Fuel Cells Module Manual

FLUENT 6.3

Fuel Cell Modules Manual

September 2006

Copyright c© 2006 by Fluent Inc.All rights reserved. No part of this document may be reproduced or otherwise used in

any form without express written permission from Fluent Inc.

Airpak, FIDAP, FLUENT, FLUENT for CATIA V5, FloWizard, GAMBIT, Icemax, Icepak,Icepro, Icewave, Icechip, MixSim, and POLYFLOW are registered trademarks of FluentInc. All other products or name brands are trademarks of their respective holders.

CHEMKIN is a registered trademark of Reaction Design Inc.

Portions of this program include material copyrighted by PathScale Corporation2003-2004.

Fluent Inc.Centerra Resource Park

10 Cavendish CourtLebanon, NH 03766

Contents

Preface UTM-1

1 PEM Fuel Cell Model Theory 1-1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.2 Electrochemistry Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.3 Current and Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . 1-6

1.4 Liquid Water Formation, Transport, and its Effects . . . . . . . . . . . . 1-7

1.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

1.6 Transient Simulation of PEM Fuel Cells . . . . . . . . . . . . . . . . . . 1-10

2 Using the PEM Fuel Cell Model 2-1

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2 Geometry Definition for the PEM Fuel Cell Model . . . . . . . . . . . . 2-2

2.3 Installing the PEM Fuel Cell Model . . . . . . . . . . . . . . . . . . . . 2-2

2.4 Loading the PEM Fuel Cell Module . . . . . . . . . . . . . . . . . . . . . 2-3

2.5 Setting Up the PEM Fuel Cell Module . . . . . . . . . . . . . . . . . . . 2-3

2.6 PEM Fuel Cell Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

2.6.1 Specifying the PEM Model Options . . . . . . . . . . . . . . . . 2-5

2.6.2 Specifying the PEM Model Parameters . . . . . . . . . . . . . . 2-6

2.6.3 Specifying Anode Properties . . . . . . . . . . . . . . . . . . . . 2-7

2.6.4 Specifying Membrane Properties . . . . . . . . . . . . . . . . . . 2-11

2.6.5 Specifying Cathode Properties . . . . . . . . . . . . . . . . . . . 2-12

2.6.6 Setting Advanced Properties for the PEM Fuel Cell . . . . . . . 2-16

2.6.7 Specifying the Solution Controls . . . . . . . . . . . . . . . . . . 2-20

2.6.8 Reporting on the Solution . . . . . . . . . . . . . . . . . . . . . . 2-21

c© Fluent Inc. July 19, 2006 i

CONTENTS

2.7 Modeling of Current Collectors . . . . . . . . . . . . . . . . . . . . . . . 2-22

2.8 PEM Fuel Cell Boundary Conditions . . . . . . . . . . . . . . . . . . . . 2-23

2.9 Solution Guidelines for the PEM Fuel Cell Model . . . . . . . . . . . . . 2-24

2.10 Postprocessing the PEM Fuel Cell Model . . . . . . . . . . . . . . . . . . 2-24

2.11 User-Accessible Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2-26

2.11.1 Compiling the Customized PEM Fuel Cell Source Code . . . . . 2-29

2.12 IV-Curve Calculations Using the Text Interface . . . . . . . . . . . . . . 2-30

3 SOFC Fuel Cell Model Theory 3-1

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1.1 Motivation for Consideration of the Electric Field . . . . . . . . 3-3

3.1.2 Overview of the Electric Field Model . . . . . . . . . . . . . . . . 3-3

3.1.3 Overview of the Electrochemical Model . . . . . . . . . . . . . . 3-4

3.2 The SOFC Modeling Strategy . . . . . . . . . . . . . . . . . . . . . . . . 3-4

3.3 Modeling Stacked Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

3.4 Modeling Fluid Flow, Heat Transfer, and Mass Transfer . . . . . . . . . 3-6

3.5 Modeling Current Transport and the Potential Field . . . . . . . . . . . 3-6

3.5.1 Treatment of the Activation Overpotential . . . . . . . . . . . . 3-7

3.5.2 Cell Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

3.6 Modeling Electrochemical Reactions . . . . . . . . . . . . . . . . . . . . 3-15

4 Using the Solid Oxide Fuel Cell Model 4-1

4.1 Installing the Solid Oxide Fuel Cell Model . . . . . . . . . . . . . . . . . 4-1

4.2 Loading the Solid Oxide Fuel Cell Module . . . . . . . . . . . . . . . . . 4-1

4.3 Solid Oxide Fuel Cell Module Set Up Procedure . . . . . . . . . . . . . . 4-2

4.4 Setting the Parameters for the SOFC Model . . . . . . . . . . . . . . . . 4-8

4.5 Setting Up the Electrode-Electrolyte Interfaces . . . . . . . . . . . . . . 4-10

4.5.1 Setting Up the Anode Electrode-Electrolyte Interface . . . . . . 4-10

4.5.2 Setting Up the Cathode Electrode-Electrolyte Interface . . . . . 4-11

4.6 Setting Up the Electric Field Model Parameters . . . . . . . . . . . . . . 4-12

ii c© Fluent Inc. July 19, 2006

CONTENTS

4.7 Setting Up the Tortuosity Parameters . . . . . . . . . . . . . . . . . . . 4-13

4.8 Setting Up the Activation Parameters . . . . . . . . . . . . . . . . . . . 4-14

c© Fluent Inc. July 19, 2006 iii

CONTENTS

iv c© Fluent Inc. July 19, 2006

Using This Manual

The Contents of This Manual

The FLUENT Fuel Cell Modules Manual tells you what you need to know to modelpolymer electrolyte membrane (PEM) fuel cells or solid oxide fuel cell (SOFC) fuel cellswith FLUENT. In this manual, you will find background information pertaining to themodels, a theoretical discussion of the models used in FLUENT, and a description of usingthe models for your CFD simulations.

Typographical Conventions

Several typographical conventions are used in this manual’s text to facilitate your learningprocess.

• An informational icon ( i ) marks an important note.

• An warning icon ( ! ) marks a warning.

• Different type styles are used to indicate graphical user interface menu items andtext interface menu items (e.g., Iso-Surface panel, surface/iso-surface com-mand).

• The text interface type style is also used when illustrating exactly what appears onthe screen or exactly what you need to type into a field in a panel. The informationdisplayed on the screen is enclosed in a large box to distinguish it from the narrativetext, and user inputs are often enclosed in smaller boxes.

• A mini flow chart is used to indicate the menu selections that lead you to a specificcommand or panel. For example,

Define −→Boundary Conditions...

indicates that the Boundary Conditions... menu item can be selected from the Definepull-down menu, and

display −→grid

indicates that the grid command is available in the display text menu.

The words before the arrows invoke menus (or submenus) and the arrows pointfrom a specific menu toward the item you should select from that menu. In this

c© Fluent Inc. July 19, 2006 UTM-1

Using This Manual

manual, mini flow charts usually precede a description of a panel or command, ora screen illustration showing how to use the panel or command. They allow youto look up information about a command or panel and quickly determine how toaccess it without having to search the preceding material.

• The menu selections that will lead you to a particular panel are also indicated(usually within a paragraph) using a “/”. For example, Define/Materials... tellsyou to choose the Materials... menu item from the Define pull-down menu.

Mathematical Conventions• Where possible, vector quantities are displayed with a raised arrow (e.g., ~a, ~A).

Boldfaced characters are reserved for vectors and matrices as they apply to linearalgebra (e.g., the identity matrix, I).

• The operator ∇, referred to as grad, nabla, or del, represents the partial derivativeof a quantity with respect to all directions in the chosen coordinate system. InCartesian coordinates, ∇ is defined to be

∂

∂x~ı+

∂

∂y~+

∂

∂z~k

∇ appears in several ways:

– The gradient of a scalar quantity is the vector whose components are thepartial derivatives; for example,

∇p =∂p

∂x~ı+

∂p

∂y~+

∂p

∂z~k

– The gradient of a vector quantity is a second-order tensor; for example, inCartesian coordinates,

∇(~v) =

(∂

∂x~ı+

∂

∂y~+

∂

∂z~k

)(vx~ı+ vy~+ vz~k

)This tensor is usually written as

∂vx∂x

∂vx∂y

∂vx∂z

∂vy∂x

∂vy∂y

∂vy∂z

∂vz∂x

∂vz∂y

∂vz∂z

UTM-2 c© Fluent Inc. July 19, 2006

Using This Manual

– The divergence of a vector quantity, which is the inner product between ∇and a vector; for example,

∇ · ~v =∂vx∂x

+∂vy∂y

+∂vz∂z

– The operator ∇ · ∇, which is usually written as ∇2 and is known as theLaplacian; for example,

∇2T =∂2T

∂x2+∂2T

∂y2+∂2T

∂z2

∇2T is different from the expression (∇T )2, which is defined as

(∇T )2 =

(∂T

∂x

)2

+

(∂T

∂y

)2

+

(∂T

∂z

)2

Technical Support

If you encounter difficulties while using FLUENT, please first refer to the section(s) ofthe manual containing information on the commands you are trying to use or the typeof problem you are trying to solve. The product documentation is available from theonline help on the documentation CD, or from the Fluent Inc. User Services Center(www.fluentusers.com).

If you encounter an error, please write down the exact error message that appearedand note as much information as you can about what you were doing in FLUENT.Then refer to the following resources available on the Fluent Inc. User Services Cen-ter (www.fluentusers.com):

• Installation and System FAQs - link available from the main page on the UserServices Center. The FAQs can be searched by word or phrase, and are availablefor general installation questions as well as for products.

• Known defects for FLUENT - link available from the product page. The defects canbe searched by word or phrase, and are listed by categories.

• Online Technical Support - link available from the main page on the User ServicesCenter. From the Online Technical Support Portal page, there is a link to theSearch Solutions & Request Support page, where the solutions can be searched byword or phrase.

The User Services Center also provides online forums, where you can discuss topics ofmutual interest and share ideas and information with other Fluent users, and the abilityto sign up for e-mail notifications on our latest product releases.

c© Fluent Inc. July 19, 2006 UTM-3

Using This Manual

Contacting Technical Support

If none of the resources available on the User Services Center help in resolving the prob-lem, or you have complex modeling projects, we invite you to call your support engineerfor assistance. However, there are a few things that we encourage you to do before calling:

• Note what you are trying to accomplish with FLUENT.

• Note what you were doing when the problem or error occurred.

• Save a journal or transcript file of the FLUENT session in which the problem oc-curred. This is the best source that we can use to reproduce the problem andthereby help to identify the cause.

UTM-4 c© Fluent Inc. July 19, 2006

Chapter 1. PEM Fuel Cell Model Theory

This chapter presents the theoretical background for the PEM fuel cell modeling capa-bilities in FLUENT.

• Section 1.1: Introduction

• Section 1.2: Electrochemistry Modeling

• Section 1.3: Current and Mass Conservation

• Section 1.4: Liquid Water Formation, Transport, and its Effects

• Section 1.5: Properties

• Section 1.6: Transient Simulation of PEM Fuel Cells

1.1 Introduction

The PEM Fuel Cell (PEMFC) module is provided as an addon module with the standardFLUENT licensed software. A special license is required to use the PEMFC module.

A fuel cell is an energy conversion device that converts the chemical energy of fuel intoelectrical energy. A schematic of a polymer electrolyte membrane (PEM) fuel cell isshown in Figure 1.1.1.

c© Fluent Inc. July 19, 2006 1-1

PEM Fuel Cell Model Theory

��

��

O + 4H + 4e −2 22H O+

−+ 4H + 4e 22H

Cooling Channel(s)

Cathode Collector

Cathode Gas Diffusion Layer

Cathode Catalyst Layer

Anode Catalyst Layer

Anode Gas Diffusion Layer

Anode CollectorCooling Channel(s)

e −

��

��

load+ +H H

2Gas Channel (H )

Gas Channel (air)

Electrolyte Membrane

Figure 1.1.1: Schematic of a PEM Fuel Cell

Hydrogen flows into the fuel cell on the anode side. It diffuses through the porous gasdiffusion layers and comes in contact with the catalyst layer. Here it forms hydrogenions and electrons. The hydrogen ions diffuse through the polymer electrolyte membraneat the center, the electrons flow through the gas diffusion layer to the current collectorsand into the electric load attached. Electrons enter the cathode side through the currentcollectors and the gas diffusion layer. At the catalyst layer on the cathode side, theelectrons, the hydrogen ions and the oxygen combine to form water.

