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Journal of Power

Sources

j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /j p o w s o u

r

Characterization of flooding and two-phase flow in polymer electrolyte membrane fuel cell stacks

G.Karimi a ,∗,F.Jafarpour a ,X.Li b

a Department of Chemical and Petroleum Engineering,Shiraz University,Shiraz 7134851154,Iran

b

Department of Mechanical and Mechatronics Engineering,University of Waterloo,Waterloo,Ontario,Canada N2L 3G1

a r t i c l e i n f o Article history:

Received 23May 2008Received in revised form 14September 2008

Accepted 19October 2008

Available online 5November 2008Keywords:PEM fuel cell stack

Water management Flooding;

Flow network analysis Mathematical modeling

a b s t r a c t

A partially flooded gas diffusion layer (GDL)model is proposed and solved simultaneously with a stack flow network model to estimate the operating conditions under which water flooding could be initiated in a polymer electrolyte membrane (PEM)fuel cell stack.The models were applied to the cathode side of a stack,which is more sensitive to the inception of GDL flooding and/or flow channel two-phase flow.The model can predict the stack performance in terms of pressure,species concentrations,GDL flooding and quality distributions in the flow fields as well as the geometrical specifications of the PEM fuel cell stack.The simulation results have revealed that under certain operating conditions,the GDL is fully flooded and the quality is lower than one for parts of the stack flow fields.Effects of current density,operating pressure,and level of inlet humidity on flooding are investigated.

©2008Elsevier B.V.All rights reserved.

1.Introduction

Polymer electrolyte membrane (PEM)fuel cells convert the chemical energy of hydrogen and oxygen directly and efficiently into electrical energy and are widely regarded as an alternative power source for stationary co-generation units,automotive and portable applications [1,2].The main characteristics of PEM fuel cells are classified as (a)those that produce water as a byprod-uct;(b)those having higher efficiency when compared with heat engines;(c)those that operate at low temperatures (up to 90◦C)allowing a fast start-up;and (d)those using a solid polymer as the electrolyte,which reduces concerns related to construction,transportation,and safety.

In a PEM fuel cell,hydrogen and oxygen react electrochemically to water,producing electricity and heat.For the proper opera-tion of a PEM fuel cell,both thermal and water management are critical to prevent the fuel cell system from overheating and performance deterioration [3].If there is not enough water,the membrane becomes dehydrated and its resistance to proton con-duction increases sharply [4].On the other hand,if too much water

∗Corresponding author.Tel.:+987112303071;fax:+9871174619.E-mail addresses:ghkarimi@shirazu.ac.ir ,gkarimi@engmail.uwaterloo.ca (G.Karimi).

is present,flooding may occur,and the pores of the gas diffusion layer,GDL,may be filled by liquid water,blocking the transport of reactants to the reaction sites,resulting in a serious performance drop,particularly at high current densities [5,6].

The problem of water management and transport in PEM fuel cells has been the subject of several theoretical and experimental studies.Bernardi [7]was the first to propose a one-dimensional model in order to study water management and to identify the humidification conditions which induce either the dehydration of the membrane or excessive flooding.Some other models were derived from the first principle,taking into account heat man-agement [8],mass transport in the gas diffusion electrode [9],or to introduce a different treatment of the electrochemical reaction [10].Bernardi et al.[11]and Springer et al.[12]also presented a one-dimensional model to investigate the factors that limit cell per-formance and to elucidate the mass transport mechanism within the complex network of gas,liquid and solid phases constituting the gas diffusion electrode.The molar changes along the gas flow channels are taken into consideration.Shimpalee et al.[13]investi-gated the performance of a 200-cm 2PEM fuel cell with serpentine flow fields at various gas flow path lengths in terms of distributions in the local temperature,water content,and current density.They concluded that the shorter path length gives more uniform cur-rent density distribution and less condensed liquid water than the longer path.Liu et al.[14]studied membrane hydration and elec-

0378-7753/$–see front matter ©2008Elsevier B.V.All rights reserved.doi:10.1016/j.jpowsour.2008.10.108

Nomenclature

Nomenclature

A cross-sectional area(m2)

C molar concentration(mol m−3)

C f wall friction coefficient

d hflow channel hydraulic diameter(m)

D diffusion coefficient(m2s−1)

D h manifold hydraulic diameter(m)

