three dimensional, two phase ﬂow mathematical model for pem fuel cell. part i. model development

8/7/2019 Three dimensional, two phase flow mathematical model for PEM fuel cell. Part I. Model development

1/22

Three dimensional, two phase flow mathematical model

for PEM fuel cell: Part I. Model development

Mingruo Hu a,b,*, Anzhong Gu a, Minghua Wang b, Xinjian Zhu b, Lijun Yu c

a Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, 1954 Huashan Road,

Shanghai 200030, PR Chinab Institute of Fuel Cell, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai, 200030, PR China

c Department of Energy, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China

Received 2 December 2002; received in revised form 2 June 2003; accepted 12 September 2003

Abstract

A full three dimensional PEM fuel cell model is developed, which considers not only the rib resistance to

the species but both the single and two phase flow and transport in the gas channels and diffusers at both

the anode and cathode sides of the PEM fuel cell. Two sets of boundary conditions, one for a conventional

flow field and the other for an interdigitated one, are presented. A detailed discussion of the numerical

techniques for the PEM fuel cell model is given with a flow diagram to provide an overview of the solutionprocedure using the FORTRAN language. A rigorous validation method is used to show good agreement

between our predicted results and the experimental data.

2003 Elsevier Ltd. All rights reserved.

Keywords: PEM fuel cell; Two phase flow; Model; Conventional flow field; Interdigitated flow field; SIMPLE algorithm

1. Introduction

As an important method to study proton exchange membrane (PEM) fuel cells, mathematicalmodels have received strong emphasis in resent years. A good PEM fuel cell model can not onlyhelp to understand the internal mechanisms, such as heat and mass transfer, but also, it can help

to improve the efficiency and save much money when designing and experimenting with fuel cells.Referring to PEM fuel cell models developed in the most recent ten years, they can be classified

Energy Conversion and Management 45 (2004) 18611882

www.elsevier.com/locate/enconman

* Corresponding author. Tel./fax: +86-21-6293-2602.

E-mail address: [email protected] (M. Hu).

0196-8904/$ - see front matter

2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.enconman.2003.09.022
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


2/22

Nomenclature

a water activityAv effective catalyst area per unit volume (m

1)cf fixed charge concentration in proton exchange membrane (mol m

3)C mass concentration fraction in gas channel and gas diffuser or mole concentration in

catalyst layer (mol m3)Cref reference mole concentration (mol m

3)D diffusivity (m2 s1)F Faraday constant (96,487 C mol1)g gravitational acceleration (m s2)io;ref reference exchange current density (A m

2)I current density (A m2)

j diffusive mass flux (kg m2s1)J capillary pressure functionk conductivity (Xm1)kp permeability of proton exchange membrane (m

2)kr relative permeabilityK permeability of gas diffuser (m2)M molecular weight (kg mol1)nd electro-osmotic drag coefficientN net mass flux in gas channel and gas diffuser (kg cm2 s1) or net mole flux in catalyst

layer and membrane (mol cm2 s1)

P total pressure (Pa)Pc capillary pressure (Pa)p partial pressure of gas (Pa)R gas constant (J mol1 K1)Rm membrane resistance (Xm

2)s liquid water saturationT temperature (K)t time (s)u intrinsic velocity vector (m s1) or intrinsic velocity in x direction (m s1)U cell working voltage (V)UO thermodynamic open circuit potential (V)

v intrinsic velocity in ydirection (m s1)Vp total pore volume in gas diffuser (m

3)Vw liquid water volume in gas diffuser (m

3)w intrinsic velocity in zdirection (m s1)x x direction coordinate (m)y ydirection coordinate (m)z zdirection coordinate (m)

1862 M. Hu et al. / Energy Conversion and Management 45 (2004) 18611882


3/22

into three categories, i.e. one dimensional models, two dimensional models and three dimensional

models.As a first step to simulate a PEM fuel cell, Bernardi and Verbrugge [1] and Springer et al. [2]

developed one dimensional models, that is, they only considered the changes across the mem-brane. These models not only presented some basic results for understanding the internal

mechanisms of a working PEM fuel cell, but more importantly, they also provided a funda-mental framework to build the multidimensional models that followed. Fuller and Newman [3],Nguyen and White [4] and Yi and Nguyen [5] started to develop two dimensional models that

considered both the changes across the membrane and in the direction of the bulk flow. In thesemodels, the authors demonstrated the important roles played by water and thermal managementin maintaining high performance of PEM fuel cells. Furthermore, Gurau et al. [6] and Um et al.[7] developed two dimensional models using the computational fluid dynamics (CFD) approach.

However, all the models mentioned above neglected the ribs between flow channels. West [8]

Greek symbols

e porosity

q density (kg m3

)l dynamic viscosity (kg m1 s1)c advection correction factort kinematic viscosity (m2 s1)/ potential (V)k relative mobility or water content in proton exchange membrane (mol H2O/equivalent

SO13 )r surface tension (N m1) or local conductivity of proton exchange membrane (Xm1)a charge transfer coefficientg overpotential (V)d coefficient thickness (m)

Subscripts and superscripts

a anode

avg averagec cathode or catalyst layereff effective

g gasin entrance of gas channelk phase

l liquidm proton exchange membrane or Naifon phase in catalyst layer

s solid phasesat saturationw watera species

M. Hu et al. / Energy Conversion and Management 45 (2004) 18611882 1863


4/22

developed a two dimensional model mainly considering the influence of rib spacing in a PEM

fuel cell and concluded that the ribs would restrict the access of fuel and oxidant gases to thecatalyst layer and the movement of water under the ribs. Recently, Shimpalee et al. [9] developed

a three dimensional model. This model used the commercially available computational fluidscode, Fluent, as a solver and presented such results as velocity, mixture density, pressure andlocal current contours under the ribs, which could not be understood in a two dimensionalmodel.