In the PEM fuel cell model in FLUENT, two electric potential fields are solved. Onepotential is solved in the membrane and catalyst layers. The other is solved in thecatalyst layers, the diffusion layers, and the current collectors. Surface reactions onthe porous catalyst region are solved and the reaction diffusion balance is applied tocompute the rates. Based on the cell voltage that you prescribe, the current densityvalue is computed. Alternatively, a cell voltage can be computed based on a prescribedaverage current density.

1-2 c© Fluent Inc. July 19, 2006

1.2 Electrochemistry Modeling


At the center of the electrochemistry is the computation of the rate of the hydrogenoxidation and the rate of oxygen reduction. In the FLUENT PEM model, these electro-chemical processes are treated as heterogeneous reactions that take place on the catalystsurfaces inside the two catalyst layers on both sides of the membrane. Such detailedtreatment has been used by other groups ([1], [2], and [8]).

The driving force behind these reactions is the surface over-potential: the difference be-tween the phase potential of the solid and the phase potential of the electrolyte/membrane.Therefore, two potential equations are solved for in the PEM model: one potential equa-tion (Equations 1.2-1) accounts for the electron transport e− through the solid conductivematerials (i.e., the current collectors and solid grids of the porous media); the other po-tential equation (1.2-2) represents the protonic (i.e., ionic) transport of H+ . The twopotential equations read,

∇ · (σsol∇φsol) +Rsol = 0 (1.2-1)

∇ · (σmem∇φmem) +Rmem = 0 (1.2-2)

where

σ = electrical conductivity (1/ohm-m)φ = electric potential (volts)R = volumetric transfer current (A/m3)

The following figure illustrates the boundary conditions that are used to solve for φsol

and φmem.

There are two types of external boundaries. Those through which there passes an elec-trical current and those through which there passes no current.

As no protonic current leaves the fuel cell through any external boundary, there is azero flux boundary condition for the membrane phase potential, φmem, on all outsideboundaries.

For the solid phase potential, φsol, there are external boundaries on the anode and thecathode side that are in contact with the external electric circuit and only through theseboundaries passes the electrical current generated in the fuel cell. On all other externalboundaries there is a zero flux boundary condition for φsol.

On the external contact boundaries, we recommend to prescribe fixed values for φsol

(potentiostatic boundary conditions). If the anode side is set to zero, the (positive) valueprescribed on the cathode side is the cell voltage. Specifying a constant flux (say on thecathode side) means to specify galvanostatic boundary conditions.



�

�

�

�

��

?

6

-�

Cathode Catalyst Layer

Cooling Channel(s)

Anode Collector

Anode Gas Diffusion Layer

Anode Catalyst Layer

Cathode Gas Diffusion Layer

Cathode Collector

Cooling Channel(s)

2H2 −→ 4H+ + 4e−

φsol = 0

φsol = Vcell or ∂φsol∂n

= constant

∂φmem∂n

= 0∂φsol∂n

= 0

∂φmem∂n

= 0

∂φmem∂n

= 0

∂φmem∂n

= 0∂φsol∂n

= 0

O2 + 4H+ + 4e− −→ 2H2O

? ?

�

+ +H H

�

Gas Channel (air)

Electrolyte Membrane

Gas Channel (H2)

Figure 1.2.1: Boundary Conditions for φsol and φmem

The transfer currents, or the source terms in Equations 1.2-1 and 1.2-2, are non-zero onlyinside the catalyst layers and are computed as:

• For the solid phase, Rsol = −Ran(< 0) on the anode side and Rsol = +Rcat(> 0) onthe cathode side.

• For the membrane phase, Rmem = +Ran(> 0) on the anode side and Rmem =−Rcat(< 0) on the cathode side.

The source terms in Equations 1.2-1 and 1.2-2 are also called the exchange current density(A/m3), and have the following general definitions:

Ran = jrefan

([H2]

[H2]ref

)γan (eαanFηan/RT − e−αcatFηan/RT

)(1.2-3)

Rcat = jrefcat

([O2]

[O2]ref

)γcat (−e+αanFηcat/RT + e−αcatFηcat/RT

)(1.2-4)



where

jref = volumetric reference exchange current density (A/m3)[ ],[ ]ref = local species concentration, reference value (kgmol/m3)γ = concentration dependence (dimensionless)α = transfer coefficient (dimensionless)F = Faraday constant (9.65× 107 C/kgmol)

The above equation is the general formulation of the Butler-Volmer function. A simpli-fication to this is the Tafel formulation that reads,

Ran = jrefan

([H2]

[H2]ref

)γan (eαanFηan/RT

)(1.2-5)

Rcat = jrefcat

([O2]

[O2]ref

)γcat (e−αcatFηcat/RT

)(1.2-6)

By default, the Butler-Volmer function is used in the FLUENT PEM model to computethe transfer currents inside the catalyst layers.

The driving force for the kinetics is the local surface over-potential, η, also known as theactivation loss. It is generally the difference between the solid and membrane potentials,φsol and φmem.

The gain in electrical potential from crossing from the anode to the cathode side canthen be taken into account by subtracting the open-circuit voltage Voc on the cathodeside.

ηan = φsol − φmem (1.2-7)

ηcat = φsol − φmem − Voc (1.2-8)

From Equations 1.2-1 through 1.2-8, the two potential fields can be obtained.



1.3 Current and Mass Conservation

The following reactions occur, respectively, at the anode and the cathode:

H2 −→ 2H+ + 2e−

O2 + 4H+ + 4e− −→ 2H2O

The volumetric source terms for the species equations (kg/m3-s) and energy equation(W/m3) are given in Equations 1.3-1–1.3-4.

SH2 = −Mw,H2

2FRan (1.3-1)

SO2 = −Mw,O2

4FRcat (1.3-2)

SH20 =Mw,H20

2FRcat (1.3-3)

Additional volumetric sources to the energy equation implemented in the FLUENT PEMmodel include ohmic heating, heat of formation of water, electric work and latent heatof water.

Sh = I2Rohm + hreaction + ηRan,cat + hphase (1.3-4)

The electrochemical reactions that take place inside the catalyst layers are consideredheterogeneous reactions that take place on the catalyst surfaces in the porous media.Therefore, the species concentrations of hydrogen and oxygen in the rate calculation,Equations 1.2-3 and 1.2-6, are the surface values. The reactions are treated as surfacereactions in the two catalyst layers, and it is assumed that the diffusive flux of anyreacting species is balanced by its rate of production.

ρDi

δ(yi,surf − yi,cent)r =

Mw,i

nFRan,cat (1.3-5)

where

Di = mass diffusivity of species i (m2/s)r = specific reacting surface area of the catalyst layer,

or surface-to-volume ratio (1/m)yi,surf = mass fraction of species i at the reacting surfaceyi,cent = mass fraction of species i at the cell centerδ = average distance between the reaction surfaces and the cell center (m)


1.4 Liquid Water Formation, Transport, and its Effects

The left hand side of Equation 1.3-5 represents the diffusive flux at the reacting surfaceand the right hand side represents the rate of mass generation. The average distancefrom the cell-center to the reacting surface is estimated as δ = 1/r . Equation 1.3-5 isused to obtain the surface values of H2 and O2 concentrations, applying a Newtoniansolution procedure. These surface, or wall, values are then used to compute the rates inEquations 1.2-3 through 1.2-6.

1.4 Liquid Water Formation, Transport, and its Effects

Since PEM fuel cells operate under relatively low temperature (<100◦C), the water vapormay condense to liquid water, especially at high current densities. While the existence ofthe liquid water keeps the membrane hydrated, it also blocks the gas diffusion passage,reduces the diffusion rate and the effective reacting surface area and hence the cell perfor-mance. To model the formation and transport of liquid water, FLUENT uses a saturationmodel based on [7],[5]. In this approach, the liquid water formation and transport isgoverned by the following conservation equation for the volume fraction of liquid water,s, or the water saturation,

∂(ερls)

∂t+∇ ·

(ρl−→V ls

)= rw (1.4-1)

where the subscript l stands for liquid water, and rw is the condensation rate that ismodeled as,

rw = cr max([

(1− s) Pwv − Psat

RTMw,H20

], [−sρl]

)(1.4-2)

where −rw is added to the water vapor equation, as well as the pressure correction (masssource). This term is not applied inside the membrane. The condensation rate constantis hardwired to cr = 100s−1. It is assumed that the liquid velocity, Vl, is equivalent to thegas velocity inside the gas channel (i.e., a fine mist). Inside the highly-resistant porouszones, the use of the capillary diffusion term allows us to replace the convective term inEquation 1.4-1:

∂(ερls)

∂t+∇ ·

[ρlKs3

µl

dpc

ds∇s]

= rw (1.4-3)

Depending on the wetting phase, the capillary pressure is computed as a function of s(the Leverett function),

pc =

σcosθc(Kε

)0.5 (1.417(1− s)− 2.12(1− s)2 + 1.263(1− s)3) θc < 90◦σcosθc(Kε

)0.5 (1.417s− 2.12s2 + 1.263s3) θc > 90◦ (1.4-4)



where ε is the porosity, σ is the surface tension (N/m2), θc is the contact angle and Kthe absolute permeability.

Equation 1.4-1 models various physical processes such as condensation, vaporization,capillary diffusion, and surface tension.

The clogging of the porous media and the flooding of the reaction surface are modeledby multiplying the porosity and the active surface area by (1− s), respectively.

1.5 Properties

The gas phase species diffusivities are given by

Di = ε1.5(1− s)rsD0i

(p0

p

)γp ( TT0

)γt(1.5-1)

where D0i is the mass diffusivity of species i at reference temperature and pressure (P0,

T0) [8]. These reference values and the exponents (γp, γt) as well as the exponent of poreblockage (rs) are defined in the PEM user defined functions (UDF) as,

p0 = 101325 N/m2

T0 = 300 K

γp = 1.0

γt = 1.5

rs = 2.5

The electrolyte membrane of the fuel cell is modeled as a porous fluid zone. Propertiessuch as membrane phase electrical conductivity, water diffusivity, and the osmotic dragcoefficient are evaluated as functions of the water content, using various correlations assuggested by [9]. To capture the relevant physics of the problem, various properties ofthe membrane are incorporated into the model as default options. You can, however,directly incorporate your own formulations and data for these properties by editing thefunctions defined in the provided source code file called pem user.c and compiling thecode yourself. For more information, see Section 2.11: User-Accessible Functions.

• Membrane Phase Electric Conductivity

σmem = βε(0.514λ− 0.326)ωe1268( 1303− 1T ) (1.5-2)

where λ is the water content. Two model constants, β and ω are introduced inFLUENT for generality. Equation 1.5-2 becomes the original correlation from [9]when β = ω = 1.


1.5 Properties

• Osmotic Drag Coefficient

αd = 2.5λ

22(1.5-3)

• Back Diffusion Flux

Jdiffw = − ρmMm

Mh20Dl∇λ (1.5-4)

where ρm and Mm are the density and the equivalent weight of the dry membrane,respectively.

• Membrane Water Diffusivity

Dl = f(λ)e2416( 1303− 1T ) (1.5-5)

• Water Content

The water content, λ, that appears in the preceding property computations areobtained using Springer et al’s correlation [9],

λ = 0.043 + 17.18a− 39.85a2 + 36a3(a < 1)

λ = 14 + 1.4(a− 1)(a > 1) (1.5-6)

here a is the water activity that is defined as,

a =Pwv

Psat

+ 2s (1.5-7)

• Water Vapor Pressure

The water vapor pressure is computed based upon the vapor molar fraction andthe local pressure,

Pwv = xH2OP (1.5-8)

• Saturation Pressure

The saturation pressure is calculated, in terms of atm, as,

log10Psat = −2.1794 + 0.02953(T − 273.17)−9.1837× 10−5(T − 273.17)2 +

1.4454× 10−7(T − 273.17)3 (1.5-9)



It is noted here that in [9], water activity is defined on the basis of total wateror super-saturated water vapor. With phase change being invoked in the presenttwo-phase model, 2s is added to the original formulation as suggested by [3].

1.6 Transient Simulation of PEM Fuel Cells

Dynamics response of PEM fuel cells to changes in operating conditions as a function oftime can be modeled in the PEMFC module. For example, a change in the cell voltage orcurrent density, or inlet mass flow rates at the anode and/or the cathode. The procedurefor setting up and solving transient PEMFC problems are the same as that used for anormal FLUENT transient problem as discussed in the FLUENT User’s Guide.