F Faraday constant(985C mole−1)

h mass transfer coefficient(m s−1)

H bipolar plate effective height(m)

J cell current density(A m−2)

lflow channel length(m)

L gas diffusion layer thickness(m)

m mass(kg)

N number of cells/loops/inlets/outlets/flow chan-nels/turns

˙N molarflow rate(mol s−1)

˙N molarflux(mol m−2s−1)

P pressure(Pa)

Re Reynolds number

RH relative humidity(%)

Sh Sherwood number

S stoichiometry

T stack temperature(K)

V average velocity(m s−1)

W bipolar plate effective width(m)

x thermodynamic quality

Greek letters

ıthickness(m)

difference

fraction offlooded GDL

Âflow direction convention

electroosmotic drag coefficient

viscosity(N s m−2)

density(kg m−3)

porosity(%)

Subscripts/superscrpts

BD back diffusion

c cathode/flow channel

cell fuel cell

CV control volume

eff effective

fflooded

EO electroosmotic drag

G gas

GDL gas diffusion layer

Gen generation

i loop number

in inlet

inlets inlet number(s)

j segment number

loop loop

L liquid

min minimum

Max maximum

outlet outlet number

Out outlet

stack stack

total total

TP two-phase

w water

b bulk

w-g water–gas

trodeflooding by developing a2D partialflooding model in which size distributions are assigned for the hydrophobic and hydrophilic pores of the GDL.The liquid water produced is considered to con-dense in hydrophilic and hydrophobic pores in sequence if the water vapor pressure is higher than the condensation pressure for the pores.The model results,under a wide range of operating con-ditions,have shown reasonable agreement with the experimental data.Experimentally,two-phaseflow and transport of reactants and products in the cathode of a transparent PEM fuel cell were studied by Tuber et al.[15].Images of water formed inside the cathode gas channels are presented to explain the phenomenon of waterflooding.Effects of air stoichiometry,temperature,air humid-ity and different characteristics of diffusion layers are discussed. Liu et al.[16]and Weng et al.[17]studied waterflooding and two-phaseflow in cathodeflow channels with differentflow paths using transparent PEM fuel cells.Their experimental results indicated the significant effects offlow channel patterns and cathode gas stoichiometry on water removal efficiency and cell performance.

Almost all of the research conducted to studyflooding in PEM fuel cells experimentally or theoretically,are limited to a single cell. The goal of this research is to add to the knowledge base to produce generic design guidelines for operating conditions andflowfields that can be applied to PEM fuel cell stacks,consisting of a more practical number of cells.It is assumed that the development of these design techniques could be a useful tool for the improvement of water management,and shed further light on its effect on fuel cell stack performance.To this end,a partiallyflooded GDL model is proposed and solved simultaneously with a stackflow network model to estimate the design and operating conditions under which waterflooding could be initiated in a PEM fuel cell stack.The model was applied to the cathode side of a stack,which is more sensitive to the inception of GDLflooding and/orflow channel two-phase flow.The model presented here is an extended version of a recent model[18,19]which can predict the stack performance in terms of pressure,species concentrations,GDLflooding,and quality distri-butions in theflowfields as well as the geometrical specifications of the PEM fuel cell stack.

2.Model formulation

A schematic diagram of a typical PEM fuel cell stack is shown in Fig.1.The individual cells,referred to as membrane electrode assemblies(MEAs),are composed of a membrane electrolyte sand-wiched in the middle of the cell,and typically contains catalyst and microporous gas diffusion layers along with gaskets as a sin-gle integrated unit.One of the gas diffusion layers is referred to as the anode,the other as the cathode.The catalyst layer at the anode separates hydrogen molecules into protons and electrons. The membrane permits ion transfer(protons),requiring the elec-trons toflow through an external circuit before recombining with protons and oxygen at the cathode to form water.This migration of electrons produces useful work.

