be affected by surface properties, roughness, etc. Also, in the model formulation, vapor pressure in the gas phase cannot go below saturation pressure even in the dry zone 共sw = 0兲, thus limiting the drying of the membrane. Future research is needed to explore these factors.

1. PCI water flux is negligible compared to the water flux due to the capillary transport, as long as the saturation is above the irreducible level. In the porous DM, liquid-water flow is first dominated by capillary flux 共funicular regime兲 until the irreducible saturation is achieved and then dominated by vapor diffusion or phase-change flux 共pendular regime兲. 2. PCI flux can enhance the heat removal by 100–200% in the porous DM/CL. 3. PCI water flux increases significantly with an increase in average temperature and temperature gradient. However, the effect of DM兩CL thermal mass is negligible on the phase-change water flux. 4. Depending on the type of membrane 共hydrophobic/ hydrophilic兲, thermo-osmotic flux can assist or oppose the diffusion flux in the polymeric membrane 共i.e., positive or negative thermoosmotic diffusion coefficient兲. The membranes used in the PEFC have a negative thermo-osmotic diffusion coefficient; thus, water moves from cold to hot side due to thermo-osmosis. At certain conditions, especially at higher average temperature and gradient, thermo-osmotic flux can be significant and can form a barrier for water transport from anode to cathode side or vice versa. By controlling the PCI water flux in DM, and net balance of thermoosmotic and diffusion flux in the membrane, the liquid water on the cathode CL layer can be minimized to enhance the fuel cell freeze performance and durability. Acknowledgments This work was partially supported by the Advanced Technology Center, Research and Development Division for Hyundai Motor Company and Kia Motors Corporation. Additional funding was provided by NSF Career Award no. 0644811. The authors thank Dr. Soowhan Kim for providing the NR images and useful discussion about thermo-osmosis, and Dr. D. S. Hussey and Dr. D. L. Jacobson for assistance in using the NR facility at NIST. List of Symbols c Cp D EW Hgl J k kr M ˙⬙ m n p Psat Q⵮ R s S t T u Greek ␧ ␭ ␮ ␪ ␳ ␴

Conclusions In this work, a transient two-phase computational model was developed to understand the water redistribution in the PEFC component after shutdown. Temperature-gradient-driven flow via thermo-osmosis in the polymeric membrane and PCI water flux in porous DM were incorporated in the analytical formulation. A quali-

molar water concentration, mol/m3 specific heat, J/kg K diffusion coefficient, m2 /s, thermo-osmotic diffusion coefficient, mol/ms K equivalent weight, kg/mol latent heat of vaporization, J/kg Leverett function permeability, m2, thermal conductivity, W/mK relative permeability molecular weight, kg/mol mass flow rate, kg/m2 s molar water flux, mol/m2 s phase pressure, Pa saturation pressure, Pa volumetric heat source, W/m3 universal gas constant, 8.314 J/mol K saturation volumetric mass source term, kg/m3 s time, s temperature, K velocity, m/s

porosity water content in Nafion 共mol H2O/mol SO−3 兲 viscosity, Pa s contact angle, degree density, kg/m3 surface tension, N/m

Subscript avg average


Journal of The Electrochemical Society, 156 共6兲 B703-B715 共2009兲 c CL DM e g irr m nw ph pm vap/cond w

capillary catalyst layer diffusion media effective gas phase irreducible membrane nonwetting phase change porous media vaporization/condensation liquid-water phase, water in Nafion

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