Assuming that the time scales associated with the electric fields are much smaller thanthose associated with the flow and thermal fields, the steady-state equations are retainedfor the two electric potentials, (i.e., Equations 1.2-1 and 1.2-2). Transient terms in allother equations such as momentum transport, energy transport, species transport, liquidwater transport, and membrane water content equations are activated.


Chapter 2. Using the PEM Fuel Cell Model

The procedure for setting up and solving PEM fuel cell problems is described in detailin this chapter. Please refer to the following sections for more information:


• Section 2.2: Geometry Definition for the PEM Fuel Cell Model

• Section 2.3: Installing the PEM Fuel Cell Model

• Section 2.4: Loading the PEM Fuel Cell Module

• Section 2.5: Setting Up the PEM Fuel Cell Module

• Section 2.6: PEM Fuel Cell Modeling

• Section 2.7: Modeling of Current Collectors

• Section 2.8: PEM Fuel Cell Boundary Conditions

• Section 2.9: Solution Guidelines for the PEM Fuel Cell Model

• Section 2.10: Postprocessing the PEM Fuel Cell Model

• Section 2.11: User-Accessible Functions

• Section 2.12: IV-Curve Calculations Using the Text Interface

2.1 Introduction

The FLUENT PEM Fuel Cell (PEMFC) model is comprised of several user-defined func-tions (UDFs) and a graphical user interface. The potential fields are solved as user-defined scalars. The liquid water saturation, s, and the water content, λ, are also solvedas user-defined scalars. The electrochemical reactions occurring on the catalyst are mod-eled through various source terms while other model parameters are handled through theuser interface. The PEMFC model can be used in parallel FLUENT as well.


Using the PEM Fuel Cell Model

2.2 Geometry Definition for the PEM Fuel Cell Model

Due to the fact that there are a number of different physical zones associated with thefuel cell, the following regions must be present in the fuel cell mesh:

• Anode flow channel

• Anode gas diffusion layer

• Anode catalyst layer

• Membrane layer

• Cathode catalyst layer

• Cathode gas diffusion layer

• Cathode flow channel

The following zones have to be identified, if present in the fuel cell mesh:

• Anode current collector

• Cathode current collector

• Coolant channel

2.3 Installing the PEM Fuel Cell Model

The PEM Fuel Cell (PEMFC) model is provided as an addon module with the standardFLUENT licensed software. A special license is required to use the PEMFC model.The module is installed with the standard installation of FLUENT in a directory calledaddons/fuelcells2.2 in your installation area. The PEMFC model consists of a UDFlibrary and a pre-compiled scheme library, which needs to be loaded and activated beforecalculations can be performed.


2.4 Loading the PEM Fuel Cell Module

2.4 Loading the PEM Fuel Cell Module

The PEM Fuel Cell (PEMFC) module is loaded into FLUENT through the text userinterface (TUI). The module can only be loaded after a valid FLUENT case file has beenset or read. The text command to load the addon module is

define −→ models −→addon-module

A list of FLUENT addon modules is displayed:

FLUENT Addon Modules:

1. MHD Model

2. Fiber Model

3. PEM Fuel Cell Model

4. SOFC Fuel Cell Model

5. Population Balance Model

Enter Module Number: [1] 3

Select the PEMFC model by entering the module number 3. During the loading processa scheme library containing the graphical and text user interface, and a UDF librarycontaining a set of user defined functions are loaded into FLUENT.

2.5 Setting Up the PEM Fuel Cell Module

The following describes an overview of the procedure required in order to use the PEMfuel cell model in FLUENT.

1. Start FLUENT.

2. Read the case file.

3. Scale the grid.

4. Use the PEM Model panel to define the fuel cell model parameters.

5. Define material properties.

6. Set the operating conditions.

7. Set the boundary conditions.

8. Start the calculations.

9. Save the case and data files.

10. Process your results.



i The PEM Model panel greatly simplifies the input of parameters and bound-ary conditions, but it does not replace the boundary conditions interface.Therefore it is a good policy, to start the set up with the PEM Model paneland do the finishing steps for boundary conditions afterwards.

2.6 PEM Fuel Cell Modeling

In order to set PEM fuel cell model parameters and assign properties to the relevantregions in your fuel cell, you need to access the PEM fuel cell graphical user interface(the PEM Model panel) using:

Define −→ Models −→PEMFC

Here, you can identify the relevant zones for the current collectors, flow channels, gasdiffusion layers, catalyst layers, and the membrane. You can specify the following inputsusing the PEM Model panel. Optional inputs are indicated as such.

1. Enable either the single-phase or the multi-phase PEM model.

2. Set the appropriate options for the PEM model (optional).

3. Set the various parameters for the PEM model.

4. Select the appropriate zones and specify the properties on the anode side.

5. Select the appropriate zones and specify the properties of the membrane.

6. Select the appropriate zones and specify the properties on the cathode side.

7. Provide input for advanced features such as contact resistivities, coolant channelproperties, or stack management settings (optional).

8. Set solution controls such as under-relaxation factors (optional).

9. Provide input to assist reporting (optional).



2.6.1 Specifying the PEM Model Options

The Model tab of the PEM Model panel allows you to turn on or off various options whensolving a PEM fuel cell problem.

Figure 2.6.1: The Model Tab of the PEM Model Panel

Several PEM model options are available in the Model Tab of the PEM Model panelincluding:

• The Joule Heating option takes into account ohmic heating. This option includesthe I2R term in the energy source term from Equation 1.3-4 in the calculations.

• The Reaction Heating option takes into account the heat generated by the chemicalreactants. This option includes the hreaction term in the energy source term fromEquation 1.3-4 in the calculations. This accounts for the latent heat of formationof water vapor.

• The Butler-Volmer Rate option (the default) is used to compute the transfer currentsinside the catalyst layers. If this option is turned off, the Tafel approximation(Equation 1.2-6) is used.

• The Membrane Water Transport option takes into account the transport of wateracross the membrane.

• The Multiphase option takes into account multiphase calculations. Use this optionif you are solving for approximate liquid transport in the gas diffusion layer of thefuel cell.

Nearly all options are turned on by default. You may wish to override the default values,depending on the problem you wish to model. For instance, if you are not concernedwith the heat generated due to chemical reaction, then you may want to turn off theReaction Heating option.



2.6.2 Specifying the PEM Model Parameters

You can use the Parameters tab of the PEM Model panel to specify the electrochemistryparameters for the PEM model, reference diffusivities for the reactants, among othermodel parameters.

Figure 2.6.2: The Parameters Tab of the PEM Model Panel

There are various parameters under Electrochemistry in the PEM Model panel. For boththe anode and the cathode, you can also set the following parameters or leave the defaultvalues.

• The Ref. Current Density corresponds to jrefan and jrefcat , the reference exchangecurrent density from Equation 1.2-3 and Equation 1.2-4.

• The Ref. Concentration corresponds to the reference concentration ([H2]ref and[O2]ref) with units of 1 kgmol/m3 (see Equation 1.2-3 and Equation 1.2-4).

• The Concentration Exponent corresponds to γ, the concentration dependence fromEquation 1.2-3.

• The Exchange Coefficient corresponds to α, the transfer coefficient from Equa-tion 1.2-3.

• The Open-Circuit Voltage corresponds to Voc in Equation 1.2-8.

Moreover, the following parameters can also be set here:

• The Reference Diffusivities correspond to Di from Equation 1.3-5, the species massdiffusivity.

• The Saturation Exponent for Pore Blockage corresponds to rs from Equation 1.5-1for multiphase PEM calculations.



2.6.3 Specifying Anode Properties

You can use the Anode tab of the PEM Model panel to specify zones and properties ofthe current collector, the flow channel, the diffusion layer, and the catalyst layer for theanode portion of the PEM fuel cell.

Specifying Current Collector Properties for the Anode

Figure 2.6.3: The Anode Tab of the PEM Model Panel With Current CollectorSelected

1. Select the Anode tab of the PEM Model panel.

2. Select Current Collector under Anode Zone Type.

3. Select a corresponding zone from the Zone(s) list. If you are modeling a fuel cellstack, then you must pick all zones of a particular type as a group.

4. Select a Solid Material from the corresponding drop-down list. Solid materials canbe customized using the Materials panel. Note that for the Electrical Conductivity,you can only choose a constant value in the Materials panel. The solid electricalconductivity value is the diffusivity of the solid phase potential in the solid zones.



Specifying Flow Channel Properties for the Anode

Figure 2.6.4: The Anode Tab of the PEM Model Panel With Flow ChannelSelected


2. Select Flow Channel under Anode Zone Type.

3. Select a corresponding zone from the Zone(s) list.



Specifying Diffusion Layer Properties for the Anode

Figure 2.6.5: The Anode Tab of the PEM Model Panel With Diffusion LayerSelected


2. Select Diffusion Layer under Anode Zone Type.



5. Specify a value for the Porosity.

6. Specify a value for the Viscous Resistance.

7. Specify a value for the Contact Angle for multiphase PEM calculations (θc inEqua-tion 1.4-4).



Specifying Catalyst Layer Properties for the Anode

Figure 2.6.6: The Anode Tab of the PEM Model Panel With Catalyst LayerSelected


2. Select Catalyst Layer under Anode Zone Type.





7. Specify a value for the Surface-to-Volume Ratio (η in Equation 1.3-5). This is thespecific surface area of the catalyst medium.




Specifying Boundary Conditions for the Anode

For each case of the anode’s current collector, diffusion layer, and catalyst layer, youassign a solid material and/or set the porosity and the viscous resistance. These settingsrepresent setting a boundary condition. With the Collective Boundary Conditions optionturned on (the default setting), this boundary condition is applied to all selected zonesin the Zone(s) list. If you want to set the boundary conditions for each zone individuallyusing the corresponding Boundary Conditions panel, you should turn off the CollectiveBoundary Conditions option.

2.6.4 Specifying Membrane Properties

You can use the Membrane tab of the PEM Model panel to specify zones and propertiesof the membrane portion of the PEM fuel cell.

Figure 2.6.7: The Membrane Tab of the PEM Model Panel

1. Select a corresponding zone from the Zone(s) list. If you are modeling a fuel cellstack, then you must pick all membrane zones as a group.

2. Select a Solid Material from the corresponding drop-down list. Solid materials canbe customized using the Materials panel.

3. Specify a value for the Equivalent Weight (Mm in Equation 1.5-4).

4. Specify a value for the Protonic Conduction Coefficient (β in Equation 1.5-2). Thisis used to calculate the membrane phase electric conductivity.

5. Specify a value for the Protonic Conduction Exponent (ω in Equation 1.5-2).



Specifying Boundary Conditions for the Membrane

When you assign a solid material to the membrane, you are setting a boundary condi-tion. With the Collective Boundary Conditions option turned on (the default setting), thisboundary condition is applied to all selected zones in the Zone(s) list. If you want toset the boundary conditions for each zone individually using the corresponding BoundaryConditions panel, you should turn off the Collective Boundary Conditions option.

2.6.5 Specifying Cathode Properties

You can use the Cathode tab of the PEM Model panel to specify zones and properties ofthe current collector, the flow channel, the diffusion layer, and the catalyst layer for thecathode portion of the PEM fuel cell.

Specifying Current Collector Properties for the Cathode

Figure 2.6.8: The Cathode Tab of the PEM Model Panel With Current Col-lector Selected

1. Select the Cathode tab of the PEM Model panel.

2. Select Current Collector under Cathode Zone Type.





Specifying Flow Channel Properties for the Cathode

Figure 2.6.9: The Cathode Tab of the PEM Model Panel With Flow ChannelSelected


2. Select Flow Channel under Cathode Zone Type.




Specifying Diffusion Layer Properties for the Cathode

Figure 2.6.10: The Cathode Tab of the PEM Model Panel With Diffusion LayerSelected


2. Select Diffusion Layer under Cathode Zone Type.








Specifying Catalyst Layer Properties for the Cathode

Figure 2.6.11: The Cathode Tab of the PEM Model Panel With Catalyst LayerSelected


2. Select Catalyst Layer under Cathode Zone Type.





7. Specify a value for the Surface-to-Volume Ratio (s in Equation 1.3-5). This is thespecific surface area of the catalyst medium.