However, all the models mentioned above assumed single phase flow in the gas channels andgas diffusers and are only valid in the absence of liquid water. During the fuel cell operation,especially at high current densities, liquid water is likely to appear in the cathode side, resulting in

two phase transport phenomena, and also liquid water will appear in the anode side if thehumidification temperature is higher than that of the PEM fuel cell or liquid is injected directly

into the anode gas channel [4,10]. Baschuk and Li [11] noted that the fraction of liquid waterexisting in a gas diffuser was small, having a magnitude of 104 when air was used as oxidant.

However, a small amount of liquid water could block some pores of the gas diffuser and influenceperformance at high current densities [11]. Predicting the formation of liquid water and mini-mizing its detrimental effect on fuel cell performance are important issues in the design and

operation of PEM fuel cells. You et al. [12] presented a two dimensional, two phase flow model to

Fig. 1. Schematic diagram of PEM fuel cells with conventional and interdigitated flow fields.



5/22

consider the transport mechanisms in the two phase flow area. However, this model, which was a

half cell model and only included the gas channel, gas diffuser and catalyst layer in the cathodeside of a PEM fuel cell, could not explain the two phase transport phenomena in the anode side of

a PEM fuel cell, could not find the water distributions in the membrane to direct the humidifi-cation of a PEM fuel cell and could not know what level of flooding happened inside a gas diffusernear the rib region.

Recently, some special focuses have been put on a new flow field called an interdigitated flow

field [13]. As shown in Fig. 1, by making the inlet and outlet gas channels dead-ended, the reactantgases in an interdigitated flow field are forced to flow into the electrode as compared to flowingover the surface of the electrode in a conventional parallel channel flow field. This design can

convert the transport of the reactant/product gases to/from the catalyst layers from a diffusionmechanism to a forced convection mechanism with a much reduced gas diffusion boundary layer

over the catalyst sites. Since forced convection is much faster than diffusion, the reaction rates atthe catalyst sites can be significantly enhanced. In addition, the shear force of the gas flow helps

remove most of the liquid water that is entrapped in the inner layer of the electrode, therebysignificantly reducing the electrode flooding problem. Yi [14] developed a cathode half cell modelto compare the performance of PEM fuel cells containing interdigitated and conventional flow

fields. However, this two dimensional model could only describe the cross flow section of PEMfuel cells, so the boundary conditions at the interfaces between the gas channels and gas diffusersmust be assumed, which actually are not known a priori and are not constant in the main flow

direction. Furthermore, this model neglected the existence of liquid water and did not explainquantitatively how well an interdigitated flow field helped to remove the liquid water as comparedto a conventional flow field.

This paper forms the first part of a two part study on a PEM fuel cell model. In this paper, a

three dimensional numerical model is developed, which considers not only the rib resistance to thespecies but also both the single and two phase flow and transport in the gas channels and diffusersat both the anode and cathode sides of the PEM fuel cell. Two sets of boundary conditions, one

for a conventional flow field and the other for an interdigitated one, are presented. A detaileddiscussion of the numerical techniques for the PEM fuel cell model is given with a flow diagram toprovide an overview of the solution procedure using the FORTRAN language. At last, a

parameter based validation method is used, and the model shows good agreement betweennumerically predicted polarization curves and experimental data.

2. Model development

Fig. 2 shows a schematic of a three dimensional PEM fuel cell model. The model regions, whichare symmetrically divided through a PEM fuel cell, consist of half gas channels separated by the

membrane and electrode assembly (MEA) that includes two diffusion layers and two catalystlayers.

The numerical model assumes that

1. The PEM fuel cell operates under steady state condition. This is because the startup or stop orany transient process of a fuel cell are not considered in this model.



6/22

2. There are no temperature changes in the PEM fuel cell, i.e. isothermal operation, This is

because, when a single cell is tested, the cell temperature is usually controlled using heatingrods and thermocouples. As a result, the operation temperature of a single fuel cell can be keptalmost constant in all the regions of the working fuel cell.

3. Laminar flow exists everywhere in the gas channels and the flow is fully developed at the exitsof the gas channels. The assumption of laminar flow is based on the small flow velocities andthe small pressure gradients through a single cell [7]. After the gases flow out of the exits of the

gas channels, there are no further chemical reactions, so there are no further changes in the con-centration of the gases, such as O2 or H2. As a result, the assumption means that, at the exits of

the gas channels, the gradients of the gas flow at the outlets are zero [7].

4. Isotropic porous media exists in the gas diffusers, catalyst layers and membrane. This assump-tion asserts that the porosity is a constant in the whole region of the gas diffusers, and the vol-ume fraction of Nafion in the catalyst layer is also constant.

5. H2, O2 and N2 are insoluble in the liquid water probably existing in the anode and cathode gaschannels and diffusers. This is an assumption for our two phase flow model. In the gas channel

and gas diffuser, there usually co-exists a gas phase and a liquid water phase. However, the sol-ubilities of H2, O2 and N2 in liquid water are quite small, and the diffusivities of H2, O2 and N2in liquid water are also quite small compared with the diffusivities of H2, O2 and N2 in the gas

phase. You [12] also considered O2 and N2 insoluble in liquid water to neglect the extremelysmall amount of gas diffusion in the liquid water.