In practice,oxygen,pure or in air,enters the cathode side of the stack through the main inlet(s),travels along the inlet manifold and is distributed into theflow channels orfields.From theflow fields,O2diffuses through the GDL towards the cathode–membrane interface where it is reduced to form water and heat which are

158G.Karimi et al./Journal of Power Sources 187(2009)

156–1

Fig.1.Schematic of a PEM fuel cell stack.

then removed from the system.Water can also be added to the cathode side due to the electroosmotic drag and be transferred from the cathode to the anode due to back diffusion.The simultaneous water and oxygen transfer,in conjunction with significant pressure variations in the cathode bipolar plate flow channels can lead to situations where liquid water flood the GDL,partially or completely,and even deposit on the flow fields.This phenomenon hinders the O 2transport to the reaction sites,and could stop fuel cell operation if the extent of flooding is significant.The focus of the present work is to develop a thermo-hydraulic model based on conservation laws to predict the state of flooding in the stack.The mathematical model consists of two parts;a stack flow model and a partially flooded GDL model,which are explained as follows.2.1.Stack flow model

The cross-section of a PEM fuel cell stack was shown in Fig.1.Fig.2illustrates,in greater detail,the structure of a typical cathode bipolar plate with inlet and outlet manifolds and three flow chan-nels arranged in a serpentine configuration.To model the extent of GDL flooding and the possibility of two-phase flow distribution in the stack,the complex oxidant flow paths consisting of the main inlet(s),inlet and outlet manifolds and the gas flow channels can be reduced into a graphical flow network as depicted in Fig.3,where each MEA is surrounded by the manifolds and the flow channels.The top manifold supplies the oxidant stream to the flow

channels

Fig.2.Illustration of a bipolar plate with serpentine flow field and three channels per plate.

and the unreacted oxygen,accompanied by water and nitrogen,exit into the bottom manifold,collect and leave the stack.

The oxygen reduction starts at the beginning of the flow chan-nels right after the oxygen,protons and electrons are brought into contact in the catalytic layer.The reaction rates vary along the flow channels as the species composition and pressure change.In a recent study,Karimi et al.[18]employed a simplified control vol-ume approach (six control volumes per cell in total)to predict the average species concentrations throughout the fuel cell stack for a wide variety of inlet-outlet topologies.In the present work,the flow paths are divided into a larger number of control volumes in the flow channels to capture,in greater detail,the variations in the species concentrations,local pressures,and the possibility of GDL flooding and/or two-phase flow inception in the flow fields.The new network model consists of an arbitrary number of loops (associated with each cell),with each loop comprised of M seg-ments:two in the inlet and outlet manifolds,and (M −2/2)in each flow channel.The interfaces at which any two segments meet are represented by nodes (•)as illustrated in Fig.3.Within each con-trol volume it is assumed that the pressure and compositions are uniform.Mass transfers due to electrochemical reactions are con-sidered to take place uniformly along the flow channels,resulting in uniform current

distribution.

Fig.3.Graphical representation of the stack flow network model.

159

Fig.4.Graphical representation of the partiallyflooded GDL model.

The total molarflow rate of O2entering the cathode side of the stack can be determined by

˙N stack

O2

=

S c N cell J A cell

4F

(1)

where S c is the cathode stoichiometry,N cell is the total number of fuel cells in the stack,J is the current density and A cell is the active area of a unit cell,respectively.Total inlet molarflow rate can be calculated by adding nitrogen and water vapor to the oxygen molar flow rate.The maximum amount of water vapor coming into the cathode corresponds to100%relative humidity.

The distribution of incoming components in the stack is gov-erned by the conservation laws.The local species concentrations are calculated based on the amount of water produced electro-chemically,and the amount that is transferred by diffusion and electroosmotic drag in the catalytic layer.This will be discussed in the next section.In order to satisfy the conservation of energy, the sum of pressure changes around each of the loops,i,should

be Fig.5.Spatial variations of operating parameters in the cathode side of a PEM fuel cell stack(a)Reynolds number,(b)Pressure,(c)GDLflooding ,and(d)mixture quality x.

Theflow channels shown on the right represent the distribution of parameters in the central bipolar plate(#16)in the stack shown on the left.(J=5000A m−2,T=353K, P out=1atm,RH in=80%).