8. Specify a value for the Contact Angle for multiphase PEM calculations (θc in Equa-tion 1.4-4).



Specifying Boundary Conditions for the Cathode

For each case of the cathode’s current collector, diffusion layer, and catalyst layer, youassign a solid material and/or set the porosity and the viscous resistance. These settingsrepresent setting a boundary condition. With the Collective Boundary Conditions optionturned on (the default setting), this boundary condition is applied to all selected zonesin the Zone(s) list. If you want to set the boundary conditions for each zone individuallyusing the corresponding Boundary Conditions panel, you should turn off the CollectiveBoundary Conditions option.

2.6.6 Setting Advanced Properties for the PEM Fuel Cell

You can use the Advanced tab of the PEM Model panel to specify the contact resistivityfor any material interface in the geometry, set parameters for coolant channels, and definefuel stack units for managing stacks of fuel cells.

Setting Contact Resistivities for the PEM Fuel Cell

Figure 2.6.12: The Advanced Tab of the PEM Model Panel for Contact Re-sistivities

1. Select the Advanced tab of the PEM Model panel.

2. Select Contact Resistivity under Advanced Setup.

3. Select any number of corresponding interfaces from the Available Zone(s) list. Thesezones are face zones over which a jump in electrical potential is caused by imperfectconduction.

4. Specify a value for the Resistivity for each specified zone.

5. To simplify the input, you can choose to use the resistivity value of the first selectedzone for all others as well by turning on the Use First Value for All option.



Setting Coolant Channel Properties for the PEM Fuel Cell

Figure 2.6.13: The Advanced Tab of the PEM Model Panel for the CoolantChannel


2. Select Coolant Channel under Advanced Setup.

3. Select any number of corresponding zones from the Zone(s) list.

4. Specify a value for the Density.

5. Specify a value for the Heat Capacity.

6. Specify a value for the Thermal Conductivity.

7. Specify a value for the Viscosity.

8. To enable the coolant channel, turn on the Enable Coolant Channel(s) option.Amongst other settings, this will change the mixture to include the coolant species,which is otherwise absent.



Managing Stacks for the PEM Fuel Cell

Figure 2.6.14: The Advanced Tab of the PEM Model Panel for Stack Man-agement

The FLUENT PEM fuel cell model allows you to model fuel cell stacks as well as individualfuel cells. In the Advanced tab of the PEM Model panel, you can define fuel cell units foreach fuel cell in a stack. A fuel cell unit consists of all zones of a single fuel cell in thestack.

i If you are only modeling a single fuel cell, then you do not need to setanything for Stack Management in the Advanced tab of the PEM Modelpanel.


2. Select Stack Management under Advanced Setup.

3. Since a fuel cell unit consists of all zones of a single fuel cell in the stack, select thecorresponding zones from the Zone(s) list.

4. Create a new fuel cell unit by clicking the Create button. The new fuel cell is listedunder Fuel Cell Unit(s) with a default name.

5. Edit a preexisting fuel cell unit by selecting it in the Fuel Cell Unit(s) list. Thezones in this fuel cell unit are automatically selected in the Zone(s) list. You canthen modify the zones that comprise the fuel cell unit and/or change its name inthe Name field and click Modify to save the new settings.

6. Remove a preexisting fuel cell unit by selecting it in the Fuel Cell Unit(s) list andclicking the Delete button.



7. If your model contains many zone names, you can use the Match Zone Name Patternfield to specify a pattern to look for in the names of zones. Type the pattern inthe text field and click Match to select (or deselect) the zones in the Zones list withnames that match the specified pattern. You can match additional characters using* and ?. For example, if you specify wall*, all surfaces whose names begin withwall (e.g., wall-1, wall-top) will be selected automatically. If they are all selectedalready, they will be deselected. If you specify wall?, all surfaces whose namesconsist of wall followed by a single character will be selected (or deselected, if theyare all selected already).

For example, in a stack there are many fuel cells, say 10 - 100, each having at least9 zones (current collector, gas channel, diffusion layer, and catalyst layer for bothanode and cathode and a membrane). Additionally, there may be coolant channels,and it may be that for mesh construction reasons each of these physical zones ismade up of more than one mesh zone. Even for small stacks, you can easily endup having hundreds of cell zones in a FLUENT mesh. Therefore, you may want toconsider numbering the fuel cells in a stack and to use the assigned fuel cell numberin the names of the mesh zones. When you set up your stacked fuel cell case, youwould use the Match Zone Name Pattern field to pick all the zones belonging to asingle fuel cell in the stack, rather than scrolling through the potentially very longlist and selecting them manually.



2.6.7 Specifying the Solution Controls

You can use the Controls tab of the PEM Model panel to influence the solution process.

Figure 2.6.15: The Controls Tab of the PEM Model Panel

The saturation source term rw in Equation 1.4-2 usually requires under-relaxation. Youcan change the default value for the under-relaxation factor by changing the value forSaturation Source.

The water content, λ, in Equation 1.5-6 also may need under-relaxation. You can changethe default value for the under-relaxation factor by changing the value for Water Content.

If you are only interested in the basic flow field throughout the fuel cell, you can turn offthe Electrochemistry option in order to suppress most effects of the PEM fuel cell model.To turn off all effects of the PEM fuel cell model, you should also turn off the MembraneWater Transport and Multiphase options in the Model tab of the PEM Model panel.



2.6.8 Reporting on the Solution

You can use the Reports tab of the PEM Model panel to set up parameters that will beuseful in reporting data relevant to the fuel cell.

Figure 2.6.16: The Reports Tab of the PEM Model Panel

The Membrane-Electrode-Assembly Projected Area field requires the projected area of theMembrane Electrolyte Assembly (MEA) and is only used to calculate the average currentdensity. The assembly consists of the membrane and the catalyst layers above and belowthe membrane. The value of the projected area can be computed from the ProjectedSurface Areas panel.

Reports −→Projected Areas...

The External Contact Interface(s) fields requires the face zones that act as external contactsurfaces for the anode and the cathode.

These inputs are used to report cell voltage. For potentiostatic boundary conditions, thisis the difference between the provided values, but for galvanostatic boundary conditions,the cell voltage is part of the solution.



2.7 Modeling of Current Collectors

In previous versions of FLUENT, user-defined scalar (UDS) equations could only be solvedin fluid zones. This restriction is now removed. As a result, the PEMFC module allowsusers to model current collectors as solid, as well as fluid zones. One advantage of usingsolids as the current collector is that the convergence of the species equations are nothindered by the potentially skewed mesh inside the current collectors.

If fluid zones are used to model solid current collectors, FLUENT automatically set ve-locities to zero and cuts off species transport into these zones. If solid zones are used,however, you need to activate the solution of the electric potential (UDS-0) in thesesolid zones (see the separate FLUENT User’s Guide for details). The value of the ElectricConductivity for the solid material needs to be assigned in the Materials panel.

Figure 2.7.1: The Electric Conductivity Field in the Materials Panel.

i Note that the UDS Diffusivity should be set to user-defined

(cond::fuelcells2.2). Do not use the defined-per-uds option.

For more information on the user-defined scalar diffusivity, see the separate FLUENTUser’s Guide.


2.8 PEM Fuel Cell Boundary Conditions

2.8 PEM Fuel Cell Boundary Conditions

The following boundary conditions need to be defined for the PEM fuel cell simulationbased on your problem specification:

• Anode Inlet

– Mass flow rate

– Temperature

– Direction specification method

– Mass fractions (e.g., h2, and h2o).

– The coolant must be set to zero if coolant channels are enabled.

– UDS-2 (Water Saturation) must be set to 0

• Cathode Inlet

– Mass flow rate

– Temperature


– Mass fractions (e.g. o2, h2o, and n2).

– The coolant must be set to zero if coolant channels are enabled.


• Coolant Inlet (if any)

– Mass flow rate

– Temperature


– Coolant mass fraction set to 1


• Pressure Outlets (all)

Realistic backflow conditions.

• Terminal Anode

– Temperature (or flux if known)

– UDS-0 (electric potential) set to ground voltage

• Terminal Cathode

– Temperature (or flux if known)



– UDS-0 (electric potential) is set to the voltage of the cathode (if solving toconstant voltage), or the UDS-0 (electric potential) flux is set to the currentdensity in A/m2(SI units) (if solving for constant current). Note that the signof the UDS-0 flux on the cathode side is negative.

2.9 Solution Guidelines for the PEM Fuel Cell Model

For potentiostatic boundary conditions, after initialization, steady state solutions arecalculated easily for cell voltages close to the open-circuit voltage. The same can besaid for galvanostatic boundary conditions and low electric current. By lowering the cellvoltage or by raising the average electric current, you can calculate subsequent stationarysolutions.

In the event of convergence problems, it is recommended to change the multigrid cycle toF-cycle with BCGSTAB (bi-conjugate gradient stabilized method) selected as the stabi-lization method for the species and the two potential equations. For the species and theuser-defined scalar equations, it may be necessary to reduce the termination (criterium)of the multigrid-cycles to 1× 10−3. For stack simulations, the termination criterium maybe reduced to ×10−7 for the two potential equations.

Also, it may be useful to turn off Joule Heating and Reaction Heating in the PEM Modelpanel (in the Model tab) for the first few (approximately 5-10) iterations after initializa-tions. This allows the two electric potentials to adjust from their initial values to morephysical values, avoiding the possibility of extreme electrochemical reactions and electriccurrents that would in turn adversely impact the solution.

2.10 Postprocessing the PEM Fuel Cell Model

You can perform post-processing using standard FLUENT quantities and by using user-defined scalars and user-defined memory allocations. By default, the FLUENT PEMfuel cell model defines several user-defined scalars and user-defined memory allocations,described in Table 2.10.1 and Table 2.10.2.

Table 2.10.1: User-Defined Scalar Allocations

UDS 0 Electric Potential (solid phase potential) (Volts)UDS 1 Protonic Potential (membrane phase potential) (Volts)UDS 2 Water Saturation (liquid saturation)UDS 3 Water Content


2.10 Postprocessing the PEM Fuel Cell Model

Table 2.10.2: User-Defined Memory Allocations

UDM 0 X Current Flux Density (A/m2)UDM 1 Y Current Flux Density (A/m2)UDM 2 Z Current Flux Density (A/m2)UDM 3 Current Flux Density Magnitude (A/m2)UDM 4 Ohmic Heat Source (W/m3)UDM 5 Reaction Heat Source (W/m3)UDM 6 Overpotential (Volts)UDM 7 Phase Change Source (kg/m3-s)UDM 8 Osmotic Drag CoefficientUDM 9 Liquid Water ActivityUDM 10 Membrane Water ContentUDM 11 Protonic Conductivity (1/ohm-m)UDM 12 Back Diffusion Mass Source (kg/m3-s)UDM 13 Transfer Current (A/m3)UDM 14 Osmotic Drag Source (kg/m3-s)

You can obtain this list by opening the Execute On Demand panel and pulling down theFunction drop-down list.

Define −→ User-Defined −→Execute On Demand...

and access the execute-on-demand function called list pemfc udf.

Alternatively, you can view the listing that appears when you first load your PEM fuelcell case, or you can type list pemfc udf in the text user interface and the listing willappear in the console window.

i When you load older PEM fuel cell cases into FLUENT 6.3, and you aremonitoring a UDS using volume or surface monitors, make sure you re-visitthe corresponding monitors panel (e.g., the Define Volume Monitor or theDefine Surface Monitor panel) to make sure that the correct UDS name isused for the appropriate monitor.



2.11 User-Accessible Functions

As noted in Section 1.5: Properties, you can directly incorporate your own formulationsand data for the properties of the fuel cell membrane using the pem user.c source codefile.

The following listing represents a description of the contents of the pem user.c sourcecode file:

• real heat apportionment factor(cell t c, Thread *t): A fraction of the en-ergy released in the chemical reaction for the formation of water is released as heat.This function gives this fraction of the maximal available energy of this reaction.

• real Get P sat(real T): Returns the value of the water vapor saturation pressureas a function of temperature (Equation 1.5-9).

• real Water Activity(real P, real T, cell t c, Thread *t): Returns the valueof water activity (Equation 1.5-7).

• real Water Content (real act): Returns the value of the membrane water con-tent at the membrane catalyst interface (Equation 1.5-6).

• real Osmotic Drag Coefficient(real P, real T, cell t c, Thread *t): Re-turns the value of the osmotic drag coefficient (Equation 1.5-3).