Actually, assumptions 14 are usually considered as standard assumptions in many papers [13,69,11,12,14,1820]. Assumption 5 is based on Ref. [12].

In this paper, a multiple phase mixture based two phase flow and transport model is developedto study the two phase flow in the gas diffusers and the gas channels in both the anode and cathode

sides. Wang and Cheng [15] first proposed a detailed multiphase mixture mathematical model formultiphase flow in the porous media. The key idea in the multiphase mixture model is to focus onthe level of a multiphase mixture, rather than on the levels of the separate phases. The multiple

phases are regarded as constituents of a multiphase mixture in the model. A mass averaged mixturevelocity and a diffusive flux representing the difference between the mixture velocity and an indi-

Fig. 2. Schematic of a PEM fuel cell model.



7/22

vidual phase velocity can describe the multiphase flow. In this definition, the phases are assumed to

be distinct and separable components with non-zero interfacial areas, and their mixture representsa single fluid with smoothly varying phase compositions [15]. The multiphase mixture model can be

derived from the classical multiphase approach without any additional assumptions [15].

2.1. Model equations

2.1.1. Governing equations in gas channels and gas diffusers

Conservation of mass

oeq

ot r equ 0 1

Momentum transport

oequ

ot r equu erP r elru eqg e2

l

Ku 2

where the last term represents the drag of the porous solid on the liquid [16].

Species transport [15]

eo

otqCa r ecaquC

a r eqDarCa r eX

k

qkskDakrC

ak

" rCa

#

r X

k

Cakjk

" #3

Eq. (3) reduces to the following for a two phase (gas and liquid) system

eo

otqCa r ecaquC

a r eqDarCa r feq1sDal rC

al qg1 sD

agrC

ag qD

arCag

r Cal Cag jl 4

The definitions of the variables in the above equations are defined in Table 1. The velocity vector u

in Eqs. (1)(4) is the intrinsic velocity vector based on the open pore area. Eqs. (1), (2) and (4) canbe used to calculate the velocity and concentration fraction distributions in both the single and

two phase zones in the gas channels and gas diffusers. When the liquid saturation s equals zero,Eq. (4) reduces to the species equation in the single phase zone. When the porosity equals unityand the permeability equals infinity, Eq. (2) reduces to the momentum equation in the gas

channel.Eq. (4) can be rewritten for the species H2, O2 and N2 at the anode and cathode sides as follows.At the anode side, we get

r ecH2quCH2 r eqDH2g rC

H2 r e1

" sqgD

H2g C

H2 rq

qg1 s

#

r qCH2

qg1 sjl

" #5



8/22

The water mass concentration fraction in the anode side can be obtained by

CH2O 1 CH2 6

At the cathode side, we get

r ecO2quCO2 r eqDO2g rC

O2 r e1

" sqgD

O2g C

O2 rq

qg1 s

#

r qCO2

qg1 sjl

" #7

Table 1

Variables used to describe the two phase flow and transport

Variables Definitions

Liquid water saturation s s VwVp

Velocity vector u qu qlul qgug

Advection correction factor ca ca qklC

al kgC

ag

qlsCal qg1 sC

ag

Mass concentration fraction for species a

in liquid or gas Cal , Cag

Cal qalq

Cag qag

qg

Mass concentration fraction for species a, Ca Ca qa

qor qCa qlsC

al qg1 sC

ag

Diffusion coefficient for species a, Da qDa qlsD

al qg1 sD

ag

Diffusive mass flux of liquid and gas within

the two phase mixture jg, jl

jg qgug kgqu; jl qlul klqu or

jl Kklkg

mrPc ql qgg and jl jg 0

The individual intrinsic phase velocities

for liquid and gas ul, ug

eqlul jl klequ; eqgug jl kgequ

Capillary pressure between liquid and gas Pc Pc Pg Pl re

K

12

Js;

Js 1:4171 s 2:1201 s2 1:2631 s3

Density q q qls qg1 s

Viscosity l l q

krl=ml krg=mg

Relative mobility for liquid and gas kl, kg kl krl=ml

krl=ml krg=mgkg

krg=mgkrl=ml krg=mg

Relative permeability for liquid and gas krl, krg krl s3 krg 1 s

3



9/22

r ecN2quCN2 r eqDN2g rC

N2 r e1

" sqgD

N2g C

N2 rq

qg1 s

#

r qCN2

qg1 sjl

" #8

The water mass concentration fraction in the cathode side can be obtained by

CH2O 1 CO2 CN2 9

In Eqs. (5), (7) and (8), we neglect the transient terms and consider the mass concentrationfractions of hydrogen, oxygen and nitrogen C

H2l , C

O2l and C

N2l in the liquid water equal to zero

according to the assumption above.

2.1.2. Governing equations in the catalyst layers

Fig. 3 shows the construction of a catalyst layer. In this layer, the oxygen or hydrogen dissolvesinto the Nafion and diffuses to the surface of the Pt/C where it reacts. The electro-chemicalreactions at both the anode and the cathode sides can be expressed by the following equations.

At the anode side

2H2 ! 4H 4e 10

At the cathode side

4H 4e O2 ! 2H2O 11

From the mass-current balance, we can get

dNdy

1nF

dIdy

12

where n 2 for H2 and n 4 for O2.A ButlerVolmer expression is used to characterize the potential dependence of the rates of

Eqs. (10) and (11) as follows [17]:

dI

dy Avio;ref

Cm

Cref

nexp

acFg

RT

exp

aaFg

RT

13

Gas diffuser

c

0

Gas pore y

Membran

e

Pt/C

Nafion

O2

or H2

H+

Catalyst layer

H+

Fig. 3. Schematic of a catalyst layer.