160G.Karimi et al./Journal of Power Sources 187(2009)

156–1

Fig.6.Effect of level of air humidification on the GDL flooding and the resulting mixture quality x in the bipolar flow channels (J =5000A m −2,T =353K,P out =1atm).

zero,

M j =1

Âi,j P i,j =0(i =1,2,3,s,N loop )

(2)

where Âi,j is a sign convention representing the direction of flow in the segment j of loop i .Âi,j is considered to be +1when fluid flows in a clockwise direction and −1if the direction is reversed.The frictional pressure drop in a segment j of the loop i can be calculated from P f,i,j =C f,i,j

l i,j D h,i,j

i,j V 2

i,j

2

(3)

where l i,j and D h,i,j are the segment (or control volume)length and hydraulic diameter, i,j is the fluid average density,and V i,j is the

average velocity in the segment j of the loop i .The friction coeffi-cient,C f,i,j ,is a function of the Reynolds number defined based on

the hydraulic diameter

Re i,j =

i,j V i,j D h,i,j

i,j

(4)

The physical properties and are calculated depending on whether single-or two-phase flow prevails in the flow channels.For two-phase flow in mm-sized flow channels,Pehlivan et al.[20]showed experimentally that the conventional homogeneous model can be used to predict two-phase pressure drops reasonably well.It is also to be noted that at high mixture qualities,the liquid vol-ume fractions in the flow fields are extremely small,so the error associated with the homogeneous model should be minimal.The

Table1

Parameters and properties used in the present PEM fuel cell stack model.

Component Parameter Value

Bipolar plate W0.15m

H0.15m

l 1.8m

d h1mm

D h10mm

N c6

Stack N cells31

N inlets2(#1,#31)

N outlet1(#16)

T353K

P out1–2atm

RH in50–100%

J1000–10000A m−2

S c 2.0

0.4

GDLıGDL250␮m

GDL40%

two-phase physical properties are calculated from:

TP=

x

G

+

1−x

L

−1

, TP=

x

G

+

1−x

L

−1

(5)

where x is the mixture quality defined as the mass fraction of gas in theflowfields:

x=

m G

m total

(6)

Subscripts TP,G,and L denote“two-phase”,“gas”and“liquid”, respectively.

2.2.Partiallyflooded GDL model

Fig.4illustrates different mechanisms for water transport through a segment of the cathode side in a PEM fuel cell stack. The network of mass transfer resistances and potentials are also included in thisfigure.The partiallyflooded GDL model presented here is based on the following assumptions:(1)Oxygen reduction occurs in the catalytic layer with negligible thickness,hence,the cathodeflooding,if any,starts at the interface of the catalytic layer and grows inside the GDL towards theflowfields.(2)The hampering of O2diffusion through the cell results in increased overpotentials.

(3)The combined electroosmotic drag and back diffusion effects are considered to be uniform along the catalytic layer.The net effect is determined by considering that the gas stream leaving the anode outlet is fully humidified.(4)The GDL pore structure is considered to be uniform throughout the cell.If GDL is fullyflooded,the extra water enters theflow channels.The liquid water moves along with the gas stream uniformly and its volume is negligible.(5)The stack operates at constant temperature.With these assumptions in place, the extent of GDLflooding is governed by the balance of water gen-eration,the net water transport due to the electroosmotic drag and back diffusion,and the rate at which water is removed by the cath-ode gas.The maximum possible water transfer through the GDL can be represented by

˙N Max =

C sat w−C w,b

L

D eff w–g

+

1

h w,b

(7)

where C sat

w

is the water saturation concentration at the cell oper-ating temperature,C w,b is the local bulk concentration in theflow channel,L is the GDL thickness,D eff w–g is the effective diffusivity for water vapor transfer through the GDL,and h w,b is the local mass transfer coefficient.h w,b can be estimated by considering mass Fig.7.Effect of inlet air humidification on the minimum x in bipolarflow channels (J=5000A m−2,T=353K,P out=1atm).

transfer in a fully developed laminarflow through a three-sided adi-abatic square duct with constant massflux applied at one surface and no-flux applied at the others[21]:

Sh=

h w,b D h

D w–g

=2.7(8)

Now,if the rate of water production at the catalytic layer–GDL inter-face is less than or equal to˙N Max,water can be easily removed from the system,otherwise the GDL will beflooded with extra water in liquid form.Under steady-state conditions the local fraction of the flooded GDL, ,can be estimated as

=L f

L

where L f=L−

C sat w−C w,b

˙N

H2O

−1

h w,b

D eff w–g(9)

where L f is the thickness of theflooded GDL.˙N

H2O

can be calculated from:

˙N

H2O

=

˙N H

2O

A CV

and˙N H

2O

=˙N Gen+˙N EO−˙N BD

Net interaction with anode

(10) where

˙N Gen=JA CV

2F

(11) where A CV is the control volume surface area available for mass transfer.