• real Membrane Conductivity(real lam, cell t c, Thread *t): Returns thevalue of the membrane’s protonic conductivity (Equation 1.5-2).

• real Water Content Diffusivity(real lam, real T, real mem mol density,

cell t c, Thread *t): Returns the value of the water content diffusivity in themembrane(Equation 1.5-5).

• real Gas Diffusivity(cell t c, Thread *t, int j spe): Returns the valueof the gaseous species diffusivities in the channels, gas diffusion layers and catalysts(Equation 1.5-2).

• real Saturation Diffusivity(real sat, real cos theta, real porosity, cell t

c, Thread *t): Returns the value of diffusivity of the liquid saturation. It com-prises the term

ρlKs3

µl

dpcds

(2.11-1)

from Equation 1.4-1.

• real Anode AV Ratio(cell t c, Thread *t): Returns the value of the surface-to-volume ratio s used in Equation 1.3-5 for the anode catalyst.



• real Cathode AV Ratio(cell t c, Thread *t): Returns the value of the surface-to-volume ratio s used in Equation 1.3-5 for the cathode catalyst.

• real Anode J TransCoef(cell t c, Thread *t): Returns the value of the anode

reaction reference current density jrefan

([H2]ref )γan used in Equation 1.2-3.

• real Cathode J TransCoef(cell t c, Thread *t): Returns the value of the cath-

ode reaction reference current densityjrefcat

([O2]ref )γcatused in Equation 1.2-4.

• real Open Cell Voltage(cell t c, Thread *t): Returns the value of the open-circuit voltage Voc used in Equation 1.2-8.

• void Set UDS Names(char uds[n uds required][STRING SIZE]): Used to renameuser defined scalars (UDSs). Note that the units of the user defined scalars cannotbe changed.

void Set_UDS_Names(char uds[n_uds_required][STRING_SIZE])

{

strncpy(uds[0], "Electric potential", STRING_SIZE-1);

strncpy(uds[1], "Protonic potential", STRING_SIZE-1);

strncpy(uds[2], "Water saturation", STRING_SIZE-1);

strncpy(uds[3], "Water content", STRING_SIZE-1);

}

If you want to change the names of UDSs, change the second argument of thestrncpy functions, recompile and link the module as with any modification topem user.c. Note that STRING SIZE is fixed in pem.h and should not be changed.

i When you load older PEM fuel cell cases into FLUENT 6.3, and you aremonitoring a UDS using volume or surface monitors, make sure you re-visitthe corresponding monitors panel (e.g., the Define Volume Monitor or theDefine Surface Monitor panel) to make sure that the correct UDS name isused for the appropriate monitor.



• void Set UDM Names(char udm[n udm required][STRING SIZE]): Used to renameuser defined memory (UDMs). Note that the units of user defined memory cannotbe changed.

void Set_UDM_Names(char udm[n_udm_required][STRING_SIZE])

{

strncpy(udm[ 0], "X Current Flux Density", STRING_SIZE-1);

strncpy(udm[ 1], "Y Current Flux Density", STRING_SIZE-1);

strncpy(udm[ 2], "Z Current Flux Density", STRING_SIZE-1);

strncpy(udm[ 3], "Mag Current Flux Density", STRING_SIZE-1);

strncpy(udm[ 4], "Ohmic heat source", STRING_SIZE-1);

strncpy(udm[ 5], "Reaction heat source", STRING_SIZE-1);

strncpy(udm[ 6], "Overpotential", STRING_SIZE-1);

strncpy(udm[ 7], "Phase change source", STRING_SIZE-1);

strncpy(udm[ 8], "Osmotic drag coeff", STRING_SIZE-1);

strncpy(udm[ 9], "Activity liquid water", STRING_SIZE-1);

strncpy(udm[10], "Membrane water content", STRING_SIZE-1);

strncpy(udm[11], "Protonic conductivity", STRING_SIZE-1);

strncpy(udm[12], "Back diffusion mass source", STRING_SIZE-1);

strncpy(udm[13], "Transfer current", STRING_SIZE-1);

strncpy(udm[14], "Osmotic drag source", STRING_SIZE-1);

}

If you want to change the names of UDMs, change the second argument of thestrncpy functions, recompile and link the module as with any modification topem user.c. Note that STRING SIZE is fixed in pem.h and should not be changed.

i When you load older PEM fuel cell cases into FLUENT 6.3, and you aremonitoring a UDM using volume or surface monitors, make sure you re-visit the corresponding monitors panel (e.g., the Define Volume Monitor orthe Define Surface Monitor panel) to make sure that the correct UDM nameis used for the appropriate monitor.



2.11.1 Compiling the Customized PEM Fuel Cell Source Code

This section includes instructions on how to compile a customized PEM fuel cell user-defined module. Note that you can also refer to the file INSTRUCTIONS-CLIENT thatcomes with your distribution (see addons/fuelcells2.2).

i It is assumed that you have a basic familiarity with compiling user-definedfunctions (UDFs). For an introduction on how to compile UDFs, pleaserefer to the separate UDF manual.

You will first want to use a local copy of the fuelcells2.2 directory in the addons

directory before you recompile the PEM fuel cell module.

Compiling the Customized Source Code Under UNIX1. Make a local copy of the fuelcells2.2 directory. Do not create a symbolic link.

2. Change directories to the fuelcells2.2/src directory.

3. Make changes to the pem user.c file.

4. Edit the makefile. You must edit the makefile located in the src/ directory andmake sure that the FLUENT INC variable correctly refers to the current FLUENTinstallation directory. Be careful not to leave any trailing spaces when you makeyour changes.

5. Change directories to the fuelcells2.2/ directory.

6. Issue the following make command:

make -f Makefile-client FLUENT_ARCH=your_arch

where your arch is lnx86 on LINUX, or ultra on the Sun operating system, etc.

Compiling the Customized Source Code Under Windows1. Open Visual Studio .NET at the DOS prompt.

2. Make sure that the $FLUENT INC environment variable is correctly set to the currentFLUENT installation directory.

3. Make a local copy of the fuelcells2.2 folder. Do not create a shortcut.

4. Enter the fuelcells2.2\src folder.

5. Make changes to the pem user.c file.



6. Return to the fuelcells2.2 folder.

7. Issue the following command in the command window:

nmake /f makefile_master-client.nt

2.12 IV-Curve Calculations Using the Text Interface

A text user interface (TUI) exists for the PEM Fuel Cell Model. The TUI allows textcommands to be typed directly in the FLUENT console window where additional infor-mation can be extracted and processed for more advanced analysis.

For valid case and data files, there are two text commands available to assist in theIV-curve calculation. These commands are set-stack-voltage (aliased as ssv) andset-stack-current-density (aliased as ssc), available from the PEM Fuel Cell textcommand menu: /define/models/pemfc/.

For fuel cells, you either prescribe the voltage and obtain the total current delivered bythe fuel cell as a result, or you specify the total current (via flux boundary conditionsmultiplied by the area) and obtain the voltage as part of the solution. The detailsof this IV-relation are specific for each single fuel cell and depend on mass and heattransport, electrochemistry and inlet conditions, outlet conditions, operating conditions,and any other parameter or material property involved in the calculation. The IV-curveis important for applications, because its product is the power delivered by the system.

As described earlier in this manual, you would start a new simulation from fairly staticconditions, i.e., high voltage/low current (which implies low species transport and lowheat generation). After convergence, you typically may be interested in solutions for newelectric boundary conditions, i.e., either for a new cell/stack voltage or current.

In such cases, simply going to the Boundary Conditions panel and changing the value ofthe electric potential (uds-0) boundary condition, typically allows only small changes,most notably for stacks. Otherwise the solution will not converge. This is where theset-stack-voltage and set-stack-current-density commands are important.

In addition to changing the boundary conditions (either to a prescribed voltage or currentdensity), these commands process the current data in order to estimate the solutionfor the new boundary conditions. Because these commands modify the data, you areprompted to save your data, if you have not already done so.

Before going into details of the commands, here are some general remarks about electricpotential boundary conditions.

For fixed voltage boundary conditions, both external contacts have a fixed value forthe electric potential (uds-0). The anode value will typically be zero, but it does nothave to be. The cathode value will be larger than the anode value and the difference(Vcathode-Vanode) is the positive cell/stack voltage.


2.12 IV-Curve Calculations Using the Text Interface

For a fixed current boundary condition, one external contact has to have a fixed valueand the other flux boundary conditions. As described earlier in the manual, typically, theanode will have a fixed (zero) value, and the cathode will be floating, however, you canalso set the cathode to a fixed zero potential, yielding a floating negative anode potential.

The set-stack-voltage command sets the effective stack voltage, i.e., the difference(Vcathode-Vanode). For fixed voltage boundary conditions for the previous solution, bound-ary conditions on both boundaries are of type fixed value and then the cathode value willbe changed accordingly. In the case of fixed current boundary conditions for the previoussolution, the flux boundary condition will be changed to a fixed value boundary condi-tion, and the value adjusted accordingly with respect to the other fixed value boundarycondition.

The set-stack-current-density command sets the current density on one boundaryto the desired value. Note that the input will be in A

cm2 , not Am2 as you would normally

have to enter in the Boundary Conditions panel. The reason for this is that average currentdensities reported in the text command interface are also in A

cm2 , and this makes it easierto choose the conditions you would like to prescribe next. Also, flux boundary conditionsentered in the Boundary Conditions panel would have to have a positive sign on the anodeside, and a negative sign on the cathode side. The input for the text interface commandis just a positive number, signs are automatically accounted for.

For fixed current boundary conditions for the previous solution, theset-stack-current-density command changes the respective flux boundary conditionaccordingly. In the case of fixed voltage boundary conditions for the previous solution,the cathode side is chosen to be changed from a fixed value to a flux boundary conditionwith the new flux.

The two commands may be mixed in an IV-curve calculation. For the type of bound-ary condition setups currently described in this manual, boundary condition changeswill consistently happen on the cathode side. However, if anode flux boundary con-ditions had been chosen initially, switching to fixed voltage boundary conditions byset-stack-voltage command and then back to fixed current boundary conditions by theset-stack-current-density command will then have flux boundary conditions on thecathode side. In this case, using the set-stack-current-density command exclusivelywill preserve the anode flux boundary condition setting.


Chapter 3. SOFC Fuel Cell Model Theory

This chapter presents an overview of theory and equations for solid oxide fuel cell (SOFC)modeling capabilities in FLUENT.


• Section 3.2: The SOFC Modeling Strategy

• Section 3.3: Modeling Stacked Fuel Cells

• Section 3.4: Modeling Fluid Flow, Heat Transfer, and Mass Transfer

• Section 3.5: Modeling Current Transport and the Potential Field

• Section 3.6: Modeling Electrochemical Reactions

3.1 Introduction

The Solid Oxide Fuel Cell (SOFC) module is provided as an addon module with the stan-dard FLUENT licensed software. A special license is required to use the SOFC module.

A fuel cell is an energy conversion device that converts the chemical energy of fuel into theelectrical energy. A schematic of a solid oxide fuel cell (SOFC) is shown in Figure 3.1.1.


SOFC Fuel Cell Model Theory

2e−

2e−2e−

CathodeElectrode

InterlayerCathode

AnodeInterlayer Anode

Electrode

Electrolyte

AnodeCurrentCollector

CathodeCurrentCollector

e−

e−

��

��

��

��

��

��

2e−

Anode Flow Channels

Cathode Flow Channels

2H O2H O

O−

2O

2H

Figure 3.1.1: Schematic of a Solid Oxide Fuel Cell

As noted in [4], a solid oxide fuel cell is typically composed of an anode, cathode, andan electrolyte. Multiple fuel cells can be connected together, or stacked, using electricalinterconnects. The electrolyte material must be solid, i.e., non-porous, and exhibit a highionic conductivity.

All components of the fuel cell must have similar thermal expansion in order to minimizethermal stresses, which may cause cracking and de-lamination during thermal cycling. Inaddition, the components must be chemically stable in order to limit chemical interactionswith other cell components.

A solid oxide fuel cell works by having electrically conducting porous ceramic elec-trodes attached on each side of an ionically conducting ceramic material. At the cath-ode/electrolyte/gas interface, also known as the triple phase boundary, oxygen is reducedto oxygen ions. The oxygen ions are conducted through the oxygen vacancies in the elec-trolyte to the anode side. At the anode/electrolyte/gas interface, oxygen ions combineto react with hydrogen at the anode electrode to form water and release electrons. Theelectrons travel through an external circuit to a load and back to the cathode electrodeto close the circuit.