10/22

where n 0:5 for H2, n 1 for O2 and g is the total electrode overpotential [6] expressed asfollows [18]:

g

/s

/m

constant

14

where /s and /m stand for the potentials of the matrix of the solid phase and the Nafion phase,respectively, in the catalyst layer.

Assuming that the matrix of the solid phase is equipotential [19], we get the expression for theoverpotential

dg

dy

I

keffm15

The hydrogen or oxygen diffusion equation can be described as follows:

N Deffm dCmdy16

where keffm e1:5mckm, D

effm e

1:5mcDm [7] in Eqs. (15) and (16).

At the interface between a gas diffuser and a catalyst layer, the hydrogen or oxygen concen-

tration fraction is related with Henrys law

Cm p

H17

where pis the partial pressure of hydrogen or oxygen on the gas phase side, and His Henrysconstant.

2.1.3. Governing equations in proton exchange membrane

The water is transported by three mechanisms in a proton exchange membrane, that is, theelectro-osmotic drag, diffusion and permeation, which are induced separately by the moving

protons, the water concentration difference and the pressure difference between the two sides ofthe membrane as shown in Fig. 4. The net water flux through the membrane is the sum of thesethree water fluxes and is expressed by the following equation.

Fig. 4. Schematic of water transport phenomena in a proton exchange membrane.



11/22

Nw ndI

F

kp;m

lCwm

dp

dy Dwm

dCwmdy

18

where the electro-osmotic drag coefficient nd is expressed by the following equation [2]:

nd 2:5

22k 19

the water mole concentration Cwm in the membrane has a form as [20]

Cwm kcf 20

and the water diffusion coefficient Dwm in the membrane can be expressed by the following equation[9]

Dwm nd5:5 1011 exp 2416

1

303

1

T21

Substituting Eqs. (19) and (20) into Eq. (18), we get

Nw 0:1136kI

F

kp;m

lkcf

dp

dy Dwmcf

dk

dy22

where cfmeans the fixed charge concentration in the membrane, and k stands for the water

content (mol H2O/equivalent SO13 ). The water content k can be expressed by [2]

k 0:043 17:81a 39:85a2 36:0a3 0 < a6 1 23

k 14:0 1:4a 1 1 < a6 3 24

k 16:8 aP 3 25

a being the water vapor activity given by

a xwP

pSatw26

where xw is the mole fraction of water.

2.1.4. Fuel cell voltage

The fuel cell voltage is given by

U UO ga

gc

IavgRm 27

where UO is the thermodynamic open circuit potential given by [1]

UO 1:23 0:9 103T 298 2:3

RT

4Fp2H2 pO2 28

The average current density is determined by

Iavg

RA

Ix; z

A29

where A is the reaction area. The membrane resistance Rm in Xm2 is given by



12/22

Rm

Zdm0

dy

rk30

where rk is the local conductivity of the membrane. Springer et al. [2] presented the experi-mental fit of conductivity for Nafion117 as follows:

rk exp 12681

303

1

T

0:5139k 0:326 31

3. Boundary conditions

The computation domains, as shown in Fig. 5, are composed of half gas channels, gas diffusersand catalyst layers at each side of the PEM fuel cell and proton exchange membrane.

3.1. Boundary conditions for gas channels and gas diffusers

At the entrances of the gas channels, constant flow rate and mass concentration fraction of each

species are specified, and at the exits, both the velocity and concentration fraction distributionsare assumed to be fully developed. The boundary conditions for the gas channels and gas diffusersin the two PEM fuel cells with conventional and interdigitated flow fields are given separately as

follows.

3.1.1. For the conventional flow field

At faces F0M0N0E0, FMNE, B0O0P0A0 and BOPA (the entrances of the half gas channels)

u uin; v 0; w 0; Ca Cain 32

at faces G0Q0R0H0, GQRH, K0S0T0L0 and KSTL (the exits of the half gas channels)

ou

ox 0;

ov

ox 0;

ow

ox 0;

oCa

ox 0 33

Fig. 5. Schematic of the computation domains.



13/22

at faces EE0D0D, BB0C0C, HH0I0I and KK0J0J (the front and end faces of the diffusion layers)

u 0; v 0; w 0;oCa

ox

0 34

at faces FGID, F0G0I0D0, CJLA and C0J0L0A0 (the side faces of the half gas channels and diffusers),having the symmetrical conditions

ou

oz 0;

ov

oz 0;

ow

oz 0;

oCa

oz 0 35

at faces FMQG, F0M0Q0G0, APTL and A0P0T0L0 (the top and bottom faces of the gas channels)

u 0; v 0; w 0;oCa

oy 0 36

at the interface between the cathode gas diffuser and catalyst layer

u 0; v NO2 NH2O

eq; w 0

NO2 ecO2qvCO2 eqDO2g

oCO2

oy e1 sqgD

O2g C

O2o

q

qg1s

h ioy

qCO2

qg1 sjl;y

NN2 ecN2qvCN2 eqDN2g

oCN2

oy e1 sqgD

N2g C

N2o

q

qg1s

h ioy

qCN2

qg1 sjl;y

37

at the interface between the anode gas diffuser and catalyst layer

u 0; v NH2 NH2O

eq; w 0

NH2 ecH2qvCH2 eqDH2g

oCH2

oy e1 sqgD

H2g C

H2o

q

qg1s

h ioy

qCH2

qg1 sjl;y 38

3.1.2. For the interdigitated flow field

At faces F0M0N0E0 and B0O0P0A0 (the entrances of the inlet gas channels)

u uin; v 0; w 0; Ca

Ca

in 39at faces FMNE and BOPA (the dead ends of the outlet gas channels)

u 0; v 0; w 0;oCa

ox 0 40

at faces GQRH and KSTL (the exits of the outlet gas channels)

ou

ox 0;

ov

ox 0;

ow

ox 0;

oCa

ox 0 41



14/22

at faces G0Q0R0H0 and K0S0T0L0 (the dead ends of the inlet gas channels)

u 0; v 0; w 0;oCa

ox 0 42

The other boundary conditions for the gas channels and gas diffusers in a PEM fuel cell withinterdigitated flow fields are the same as Eqs. (34)(38).