Eq.(9)can be solved in conjunction with the recently developed flow network solution algorithm[18,19]to obtain the distribution in the extent of waterflooding in the GDL under a variety of geo-metrical and operating conditions.

3.Numerical procedure

Theflow network solution algorithm adopted in this work is based on the modified Hardy Cross method that has recently been reported[18].Numerical solution begins with assigning a tempo-raryflow direction to theflow network.The sum of the molarflow rates from all the inlets(with known compositions)is then divided uniformly among the gasflow channels.The molarflow rates and compositions in the downstream sections of theflow channels are calculated by subtracting the consumed reactants and adding the produced or transported components.Water transport through the membrane due to the combined effects of electroosmotic drag and back diffusion is also considered.The conservation of mole equation is used to calculate the molarflow rates in the inlet and outlet man-ifolds based on the predefined directions and the assumed molar

162G.Karimi et al./Journal of Power Sources 187(2009)

156–1

Fig.8.Effect of the stack pressure on the GDL flooding and the resulting mixture quality in the bipolar flow channels (J =5000A m −2,T =353K,RH in

=80%).

Fig.9.Effect of stack pressure on the minimum x in bipolar flow channel (J =5000A m −2,T =353K,RH in =80%).

flow rates in the flow channels.To satisfy the conservation of energy equation,the pressure drop is calculated for each segment based on the local mixture quality and appropriate pressure drop equa-tions.The procedure is repeated until correct flow directions and molar flow rates are obtained for the whole flow network.The local species concentrations are used in Eq.(9)to calculate the fraction of GDL that is flooded with liquid water, .If GDL is fully flooded at some point along the flow fields,the extra water is added into the cathode stream,changing the local quality,x ,and hence the result-ing pressure drop is calculated based on two-phase flow properties defined in Eq.(5).

4.Results and discussion

The input parameters for the fuel cell and stack flow model are classified as operating and design parameters.The design parameters are the fuel cell size,stack manifold and flow channel dimensions and configuration.Operating parameters include the stack current density,temperature,pressure,stoichiometry and the

163

Fig.10.Effect of the current density on the GDLflooding and the resulting mixture quality in the bipolarflow channels(T=353K,P out=1atm,RH in=80%).

reactant composition at the stack inlet.Table1lists a summary of the operating and relevant design parameters used in the present study for the cathode side of the PEM fuel cell stack.

The stack is considered to be composed of31cells,with two oxi-dant inlets at the endpoints of the inlet manifold(#1and#31)and one outlet at the middle of the exit manifold(#16).This symmet-ric double-inlet-single-outlet configuration was shown to be the most effective scheme with minimal cell-to-cell voltage variations and parasitic losses[18].Humidified air at different pressures and water contents was injected into the stack.

Fig.5a–d shows variations in different operating parameters in the cathode side of a fuel cell stack at a current density of 5000A m−2.Thefigures on the left side display an overall view of the stack consisting of the inlet and outlet manifolds and theflow channels.Thefigures on the right side show details of the oper-ating parameters inside the central bipolar plate(#16)with three serpentineflow channels.Fig.5a shows the variation of Re number in the manifolds and across theflow channels.The Reynolds num-ber decreases along the inlet manifold from the endpoints towards the middle channels as the reactants are delivered to theflow chan-nels.Theflow division is almost uniform due to the large pressure drop in theflow channels compared with that of the inlet manifold, resulting in a linear reduction in the Reynolds number.This phe-nomenon is repeated for the outlet manifold as the unreacted O2, N2and H2O are collected and leave the stack.Inside theflow chan-nels,Re number varies as the local velocity,density and viscosity change.However,theflow remains in the laminar region.