3.1 Introduction

The FLUENT SOFC model provides the following features:

• Local electrochemical reactions coupling the electric field and the mass, species,and energy transport.

• Electric field solution in all porous and solid cell components, including ohmicheating in the bulk material.

• The ability to handle H2 and combined CO/H2 electrochemistry.

• Inclusion of tortuosity for porous regions

• The ability to treat an arbitrary number of electrochemical cells arrayed as a stack.

• Significant geometric flexibility for treating planar, tubular, and other nonstandardSOFC configuration.

3.1.1 Motivation for Consideration of the Electric Field

Electrical conduction throughout the entire fuel cell assembly can cause a significantchange to electrochemical operation of the fuel cell. Normally, the fuel cell is thoughtof as an idealized planar or cylindrical surface of reaction. Local species concentrations,pressure, and temperature influence the distribution of current along the electrolyte. Inaddition, ohmic losses other than those at the electrolyte influence the current densityas well. As electrolyte thicknesses are reduced and the electrolyte resistance becomesless dominate, these external resistive losses become important. When the electroderesistance become significant compared to that of the electrolyte, you can expect the sizeand location of the current collector to have an impact on the current density. This cancertainly become the case for widely-spaced current collectors or thin electrodes.

3.1.2 Overview of the Electric Field Model

The existing functionality of solving a user defined scalar in FLUENT has been extendedin order to solve for the potential field in the solid oxide fuel cell. First, additionalfunctionality has been added to the SOFC user-defined function that enables the solu-tion of a user defined scalar in solid zones. Second, to handle the electrolyte interfacesand contact resistance interfaces, a more general flux computation needs to be definedand incorporated into the FLUENT UDS solution (see Section 3.5.2: Treatment of theElectrolyte Interface).



3.1.3 Overview of the Electrochemical Model

The local Nernst potential is a function of the composition of the flow immediatelyadjacent to the electrolyte in the anode and cathode. The activation and ohmic losses inthe electrolyte is a function of the current density. Likewise, species and energy fluxes aredetermined by the current density. Therefore, any external effects can have a tangibleeffect not only on the current density distribution but also on the transport behavior ofthe fuel cell.

In the electrochemical model, the two sides of the electrolyte is separated by a localpotential difference. This difference is determined by three physical effects: Nernst po-tential, ohmic losses in the electrolyte, and activation losses at the electrode-electrolyteinterfaces.

δV = N − iρeletele − ηact(i) (3.1-1)

where i is the local current density. For the idealized case with uniform contact betweenthe electrodes and highly conductive current collectors, you can specify a cell voltage andsolve for the current density distribution using this relation.

3.2 The SOFC Modeling Strategy

To model solid oxide fuel cells (SOFC), you need to perform the following:

• Capture the fluid flow, heat transfer, and the mass transfer in the flow channelsand in the porous anode and cathode electrodes.

• Model the transport of the current and the potential field in the porous electrodesand in the solid conducting regions.

• Model the electrochemical reactions that take place at the electrolyte/electrode/gaseousspecies interface.


3.3 Modeling Stacked Fuel Cells

FLUENT

Species

Momentum

Energy

Electric Potential Field

SOFC Model

Nernst Voltage

Boundary ConditionsElectric Potential Field

Electrolyte and Overpotentials atCurrent Distribution

and temperatureLocal species concentration

Species and heat fluxat the boundaries

Figure 3.2.1: How the SOFC Model Works in FLUENT

3.3 Modeling Stacked Fuel Cells

Typically in fuel cell modeling, the cell current at a specified cell voltage is calculated.When modeling stacked fuel cells with this approach, you must insure the conservationof charge is satisfied at each cell in the stack.. To simplify stack modeling, the FLUENTSOFC model calculates the cell voltage at a specified cell current. Specifying the fuelcell current as an input insures that the total current flux through each cell in a stack isconserved.



3.4 Modeling Fluid Flow, Heat Transfer, and Mass Transfer

All aspects of fluid flow, heat transfer, and mass transfer in the flow channels and porouselectrodes are handled by FLUENT.

The default multicomponent diffusion model in FLUENT is used to calculate the massdiffusion coefficient of species i in the mixture.

To account for the effect of porosity on the multicomponent mass diffusion coefficient

Dij,eff =ε

τDij (3.4-1)

where ε is the porosity and τ is the tortuosity (i.e., the average path length over theactual length).

3.5 Modeling Current Transport and the Potential Field

Solving for three-dimensional electrical conduction is directly analogous to the calculationof heat transfer. The potential field throughout the conductive regions is calculated basedon the conservation of charge.

∇ · i = 0 (3.5-1)

where

i = −σ∇φ (3.5-2)

and σ is the electrical conductivity and φ is the electrical potential. Therefore, thegoverning equation for the electric field is the Laplace equation:

∇ · (σ∇φ) = 0 (3.5-3)

The electric field potential calculation combines the following attributes:

• Ohmic losses in all the conducting materials, including the electrolyte, electrodes,and current collectors.

• Contact resistance at the appropriate interfaces.

• Ohmic heating through conduction materials as the result of ohmic losses, and thecurrent density throughout the domain.



3.5.1 Treatment of the Activation Overpotential

Since the Butler-Volmer relation for activation overpotential is nonlinear, it is usefulto build a locally linearized form. This allows for a more complete coupling of theelectrochemistry and ultimately better model stability and improved convergence. Usinga previous value for the local current density, iold, a Taylor series expansion can beconstructed:

ηact(i) ≈ ηo − η1(i− iold) (3.5-4)

ηo = ηact(iold) (3.5-5)

η1 = −∂ηact

∂i|i=iold

(3.5-6)

With this approximation, the formula for the voltage jump across the electrolyte becomes:

V ≈ [N − ηo + η1iold]− i[ρeletele + η1] (3.5-7)

The first term in brackets is now a constant. The second term is linear with local currentdensity. Here, the activation overpotential is included as a direct reduction of the voltageand as an additional effective resistance. During the calculation, this form allows theactivation overpotential to respond to changes in local current density. Even though theButler-Volmer relation is nonlinear, the linearized form becomes more accurate as thesolution reaches convergence and is exact at convergence.

3.5.2 Cell Potential

The electrode reactions are assumed to take place in a single step. The charge transferreaction acts as the rate limiting step for the electrode reactions.

Oxygen is electrochemically reduced at the triple phase boundary at the cathode elec-trode:

1

2O2,cathode + 2e− ⇔ O2− (3.5-8)

Oxygen is electrochemically re-oxidized at the triple phase boundary at the anode elec-trode:

O2− ⇔ 1

2O2,anode + 2e− (3.5-9)



In the absence of an electrical load, the oxygen activity on both sides of the electrolyteis fixed and given by their respective chemical potentials. Under equilibrium, the elec-tromotive force, or reversible cell voltage, is given by the Nernst equation:

φideal =RT

4FlnpO2,cathode

pO2,anode

(3.5-10)

If is present at the anode electrode, then the cell reaction becomes:

H2 +O2− ⇔ H2O + 2e− (3.5-11)

At equilibrium, the cell voltage is given by the Nernst equation:

φideal = φo +RT

2FlnpH2p

12O2

pH2O

(3.5-12)

The cell potential measured at equilibrium (i.e., no load), is called the open circuitvoltage. The open circuit voltage should be equivalent to the Nernst potential at noload, unless there is leakage across the electrolyte. When the external circuit is closed,then cell voltage drops due to polarization losses at the electrodes.

The electric field and the electrochemistry interact solely at the electrolyte interface.FLUENT treats the electrolyte interface as an impermeable wall. The potential field musthave a “jump” condition applied to the two sides of this wall to account for the effectof the electrochemistry. To closely couple the electrochemical behavior to the potentialfield calculation, you need to include all of the electrochemical effects into this jumpcondition. It encapsulates the voltage jump due to Nernst, the voltage reduction due toactivation, the Ohmic losses due to the resistivity of the electrolyte, and a linearized forvoltage reduction due to activation. This interface condition relates the potential on theanode side and the cathode side of the electrolyte and has the following form:

φcell = φjump − ηs (3.5-13)

where

φjump = φideal − ηele − ηact,a − ηact,c (3.5-14)

where ηele represents the ohmic overpotential of the electrolyte, and ηact,a ηact,c representthe activation overpotential of the anode and the cathode. ηs represent ohmic losses inthe solid conducting regions. φideal represents the Nernst potential.

FLUENT solves the Laplace equation for the potential field by subdividing the compu-tational domain into control volumes and enforcing flux conservation on each cell. For



cells away from the electrolyte boundary, the flux from a cell to its neighbor is given byOhm’s law:

−→i = −σ∇φ (3.5-15)

This gradient is approximated using the numerical difference of the values of φ in eachof the cells.

At the electrolyte interface, Ohm’s law along with the relation for voltage differencemust be used. In that case, for normal flux, there are three separate voltage relations.That coupling the anode computational cell center voltage to the anode/electrolyte facevoltage, the cathode cell center voltage to the cathode/electrolyte face voltage, and theelectrochemistry jump condition:

φanodecentroid = φanodewall − iρanodeDanode (3.5-16)

φcathodecentroid = φcathodewall − iρcathodeDcathode (3.5-17)

φanodewall = φcathodewall − i(ρt)eff (3.5-18)

where Danode and Dcathode are the distances from the electrode computational cell centerto the electrolyte. Together, these three linear equations can be solved to compute thenormal current flux, i, based on the values of the electric field in the anode and cathodecomputational cells.

By incorporating this linearized form of the electrochemistry model in this way, thesolution of the electric field directly includes variations in species concentrations viaNernst variations, resistive response of the electrolyte, and an approximate response tochanges in activation losses.

Treatment of the Electrolyte Interface

Allowing the solution of the electric field in the solid and the inclusion of electrolyte andcontact resistance interfaces requires non-standard modifications to be made to the FLU-ENT UDS solution procedure. FLUENT imposes a flux balance between computationalcells that share a common face. It is the calculation of this flux that is the cornerstoneof the solution procedure and where the special interface modifications are to be made.This section presents an overview of how FLUENT manages the unique conditions at theelectrolyte interface.

For two neighboring computational cells, C0 and C1, that share a face F , you can writeformulae for the flux (or current density). Let the value of the potential field at thecell centroids be denoted as φ0 and φ1. Also, let the values of the potential field at theface centroids be φf0 and φf1. Note that though both values are located at the samephysical location, the presence of contact resistance and/or electrolytic reactions can



cause a discontinuity in the potential field. These two face values represent the values ofthe potential on each side of the discontinuity.

Each computational cell can possess its own electrical conductivity, σ0 and σ1. The facecan have a potential jump, δV , and a contact resistance, Rc. If we assume Cartesian cellsfor simplicity, the current flux from the C0 centroid to the face centroid is:

i = −σ0Aface

dx0

(φf0 − φ0) (3.5-19)

The current from the face centroid to the C1 centroid is:

i = −σ1Aface

dx1

(φ1 − φf1) (3.5-20)

Now, at the discontinuity, the face values of potential change discontinuously. The currentcan be deduced from the following relation:

φf1 = φf0 + δV − iRc (3.5-21)

This can be rearranged into a form similar to the previous two equations:

i =1

Rc

(φf0 − φf1 + δV ) (3.5-22)

You can define connection parameters to simplify these:

i = −λ0(φf0 − φ0) (3.5-23)

where

λ0 =σ0Aface

dx0

(3.5-24)

andi = −λ1(φ1 − φf1) (3.5-25)

where

λ1 =σ1Aface

dx1

(3.5-26)

andi = λc(φf0 − φf1 + δV ) (3.5-27)



where

λc =1

Rc

(3.5-28)

These equations can then be manipulated to solve for the current and eliminate the facepotential values:

i =λ0λ1λc

λ0λ1 + λ0λc + λ1λc

(φ0 − φ1 + δV ) (3.5-29)

This definition of the current flux can then be used in FLUENT to build the system oflinear equations to solve for the potential field. This formulation is valid for any interfacethat has a contact resistance and voltage jump. It is trivial to set the voltage jump to zerofor the case of pure contact resistance, as would occur between electrochemically inertconductive layers. For cases where no contact resistance is present, the above formula isvalid if you take the limit as λc −→∞.

i =λ0λ1

λ0 + λ1

(φ0 − φ1 + δV ) (3.5-30)

Inclusion of these general flux computations into the FLUENT UDS solution schemeprovides the ability to directly compute jump and contact resistance interfaces.