The net mass fluxes of species at the interface between the gas diffusers and the catalyst layers in

Eqs. (37) and (38) are given by

NH2 MH2 I

2F; NH2Oa MH2ON

w 43

NO2 MO2 I

4F; NN2 0; NH2Oc

MH2OI

2F NH2Oa 44

Here, we do not present the boundary conditions at the surfaces of ribs because they are closely

related with the solution techniques introduced in the next section.

3.2. Boundary conditions for the catalyst layer and proton exchange membrane

As the current density distribution at the interface between a catalyst layer and a membraneand the reactant concentration fraction distributions at the interface between a gas diffuser and acatalyst layer are not known a priori, the total electrode overpotential in the catalyst layer pre-

sented in Eq. (13) is initially assumed [6].Referring to Fig. 3, the boundary conditions for a membrane are given by

at y

0;

k

k

0

and at y

dm;

k

kd

m 45

4. Numerical techniques and procedures

The model equations developed above are coupled closely, so the whole set of equations must

be solved simultaneously and iteratively. In the gas channels and gas diffusers, the model equa-tions, including the continuity, momentum and species equations, are extremely similar to each

other as mentioned above, so these two regions can be solved together, and the boundary con-ditions between these two layers are omitted. The SIMPLE algorithm [21] is used to solve thetransport equations in the gas channels and gas diffusers. However, the ribs between two

neighboring channels, as shown in Fig. 5, cause some difficulties when using the SMPLE algo-rithm to solve such non-rectangular boundaries, excluding the ribs.

In order to solve the above difficulties, we consider the ribs as a special fluid, which has an

extremely large viscosity (for example, l 1030) and unity mass concentration fraction. First, weconsider the region F0G0GFDII0D0 or A0L0LACJJ0C0 in Fig. 5, including a gas diffuser, two halfgas channels and a rib, as an integrated domain for calculation. Second, we must give the

additional boundary conditions not mentioned in the above section, that is, at faces MQQ0M0 andPTT0P0 shown in Fig. 5, that is, the interfaces between the ribs and the plates

u 0; v 0; w 0; Ca 0 a O2; N2; H2; CRib 1 46



15/22

at faces MM0N0N, OO0P0P, QQ0R0R and SS0T0T, the front and end faces of the anode and cathoderibs

u 0; v 0; w 0; Ca 0 a O2;N2;H2; CRib 1 47

The large viscosity set purposely in a rib region and the zero velocities set on the three surfacesof the rib make this region static. At last, the large source term [21] is used to freeze the reactantconcentration fractions at zero in the whole rib region, and the rib concentration fraction at unity

in the rib region and at zero in the gas channels and gas diffusers. Hence, the mass concentrationfractions are coupled automatically at the interfaces between the gas channels and the rib, and so,it is no use presenting the boundary conditions for faces MNRQ, M0N0R0Q0, OPTS and O0P0T0S0

in Fig. 5.The shooting method and fourth order RungeKutta method are used separately to solve the

equations in the catalyst layers and equation in the membrane.

Yes

No

Begin

exp


16/22

Fig. 6 presents the coupled algorithms to solve the PEM fuel cell model. The solution is

considered to be convergent when the relative error between two consecutive iterations is less thana small value.

5. Model validation

Because of the small size of the gas channels and the small thickness of the gas diffuser, catalystlayer and membrane, it is very difficult to measure the flow field and concentration field in the gas

channel and gas diffuser, and it is also very difficult to measure the concentration field and localcurrent density distributions in the catalyst layer and in the membrane [22]. As a result, few paperscan be found to address measurement of such fields or distributions. However, voltage vs. current

density curves, which are highly related to the distributions in a PEM fuel cell, can be measureddirectly and accurately in experiments [23]. As a result, all PEM fuel cell models published pre-viously have used experimental voltage vs. current curves to validate the reliability of these

models, with some parameters supplied by the experiments and others by the models [1,6,7,22,25].To validate the current three dimensional, two phase flow model, comparisons are made to the

experimental data of Ticianelli et al. [23] for a single cell. The schematic of the computation

domains are presented in Fig. 5, and the basic parameters used in the model validation are givenin Table 2. The grid numbers of 42 42 42, used separately in the anode and cathode gas

channels and gas diffusers, provides satisfactory results.In Table 2, the parameters for fuel cell operating conditions, such as cell temperature and

pressure, were supplied in Ref [23]. However, the parameters, such as the exact fuel cell geometryand stoichiometric flow ratio are unknown in Ref. [23]. Usually, papers presenting experimental

data do not supply the fuel cell geometries. However, this is not a problem for using theseexperimental data to validate a new model through the following steps. The common method forvalidation is as follow [1,6,7,22,25]:

First, supply a set of geometry parameters, stoichiometric flow ratios or other operating con-

ditions for the PEM fuel cell model if such parameters are not given with the experimental data.As a result, Avi

refo;c and Avi

refo;a are the only two adjustable parameters.