Pressure distribution in the stack is illustrated in Fig.5b.There are two factors contributing to the pressure variation along the flow path in the stack.First,the total molarflow rate is increased along theflow channels because two moles of H2O are produced for every mole of O2consumed in the catalytic layer;this will increase1G.Karimi et al./Journal of Power Sources187(2009)156–1

the local pressure.Second,pressure along the channels is reduced due to friction.At constant current density,water concentration is increased along theflow channels.This increases the possibility of waterflooding in the GDL and even condensation of water vapor inside theflow channels.The latter could initiate two-phaseflow in the lower parts of the stack.This is evident from Fig.5c and d where the percentage offlooded GDL, ,and mixture quality,x,along the flow channels are depicted.As expected,the GDL must be com-pletelyflooded before the mixture qualities start to decrease from 1(single-phase)to lower values(e.g.0.97),for which the onset of two-phaseflow in lower parts of the stack could be initiated.How-ever,the resulting quality distributions in Fig.5d are insufficient to incept a major two-phaseflow in the stack.This is due to very large liquid-to-gas density ratio,which leads to gas fractions of greater than99%in theflow channels.

Fig.6a–f illustrates the effect of the level of air humidification on the GDLflooding and the resulting mixture quality in the bipolar flow channels.As seen from thesefigures,as the inlet air rela-tive humidity,RH in,is increased,a larger portion of the GDL will beflooded and lower qualities are observed in theflow channels. Although the extent of GDLflooding does have a significant effect on the mass transfer overpotential,the lower x values may not be important as pointed out earlier.

It is quite useful to estimate the minimum quality,x min,which prevails in the bipolarflow channels and how the inlet air humidity, stack pressure and current density affect this value.In fact,x min is a direct indication of the extent of GDLflooding because the GDL must be completelyflooded before the x values drop below one. Fig.7indicates the effect of RH in on the x min in theflow channels. As seen from thisfigure,portions of GDL start toflood if the RH in exceeds68%.Theflooded regions grow as RH in increases and GDL is fullyflooded when RH in=73%.Numerical results showed that at inlet relative humidities of70%and72%,GDL wasflooded by5% and92%,respectively.At larger RH in values,the extra water per-meates through theflow channels and results in lower x values. The minimum x observed for this case was about90%when a fully humidified air enters the stack.

The effect of the stack pressure on the GDLflooding and the resulting quality in the bipolarflow channels are shown in Fig.8a–f. As indicated,by increasing the stack pressure,the partial pressure of the produced water is augmented and water tends to condensate in the GDL pores.As a result,the quality of the gasflow is also decreased.Fig.9indicates that the stack operating pressure has a more noticeable effect on GDLflooding than the inlet air humidity. As shown in thisfigure,increasing the stack outlet pressure from 1atm to2atm will result in a minimum x value of about84%.

Fig.10a–f illustrates the effect of the stack current density on the GDLflooding and the resulting quality in the bipolarflow channels. By increasing the stack operating current density,a larger amount of oxidant stream needs toflow in the bipolarflowfields,resulting in significantly low pressures,particularly in the vicinity of theflow channel exits.The lower pressures there enhance water transport through the GDL,reducing the extent offlooding.The quality in the flow channels are also affected.Figs.10and11indicate that the possibility of GDLflooding and the inception of two-phaseflow in theflowfields is reduced at high stack current densities.

The numerical results shown in Figs.5–11indicate the GDL in the cathode side of the PEM fuel cell stack can easily beflooded and the quality of gasflow in theflowfields could decrease from one.Although GDLflooding and the inception of two phaseflow are among the common issues in PEM fuel cells and have been reported in numerous papers,the extent of these issues are alleviated by the heat of the reaction which is released during the oxygen reduction at the catalyst layer,and by significant reduction in the O2diffusion through the GDL.Also,as pointed out earlier,a minimum x value

of Fig.11.Effect of operating current density on the minimum x in bipolarflow chan-nels(T=353K,P out=1atm,RH in=80%).

about84%,can not be responsible for the inception of two-phase flow in the system.Because the corresponding void fraction is esti-mated to be very close to1(e.g.99.9%)for the worst case scenario reported here.

5.Conclusions

Water management is one of the most critical issues for high-performance polymer electrolyte membrane(PEM)fuel cells.A partiallyflooded GDL model is proposed and used in conjunction with a cathodeflow network model to predict the conditions under which the GDL could beflooded in a PEM fuel cell stack.Effects of current density,operating pressure,and level of inlet humidity on GDLflooding are studied.The simulation results have revealed that although under certain operating conditions the GDL is fully flooded,and water can be liquified in parts of the stackflowfields, the amount of the liquified water in the stack is not significant enough to cause a major pressure drop or an appreciable change in the species concentrations.

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