Treatment of the Energy Equation at the Electrolyte Interface

For an incompressible flow, the energy equation that FLUENT solves for within eachcomputational cell is given by the following:

∂

∂t(ρE) +∇ · (~v(ρE + p)) = ∇ ·

keff∇T −∑j

hj ~Jj + (τ eff · ~v)

+ Sh (3.5-31)

where Sh is the volumetric source or sink of energy and where

E = h− p

ρ+v2

2(3.5-32)

and

h =∑j

Yjhj (3.5-33)



In all electrically conducting zones (e.g., electrodes, current collectors, interconnects),ohmic heating, i2 ∗ Rohmic, is added to the energy equation as a source term. In otherwords,

Sh = i2 ∗Rohmic (3.5-34)

In addition, the energy equation needs treatment at the electrode-electrolyte interface toaccount for the heat generated or lost as the result of electrochemistry and the overpo-tentials (i.e., activation overpotential and ohmic loss through the electrolyte).

cathode side

anode side

electrolyte interface

50 % into cell source term

50 % into cell source term

energy balance

Figure 3.5.1: Energy Balance at the Electrolyte Interface

The total energy balance on the electrolyte interface is computed by enumerating theenthalpy flux of all species, including the heat of formation (sources of chemical energyentering the system), and then subtracting off the work done (leaving the system) whichis simply the local voltage jump multiplied by the local current density. What remainsis the waste heat due to irreversibilities. For hydrogen reaction, the balance would be

Q′′

= h′′

H2+ h

′′

O2− h′′H20 − i∆V (3.5-35)

where Q is the heat generation (W) and h is the total enthalpy of species (J/s) composedof the sensible enthalpy in addition to the enthalpy of formation.



The heat of formation is

hH2 = mH2[∫ T

Tref

CpdT + h0] (3.5-36)

The source term is then added in the cell energy equation by taking Sh = QV olume

.

One half of this value is applied as a source term to the energy equation of the anodecomputational cell adjacent to the electrolyte and the other half is applied as a sourceterm to the energy equation for the cathode cell adjacent to the electrolyte. The equaldistribution of the heat generation/destruction is purely arbitrary. Note that by usingthe work term, the effect from all overpotentials are taken into account.

Ohmic Overpotential

Ohmic polarization involves ionic losses through the electrolyte, electrical resistance in theconducting porous electrodes and solid collectors. The ohmic polarization also includesthe electrical resistance at the interface of the current collectors and the electrodes or theelectrodes and the membrane (i.e., the contact resistance).

ηohmic = i ·R (3.5-37)

Activation Overpotential

The general electrochemical reaction is, according to [6],

∑j

NajAj ⇔ ne− (3.5-38)

where aj is the stoichiometric coefficient of species j, Aj is the chemical species, and nis the number of electrons.

The reaction rate is:

r =i

nF= kae

αanFRT

φ∏i

cipi − kce−

αcnFRT

φ∏i

ciqi (3.5-39)

where φ is the voltage, ka and pi are the rate constant and the reaction order for theanodic direction, kc and qi are the rate constant and the reaction order for the cathodicdirection, αa is the anodic transfer coefficient, αc is the cathodic transfer coefficient, andn is the number of electrons that are released.



At equilibrium, the forward and the backward reaction rates are the same, therefore:

i0nF

= kaeαanFRT

φ0∏i

cipi = kce

−αcnFRT

φ0∏i

ciqi (3.5-40)

where i0 is the exchange current density.

The reaction rate (i.e., current) can be written in terms of the exchange current densityi0 to obtain the Butler-Volmer formulation [6]:

i = i0

[eαan(φ−φ0)F

RT − e−αcn(φ−φ0)F

RT

](3.5-41)

The activation overpotential is the energy lost due to the slowness of electrochemicalreactions at the anode and the cathode electrodes.

ηact = φ− φ0 (3.5-42)

Using this relation, the Butler-Volmer equation can be written as:

i = i0eff

[eαanηactF

RT − e−αcnηactF

RT

](3.5-43)

where

i0eff = i0,ref(Yj)γ (3.5-44)

with i0,ref being the exchange current density at the reference condition, (Yj) is the molefraction and γ is the concentration exponent.

Given values for αa and αc. the full version of the Butler-Volmer equation can be solvedusing the Newton method, therefore finding the activation overpotential at the anode(ηact,a) and the cathode (ηact,c).


3.6 Modeling Electrochemical Reactions

3.6 Modeling Electrochemical Reactions

The rate of species production and destruction is:

S = − ai

nF(g −mole/m2/sec) (3.6-1)

where S is the source or sink of the species (molar flux), a is the stoichiometric coefficient,i is the current density (A/m2), n is the number of electrons per mole of fuel, and F isthe Faraday constant.

Using the local current information, the FLUENT SOFC model applies species fluxes tothe electrode boundaries. By convention [6], the current density is positive when it flowsfrom the electrode into the electrolyte solution. The current densities are positive at theanodes and negative at the cathodes.

The reaction at the cathode electrode is:

1

2O2,cathode + 2e− ⇔ O2− (3.6-2)

or

SO2 = −−(1

2)(−i)

2F= − i

4F(3.6-3)

S2−O = −(−i)

2F= − i

2F(3.6-4)

The reaction at the anode electrode is:

H2 +O2− ⇔ H2O + 2e− (3.6-5)

SH2 = − i

2F(3.6-6)

S2−O = − i

2F(3.6-7)

SH2O = −−(1)(i)

2F=

i

2F(3.6-8)


Chapter 4. Using the Solid Oxide Fuel Cell Model

The procedure for setting up and solving solid oxide fuel cell problems is described indetail in this chapter. Please refer to the following sections for more information:

• Section 4.1: Installing the Solid Oxide Fuel Cell Model

• Section 4.2: Loading the Solid Oxide Fuel Cell Module

• Section 4.3: Solid Oxide Fuel Cell Module Set Up Procedure

• Section 4.4: Setting the Parameters for the SOFC Model

• Section 4.5: Setting Up the Electrode-Electrolyte Interfaces

• Section 4.6: Setting Up the Electric Field Model Parameters

• Section 4.7: Setting Up the Tortuosity Parameters

• Section 4.8: Setting Up the Activation Parameters

4.1 Installing the Solid Oxide Fuel Cell Model

The Solid Oxide Fuel Cell (SOFC) model is provided as an addon module with the stan-dard FLUENT licensed software. A special license is required to use the SOFC model.The module is installed with the standard installation of FLUENT in a directory calledaddons/sofc1.2 in your installation area. The SOFC module consists of a UDF li-brary and a pre-compiled scheme library, which needs to be loaded and activated beforecalculations can be performed.

4.2 Loading the Solid Oxide Fuel Cell Module

The Solid Oxide Fuel Cell (SOFC) module is loaded into FLUENT through the text userinterface (TUI). The module can only be loaded after a valid FLUENT case file has beenset or read. The text command to load the addon module is

define −→ models −→addon-module


Using the Solid Oxide Fuel Cell Model

A list of FLUENT addon modules is displayed:

FLUENT Addon Modules:

1. MHD Model

2. Fiber Model

3. PEM Fuel Cell Model

4. SOFC Fuel Cell Model

5. Population Balance Model

Enter Module Number: [1] 4

Select the SOFC model by entering the module number 4. During the loading processa scheme library containing the graphical and text user interface, and a UDF librarycontaining a set of user defined functions are loaded into FLUENT.

4.3 Solid Oxide Fuel Cell Module Set Up Procedure

The following describes an overview of the procedure required in order to use the SOFCmodel in FLUENT.

1. Start FLUENT.

You must start FLUENT in 3d double-precision mode. Note that the SOFC modelis only available in 3d.

2. Read the case file.

File −→ Read −→Case...

3. Scale the grid.

Grid −→Scale...

4. Define various model parameters for the simulation.

(a) Open the Solver panel.

Define −→ Models −→Solver...

i. Enable Pressure Based under Solver.

ii. Enable Implicit under Formulation.

iii. Enable Steady under Time.

iv. Enable Absolute under Velocity Formulation.

(b) Open the Energy panel and enable the Energy option (if it is not alreadyenabled).

Define −→ Models −→Energy...



(c) Open the Viscous Model panel and enable the Laminar option.

Define −→ Models −→Viscous...

(d) Open the Species Model panel.

Define −→ Models −→ Species −→Transport & Reaction...

i. Enable the Species Transport option.

ii. Enable the Diffusion Energy Source option under Options.

iii. Enable the Full Multicomponent Diffusion option under Options.

iv. Enable the Thermal Diffusion option under Options.

(e) Set the parameters for the SOFC model.

Define −→ Models −→ SOFC −→SOFC Model Control Panel...

i. Set the Current Underrelaxation Factor to a value of either 0.3 or 0.4.

ii. Set the Electrochemical and Electrical Parameters according to your problemspecification.

(f) Set the anode interface components for the SOFC model.

Define −→ Models −→ SOFC −→Anode Interface Setup...

i. Specify a zone in the Zone(s) list.

ii. Enable Anode Interface if it is applicable.

(g) Set the cathode interface components for the SOFC model.

Define −→ Models −→ SOFC −→Cathode Interface Setup...


ii. Enable Cathode Interface if it is applicable.

(h) Set the tortuosity parameters for the SOFC model.

Define −→ Models −→ SOFC −→Tortuosity Setup...


ii. Turn on Enable Tortuosity if it is applicable.

(i) Set the parameters for the electric field model.

Define −→ Models −→ SOFC −→Electric Field Model Input Panel...

i. Specify up to 5 conductive regions.

ii. Specify up to 3 contact surfaces.

iii. Specify a voltage tap surface.

iv. Specify a current tap surface.



(j) Set the activation parameters.

Define −→ Models −→ SOFC −→Activation Panel...

i. Enter values for the Constant Exchange Current Densities for the anode andthe cathode. These are the i0 values in the Butler-Volmer equation for theanodic and cathodic reactions. By default, the Anode Exchange CurrentDensity is set to 1000 Amps and the Cathode Exchange Current Density isset to 100 Amps.

ii. Enter values for the Mole Fraction Reference Values.

These are the species concentrations at which the exchange current den-sities were taken. These are used to adjust the i0 values of reactantspecies that are depleted. By default, the H2 Reference Value is set to0.8 moles/moles, the 02 Reference Value is set to 0.21 moles/moles, andthe H2O Reference Value is set to 0.2 moles/moles.

iii. Enter values for the Stoichiometric Exponents by setting values for the H2Exponent, the H20 Exponent, and the O2 Exponent (defaulted to 0.5).These are the stoichiometric factors in the electrochemical reaction equa-tion. They are used as exponents as part of the i0 scaling.

iv. Enter values for the Butler-Volmer Transfer Coefficients by setting valuesfor the Anodic Transfer Coefficient and the Cathode Transfer Coefficientfor both the anode reaction and the cathode reaction (defaulted to 0.5).These are the alpha values in the Butler-Volmer equation. They representthe forward and backward rates of reaction at both the anode and cathode.

v. Enter a values for the Temperature Dependent Exchange Current Densityby turning on the Enable Temperature Dependent I 0 option and settingvalues for A and B. These two coefficients allow the i0 for the cathode tovary as a function of temperature.

5. Define material properties.

(a) Define a user-defined scalar.

i. Open the User-Defined Scalars panel.

Define −→ User-Defined −→Scalars...

ii. Change the Number of User-Defined Scalars to 1.

iii. Indicate none as the Flux Function and click OK to close the User-DefinedScalars panel.

(b) Define 12 user-defined memory locations.

i. Open the User-Defined Memory panel.

Define −→ User-Defined −→Memory...



ii. Change the Number of User-Defined Memory Locations to 12.

iii. Click OK to close the User-Defined Memory panel.

(c) Define user-defined function hooks.

i. Open the User-Defined Function Hooks panel.

Define −→ User-Defined −→Function Hooks...

ii. Change the Adjust function to adjust function.

iii. Click OK to close the User-Defined Function Hooks panel.

(d) Create or re-define new solid materials as appropriate for the anode, the cath-ode, and the electrolyte according to your problem specification.

i Note that since the FLUENT SOFC model does not currently support theshell conduction model, you cannot use it to take into account the transver-sal conductive heat inside the membrane.