Second, adjust Avirefo;c and Avi

refo;a to fit the numerically predicted curve to its corresponding

experimental voltage vs. current density curve for the PEM fuel cell.Usually, based on the above two steps, a model can be proved to be a good one by only fitting

one predicted curve to its corresponding experimental voltage vs. current density curve throughadjusting A

viref

o;cand A

viref

o;a[1,6,22,25]. However, here, a third step is used to eliminate the possible

inaccuracy of model validation, which is derived from the supplied parameters in step one. For

example, through a special choice for the geometry parameters and adjusting Avirefo;c and Avi

refo;a, a

bad model possibly can show good agreement between predicted results and experimental data,but such parameters and Avi

refo;c and Avi

refo;a are closely related to this experimental voltage vs. current

density curve and cannot be used in other experimental voltage vs. current density curves for thesame fuel cell.

Third, choose another experimental voltage vs. current density for the same fuel cell, for

example, choosing an experimental voltage vs. current density at a different operating pressure. Ifthe same value for Avi

refo;c and Avi

refo;a, which is determined in step 2, can make the numerically



17/22

predicted curve show good agreement with the newly selected experimental voltage vs. currentdensity curve for the same fuel cell, then the model is proved to be an accurate model, and thepossibility of inaccuracy of model validation, which is derived from the supplied parameters, is

eliminated. Furthermore, because irefo;c is related to temperature [26], when choosing an experi-mental voltage vs. current density at a different operating temperature, a further step should be

done as follows: While Av is a constant for a certain electrode [18], Avirefo;c for other temperaturesshould be calculated based on Eq. (48), which is a correlation of the exchange current density and

the fuel cell temperature presented in Ref. [26]. Then, if the newly calculated Avirefo;c can make the

numerically predicted curve show good agreement with the newly selected experimental voltagevs. current density at the different operating temperature, then the model is proved to be an

accurate model.

log10irefo;c 3:507

4001

T48

Table 2

Basic parameters for PEM fuel cells

Quantity Value Source

Gas channel length FG, BK 7.11 102 m [6,7]Gas channel height FE, AB 7.62 104 m [6,7]

Half gas channel width F0M0, MF or B0O0, OB 5 104 m

Rib width M0M or P0P 1 103 m

Gas diffuser height ED, BC 2.54 104 m [6,7]

Catalyst layer thickness dc 2.87 105 m [6,7]

Membrane thickness for Nafion 117 dm 1.75 104 m [23]

Gas diffuser porosity e 0.4 [6,7]

Volume fraction of Nafion in the catalyst layer em;c 0.4 [6,7]

Cell Temperature T 353 K [23]

Anode and cathode side inlet pressure Pa=Pc 3.03 105/5.05 105 Pa [23]

Anode fuel stoichiometric flow ratio 3

Cathode oxidant stoichiometric flow ratio 4

Humidification temperature for anode inlet fuel 358 K [7]

Relative humidity of anode inlet fuel 100% [7]

Relative humidity of cathode inlet air 0% [7]

Fixed charge concentration in the membrane cf 1.2 103 mol/m3 [1,6,7]

Oxygen diffusivity in gas DO2g 5.22 106 m2/s [24]

Nitrogen diffusivity in gas DN2g 2.2 106 m2/s [24]

Hydrogen diffusivity in gas DH2g 3.76 106 m2/s [24]

Oxygen diffusivity in Nafion DO2m 2.0 108 m2/s [7]

Hydrogen diffusivity in Nafion DH2m 2.59 1010 m2/s [7]

Ionic conductivity of Nafion in catalyst layer km 17 S/m [18]

Permeability of gas diffuser K 1.76 1011 m2 [1,6,7]

Permeability of membrane kp 1.8 1018 m2 [1,6,7]

Anodic transfer coefficient aa 2 [1,6,7]Cathodic transfer coefficient ac 2 [1,6,7]

Reference exchange current density multiplied by area of anode Avirefo;a 5.2 10

8 A/m3

Reference exchange current density multiplied by area of cathode Avirefo;c 110 A/m

3



18/22

As a result, the probably curve related inaccuracy, which is derived from the selected para-

meters, is omitted using the third step. The same method was used in Ref. [7] to present anaccurate model.

The parameters that are additionally required to apply our model when using the experimentaldata presented in Ref. [23] are the fuel cell geometry and stoichiometric flow ratio, leaving theAvi

refo;c and Avi

refo;a to be adjusted. In order to compare the current model with Ref. [7], which also

used the same experimental data presented in Ref. [23], our model shares parameters for the

geometry and stoichiometric flow ratio with Ref. [7] and presents half gas channel width and ribwidth for the three dimensional property of our model. Based on the three steps mentioned above,a good agreement is reached between the predicted results and the experimental data at two

operating temperatures, but with larger values for Avirefo;c than those presented in Ref. [7]. When

considering the half gas channel width and rib width as the only differences between the supplied

parameters listed in Table 2 and those presented in Ref. [7], it is deduced that the increased valueofAvi

ref

o;cis mainly derived from the addition of such three dimensional geometry parameters. In

consequence, the cathode oxidant stoichiometric flow ratio is increased to 4, an increase of 33%compared with the 3 presented in Ref. [7], and then, the new value found for Avi

refo;c at 353 K is

almost the same as that presented in Ref. [7]. The newly predicted curves with an oxidant stoi-

chiometric flow ratio of 4 show good agreement with the experimental data at two operatingtemperatures, as shown in Fig. 7. The main reason for the above operation can be explained asfollows: First, referring to Eq. (13), it is found that the oxygen mole concentration Cm and Avi

refo;c

are in direct proportion to the current density. Second, the model presented in Ref. [7] is a two

dimensional model and neglects oxygen transport in the zdirection, as shown in Fig. 2. As aresult, the oxygen mole concentration Cm at the interface between the catalyst layer and the gas

diffusion layer is always over predicted because a two dimensional model considers a uniform

distribution of oxygen in the zdirection [6,7], and then, a smaller value for Avirefo;c is acquired whenfitting the modeling results to experimental data because of the relation in Eq. (13), while with thesame stoichiometric flow ratio of 3, in order to fit our model results to the same experimental data,

a larger value for Avirefo;c is acquired in the current model because of the decreasing value ofCm in

the zdirection as shown in Ref. [22]. As a result, an increase of stoichiometric flow ratio, whichcan supply much more oxygen to the region above the rib as shown in Fig. 2, increases the value

Fig. 7. Comparison of numerically predicted polarization curves with experimental data.



19/22

for Cm, and, hence, decreases the adjustable value for Avirefo;c in our model to show good agreement

with the same experimental data in Fig. 7.Furthermore, Fig. 8 shows comparisons of the predicted polarization curves in Fig. 7 with the

predicted curves that are calculated by the current model but using the same oxidant stoichio-metric flow ratio taken from Ref. [7]. As shown in Fig. 8, when the smaller stoichiometric flowratio presented in Ref. [7] is used in our three dimensional model (the other parameters in Table 2are not changed), the current model under predicts the fuel cell performance. Based on the reasons

mentioned above, it can be concluded that the two dimensional, single phase model presented inRef. [7], which did not take into account the rib resistance to the cross diffusion of oxygen and didnot take into account the liquid water partial flooding in the gas diffuser, always over predicted

the performance of the PEM fuel cell with a smaller inlet velocity. On the contrary, with such asmaller inlet velocity, the fuel cell performance presented in Ref. [23] cannot be reached because of

the lack of enough oxygen in the region above the rib, as shown in Fig. 2, where the currentdensity is lowest for a conventional flow field as shown in Ref. [22]. As a result, the three

dimensional, two phase flow model developed above is superior to the two dimensional, singlephase models developed before [6,7].

The slight differences between the predicted curves and the experimental data that are shown

in Fig. 7 and can also be seen in Ref. [7] are mainly derived form Eq. (48). Actually, based onthe first two steps mentioned above, a predicted curve in our model can show better agreementwith its corresponding experimental voltage vs. current density at an operating temperature by

adjusting Avirefo;c and Avi

refo;a. In order to further validate the accuracy of the PEM fuel cell model

at different temperature, Eq. (48) should be used after acquiring Avirefo;c at a temperature to

calculate another Avirefo;c at another temperature. Although Eq. (48) was proved to be accurate

based on the experimental data of Parthasarathy [26], it probably shows a little error when used

in other cases [7]. In consequence, a little accuracy for Avirefo;c at a temperature should be sac-rificed in order to acquire more accuracy for Avi

refo;c at another temperature to get totally good

agreement between the predicted curves with the experimental data at two operating temper-

atures. As a result, the slight differences shown in Fig. 7 are derived from Eq. (48), instead ofthe model itself.

Fig. 8. Comparisons of predicted polarization curves in Fig. 7 with predicted curves using 2D parameters taken from

Ref. [7].



20/22

At last, based on the validation result at 353 K, as shown in Fig. 7, a comparison between thepredicted performances of PEM fuel cells with a conventional flow field and an interdigitated oneis presented in Fig. 9. The parameters presented in Table 2 are also used in modeling the PEM fuel

cell with the interdigitated flow field. As a result, the performance difference between these twofuel cells is only derived from the different structures of these two flow fields. As shown in Fig. 9,

the performance of the PEM fuel cell with interdigitated flow field is poorer than that of a PEMfuel cell with conventional flow field when no water is added to humidify the cathode inlet air(base case shown in Table 2), a case that was verified by the experiment of Wood et al. [10]. The

reason for this phenomenon is mainly derived from a larger ohmic overpotential in the membraneof a PEM fuel cell with interdigitated flow field than that in the membrane of a PEM fuel cell with

conventional flow field, and it will be discussed in detail in Part II of this series [27].

6. Conclusion

A new PEM fuel cell model has been developed, which considers species transports or reactionsin both the anode and cathode gas channels, gas diffusers and catalyst layers and in the proton

exchange membrane, that considers the impact of ribs on the species transport to extend thismodel to a three dimensional one and considers two phase flow and transport mechanisms of the

species in both the anode and cathode gas channels and gas diffusers. Two sets of boundaryconditions, one for a conventional flow field and the other for an interdigitated one, have beenpresented. A SMPLE algorithm coupled with the shooting method and fourth order Runge

Kutta method has been used to solve the equations and show a good agreement between thepredicted results and experimental data. Furthermore, when applying the same oxidant stoichi-

ometric flow ratio taken from Ref. [7] to the current model, it is found that a two dimensional,single phase model always over predicts the performance of the same PEM fuel cell. A comparisonbetween the predicted performances of PEM fuel cells with a conventional flow field and aninterdigitated one shows the performance of a PEM fuel cell with interdigitated flow field is poorer

than that of a PEM fuel cell with conventional flow field when no water is added to humidify thecathode inlet air.