(e) Edit the mixture-template mixture material.

i. In the Materials panel, click Fluent Database... to open the Fluent DatabaseMaterials panel.

ii. In the Fluent Database Materials panel, make a copy of the h2 fluid mate-rial.

iii. In the Materials panel, change the Material Type to mixture and clickEdit... for the Mixture Species.

iv. In the Species panel, arrange the materials under Selected Species in thefollowing order: h2o, o2, h2, and n2.

v. In the Materials panel, change the Thermal Conductivity and the Viscosityto ideal-gas-mixing-law.

vi. Change the Viscosity to ideal-gas-mixing-law.

vii. Change the Mass Diffusivity to user-defined and select diffusivity::sofc1.2 asthe corresponding user-defined function.

viii. Change the UDS Diffusivity to user-defined and select E Conductivity::sofc1.2as the corresponding user-defined function.

ix. Retain the default values for the other parameters and click Change/Create.

6. Set the operating conditions.

Define −→Operating Conditions...

Retain the default values and click OK in the Operating Conditions panel.



7. Set the boundary conditions.

Define −→Boundary Conditions...

Define the conditions at the anode, the cathode, the interface between the anodeand current collector, the interface between the cathode and the current collec-tor, the anode inlet, and the cathode inlet boundaries according to your problemspecification.

Note that sources only need to be hooked to the mass, species, and energy equationin a single fluid zone in order for the FLUENT SOFC model to function properly(but can be hooked to all zones if you so choose). FLUENT does not rely on cell orthread referencing in the sources in order to cover the solution domain.

8. Set the multigrid control parameters.

Solve −→ Controls −→Multigrid...

(a) Set the cycle type for h2, h2o, and o2 to V-cycle. For serial calculations, setthe cycle type for Energy and User-defined Scalar-0 to W-cycle or F-cycle. Forparallel calculations, select the F-cycle option for both.

(b) Set the Max Cycles to 50.

(c) Retain the default values for the rest of the parameters.

9. Define the convergence criteria.

Solve −→ Monitors −→Residuals...

In the Residual Monitors panel, set the Convergence Criterion for all equations to1e-08.

10. Initialize the flow field.

Solve −→ Initialize −→Initialize...

In the Solution Initialization panel, retain the default values for all parameters andclick the Init button.

11. In the Define SOFC Model Parameters panel, turn on the FLUENT SOFC model byactivating the Enable SOFC Model option, the Enable Surface Energy Source option,the Enable Species Source option, and the Disable CO Electrochemistry option.

If the electrolyte resistivity changes as a function of temperature, then turn on theEnable Electrolyte Conductivity Submodel option. If there is CO in the fuel line, thenyou should turn off the Disable CO Electrochemistry option.

12. Run the simulation until all residuals are decreasing.

13. Turn on the Enable Volumetric Energy Source option and continue the simulationuntil convergence is achieved.

14. Save the case and data files.



15. Perform post-processing using standard quantities and by using the user-definedmemory allocations. By default, the FLUENT SOFC model defines the followinguser-defined memory allocations:

Table 4.3.1: User-Defined Memory Allocations

UDM-0 Current Density (A/m2)UDM-1 Nernst voltage (Volts)UDM-2 Activation Overpotentail (Volts)UDM-3 Energy Source due to Ohmic Heating in Conduction Zones (W/m3)UDM-4 x Component of the Current Density (A/m2)UDM-5 y Component of the Current Density (A/m2)UDM-6 z Component of the Current Density (A/m2)UDM-7 Voltage jumpUDM-8 Electrolyte Resistivity (Ohm-m)UDM-9 Effective ResistanceUDM-10 Anode Activation PolarizationUDM-11 Cathode Activation Polarization

Note that UDM-7 and UDM-8 are the linearized values that FLUENT uses to solve thepotential field and electrochemical coupling. They are necessary to the calculationsbut do not contain any real physical meaning.

Note that UDM-10 and UDM-11 contain the disaggregated activation polarizationsat the anode and cathode.

The current density in UDM-0 (Equation 3.5-29) is a conservative flux of electricity(A/m2). Performing an area integral over the electrolyte surface will sum to the to-tal current (A). That value is computed at the electrolyte faces during the electric fieldsolution. The values in UDM-4 - UDM-6 contain cell-centered values of the current den-sity vector. These are not, and cannot be, conservative, so depending on the materialconductivity and the geometric configuration, these can sometimes unavoidably producevalues that do not match the UDM values. The same issues exist when interpolatingvelocity values to obtain the mass fluxes.



4.4 Setting the Parameters for the SOFC Model

You can specify the general settings for the SOFC model using the Define SOFC ModelParameters panel.

Define −→ Models −→ SOFC −→SOFC Model Control Panel...

Figure 4.4.1: The Define SOFC Model Parameters Panel

From this panel, you can set the various parameters for the SOFC model such as totalsystem current, electrolyte thickness, electrolyte resistivity, etc.

The Enable Electrolyte Conductivity Submodel option allows the ionic conductivity (orresistivity) of the electrolyte to change as a function of temperature. At the moment,there is one correlation that provides ionic conductivity (or resistivity) of the electrolyteas function of temperature.

resistivity =0.3685 + 0.002838e

10300T

100(4.4-1)

i Note that this is valid only for temperatures ranging from 1073 K to 1373 K.


4.4 Setting the Parameters for the SOFC Model

By turning off the Enable Surface Energy Source option, FLUENT excludes the heat ad-dition due to electrochemistry and all the reversible processes. This option should beturned on at all times.

The Enable Volumetric Energy Source option includes the ohmic heating throughout theelectrically conducting zones. This option is typically turned on when the potential fieldsolution is close to convergence.

The Disable CO Electrochemistry is enabled if there is carbon monoxide (CO) in the fuelline and if you want to include the CO in the electrochemistry.

Since the calculations are very sensitive to large current fluctuations early in the solutionprocess, it is recommended to use 0.3 or 0.4 for the Current Underrelaxation Factor for amore effective solution.

The leakage current is the total amount of current due to the leakage of oxidizer to thefuel side (through the electrolyte) and the electric current across the electrolyte due toany short circuit. You can specify a value for the leakage current under Leakage CurrentDensity.

The Converge to Specified System Voltage is used when the you want to specify a systemvoltage instead of system current as an input.



4.5 Setting Up the Electrode-Electrolyte Interfaces

You can apply specific settings for both your anode and your cathode interfaces usingthe FLUENT SOFC model..

4.5.1 Setting Up the Anode Electrode-Electrolyte Interface

You can specify the anode settings for the SOFC model using the Anode Interface Setuppanel.

Define −→ Models −→ SOFC −→Anode Interface Setup...

Figure 4.5.1: The Anode Interface Setup Panel

In the Anode Interface Setup panel, you select the surface that represents the anodeelectrode interface with the electrolyte from the Zone(s) list, turn on the Anode Interfaceoption and click Apply. You can do this for as many zones as you need. When you arefinished setting up the anode interface, click the OK button.


4.5 Setting Up the Electrode-Electrolyte Interfaces

4.5.2 Setting Up the Cathode Electrode-Electrolyte Interface

You can specify the cathode settings for the SOFC model using the Cathode InterfaceSetup panel.

Define −→ Models −→ SOFC −→Cathode Interface Setup...

Figure 4.5.2: The Cathode Interface Setup Panel

In the Cathode Interface Setup panel, you select the surface that represents the cathodeelectrode interface with the electrolyte from the Zone(s) list, turn on the Cathode Interfaceoption and click Apply. You can do this for as many zones as you need. When you arefinished setting up the cathode interface, click the OK button.



4.6 Setting Up the Electric Field Model Parameters

You can set up the details of the electric field model by opening the Define Electric FieldModel Parameters panel.

Define −→ Models −→ SOFC −→Electric Field Input Panel...

Figure 4.6.1: The Define Electric Field Model Parameters Panel

Here, you can designate conductive regions and specify their conductivity, assign bound-aries to be contact surfaces and specify the contact resistances, as well as specify whichsurfaces are grounded and which surfaces exhibit a current.

The Voltage Tap Surface is the surface that is grounded (i.e., V = 0). The Current TapSurface is the surface that current is being drawn from (i.e., i is assigned a value).


4.7 Setting Up the Tortuosity Parameters

4.7 Setting Up the Tortuosity Parameters

In fuel cell modeling, tortuosity is thought of as an effective diffusive path length factor.You can specify tortuosity settings for the SOFC model using the Tortuosity Setup panel.

Define −→ Models −→ SOFC −→Tortuosity Setup...

Figure 4.7.1: The Tortuosity Setup Panel

In the Tortuosity Setup panel, you select an appropriate zone from the Zone(s) list, turnon the Enable Tortuosity option and click Apply. You can do this for as many zones asyou need. When you are finished setting up the cathode interface, click the OK button.

You can assign tortuosity values to any porous zones in your simulation. In a porouszone, the mass diffusion coefficient is reduced as follows due to the porosity effect:

Deff =porosity

tortousityD (4.7-1)

There typically are no standard means of measuring tortuosity as it either needs to bemeasured experimentally or tuned to match other experimental data. Tortuosity valuemay typically be in the range of 2 to 4, although you can use much higher values.



4.8 Setting Up the Activation Parameters

You can specify activation settings for the SOFC model using the Define Activation Pa-rameters panel.

Define −→ Models −→ SOFC −→Activation Panel...

Figure 4.8.1: The Define Activation Parameters Panel

In the Define Activation Parameters panel, you can set the anode and cathode exchangecurrent density, the anode and cathode mole fraction, the stoichiometric coefficients,the Butler-Volmer transfer coefficients, and the temperature-dependent exchange currentdensity.

You can specify a value for the exchange current density at the anode (the default valueis 1000 Amps) using the Anode Exchange Current Density field. Likewise, you can specifya value for the exchange current density at the cathode (the default value is 100 Amps)using the Cathode Exchange Current Density field.

You can also specify Mole Fraction Reference Values for the fuel cell reactants in the DefineActivation Parameters panel. By default, the reference value for H2 is 0.8, the referencevalue for O2 is 0.21, and the reference value for H2O is 0.2.

The Butler-Volmer transfer coefficients can be set in the Define Activation Parameterspanel as well. These coefficients are the αa and αc from Equation 3.5-41 for both theanode and the cathode reactions.

i = i0

[eαan(φ−φ0)F

RT − e−αcn(φ−φ0)F

RT

](4.8-1)


4.8 Setting Up the Activation Parameters

Remember that α has anodic and cathodic values at both the cathode and the anode.By default, the value of α is set to 0.5 because of the nearly universal assumption thatthere is a symmetric balance between the forward and backward reactions. In most cases,these default values will be sufficient.

If you find yourself changing the Butler-Volmer transfer coefficients, or if you have someother rate-limiting reaction in your fuel cell simulation, you may also want to considerchanging the exponents for the stoichiometric coefficients for the fuel cell reactants. Theseexponents can be specified in the Define Activation Parameters panel. By default, theexponent values for H2, O2, and H2O are 0.5.

The Enable Temperature Dependant I 0 option allows the exchange current density tochange as a function of temperature in an exponential fashion

i0 = AeBT (4.8-2)

where A and B are constants provided by the user.


Bibliography

[1] J. Divisek A.A. Kulikovsky and A.A. Kornyshev. J. Electrochemical Society,146(11):3981–3991, 1999.

[2] J. V. Cole and S. Mazumder. J. Electrochemical Soc., 150:A1510, 2003.

[3] Brandon M. Eaton. One dimensional, Transient Model of Heat, Mass, and ChargeTransfer in a Proton Exchange Membrane. M.S. Thesis, 2001.

[4] Parsons Inc. EG&G Services and Science Applications International Corporation.Fuel Cell Handbook (5th Edition). 2000.

[5] J.H. Nam and M. Karviany. Effective Diffusivity and Water-Saturation Distributionin Single- and Two-layer PEMFC Diffusion Medium. Int. J. Heat Mass Transfer,2003.

[6] J. S. Newman. Electrochemical Systems. Prentice Hall, Englewood Cliffs, New Jersey,1973.

[7] T. V. Nguyen. Electrochemical Soc. Proceedings, 99(14):222–241.

[8] C.Y. Wang Sukkee Um and K.S. Chen. J. Electrochemical Society, 147(12):4485–4493,2000.

[9] T.A. Zawodzinski T.E. Springer and S. Gottesfeld. J. Electrochemical Soc., 138.

c© Fluent Inc. July 19, 2006 Bib-1

Fuel Cells Module Manual

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