Fig. 9. Comparison of numerically predicted polarization curves for PEM fuel cells using conventional and inter-

giditated flow fields without cathode humidification at an operating temperature of 353 K.



21/22

The internal transport mechanisms in PEM fuel cells, which are derived from the above cal-

culation as shown in Fig. 9, will be discussed in Part II of this series [27].

Acknowledgements

This work was supported by the 985 Funds of Shanghai Jiao Tong University, Shanghai,China and by National Nature Science Foundation of China (No. 50206012). The authors would

like to thank Dr. Hongtan Liu, Director of Dorgan Solar Energy and Fuel Cell Laboratories,University of Miami, USA, for his constructive advices for revising this paper and his kindnessand patience.

References

[1] Bernardi DM, Verbrugge MW. A mathematical model of the solid-polymer-electrolyte fuel cell. J Electrochem Soc

1992;139(9):247791.

[2] Springer TE, Zawodzinski TA, Gottesfeld S. Polymer electrolyte fuel cell model. J Electrochem Soc

1991;138(8):233442.

[3] Fuller TF, Newman J. Water and thermal management in solid-polymer-electrolyte fuel cells. J Electrochem Soc

1993;140(5):121824.

[4] Nguyen TV, White RE. A water and heat management model for proton-exchange-membrane fuel cells.

J Electrochem Soc 1993;140(8):217886.

[5] Yi JS, Nguyen TV. An along-the-channel model for proton exchange membrane fuel cells. J Electrochem Soc

1998;145(4):114959.

[6] Gurau V, Liu H, Kakac S. Two-dimensional model for proton exchange membrane fuel cells. AIChE

J 1998;44(11):241022.

[7] Um S, Wang CY, Chen KS. Computational fluid dynamics modeling of proton exchange membrane fuel cells.


[8] West AC. Influence of rib spacing in proton-exchange membrane electrode assemblies. J Appl Electrochem

1996;26:55765.

[9] Dutta S, Shimpalee S, Van Zee JW. Three-dimensional numerical simulation of straight channel PEM fuel cells.

J Appl Electrochem 2000;30:13546.

[10] Wood III DL, Yi JS, Nguyen TV. Effect of direct liquid water injection and interdigitated flow field on the

performance of proton exchange membrane fuel cells. Electrochim Acta 1998;43(24):3795809.

[11] Baschuk JJ, Li X. Modelling of polymer electrolyte membrane fuel cells with variable degrees of water flooding.

J Power Source 2000;86(1):18196.

[12] You L, Liu H. A two-phase flow and transport model for the cathode of PEM fuel cells. Int J Heat Mass Transfer

2002;(45):227787.[13] Nguyen TV. A gas distributor design for proton-exchange-membrane fuel cells. J Electrochem Soc 1996;

143(5):L1035.

[14] Yi JS, Van Nguyen T. Multicomponent transport in porous electrode of proton exchange membrane fuel cells

using the interdigitated gas distributors. J Electrochem Soc 1999;146(1):3845.

[15] Wang CY, Cheng P. A multiphase mixture model of multiphase, multicomponent transport in capillary porous

mediaI. Model development. Int J Heat Mass Transfer 1996;39(10):360718.

[16] Wang CY, Gu WB. Micro-macroscopic coupled modeling of batteries and fuel cellsI. Model development.


[17] Weisbrod KR, Grot SA, Vanderborgh NE. Through-the-electrode model of a proton exchange membrane fuel cell.

Electrochem Soc Proc 1995;9523:152.



22/22

[18] Marr C, Li X. Composition and performance modeling of catalyst layer in a proton exchange membrane fuel cell.

J Power Source 1999;77(1):1727.

[19] You L, Liu H. A parametric study of the cathode catalyst layer of PEM fuel cells using a pseudo-homogeneous

model. Int J Hydrogen Energy 2001;26(9):9919.

[20] Ge S, Yi B, Xu H. Model of water transport for proton-exchange membrane fuel cell (PEMFC). J Chem Ind Eng

(China) 1999;50(1):3947.

[21] Patankar SV. Numerical heat transfer and fliud flow. New York: Hemisphere; 1980.

[22] Berning T, Lu DM, Djilali N. Three-dimensional computational analysis of transport phenomena in a PEM fuel

cell. J Power Source 2002;106:28494.

[23] Ticianelli EA, Derouin CR, Srinivasan S. Localization of platimun in low catalyst loading electrodes to attain high

power densities in SPE fuel cells. J Electroanal Chem 1988;251:27595.

[24] Cussler EL. Diffusion mass transfer in fluid systems. New York: Cambridge University Press; 1984.

[25] Rowe A, Li X. Mathematical modeling of proton exchange membrane fuel cells. J Power Source 2001;102(12):

8296.

[26] Parthasarathy A, Srinivasan S, Appleby J. Temperature dependence of the electrode kinetics of oxygen reduction at

the platinum/Nafion interfacea microelectrode investigation. J Electrochem Soc 1992;(139):2530.

[27] Hu M, Zhu X, Wang M, Gu A. Three dimensional, two phase flow mathematical model for PEM fuel cell: Part II.Analysis and discussion of the internal transport mechanisms. Energy Convers Manage, doi:10.1016/j.encon-

man.2003.09.023.

http://dx.doi.org/10.1016/j.enconman.2003.09.023http://dx.doi.org/10.1016/j.enconman.2003.09.023http://dx.doi.org/10.1016/j.enconman.2003.09.023http://dx.doi.org/10.1016/j.enconman.2003.09.023

three dimensional, two phase ﬂow mathematical model for pem fuel cell. part i. model development

Documents